QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE F

INANCE RESEARCH CENTRE

QUANTITATIVE FINANCE RESEARCH CENTRE

Research Paper 374 August 2016

A Penny Saved is a Penny Earned: Less Expensive Zero Coupon Bonds Alessandro Gnoatto, Martino Grasselli and Eckhard Platen

ISSN 1441-8010 www.qfrc.uts.edu.au

A Penny Saved is a Penny Earned:Less Expensive Zero Coupon Bonds∗

Alessandro Gnoatto† Martino Grasselli‡ Eckhard Platen§

August 15, 2016

Abstract

This paper introduces a more general modeling world than available under the classical no-arbitrage paradigm in finance. New research questions and interesting related econometric studiesemerge naturally. To explain in this paper the new approach and illustrate first important conse-quences, we show how to hedge a zero coupon bond with a smaller amount of initial capital thanrequired by the classical risk neutral paradigm, whose (trivial) hedging strategy does not suggest toinvest in the risky assets. Long dated zero coupon bonds we derive, invest first primarily in risky secu-rities and when approaching more and more the maturity date they increase also more and more thefraction invested in fixed income. The conventional wisdom of financial planners suggesting investorto invest in risky securities when they are young and mostly in fixed income when they approachretirement, is here made rigorous. The main reason for the existence of less expensive zero couponbonds is the strict supermartingale property of benchmarked savings accounts under the real worldprobability measure, which the calibrated parameters identify under the proposed model. We provideintuition and insight on the strict supermartingale property. The less expensive zero coupon bondsprovide only one first example that is indicative for the changes that the new approach offers in themuch wider modeling world. The paper provides a strong warning for life insurers, pension fundmanagers and long term investors to take the possibility of less expensive products seriously to avoidthe adverse consequences of the low interest rate regimes that many developed economies face.

Key words: Forex, benchmark approach, benchmarked risk minimization, stochastic volatility, longterm securities.

JEL Classification: C6, C63, G1, G12, G13

1 Introduction

This paper aims to draw the attention to a more general modeling world than available under the classicalno-arbitrage paradigm in finance. The modeling of new, unexplored phenomena becomes possible withtheoretically interesting and practically important consequences. A wide range of new research questionsand related econometric studies emerges naturally. To explain the new approach and illustrate firstimportant consequences, the less expensive pricing of long dated zero coupon bonds will be demonstratedin this paper. Under the benchmark approach, see Platen and Heath (2010), many payoffs can be less

∗Acknowledgments: we are grateful to the participants of the Quantitative Methods in Finance congress (Sydney,December 2015) and the 9-th World Congress of the Bachelier Finance Society (New York, July 2016) for useful discussions.The usual disclaimer applies.†Department of Mathematics, LMU University. Theresienstrasse 39, D-80333 Munich, Germany. Email alessan-

dro@alessandrognoatto.com‡Dipartimento di Matematica, Universita degli Studi di Padova (Italy) and Leonard de Vinci Pole Universitaire, Research

Center, Finance Group, 92 916 Paris La Defense, France. Email: grassell@math.unipd.it.§University of Technology, Sydney, Finance Discipline Group and School of Mathematical and Physical Sciences, PO

Box 123, Broadway, NSW, 2007, Australia, and University of Cape Town, Department of Actuarial Science. Email eck-hard.platen@uts.edu.au.

1

expensively produced than current theory and practice suggest. The intuitive verbal advice of financialplanners to invest in risky securities when the investor is young and mostly in fixed income when he/sheapproaches retirement, is made rigorous. The long dated zero coupon bonds we derive, invest firstprimarily in risky securities and, when approaching more and more the maturity date, they increase alsomore and more the fraction invested in fixed income. These less expensive zero coupon bonds provideonly a first example that is indicative for the changes that the new approach offers in a much widermodeling world than the classical one.Historically, Long (1990) was the first who observed that one can rewrite the risk neutral pricing formulainto a pricing formula that takes its expectation under the real world probability measure and employs the,so called, numeraire portfolio (NP) as numeraire. The benchmark approach assumes only the existenceof the NP and no longer the rather restrictive classical no-arbitrage assumptions, which are equivalentto the existence of an equivalent risk neutral probability measure. Under this much weaker assumption,one can still perform all essential tasks of valuation and risk management. The only condition imposed isthat the NP, which is the in the long run pathwise best performing portfolio, remains finite in finite time.Obviously, when this assumption is violated for a model, then some economically meaningful arbitragemust exist causing the candidate for the NP to explode. In this case the respective model makes not muchtheoretical and practical sense. Note that when a finite NP exists, various forms of classical arbitragemay be present in the market; see e.g. Loewenstein and Willard (2000) for various examples.The current paper illustrates the divergence of the benchmark approach from the classical approach byfocusing on the currency market, which is one of the most active markets. By analysing observed foreignexchange (FX) derivative prices and calibrating these to a model that can be inside or outside the classicalframework, situations are identified where the classical assumptions cannot be verified. By asking whatminimal possible price a long dated zero coupon bond can have under such circumstances, it turns outthat it can be substantially lower priced and hedged than what risk neutral pricing would suggest. Asa consequence, long dated zero coupon bonds can be produced less expensively, which we empiricallydemonstrate.This surprising result can explain the effect that has been intuitively exploited in financial planning,where one uses the extra growth present in risky securities to accumulate over long time periods morewealth with little fluctuations at maturity than by investing over the entire period in fixed income, see e.g.Gourinchas and Parker (2002). The possibility of less expensive production costs for targeted long datedpayoffs has major practical implications for annuities, life insurance contracts, pensions and many otherlong term contracts. The aim of the paper is to encourage systematic research that identifies and studiesthese and other exploitable deviations from classical theory that become possible under the benchmarkapproach.The paper is structured as follows: In Section 2 we introduce the general multi-currency modeling frame-work and recall some notions from the benchmark approach. Section 3 motivates and formally introducesthe 4/2 model as a unifying framework for stochatic volatility models driven by the CIR process as theHeston-based model of De Col et al. (2013) and the 3/2-based model of Baldeaux et al. (2015b). Theanalytical tractability of the 4/2 model is demonstrated in Section 4, which constitutes a prerequisitefor an efficient model calibration, presented in Section 5. A practical consequence of the calibration isanalyzed in Section 6, where we perform hedging in an incomplete market setting without the existenceof a risk neutral probability measure. In particular, in the absence of a risk neutral probability measure,we follow the concept of benchmarked risk minimization of Du and Platen (2016). We show how to hedgelong term products significantly less expensively than the classic risk neutral paradigm permits.

2 General Setup

In this section we present the general modeling framework of the benchmark approach for a foreignexchange (FX) market. Subsection 2.1 provides a general setup driven by a multi-dimensional diffusionprocess.

2

2.1 Specification of the Currency Market

We use superscripts to reference different currencies, and employ bold letters for vectors and subscriptsfor elements thereof. Unless specified by a suitable superscript, all expectations are considered withrespect to the real world probability measure P. We model the currency market on a probability space(Ω,FT ,P), where T < ∞ is a finite time horizon. On this space we introduce a filtration (Ft)0≤t≤Tto model the evolution of information, satisfying the usual assumptions. The above filtered probabilityspace supports a standard d-dimensional P-Brownian motion Z=Z(t) = (Z1(t), . . . , Zd(t)), 0 ≤ t ≤ Tfor modeling the traded uncertainty. The constant N denotes the number of currencies in the model,whereas d is the number of traded risky factors we employ.

In each economy we postulate the existence of a money market account, i.e. the i-th money marketaccount,when denominated in units of the i-th currency, evolves according to the relation

dBi(t) = Bi(t)ri(t)dt, Bi(0) = 1, 0 ≤ t ≤ T ; (2.1)

with the R-valued, adapted i-th short rate process ri =ri(t), 0 ≤ t ≤ T

. We denote by Si,j=

Si,j(t), 0 ≤ t ≤ T

the continuous exchange rate process between currency i and j. Here Si,j(t) denotesthe price of one unit of currency i in units of currency j, meaning that, e.g. for i = USD and j = EURand Si,jt = 0.92 we have, in line with the standard FORDOM convention, that the price of one USD is0.92 EUR at time t.Let us follow Platen and Heath (2010) and Heath and Platen (2006) and introduce a family of primarysecurity account processes via

Bi,j(t) = Si,j(t)Bj(t), 0 ≤ t ≤ T

for i 6= j. Obviously, for i = j we have Bi,i(t) = Bi(t). We take the perspective of a generic currencyreferenced with superscript i and introduce the vector of money market accounts of the form Bi(t) =(Bi,1(t), ..., Bi,N (t)

), i = 1, ..., N . Given this vector of primary security accounts, an investor may trade

on them. This is represented by introducing a family of predictable Bi-integrable stochastic processesδ =

δ(t) = (δ1(t), ..., δN (t)) , 0 ≤ t ≤ T

for i = 1, ..., N , called strategies. Each δj(t) ∈ R denotes the

number of units that an agent holds in the j-th primary security account at time t. Let us introducethe process Vi,δ =

Vi,δ(t), 0 ≤ t ≤ T

, which describes the value process in i-th currency denomination

corresponding to the portfolio strategy δ, i.e.

Vi,δ(t) =

N∑j=1

δj(t)Bi,j(t). (2.2)

The strategy δ is said to be self-financing if

dVi,δ(t) =

N∑j=1

δj(t)dBi,j(t). (2.3)

In line with Platen and Heath (2010), we assume limited liability for all investors. For this purpose,we introduce V+ as the set of all self-financing strategies forming strictly positive portfolios. For ourpurposes, we will be interested in a particular strategy δ? ∈ V+, which yields the growth optimalportfolio (GOP), which can be shown to be equivalent to the numeraire portfolio (NP), and is definedas follows:

Definition 2.1. A solution δ? of the maximization problem

supδ∈V+

E[log

(Vi,δ(T )

Vi,δ(0)

)],

for all i = 1, . . . , N and 0 ≤ T ≤ T is called a growth optimal portfolio strategy.

3

It has been shown in Platen and Heath (2010) that the GOP value process is unique in an incompletejump-diffusion market setting. We summarize the discussion above in the following assumptions.

Assumption 2.1. We assume the existence of the growth optimal portfolio (GOP), and denote byDi =

Di(t) ∈ (0,+∞), 0 ≤ t ≤ T

, i = 1, ..., N the value of the GOP denominated in the i-th currency.

The dynamics of the GOP are given by

dDi(t)

Di(t)= ri(t)dt+ 〈πi(t),πi(t)dt+ dZ(t)〉, Di(0) > 0 (2.4)

for t ∈ [0, T ] and i = 1, . . . N , where for N, d ∈ N, the N -dimensional family of predictable, Rd-valuedstochastic processes π =

πi(t) = (πi1(t), . . . , πid(t)), 0 ≤ t ≤ T

represent the market prices of risk with

respect to the i-th currency denomination. The processes πi are assumed to be integrable with respect tothe d-dimensional standard Brownian motion Z.

The GOP can be shown to be in many ways the best performing portfolio. In particular, in the long runits value outperforms almost surely those of any other strictly positive portfolio. Here we assume thatit remains finite in finite time in all currency denominations. If we were to consider a model where theGOP explodes in any of the currency denominations, then the model would allow an obvious form ofeconomically meaningful arbitrage, since one could generate in that currency denomination in finite timeunbounded wealth from finite initial capital. Given the uniqueness of the GOP, all exchange rates Si,j(t)can be uniquely determined as ratios of different denominations of the GOP in the respective currencies.

Assumption 2.2. The family of exchange rate processes Si,j =Si,j(t), 0 ≤ t ≤ T

is determined by

the ratios

Si,j(t) =Di(t)

Dj(t)(2.5)

for 0 ≤ t ≤ T and i, j = 1, . . . , N .

Given Assumption 2.2, it is immediate to compute via a direct application of the Ito formula the dynamicsof all exchange rates and all primary security accounts in all currency denominations.

Lemma 2.1. The exchange rate Si,j(t) under the real world probability measure P evolves according tothe dynamics

dSi,j(t)

Si,j(t)= (ri(t)− rj(t))dt+ 〈πi(t)− πj(t),πi(t)dt+ dZ(t)〉,

Si,j(0) = si,j > 0,

(2.6)

and the generic j-th primary security account Bi,j, in i-th currency denomination and under the realworld probability measure P, evolves according to the dynamics

dBi,j(t)

Bi,j(t)= ri(t)dt+ 〈πi(t)− πj(t),πi(t)dt+ dZ(t)〉,

Bi,j(0) = bi,j ,

(2.7)

for i, j = 1, . . . , N and t ∈ [0, T ].

2.2 The Benchmark Approach

In the present paper we evaluate contingent claims under the benchmark approach of Platen and Heath(2010). Under this approach, price processes denominated in terms of the GOP are called benchmarkedprice processes. More precisely, for i, j = 1, . . . N , let us introduce the benchmarked price process

Bj =

Bj(t) :=

Bi,j(t)

Di(t), 0 ≤ t ≤ T

.

4

We call Bj the benchmarked j-th primary security account. Note that Bj does not depend on theindex i of the currency denomination we started from. Given (2.7) and (2.4), upon an applicationof the Ito formula, it is immediate to conclude that all benchmarked price processes Bj form P-localmartingales. Even more, they are non-negative P-local martingales. Hence, due to Fatou’s lemma, theyare also P-supermartingales. Analogously, we also have that benchmarked non-negative portfolio valuesVδ(t) := Vi,δ(t)/Di(t) form P-supermartingales. Besides the exclusion of forms of economically meaningfularbitrage, which are equivalent to the explosion of the GOP, forms of classical arbitrage that are excludedunder classical no-arbitrage assumptions may exist in our model, see e.g. Loewenstein and Willard (2000).Let us now introduce for the i-th currency denomination the Radon-Nikodym derivative process, denotedby Λi =

Λi(t), 0 ≤ t ≤ T

, by setting

Λi(t) =Bi(t)

Bi(0), i = 1, . . . , N. (2.8)

This is the risk neutral density for the putative risk-neutral measure Qi of the i-th currency denomination.It arises e.g. when we consider replicable claims and assume the existence of an equivalent risk neutralprobability measure Qi. As each Λi equals the corresponding benchmarked savings account Bi (up to aconstant factor), it is clear that Λi is a P-local martingale, for i = 1, . . . , N . The classical assumption inthe foreign exchange literature that there exists an equivalent risk-neutral probability measure for eachcurrency denomination, corresponds to the requirement that each process Λi is a true martingale fori = 1, . . . , N . Such a requirement is rather strong and may be empirically rejected, see e.g. Heath andPlaten (2006), the findings in Baldeaux et al. (2015b) and the necessary and sufficient conditions of Hulleyand Ruf (2015). Hence, in the present paper, we shall allow each Λi to be either a true martingale or astrict local martingale. To work in such a generalized setting requires a more general pricing concept thanthe one provided under the classical risk neutral paradigm. In the following, we will employ the notionof real world pricing : a price process Vi =

Vi(t), 0 ≤ t ≤ T

, here denominated in i-th currency, is said

to be fair if, when expressed in units of the GOP Di, forms a P-martingale, this means its benchmarkedvalue forms a true P-martingale, see Definition 9.1.2 in Platen and Heath (2010). For a fixed maturityT ∈ [0, T ], we let H(T ) = V(T ) be an FT -measurable non-negative benchmarked contingent claim, suchthat when expressed in units of the i-th currency denomination as Hi(T ) = Di(T )H(T ) we have

E [H(T )| Ft] = E[Hi(T )

Di(T )

∣∣∣∣Ft] <∞for all 0 ≤ t ≤ T ≤ T , i = 1, ..., N . The benchmarked fair price V (t) = V i(t)/Di(t) of this contingentclaim is the minimal possible price and given by the following conditional expectation under the real-worldprobability measure P:

V(t) = E[H(T )

∣∣∣Ft] , (2.9)

which is known in the literature as real-world pricing formula, see Corollary 9.1.3 in Platen and Heath(2010). Note that benchmarked risk minimisation, described in Du and Platen (2016), gives (2.9) gen-erally. In case Λi is a true martingale we obtain, by changing in (2.9) from the real world probabilitymeasure P to the equivalent risk neutral probability measure Qi, the risk neutral pricing formula

Vi(t) = E[Bi(t)

Bi(T )

Bi(T )

Bi(t)

Di(t)

Di(T )Hi(T )

∣∣∣∣Ft]= E

[Λi(T )

Λi(t)

Bi(t)

Bi(T )Hi(T )

∣∣∣∣Ft] = EQi[Bi(t)

Bi(T )Hi(T )

∣∣∣∣Ft] . (2.10)

This shows that in this case the real-world pricing formula generalises the classical risk-neutral valuationformula and the Radon-Nikodym derivative for the respective risk neutral probability measure is givenby (2.8). In general, due to the supermartingale property of benchmarked price processes in the case

5

when Λi is a strict supermartingale, a formally obtained risk neutral price is greater than the real worldprice, see Du and Platen (2016).

Later on we consider the pricing of zero coupon bonds. According to (2.9), the minimal possible priceP i(t, T ) at time t of a zero coupon bond, denominated in the i-th currency and paying at maturity Tone unit of the i-th currency, is given by the formula

P i(t, T ) = Di(t)E[

1

Di(T )

∣∣∣∣Ft] . (2.11)

All benchmarked non-negative price processes are supermartingales. Therefore, as already mentioned,the fair price process of a contingent claim is the minimal possible price process. There may exist moreexpensive price processes that deliver the same payoff. As we will show later on, over long periods of timea formally obtained risk neutral price can be significantly more expensive than the fair one. This paperfocuses on the emerging possibility to potentially produce long dated zero coupon bonds less expensivelythan suggested under the classical paradigm. This permits, e.g. the less expensive production of contractsinvolving long dated zero coupon bonds as building blocks as in the case for pensions, annuities and lifeinsurances.

3 The 4/2 Model

To demonstrate the fact that in reality there may exist hedgeable long dated zero coupon bond priceprocesses that are significantly less expensive than respective risk neutral price processes, we need somemodel that could capture this phenomenon when it is present in the market. In this section we providesuch model, called the 4/2 model. Below we describe the 4/2 model that unifies naturally several well-known models. In Subsection 3.3 we state the precise conditions under which the crucial martingaleproperty of the benchmarked savings account fails for the 4/2 model.

3.1 The 4/2 Model as a Unifying Framework

In this subsection, we provide a specification of the volatility dynamics of the exchange rates. In particular,we consider, for simplicity, two currencies with D1(t) denoting the GOP in domestic currency and D2(t)denoting the GOP in foreign currency. For example, S1,2(t) = D1(t)/D2(t) can follow a stochasticvolatility model of Heston type (see Heston (1993)), where

dS1,2(t)

S1,2(t)=(r1(t)− r2(t)

)dt+

√V (t) (dZ(t) + λ(t)dt) ,

S1,2(0) = s1,2 > 0,

dV (t) =κ(θ − V (t))dt+ σV (t)1/2(ρdZ(t) +

√1− ρ2dZ⊥(t)

),

V (0) = v > 0.

(3.1)

Here Z⊥ =Z⊥(t), 0 ≤ t ≤ T

is a P-Brownian motion independent of Z, κ > 0, θ > 0, ρ ∈ [−1, 1] with

the predictable processes r1, r2 and λ.In the remainder of the paper, we will repeatedly employ the following terminology: Heston (type) model,3/2 (type) model, 4/2 (type) model. Let us now clarify the respective models. In the following, we considera, b ∈ R, and let Di =

Di(t), 0 ≤ t ≤ T

denote a generic place-holder for a GOP process satisfying a

scalar diffusive stochastic differential equation (SDE). Moreover, let V =V (t), 0 ≤ t ≤ T

be a square

root process as given in (3.1). A model is said to be of Heston type (resp. of 3/2 type, resp. of 4/2 type)if the diffusion coefficient in the dynamics of the GOP Di is proportial to a

√V (t) (resp. b/

√V (t), resp.

(a√V (t) + b/

√V (t))).

6

The question we would like to address is the following: Given a specification of the market price of riskprocess λ for the domestic currency denomination of securities, what are the associated dynamicsof the domestic and foreign specifications of the GOP?

Lemma 3.1. Consider a two-currency model, where the exchange rate model for S1,2 is of Heston type(3.1). The following statements hold true:

1. If λ(t) = a√V (t), a ∈ R, then the GOP denominations D1 and D2 follow both Heston-type models.

2. If λ(t) = b√V (t)

, b ∈ R, then the GOP denomination D1 follows a 3/2 model, whereas D2 follows a

4/2 model.

3. If λ(t) = a√V (t) + b√

V (t), a, b ∈ R, then the GOP denominations D1 and D2 follow both 4/2 type

models.

The proof for this result is given in Appendix A.Note that if we had started in (3.1) with the volatility 1/

√V (t), then for λ(t) = b√

V (t)we would have

always fallen into the world of 4/2 type models.Lemma 3.1 highlights an interesting interplay between several well-known financial models. It shows thatthe 4/2 model arises naturally from a standard Heston model when the market price of risk belongs tothe essentially affine class (see Duffee (2002)). Furthermore, it demonstrates that the 4/2 model providesa general framework that nests other popular model choices.Different specifications of the market price of risk do not only impact on the shape of the GOP dynamics.In fact, depending on the calibrated values of the model parameters, we may incur situations whereclassical risk neutral pricing is no longer possible because an equivalent risk neutral probability measuredoes not exist. To see this, we observe that from a direct inspection of the dynamics in (3.1) in theHeston model setting, it is tempting to define the following two continuous processes

ZQ1

(t) := Z(t) +

∫ t

0

λ(s)ds

ZQ2

(t) := Z(t) +

∫ t

0

(λ(s)−

√V (s)

)ds,

(3.2)

which, if the assumptions of the Girsanov theorem were in both cases fulfilled, would then be Q1- (resp.Q2-) Brownian motions. Let us assume that under the real world probability measure P the Fellercondition (see Karatzas and Shreve (1991), Section 5.5) is fulfilled by the parameters of the volatilityprocess V , i.e. we have 2κθ − σ2 ≥ 0, so that the square root process V remains strictly positive P-a.s.for all t ∈ [0, T ]. The following lemma shows that, depending on the specification of λ, it is possibleto obtain a variance process V under the putative risk neutral measure that may not satisfy the Fellercondition, implying that the putative risk neutral measure may fail to be equivalent to the real worldprobability measure.

Lemma 3.2. Consider a two-currency model, where the dynamics of the exchange rate S1,2 is of theHeston type (3.1), such that the variance process V fulfills the Feller condition, i.e. 2κθ − σ2 ≥ 0. Let

ZQ1

, ZQ2

, as in (3.2), be the candidate Brownian motions under the putative risk neutral measures Q1

and Q2, respectively. The following holds:

1. For the putative risk neutral measure Q1 we get:

(a) If λ(t) = a√V (t), a ∈ R, then the drift of the variance process V under Q1 is

κ (θ − V (t))− σρaV (t)

and the Feller condition is always satisfied under Q1, which is then a true equivalent martingalemeasure.

7

(b) If λ(t) = a√V (t) + b√

V (t), a, b ∈ R, then the drift of the variance process V under Q1 equals

κ (θ − V (t))− σ√V (t)ρ

(a√V (t) +

b√V (t)

)

and the Feller condition may be violated, implying that Q1 may not be equivalent to P.

(c) If λ(t) = b√V (t)

, b ∈ R, then the drift of the variance process V under Q1 is

κ (θ − V (t))− σρb

and the Feller condition may be violated, implying that Q1 may not be equivalent to P.

2. For the putative risk neutral measure Q2 we have

(a) If λ(t) = a√V (t), a ∈ R, then the drift of the variance process V under Q2 equals

κ (θ − V (t))− σρ(a− 1)V (t)

and the Feller condition is always satisfied under Q2, which is then a true equivalent martingalemeasure.

(b) If λ(t) = a√V (t) + b√

V (t), a, b ∈ R, then the drift of the variance process V under Q2 equals

κ (θ − V (t))− σ√V (t)ρ

((a− 1)

√V (t) +

b√V (t)

)

and the Feller condition may be violated, implying that Q2 would be in such case not equivalentto P.

(c) If λ(t) = b√V (t)

, b ∈ R, then the drift of the variance process V under Q2 is

κ (θ − V (t))− σρb+ σρV (t)

and the Feller condition may be violated, implying that Q2 would be in such case not equivalentto P.

The proof of these statements is straightforward using our previous notation and relationships and,therefore, omitted.

3.2 Formal Presentation of the 4/2 Model

To provide in the case of more than two currencies a concrete specification of the market prices of risk, weproceed to introduce the Rd-valued nonnegative stochastic process V =

V(t) = (V1(t), ..., Vd(t)), 0 ≤ t ≤ T

,

called the volatility factor process. The k-th component Vk of the vector process V is assumed to solvethe SDE

dVk(t) =κk(θk − Vk(t))dt+ σkVk(t)1/2dWk(t),

Vk(0) = vk > 0,(3.3)

for t ∈ [0, T ], where the parameters κk > 0, θk > 0, σk > 0, are admissible in the sense of Duffie et al.(2003), k = 1, ..., d. In addition, to avoid zero volatility factors, we impose the following assumption.

Assumption 3.1. For every k = 1, . . . , d, the parameters in (3.3) satisfy the relation

2κkθk − σ2k ≥ 0. (3.4)

8

We also allow for non-zero correlation between assets and their volatilities via the following condition:

Assumption 3.2. The Brownian motions Z and W have a covariation satisfying

d〈Wk, Zl〉(t)dt

= δklρk, k, l = 1, ..., d, (3.5)

where δkl denotes the Dirac delta function for the indices k and l.

We then proceed to provide a general specification for the family of market prices of risk.

Assumption 3.3. We assume that the i-th market price of risk vector πi(t) is a projection of thecommon volatility factor V, along a direction parametrized by a constant vector ai ∈ Rd and a projectionof the inverted elements of V along another direction parametrized by bi ∈ Rd, according to the followingrelations

πi(t) = Diag1/2(V (t))ai +Diag−1/2(V (t))bi, i = 1, ..., N, (3.6)

where Diag1/2(u) denotes the diagonal matrix whose diagonal entries are the respective square roots ofthe components of the vector u ∈ Rd. The family of short-rate processes ri, i = 1, ..., N is assumed to begiven in the form

ri(t) = hi + 〈Hi,V(t)〉+ 〈Gi,V−1(t)〉 (3.7)

where V−1 is a vector whose components are the inverses of those of V.

Under Assumption 3.3, we can express the dynamics of the GOP as

dDi(t)

Di(t)=(ri(t) + (ai)>Diag(V (t))ai + (bi)>Diag−1(V (t))bi + 2(ai)>bi

)dt

+ (ai)>Diag1/2(V (t))dZ(t) + (bi)>Diag−1/2(V (t))dZ(t).

Here we suppress the explicit formulation of the dependence of ri on V . Consequently, the dynamics ofthe exchange rate Si,j is given by the SDE

dSi,j(t)

Si,j(t)=(

(ri(t)− rj(t)) + 2(ai)>bi − (ai)>bj − (aj)>bi

+ (ai)>Diag(V (t))(ai − aj) + (bi)>Diag−1(V (t))(bi − bj))

dt

+ ((ai − aj)>Diag1/2(V (t)) + (bi − bj)>Diag−1/2(V (t)))dZ(t).

(3.8)

Notice that the dynamics of the exchange rates are fully functionally symmetric w.r.t. the constructionof product/ratios thereof.

3.3 Strict Local Martingality

In Section 3.1, we observed that, for a sufficiently general specification of the market price of risk for agiven currency, we can have a situation where an equivalent risk neutral probability measure may fail toexist, due to the behavior of the variance process near zero under the putative risk neutral measure.In this subsection, we investigate the conditions under which the i-th benchmarked savings account,

Bi(t) = Bi(t)Di(t) , is a strict P-local martingale, i = 1, ..., N . As observed in Section 2.2, Bi(t), after

normalization to one at the initial time, corresponds to the Radon-Nikodym derivative for the putativerisk neutral measure of the i-th currency denomination. Should Bi(t) be a strict P-local martingale, wenote that classical risk neutral pricing is not applicable. However, real world pricing in line with (2.9) isstill applicable, see Platen and Heath (2010), and provides the minimal possible price.

9

Given (2.7) and (2.4), the dynamics of Bi(t) are given by the SDE

dBi(t) = −Bi(t)((ai)>(Diag(V (t)))1/2dZ(t)

+ (bi)>(Diag(V (t)))−1/2dZ(t)).(3.9)

Upon integration of the above SDE we obtain

E[Bi(t)

]= Bi0

d∏k=1

E[ξik(t)

],

where we define the exponential local martingale process ξik =ξik(t) , t ≥ 0

via

ξik(t) := exp

−ρk

∫ t

0

(aikVk(s)1/2 + bikVk(s)−1/2)dWk(s)

−1

2ρ2k

∫ t

0

(aikVk(s)1/2 + bikVk(s)−1/2)2ds

.

(3.10)

The putative change of measure with respect to the i-th currency denomination involves

dWk(t) = dWk(t) + ρk(aikVk(t)1/2 + bikVk(t)−1/2)dt,

where under classical assumptions Wk should be a Wiener process under the putative risk neutral measureQi. Under this measure the process Vk would then solve the SDE

dVk(t) = κk(θk − Vk(t))dt− ρkσk(aikVk(t) + bik)dt+ σkVk(t)12 dWk(t)

= (κkθk − ρkσkbik)dt− κk(1 +ρkσka

ik

κk)Vk(t)dt+ σkVk(t)

12 dWk(t). (3.11)

Under P, the process Vk does not reach 0 if the Feller condition is satisfied, i.e.

2κkθk ≥ σ2k,

while under the putative risk neutral measure the process Vk would not reach 0 if the corresponding Fellercondition would be satisfied, that is

2κkθk ≥ σ2k + 2ρkσkb

ik.

Therefore, the process Vk would have a different behavior at 0 under the two measures, provided that

σ2k ≤ 2κkθk < σ2

k + 2ρkσkbik. (3.12)

In this case the putative risk neutral measure would not be an equivalent probability measure and clas-sical risk neutral pricing would not be well-founded.

In order to get an intuition of what is the typical path behavior when dealing with true and strict localmartingales, we simulate some paths of the Radon-Nikodym derivative for the putative risk neutral mea-

sure of the i-th currency denomination Bi(t) = Bi(t)Di(t) according to the corresponding SDE (3.9), together

with the respective quadratic variation processes, for time horizon t = 10 years. In this illustrationwe consider a one factor specification of the 4/2 model (i.e. d = 1) and fix the parameters as follows:κ = 0.49523; θ = 0.53561;σ = 0.67128;V (0) = 1.4338; ρ = −0.89728; a = 0.047360. We let the param-eter b range in the interval [−0.4, 0.4] in order to generate situations in which the process Bi is a truemartingale (b positive) or a strict local martingale (b negative). We see in Figure 1 that the quadraticvariation of the strict local martingale process almost explodes from time to time and increases throughthese upward jumps visually much faster than in the case corresponding to the true martingale process,in line with the well-known unbounded expected quadratic variation process for square integrable strict

10

local martingales, see e.g. Platen and Heath (2010).

To conclude this section, let us observe that we can compute the prices of zero coupon bonds for allcurrency denominations, meaning that it is a priori possible to devise a model for long-dated FX products,in the spirit of Gnoatto and Grasselli (2014), where a joint calibration to FX surfaces and yield curvesis performed. Depending on the parameter values, our general framework may be interpreted both fromthe point of view of real world pricing and classical risk-neutral valuation, respectively:

• Should market data imply the existence of a risk-neutral probability measure for the i-th currencydenomination, then it would be possible to equivalently employ the i-th money market account asnumeraire.

• In the other case, i.e. when risk neutral pricing is not possible for the i-th currency denominationdue to the strict local martingale property of the i-th benchmarked money-market account, thendiscounting should be performed via the GOP.

4 Valuation of Derivatives

In the present section we solve the valuation problem for various contingent claims. The general valuationtool will be given by the real-world pricing formula (2.9). Subsection 4.1 concentrates on plain vanillaEuropean FX options, for which a semi-closed form valuation is available by means of Fourier techniques.These require the knowledge of the characteristic function of the log-underlying, given below in Theorem4.1, which provides as a by-product a closed form valuation formula for benchmarked zero-coupon bonds.

4.1 FX options

We first provide the calculation of the discounted conditional Fourier/Laplace transform of xi,j(t) :=ln(Si,j(t)), which will be useful for option pricing purposes. Let us consider a European call optionC(Si,j(t),Ki,j , τ) at time t, i, j = 1, ..., N, i 6= j, on a generic exchange rate process Si,j with strikeKi,j , maturity T = t + τ and face value equal to one unit of the foreign currency. We denote viaY i(t) = log(Di(t)) the logarithm of the GOP in i-th currency denomination. Hence the log-exchangerate may be written as xi,j(t) = log(Si,j(t)) = Y i(t)− Y j(t). Let us introduce the following conditionalexpectation

φi,jt,T (z) = Di(t)E[

1

Di(T )eizx

i,j(T )

∣∣∣∣Ft]= eY

i(t)E[e−Y

i(T )+iz(Y i(T )−Y j(T ))∣∣∣Ft] (4.1)

for i =√−1. For z = u ∈ R we will use the terminology of a discounted characteristic function,

whereas for z ∈ C when the expectation exists, the function φi,jt,T will be called a generalized discountedcharacteristic function. If we denote by Ψt,T (z) the joint conditional (generalized) characteristic functionof the vector of GOP denominations Y (T ) = (Y 1(T ), ..., Y N (T )), that is

Ψt,T (ζ) := E[ei〈ζ,Y (T )〉

∣∣∣Ft] , ζ ∈ CN , (4.2)

then we have

φi,jt,T (z) = Di(t)Ψt,T (ζ), (4.3)

for ζ being a vector with ζi = z + i, ζj = −z and all other entries being equal to zero. Now, from thereal-world pricing formula (2.9), the time t price of a call option can be written as the following expectedvalue:

C(Si,j(t),Ki,j , τ) = Di(t)E[

1

Di(T )

(Si,j(T )−Ki,j

)+∣∣∣∣Ft] .11

Following Lewis (2001), we know that option prices may be interpreted as a convolution of the payoffand the probability density function of the (log)-underlying. As a consequence, the pricing of a derivativemay be solved in Fourier space by relying on the Plancherel/Parseval identity, see Lewis (2001), wherewe have for f, g ∈ L2(R,C) ∫ ∞

−∞f(x)g(x)dx =

1

2π

∫ ∞−∞

f(u)g(u)du

for u ∈ R and f , g denoting the Fourier transforms of f, g, respectively. Applying the reasoning abovein an option pricing setting requires some additional care. In fact, most payoff functions do not admit aFourier transform in the classical sense. For example, it is well-known that for the call option one has

Φ(z) =

∫Reizx

(ex −Ki,j

)+dx = −

(Ki,j

)iz+1

z(z − i),

provided we let z ∈ C with Im(z) > 1, meaning that Φ(z) is the Fourier transform of the payoff functionin the generalized sense. Such restrictions must be coupled with those that identify the domain wherethe generalized characteristic function of the log-price is well defined. The reasoning we just reportedis developed in Theorem 3.2 in Lewis (2001), where the following general formula is presented (here wewrite φi,j for φi,jt,T in order to simplify notation):

C(Si,j(t),Ki,j , τ) =1

2π

∫Zφi,j(−z)Φ(z)dz, (4.4)

with Z denoting the line in the complex plane, parallel to the real axis, where the integration is per-formed. The article Carr and Madan (1999) followed a different procedure by introducing the concept ofa dampened option price. However, as Lewis (2001) and Lee (2004) point out, this alternative approachis just a particular case of the first one. In Lee (2004), the Fourier representation of option prices isextended to the case where interest rates are stochastic. Moreover, the shifting of contours, pioneeredby Lewis (2001), is employed to prove Theorem 5.1 in Lee (2004). There the following general optionpricing formula is presented:

C(Si,j(t),Ki,j , τ)

= R(Si,j(t),Ki,j , α

)+

1

π

∫ ∞−iα0−iα

Re

(e−izk

i,j φi,j(z − i)

−z(z − i)

)dz.

(4.5)

Here ki,j = log(Ki,j), α denotes the contour of integration and the term coming from the application ofthe residue theorem is given by

R(Si,j(t),Ki,j , α

)=

φi,j(−i)−Ki,jφi,j(0), if α < −1

φi,j(−i)− Ki,j

2 φi,j(0), if α = −1

φi,j(−i) if − 1 < α < 012φ

i,j(−i) if α = 0

0 if α > 0.

(4.6)

The following theorem provides the explicit computation of the generalized discounted characteristicfunction.

12

Theorem 4.1. The joint conditional generalized characteristic function Ψt,T (·) in (4.2) is given by

Ψt,T (ζ) = exp

N∑i=1

iζi(Y i(t) + hi(T − t)

)

×d∏k=1

exp

N∑i=1

[(T − t)

(1− ρ2

k

2

N∑j=1

iζiiζj(aikbjk + ajkb

ik)

+ iζiaikbik + iζi

κkρkσk

(bik − θkaik

))

− Vk(t)iζiρka

ik

σk− iζi

ρkbik

σklog(Vk(t))

]

×(βk(t, Vk)

2

)mk+1

Vk(t)−κkθk

σ2k (λk +Kk(t))

−(

12 +

mk2 −αk+

κkθkσ2k

)

× e1

σ2k

(κ2kθk(T−t)−

√AkVk(t) coth

(√Ak(T−t)

2

)+κkVk(t)

)Γ(

12 + mk

2 − αk + κkθkσ2k

)Γ(mk + 1)

× 1F1

(1

2+mk

2− αk +

κkθkσ2k

,mk + 1,β2k(t, Vk)

4(λk +Kk(t))

),

(4.7)

where

mk =2

σ2k

√(κkθk −

σ2k

2

)2

+ 2σ2kνk,

Ak = κ2k + 2µkσ

2k,

βk(t, x) =2√Akx

σ2k sinh

(√Ak(T−t)

2

) ,Kk(t) =

1

σ2k

(√Ak coth

(√Ak(T − t)

2

)+ κk

),

and

αk = −ρkσk

N∑i=1

iζibik (4.8)

λk = −ρkσk

N∑i=1

iζiaik (4.9)

µk = −N∑i=1

iζiHik +

iζi

2(aik)2 +

1− ρ2k

2

N∑j=1

iζiiζjaikajk + iζiρka

ik

κkσk

(4.10)

νk = −N∑i=1

iζiGik +iζi

2(bik)2 +

1− ρ2k

2

N∑j=1

iζiiζjbikbjk

−iζiρkb

ik

σk

(κkθk −

σ2k

2

)).

(4.11)

13

Let be given the functions

f1k (−Im(ζ)) := κ2

k + 2σ2k

(−

N∑i=1

[−Im(ζi)Hi

k

− Im(ζi)

2(aik)2 +

1− ρ2k

2

N∑j=1

Im(ζi)Im(ζj)aikajk − Im(ζi)ρka

ik

κkσk

)

f2k (−Im(ζ)) :=

(κkθk −

σ2k

2

)2

+ 2σ2k

(−

N∑i=1

[−Im(ζi)Gik −

Im(ζi)

2(bik)2

+1− ρ2

k

2

N∑j=1

Im(ζi)Im(ζj)bikbjk +

Im(ζi)ρkbik

σk

(κkθk −

σ2k

2

))

f3k (−Im(ζ)) :=

κkθk +σ2k

2 +√f2k (−Im(ζ))

σ2k

f4k (−Im(ζ)) :=

ρkσk

N∑i=1

Im(ζi)aik +

√f2k (−Im(ζ)) + κk

σ2k

,

in conjunction with the following conditions

(i) f1k (−Im(ζ)) > 0, ∀k = 1, . . . , d;

(ii) f2k (−Im(ζ)) ≥ 0, ∀k = 1, . . . , d;

(iii) f3k (−Im(ζ)) > 0, ∀k = 1, . . . , d;

(iv) f4k (−Im(ζ)) ≥ 0, ∀k = 1, . . . , d.

The transform formula (4.7) is well defined for all t ∈ [0, T ] when the complex vector iζ belongs to thestrip Dt,+∞ = At,+∞ × iRN ⊂ CN , where the convergence set At,+∞ ⊂ RN is given by

At,+∞ :=−Im(ζ) ∈ RN

∣∣ f lk(−Im(ζ)), l = 1, . . . , 4 satisfying (i)-(iv)

Moreover, for iζ ∈ Dt,t? = At,t? × iRN with

At,t? :=−Im(ζ) ∈ RN

∣∣ f lk(−Im(ζ)), l = 1, . . . , 3 satisfying (i)-(iii) and

f4k (−Im(ζ)) < 0 for some k

⊃ At,+∞

the transform is well defined until the maximal time t? given by

t? = mink s.t. f4

k(−Im(ζ))<0

1√Ak

log

(1− 2

√Ak

κk + σkρk∑Ni=1 Im(ζi)aik +

√Ak

). (4.12)

Proof. See Appendix B.

The general transform formula above is a powerful tool, however, checking the validity of (4.7), maynot be very practical in a calibration setting. For this reason, we provide a simple, yet handy, criterion.Recall that we introduced in (2.11) the price P i(t, T ) at time t ∈ [0, T ], 0 ≤ T ≤ T of a zero coupon bondfor one unit of the i-th currency to be paid at T , i = 1, . . . , N , via the following conditional expectation

P i(t, T ) := Di(t)E[

1

Di(T )

∣∣∣∣Ft] = φi,jt,T (0). (4.13)

The criterion is provided by the next lemma.

14

Lemma 4.1. Let −1 < α < 0 and z ∈ C with z = u+ iα. Assume

P i(t, T ) ∨ P j(t, T ) <∞,

then

Di(t)E[

1

Di(T )

(Si,j(T )

)−α∣∣∣∣Ft] <∞,moreover, the discounted characteristic function φi,j(z) admits an analytic extension to the strip

Z = z ∈ C| z = u+ iα, α ∈ (−1, 0) .

Proof. See Appendix C.

Given the result in Lemma 4.1, we shall proceed to calibrate the model by employing the generalizedCarr-Madan formula of Lee, i.e. (4.5), by setting

R(Si,j(t),Ki,j , α

)= φi,j(−i).

5 Model Calibration to FX Triangles

In line with De Col et al. (2013), Gnoatto and Grasselli (2014) and Baldeaux et al. (2015b), we perform ajoint calibration to a triangle of FX implied volatility surfaces. More specifically, we consider the data setemployed in Gnoatto and Grasselli (2014), featuring implied volatility surfaces for EURUSD, USDJPYand EURJPY as of July 22nd 2010. We choose such a date so as to obtain a calibration that can beapproximately compared with the one of De Col et al. (2013), based on data as of July 23rd 2010. Weperform our calibration to options with expiry dates ranging from one up to 18 months and moneynessranging from 15 delta put up to 15 delta call, and we consider a total of 126 contracts. The model weconsider for the calibration is the full 4/2 stochastic volatility model, i.e. both the Heston and the 3/2effects are simultaneously considered. As we calibrate options with maturity up to 18 months, we do notconsider stochastic interest rates due to the limited interest rate risk, see Gnoatto and Grasselli (2014).In line with the references above, we choose the following penalty function∑

i

(σimpi,mkt − σ

impi,model

)2

,

where σimpi,mkt is the i-th observed market volatility and σimpi,model is the i-th model-derived implied volatility.

For each option contract, σimpi,model is constructed along the following steps: first, given a set of modelparameters, (4.5) for −1 < α < 0 is employed so as to obtain the corresponding model derived price,secondly, the obtained price is converted into σimpi,model, via a standard implied volatility solver. As far asthe implementation of (4.5) is concerned, we approximated the integral via a 4096-point FFT routine,with grid spacing equal to 0.1, so that the improper integral is truncated at the point e409 . Thecorresponding strike range is then given by

[e−31.4159, e31.4159

]and Simpson’s rule weights are introduced

for increased accuracy, see Carr and Madan (1999). The FFT returns then a vector of option prices fora fixed grid of strikes. Option prices for the strikes of interest are obtained via a linear interpolation.We assume that the model is driven by two square root factors. The parameters we need to calibrateare given by those appearing in the dynamics of each square root process, i.e. κk, θk, σk, Vk(0), k = 1, 2,coupled with a two-dimensional vector of correlations and six two-dimensional vectors of projections foreach currency area, i.e. ai, bi, i = 1, 2, 3, meaning that we proceed to estimate a total of 22 parameters.Clearly, in order to prevent instability and over-parametrization issues, simplified versions of the modelmay be considered.The result of the calibration is presented in Figure 2a for July 22nd, 2010, while the corresponding pa-rameters are reported in Table 1. We obtain a good fit over all three surfaces we consider, in line with

15

De Col et al. (2013). This shows that a satisfactory calibration of the model can be achieved. It allows usto perform the following analysis, which constitutes an interesting empirical result of the current paper.Given the set of parameters we obtain from the calibration, we can try to analyze whether market data ofFX options are supporting the common use of classical risk neutral pricing. Our approach is so flexible,that we can, in the setting of a single model, span both the risk-neutral valuation and the pricing underthe real-world measure. Such an analysis is summarized in Table 2. We consider different measures forpricing: the real world probability measure P, and the putative risk neutral measures Qusd, Qeur andQjpy. For each measure we compute the corresponding Feller condition for each square root process.

Under the real world probability measure P, we observe that the Feller condition 2κkθk ≥ σ2k is satisfied

by both V1 and V2. We next proceed to perform the same analysis under the two putative risk neutralmeasures Qusd and Qjpy, respectively. We observe that for the first one we still have that both processesdo never reach zero, whereas for Qjpy we have that the Feller condition is not satisfied by the secondcomponent. As discussed in Section 3.3, if at least one of the square root processes has a different be-havior under the putative risk-neutral measure, then we have that classical risk-neutral pricing is notwell founded. In summary, we have a situation where market data suggest that for the USD currencydenomination risk-neutral pricing is potentially applicable, while in the JPY denomination it is not the-oretically founded.

We also perform a second calibration experiment. The structure of the sample of the dataset is the sameas in the previous case and market data were provided as of Feb 23rd, March 23rd, April 22nd, May22nd and June 22nd, 2015. Essentially, we are taking the perspective of a derivative desk following themarket practice that involves a periodic model re-calibration across different trading dates. Such analysisallows us to provide some first evidence regarding the stability of the parameter estimates we obtain. Bylooking at Table 5, we observe a satisfactory stability of the calibrated parameters. A relevant changein the estimates is observed only between the February and March calibration. The quality of the fit iscomparable with the one obtained in our first calibration and the above mentioned papers of Baldeauxet al. (2015b) and De Col et al. (2013).Calibrated parameter values are listed in Table 5, whereas the Feller condition under all measures isreported in Table 6. We observe in this case a violation of the Feller condition for the second factor underthe Qusd putative risk neutral measure, whereas for the Qjpy measure the condition is passed. For Qeur,instead, we observe that the condition is initially passed and then, starting from April 22nd, we haverepeated violations. The overall results of our analysis allow us to suggest that markets are subject towhat we may term as regime switches in pricing between the classical risk neutral and the more generalreal world pricing approach. Such a feature would clearly provide a strong motivation for the introductionof models that are able to accomodate both valuation frameworks, like the 4/2 type specification that wepropose.

6 Pricing and Hedging of Long-Dated Zero Coupon Bonds

In the previous section we obtained a prototypical calibration to market data that shows the coexistenceof pricing under the risk-neutral and the more general benchmark approach. In the present section, wetake the calibrated values as given and consider the problem of hedging contingent claims under the 4/2model. We start in Subsection 6.1 by providing the necessary background on quadratic hedging and, inparticular, on benchmarked risk minimization. We restrict our attention to a very simple contingent claim,namely a zero coupon bond. Such an experiment is simple and yet extremely powerful in showing howthe benchmark approach allows for the hedging of contingent claims for a lower cost. The constructionof the hedging scheme requires a martingale representation for the claim under consideration, where theinitial price represents the starting point of the value of the strategy. In Subsection 6.2 we analyze thevaluation formula for a zero coupon bond and explicitly highlight the consequences of the failure of themartingale property from an analytical point of view. Finally, in Subsection 6.3 we explicitly computethe dynamic hedging scheme.

16

6.1 Benchmarked Risk Minimization

The 4/2 model is a stochastic volatility model. Hence, due to the presence of the instantaneous volatilityuncertainty, we have an example of an incomplete market. Hedging claims in an incomplete market settingis a non-trivial task, which may be performed by means of different possible criteria. Incompletenessmeans that, in general, it is not possible to construct a self-financing trading strategy that delivers atmaturity T the final payoff H(T ) almost surely. According to the survey paper of Schweizer (2001), inan incomplete market setting one may relax the requirements on the family of possible trading strategiesin either two directions:

1. A first possibility is to relax the requirement that the terminal value process of the strategy reachesthe final payoff H(T ) a.s., and hence one is induced to minimize the expected quadratic hedgingerror at maturity over a suitable set of self-financing trading strategies. This first approach is calledmean-variance hedging, and is presented in Bouleau and Lamberton (1989), Duffie and Richardson(1991) and Schweizer (1994).

2. Alternatively, one may relax the self-financing requirement and insist on attaining H(T ) a.s. atmaturity T . In this second case one is induced to minimize a quadratic function of the costprocess of the (non-self-financing) trading strategy. This second approach, referred to as local riskminimization, was introduced in Follmer and Sondermann (1986) and then generalized in Schweizer(1991) in a general semimartingale setting assuming the existence of a minimal equivalent martingalemeasure.

In this paper we adopt some type of local risk minimization. It is important to stress that we need toconsider a generalized concept of local risk minimization in the context of the benchmark approach. Afirst generalization in this sense was provided by Biagini et al. (2014). In the current paper, we adopt themore general approach of Du and Platen (2016), known as benchmarked risk minimization, which doesnot require second moment conditions.In Section 2 we introduced the notion of a self-financing trading strategy. We need to generalize ournotation, since we will consider, in general, also non-self-financing trading strategies. To this end, inline with Definition 2.1 of Du and Platen (2016), we will call a dynamic trading strategy, initiated at

t = 0, an RN+1-valued process of the form v =v(t) =

(η(t), ϑ1(t), . . . , ϑN (t)

)>, 0 ≤ t ≤ T ≤ T

for

a predictable B-integrable process ϑ =ϑ(t) = (ϑ1(t), . . . , ϑN (t)), 0 ≤ t ≤ T ≤ T

, which describes the

units invested in the benchmarked primary security accounts B and forms the self-financing part of theassociated portfolio. The corresponding benchmarked value process is then Vv = ϑ(t)>B(t)+η(t), whereη =

η(t), 0 ≤ t ≤ T ≤ T

with η(0) = 0 monitors the non-self-financing part, so that we can write

Vv(t) = Vv(0) +

∫ t

0

ϑ(s)>dB(s) + η(t).

The process η monitors the inflow/outflow of extra capital and hence measures the cost of the strategy,see Corollary 4.2 in Du and Platen (2016). When the monitoring process is a local martingale, we saythat the strategy v is mean-self-financing, see Definition 4.4 in Du and Platen (2016). Moreover, when

the monitoring part η is orthogonal to the primary security accounts, in the sense that η(t)B(t) formsa vector local martingale, we say that the trading strategy v has an orthogonal benchmarked profit andloss, see Definition 5.1 in Du and Platen (2016). Given VH(T ), the set of all mean-self-financing trading

strategies which deliver the final benchmarked payoff H(T ) P-a.s., with an orthogonal benchmarked profitand loss, see Definition 5.2 in Du and Platen (2016), we say that a strategy v ∈ VH(T ) is benchmarked risk

minimizing if, for all strategies v ∈ VH(T ), we have that Vv(t) ≤ Vv(t) P-a.s. for every 0 ≤ t ≤ T ≤ T ,

see Definition 5.3 in Du and Platen (2016).The practical application of the concept of benchmarked risk minimization necessitates the availability ofmartingale representations for benchmarked contingent claims, which will be given under the real-world

17

probability measure P. In the tractable Markovian setting of the 4/2 model, such representation can be

explicitly obtained, so that one represents a benchmarked contingent claim H(T ) in the form

H(T ) = E[H(T )

∣∣∣Ft]+

∫ T

t

ϑ(s)>dB(s) + η(T )− η(t), (6.1)

see Equation (6.1) in Du and Platen (2016), and there exists a benchmarked risk minimizing strategy v

for H(T ). In summary, to determine the strategy, one has first to compute the conditional expectationin (6.1), which will be straightforward in our case due to the analytical tractability of the 4/2 model.In a second step, an application of the Ito formula allows for the identification of the possible holdingsϑ in the self-financing part of the price process Vv. Finally, the monitoring part is given by η(t) =

Vv(t)−ϑ(t)>B(t), which when multiplied with B(t) needs to have zero drift. In the next subsections, wecarry out this procedure for a zero-coupon bond in the Japanese currency denomination, where, based onthe calibration results discussed in Section 5, we observed a failure of the risk neutral paradigm. Noticethat the calibration of Section 5 refers to a model where the short rate is assumed to be constant. We dorecognize that this represents a simplification. Note, instead one could also use as payoff one unit of thesaving accounts to demonstrate the consequences of the failure of the risk neutral approach. However,taking the short rate constant allows us to illustrate easily the impact of the violation of the risk neutralparadigm on zero coupon bonds. Introducing a stochastic short rate can be easily achieved by allowingfor non-zero projection vectors Hi,Gi in (3.7). For an example of a hedging scheme in the presence ofstochastic interest rates we refer to Baldeaux et al. (2015a).

6.2 Pricing of a Long-Dated Zero Coupon Bond

In this subsection we first quantify the impact of the violation of the classical risk neutral assumption onthe pricing of a zero coupon bond written on JPY as domestic currency when assuming a constant interestrate rJPY . If risk neutral valuation were possible in this market, then pricing as well as hedging of thiselementary security would be trivial: It would consist in keeping the JPY amount exp(−rJPY (T − t)) inthe domestic bank account BJPY and nothing in any other asset.However, for the calibrated market we showed that under the 4/2 model the risk neutral probabilitymeasure most likely does not exist for the JPY currency denomination, so that risk neutral pricing isnot allowed. As a consequence, we price the zero coupon bond under the real world probability measureusing the real world pricing formula (2.11). Recall that in our stochastic volatility context the marketis incomplete, so that perfect hedging is not possible. As already indicated, we adopt benchmarked riskminimization, developed in Du and Platen (2016), and find the corresponding mean-self-financing hedgingstrategy that delivers the payoff.The first step consists in finding the real-world price of the zero coupon bond by using the real-worldpricing formula. Assume that the domestic currency is indexed by i = 1 and set d = 2. Then we focuson the zero coupon bond value in domestic currency paying one unit of domestic currency:

P 1(t, T ) = D1(t)E[

1

D1(T )

∣∣∣∣Ft]= D1(t)Ψt,T (i, 0),

18

where Ψt,T (i, 0) = Ψt,T (i, 0, ..., 0), see (4.2). Now, from Theorem 4.1 we have for k = 1, ..., d that

αk = −ρkσkb1k;

λk = −ρkσka1k;

µk = ρka1k

(κkσk

+ρka

1k

2

);

νk = ρkb1k

(−κkθk

σk+ρkb

1k

2+ρkσkb

1k

2

);

Ak = (κk + σkρka1k)2

and the argument of the hypergeometric function in (4.7) becomes

β2k(t, Vk)

4(λk +Kk(t))= Vk

2

σ2k

(κk + σkρka1k)(e√Ak(T−t) − 1

)−1

.

The parameter mk becomes

mk =2

σ2k

|κkθk −σ2k

2− ρkσkb1k|.

Since we consider the case where the JPY is the domestic currency, we have according to Table 1

m1 =2

σ21

(κ1θ1 −

σ21

2− ρ1σ1b

11

)> 0;

m2 = − 2

σ22

(κ2θ2 −

σ22

2− ρ2σ2b

12

)> 0.

Notice that for k = 1 we have

1

2+m1

2− α1 +

κ1θ1

σ21

=2

σ21

(κ1θ1 − ρ1σ1b

11

)= m1 + 1,

which implies that the confluent hypergometric function becomes a simple exponential and the factorinvolving the term k = 1 in the expression of Ψ simplifies to 1, so that no contribution to the formulacomes from the term k = 1. This is not surprising, as it is related to a volatility factor for which the riskneutral measure exists, related to the martingale part that simplifies in the conditional expectation.The situation is different for k = 2, where it turns out that

1

2+m2

2− α2 +

κ2θ2

σ22

= 1.

19

Now the expression in (4.7) becomes

Ψt,T (i, 0) = e−Y1(t)

exp(−ρ2

2a12b

12τ −

ρ2κ2

σ2(b12 − a1

2θ2)τ +κ22θ2σ22τ)(√

A2

σ22

)m2

(coth

(√A2τ2

)+ 1)(

sinh(√

A2τ2

))1+m2

Γ(1 +m2)

.V m22 exp

(V2

√A2τ

2

(1− coth

(√A2τ

2

))).1F1

(1, 1 +m2, V2

2√A2

σ22

(e√A2τ − 1

)−1)

=exp

(−ρ2

2a12b

12τ −

ρ2κ2

σ2(b12 − a1

2θ2)τ +κ22θ2σ22τ)(√

A2

σ22

)m2

(coth

(√A2τ2

)+ 1)(

sinh(√

A2τ2

))1+m2

Γ(1 +m2)

.V m22 exp

(−V2

2√A2

σ22

(e√A2τ − 1

)−1)

.1F1

(1, 1 +m2, V2

2√A2

σ22

(e√A2τ − 1

)−1).

Using the Kummer transformation e−z1 F1(a, b, z) =1 F1(b−a, b,−z), see Equation (13.1.27) in Abramowitzand Stegun (1965), we get

Ψt,T (i, 0) = e−Y1(t)

exp(−ρ2

2a12b

12τ −

ρ2κ2

σ2(b12 − a1

2θ2)τ +κ22θ2σ22τ)(√

A2

σ22

)m2

(coth

(√A2τ2

)+ 1)(

sinh(√

A2τ2

))1+m2

Γ(1 +m2)

.V m22 1F1

(m, 1 +m2,−V2

2√A2

σ22

(e√A2τ − 1

)−1).

We now use the relation 1F1(a, a + 1,−z) = az−aγ(a, z), see Equation (13.6.10) in Abramowitz andStegun (1965), where γ(a, z) denotes the (lower) incomplete Gamma function defined as

γ(a, z) :=

∫ z

0

ta−1e−tdt.

After some computations we finally obtain the price of a zero coupon bond as

P 1(t, T ) = e−r1τ

γ

(m2, V2

2√A2

σ22

(e√A2τ − 1

)−1)

Γ(m2), (6.2)

where we recall that√A2 = κ2 + σ2ρ2a

12 and τ = T − t.

We observe that the function Ψt,T (i, 0) is decreasing with respect to τ = T−t, so that we can analyticallyconfirm that the failure of the risk neutral assumption has a direct impact on the price of long datedsecurities such as zero coupon bonds. In Table 4 we compute the price of a zero coupon bond in theJPY market for which the risk neutral probability measure does not exist. We emphasize that we obtainprices that are markedly lower than those typically given by the risk neutral approach. As a comparison,in Table 3 we compare, for the USD currency denomination, the price obtained from the (trivial) riskneutral formula for deterministic short rate, and the corresponding application of the real-world pricingformula for the zero-coupon bond, which follows from Theorem 4.1. Since, according to our calibrationfor the USD currency denomination, a risk-neutral measure Qusd exists, (see the corresponding Feller testin Table 2) we observe that the risk neutral and the real-world pricing formula coincide, thus, providingus with a concrete example of the correspondence we highlighted in (2.10).

20

Notice in (6.2) that P 1(t, T ) < e−r1τ , meaning that under the benchmark approach, hedging the unit

amount of JPY turns out to be less expensive in comparison to the trivial risk neutral strategy consistingin putting the amount e−r

1τ in the domestic money market account B1 as suggested under the riskneutral paradigm. On the other hand, benchmarked risk minimization requires a more sophisticatedhedging strategy. In Subsection 6.3 below, we describe such hedging involving the GOP and two moneymarket accounts. We are not introducing a sort of inconsistency in the pricing procedure since thebenchmark approach allows to associate formally the amount

e−r1τ

to the price of a zero coupon bond in our constant interest rate setting. Note that this is not the minimalpossible price, as there exists no equivalent risk neutral probability measure. On the other hand, thereal-world pricing formula provides under the given model the minimal possible price. The existence ofthose two self-financing price processes does not present an economically meaningful arbitrage opportu-nity because the best performing portfolio in the long run of this market, the GOP, does not explode infinite time. As pointed out by Loewenstein and Willard (2000), the classical no-arbitrage concept that isequivalent to the existence of an equivalent risk neutral probability measure is too restrictive.

One may wonder if this from the classical theory perspective observed ”pricing puzzle” represents apathological or marginal situation. To give an answer to this question, we repeat the pricing experiment,by employing the calibrated parameters of Table 5. Tables 7 - 11 report the values of the correspondingzero coupon bond prices for all the maturities available from a major provider. Note that also in thisexperiment, starting from maturities longer than ten years, the differences between the classical and thebenchmark approach become substantial in so far as benchmarked risk minimization leads to much lowerprices. By comparing the violations of the Feller test and the pricing results, we observe a one to onecorrespondence between the two phenomena.

6.3 Dynamic Hedging of a Long-Dated Zero Coupon Bond

In this subsection, we develop the second part of our analysis that involves the determination of thedynamic part of the benchmarked risk minimizing strategy, namely the holdings in the benchmarkedprimary security accounts and the non-self-financing/monitoring part. First of all we compute the SDEfor the martingale representation associated to the benchmarked price of the zero coupon bond, that wedenote by P 1(t, T ) = P 1(t, T )/D1(t). We obtain

dP 1(t, T ) = −Ψt,T (i, 0)

(a1

1

√V1(t) +

b11√V1(t)

)dZ1(t)

−Ψt,T (i, 0)

(a1

2

√V2(t) +

b12√V2(t)

)dZ2(t)

+e−r

1τ

D1(t)Γ(m2)

(2√A2

σ22

(e√A2τ − 1

))m2

.eV2

2√A2

σ22

(e√A2τ−1

)−1

Vm2−1/22 σ2

(ρ2dZ2(t) +

√1− ρ2

2dZ⊥2 (t)

),

which we rewrite, in line with Du and Platen (2016), Proposition 7.1, as

dP 1(t, T ) = x1(t)dZ1(t) + x2(t)dZ2(t) + x3(t)dZ⊥2 (t), (6.3)

21

with the implicit introduction of the respective shorthand notations x1, x2, x3. Let us recall that thecomponents of the vector of primary benchmarked assets B are

B1(t) =B1(t)

D1(t),

B2(t) =B2(t)S1,2(t)

D1(t)=B2(t)

D2(t),

B3(t) =B3(t)S1,3(t)

D1(t)=B3(t)

D3(t),

where, for i = 1, 2, 3 we have

dBi(t) = −Bi(t)

(ai1√V1(t) +

bi1√V1(t)

)dZ1(t)

− Bi(t)

(ai2√V2(t) +

bi2√V2(t)

)dZ2(t).

We introduce the 3× 3 matrix Φi,kt as in Equation (7.2) of Du and Platen (2016) in the form

Φi,k(t) =

aik√Vk(t) +

bik√Vk(t)

, k = 1, 2; i = 1, 2, 3

1, k = 3; i = 1, 2, 3.(6.4)

We now define the vector ξ(t) = (−x1(t),−x2(t),Ψt,T (i, 0)− x3(t))>

, so that the investment in theself-financing part is given as follows:

ϑ1(t) = Diag−1(B1(t))

((Φi,kt

)>)−1

ξ(t), (6.5)

see Equation (7.4) in Du and Platen (2016). The non-self-financing or monitoring part of the strategy isthen given by the local martingale

η1(t) =

∫ t

0

x3(s)dZ⊥2 (s), (6.6)

which is orthogonal to B. Thus, the conditional expectation (6.2), the vector (6.5) and the local mar-tingale (6.6) fully determine the benchmarked risk minimizing strategy for the zero coupon bond in theJapanese currency denomination.The vector ϑ1(t) describes the number of units held in the primary security accounts, and the non-hedgeable part η1(t) in (6.6) forms a local martingale orthogonal to the benchmarked primary security

accounts. Recall that orthogonality means that the product η1(t)B(t) forms a vector local martingale.In this sense the hedge error η1(t) is minimised. A special case of the 4/2 model is the 3/2 model orminimal market model, where one can see that in this case the hedge is involving only the GOP and thedomestic money market account, see Platen and Heath (2010). As shown in Baldeaux et al. (2015b), formany years to maturity the above strategy invests mainly in the GOP and slides later more and moreinto investing in the domestic money market account. Thus, the above strategy makes financial planningrigorous under the 4/2 model. Similarly, one could now study real-world pricing and hedging of Europeanput options on the GOP and other derivatives, which may become significantly less expensive than theirformally risk neutral priced counterparts.

7 Conclusion

In this paper we introduced a more general modeling world than available under the classical no-arbitrageparadigm in finance. In the context of a flexible model for exchange rates calibrated to market data, we

22

showed how to hedge a zero coupon bond with a smaller amount of initial capital than required under theclassical risk neutral paradigm. Moreover, the corresponding hedging strategy is no longer only investingin fixed income as in the risk neutral classical world: on the contrary it suggests to invest first primarilyin the GOP, that is risky securities and, when approaching more and more the maturity date, it increasesalso more and more the fraction invested in fixed income. Based on the benchmark approach, this givesa quantitative validation to the conventional wisdom of financial planners. Their rule of thumb has beentypically described in the economic literature via models of optimal life-cycle consumption expendituresand saving, see e.g. Gourinchas and Parker (2002), whereas our approach can be seen as a market impliedalternative. It does not postulate a particular form for the utiliy function of a representative agent andonly asks for the minimal possible price process that can be reasonably hedged to deliver the bond payoff.We also showed that the risk neutral price of a long dated zero coupon bond can be significantly higherthan the minimal possible price obtained using the benchmark approach under the real world probabilitymeasure.The main mathematical phenomenon underlying these surprising effects is the potential strict super-martingale property of benchmarked savings accounts under the real world probability measure, as sug-gested in several cases by our calibration exercise on the foreign exchange option market. The presentedresults represent only some example for new phenomena that can be captured under the benchmarkapproach.There is ample room for many new research questions and interesting related econometric studies. Ourstochastic volatility model allows for a non linear market price of volatility risk, a desiderable property inorder to explain non-linear effects under the real-world probability measure for the risk factors. Second,we performed our calibrations at some particular trading dates. However, much more information couldbe extracted from a statistical estimation on a time series of option prices. The proposed 4/2 modelallows for the possibility of a failure of the classical risk neutral pricing assumption and deserves moretheoretical and numerical investigation. In view of this, our paper aims to stimulate further econometricstudies.

A Proof of Lemma 3.1

Let us first observe from (3.1) and (3.8) that in the present setting we have π1(t) = λ(t) and π2(t) =λ(t)−

√V (t).

Given the general specification of the GOP dynamics, we immediately have for Case 1:

dD1(t)

D1(t)= r1(t)dt+ a

√V (t)

(a√V (t)dt+ dZ(t)

),

dD2(t)

D2(t)= r2(t)dt+ (a− 1)

√V (t)

((a− 1)

√V (t)dt+ dZ(t)

),

which are both Heston-type models.For Case 2., we have

dD1(t)

D1(t)= r1(t)dt+

b√V (t)

(b√V (t)

dt+ dZ(t)

),

dD2(t)

D2(t)= r2(t)dt+

(b√V (t)

−√V (t)

)((b√V (t)

−√V (t)

)dt+ dZ(t)

),

so that D1 follows a 3/2 model (see Heston (1997), Platen (1997)) whereas D2 follows a 4/2 type model

23

(see Grasselli (2016)). On the other hand, for Case 3., we have

dD1(t)

D1(t)= r1(t)dt+

(a√V (t) +

b√V (t)

)((a√V (t) +

b√V (t)

)dt+ dZ(t)

),

dD2(t)

D2(t)= r2(t)dt+

((a− 1)

√V (t) +

b√V (t)

)

×

(((a− 1)

√V (t) +

b√V (t)

)dt+ dZ(t)

),

which are both 4/2 type processes.

B Proof of Theorem 4.1

We start by directly considering the conditional expectation for Ψt(ζ). We perform several manipulationsso that the results of Grasselli (2016) can be directly employed. To this end, we parametrize the correlationstructure from Assumption 3.2 by writing

Z(s) = Diag(ρ)W (s) +√Id −Diag(ρ)2W⊥(s),

24

so that we have

Ψt(ζ) = exp

N∑i=1

iζi(Y i(t) + hi(T − t)

)

× E

[exp

N∑i=1

(iζi∫ T

t

〈Hi,V (s)〉+ 〈Gi,V (s)−1〉ds

+iζi

2

∫ T

t

πi(s)>πi(s)ds+ iζi∫ T

t

πi(s)>dZ(s)

)∣∣∣∣∣Ft]

= exp

N∑i=1

iζi(Y i(t) + hi(T − t)

)

× E

[exp

N∑i=1

(iζi∫ T

t

〈Hi,V (s)〉+ 〈Gi,V (s)−1〉ds

+iζi

2

∫ T

t

πi(s)>πi(s)ds+

∫ T

t

iζiπi(s)>Diag(ρ)dW (s)

)

×E

[exp

N∑i=1

∫ T

t

iζiπi(s)>√Id −Diag(ρ)2dW⊥(s)

∣∣∣∣∣V]∣∣∣∣∣Ft

]

= exp

N∑i=1

iζi(Y i(t) + hi(T − t)

)

× E

[exp

N∑i=1

(iζi∫ T

t

〈Hi,V (s)〉+ 〈Gi,V (s)−1〉ds

+iζi

2

∫ T

t

πi(s)>πi(s)ds+

∫ T

t

iζiπi(s)>Diag(ρ)dW (s)

+1

2

∫ T

t

(N∑i=1

iζiπi(s)>

)(Id −Diag(ρ)2

)( N∑i=1

iζiπi(s)

)ds

)∣∣∣∣∣Ft]

= exp

N∑i=1

iζi(Y i(t) + hi(T − t)

)

25

×d∏k=1

E

[exp

N∑i=1

(iζi∫ T

t

Hik, Vk(s) +Gik, Vk(s)−1ds

+iζi

2

∫ T

t

πik(s)>πik(s)ds+

∫ T

t

iζiπik(s)>ρkdlWk(s)

+1− ρ2

k

2

∫ T

t

(N∑i=1

iζiπik(s)

)2

ds

∣∣∣∣∣∣Ft .

Let us rewrite the final term in the exponent as follows

1− ρ2k

2

∫ T

t

(N∑i=1

iζiπik(s)

)2

ds

=1− ρ2

k

2

∫ T

t

(N∑i=1

iζiaik√Vk(s)

)2

+

N∑i=1

N∑j=1

iζiiζj(aikbjk + ajkb

ik)

(N∑i=1

iζibik1√Vk(s)

)2

ds

=

1− ρ2k

2

∫ T

t

Vk(s)ds

N∑i=1

N∑j=1

iζiiζjaikajk

+

N∑i=1

N∑j=1

iζiiζj(aikbjk + ajkb

ik)

+

∫ T

t

1

Vk(s)ds

N∑i=1

N∑j=1

iζiiζjbikbjk

.

Hence we have

Ψt(ζ) = exp

N∑i=1

iζi(Y i(t) + hi(T − t)

)

×d∏k=1

exp

1− ρ2k

2

N∑i=1

N∑j=1

iζiiζj(aikbjk + ajkb

ik)(T − t) +

N∑i=1

iζiaikbik(T − t)

× E

[exp

N∑i=1

(iζi∫ T

t

HikVk(s) +

GikVk(s)

ds+iζi

2(aik)2

∫ T

t

Vk(s)ds

+iζi

2(bik)2

∫ T

t

1

Vk(s)ds+ iζiρka

ik

∫ T

t

√Vk(s)dWk(s)

+ iζiρkbik

∫ T

t

1√Vk(s)

dWk(s) +1− ρ2

k

2

∫ T

t

Vk(s)ds

N∑i=1

N∑j=1

iζiiζjaikajk

+

∫ T

t

1

Vk(s)ds

N∑i=1

N∑j=1

iζiiζjbikbjk

∣∣∣∣∣∣Ft

= exp

N∑i=1

iζi(Y i(t) + hi(T − t)

)

26

×d∏k=1

exp

1− ρ2k

2

N∑i=1

N∑j=1

iζiiζj(aikbjk + ajkb

ik)(T − t) +

N∑i=1

iζiaikbik(T − t)

E

exp

N∑i=1

iζiHik +

iζi

2(aik)2 +

1− ρ2k

2

N∑j=1

iζiiζjaikajk

∫ T

t

Vk(s)ds

+

N∑i=1

iζiGik +iζi

2(bik)2 +

1− ρ2k

2

N∑j=1

iζiiζjbikbjk

∫ T

t

1

Vk(s)ds

+

N∑i=1

(iζiρka

ik

∫ T

t

√Vk(s)dWk(s) + iζiρkb

ik

∫ T

t

1√Vk(s)

dWk(s)

)∣∣∣∣∣Ft].

Let us recall the well-known relations∫ T

t

√Vk(s)dWk(s) =

1

σk(Vk(T )− Vk(t))−

∫ T

t

κkσk

(θk − Vk(s))ds

and ∫ T

t

1√Vk(s)

dWk(s) =1

σklnVk(T )

Vk(t)−∫ T

t

κkθkσk− σk

2

Vk(s)ds+

κkσk

(T − t).

27

Substitution of the above relations yields

Ψt(ζ) = exp

N∑i=1

iζi(Y i(t) + hi(T − t)

)

×d∏k=1

exp

1− ρ2k

2

N∑i=1

N∑j=1

iζiiζj(aikbjk + ajkb

ik)(T − t) +

N∑i=1

iζiaikbik(T − t)

× E

exp

N∑i=1

iζiHik +

iζi

2(aik)2 +

1− ρ2k

2

N∑j=1

iζiiζjaikajk + iζiρka

ik

κkσk

∫ T

t

Vk(s)ds

+

N∑i=1

iζiGik +iζi

2(bik)2 +

1− ρ2k

2

N∑j=1

iζiiζjbikbjk −

iζiρkbik

σk

(κkθk −

σ2k

2

)∫ T

t

1

Vk(s)ds

+

N∑i=1

[iζiρka

ik

(Vk(T )− Vk(t)

σk− κkθk

σk(T − t)

)+ iζiρkb

ik

(1

σklog

(Vk(T )

vk(t)

)+κkσk

(T − t))]∣∣∣∣Ft]

= exp

N∑i=1

iζi(Y i(t) + hi(T − t)

)

×d∏k=1

exp

N∑i=1

(T − t)

1− ρ2k

2

N∑j=1

iζiiζj(aikbjk + ajkb

ik)

+iζiaikbik + iζi

κkρkσk

(bik − θkaik

))− Vk(t)iζi

ρkaik

σk− iζi

ρkbik

σklog(Vk(t))

× E

[Vk(T )−αke

−λkVt(T )−µk∫ TtVk(s)ds−νk

∫ Tt

1Vk(s)

ds∣∣∣Ft] ,

where we introduced (4.8)-(4.11). The result is obtained via an iterated application of Theorem D.1 from

Grasselli (2016) with a = κkθk, b = κk, σ =σ2k

2 and τ = T − t

E[Vk(T )−αke

−λkVk(T )−µk∫ TtVk(s)ds−νk

∫ Tt

dsVk(s)

]=

(βk(τ, Vk(t))

2

)mk+1

Vk(t)−κkθk

σ2k (λk +Kk(τ))

−(

12 +

mk2 −αk+

κkθkσ2k

)

× e1

σ2k

(κ2kθk(T−t)−

√AkVk(t) coth

(√Ak(T−t)

2

)+κkVk(t)

)Γ(

12 + mk

2 − αk + κkθkσ2k

)Γ(mk + 1)

× 1F1

(1

2+mk

2− αk +

κkθkσ2k

,mk + 1,β2k(τ, Vk(t))

4(λk +Kk(τ))

),

(B.1)

28

with

mk =2

σ2k

√(κkθk −

σ2k

2

)2

+ 2σ2kνk, (B.2)

Ak = κ2k + 2µkσ

2k, (B.3)

βk(τ, Vk(t)) =2√AkVk(t)

σ2k sinh

(√Ak(T−t)

2

) , (B.4)

Kk(τ) =1

σ2k

(√Ak coth

(√Ak(T − t)

2

)+ κk

). (B.5)

Concerning the convergence set, we have

E[∣∣∣ei〈ζ,Y (T )〉

∣∣∣∣∣∣Ft] = E[e−〈Im(ζ),Y (T )〉

∣∣∣Ft]so that we are induced to study the regularity at the point −Im(ζ). Now, for each k = 1, . . . , d Equation(D.3) up to (D.6) from Grasselli (2016) allow us to obtain conditions i-iv on f lk(−Im(ζ)), l = 1, . . . , 4.Also, (4.12) can be directly inferred from (D.13). The conclusions on the convergence set follow alongthe arguments of Grasselli (2016).

C Proof of Lemma 4.1

Let −1 < α < 0, then we have

φi,j(z) = Di(t)E[

1

Di(T )eizx

i,j(T )

∣∣∣∣Ft]≤ Di(t)E

[1

Di(T )e−αx

i,j(T )

∣∣∣∣Ft] = Di(t)E

[1

Di(T )

(Di(T )

Dj(T )

)−α∣∣∣∣∣Ft].

We introduce, in line with Section 7.3.3 of Baldeaux and Platen (2013), a forward measure PT withassociated numeraire given by the zero coupon bond price process P i(., T ) =

P i(t, T ), 0 ≤ t ≤ T

.

Recall that the minimal possible benchmarked price of a derivative is a martingale under the benchmarkapproach, hence the benchmarked zero coupon bond price allows us to define the following density process

∂PT

∂P

∣∣∣∣Ft

= Zt :=P i(t, T )Di(0)

P i(0, T )Di(t), t ≤ T.

Using such a forward measure, in conjunction with Jensen’s inequality for concave functions, allows usto write

Di(t)E

[1

Di(T )

(Di(T )

Dj(T )

)−α∣∣∣∣∣Ft]

= P i(t, T )EPT[(

Di(T )

Dj(T )

)−α∣∣∣∣∣Ft]

≤ P i(t, T )EPT[(

Di(T )

Dj(T )

)∣∣∣∣Ft]−α = P i(t, T )

(Di(t)

P i(t, T )

)−αE[(

1

Dj(T )

)∣∣∣∣Ft]−α= P i(t, T )

(Di(t)

Dj(t)

)−α(P j(t, T )

P i(t, T )

)−α= P i(t, T )

(Si,j(t)

)−α(P j(t, T )

P i(t, T )

)−α.

which proves the finiteness of the characteristic function. The analytic extension to the set Z is then adirect consequence of Theorem 7.1.1 in Lukacs (1970).

29

D The General Transform of the CIR Process

For convenience we recall the formula in Grasselli (2016) giving the general transform for the CIR processX = Xt, t ∈ [0, T ] that extends formulas in Craddock and Lennox (2009) to the case

Et[X−αT e−λXT−µ

∫ TtXsds−ν

∫ Tt

dsXs

]. (D.1)

Theorem D.1. (Grasselli (2016)) Assume that Xt satisfies the SDE

dXt = (a− bXt)dt+√

2σXtdWt, X0 = x > 0, (D.2)

with a, b, σ > 0 and a > σ (Feller condition). Given α, µ, ν, λ such that

b2 + 4µσ ≥ 0 (D.3)

(a− σ)2 + 4σν ≥ 0 (D.4)

1

2+

a

2σ+

1

2σ

√(a− σ)2 + 4σν > α, (D.5)

λ ≥ −√b2 + 4µσ + b

2σ(D.6)

then the transform (D.1) is well defined for all t ≥ 0 and is given by

E[X−αt e−λXt−µ

∫ t0Xsds−ν

∫ t0dsXs

]=

(β(t, x)

2

)m+1

x−a2σ (λ+K(t))−( 1

2 +m2 −α+ a

2σ )

× e12σ

(abt−

√Ax coth

(√At2

)+bx

)Γ(

12 + m

2 − α+ a2σ

)Γ(m+ 1)

× 1F1

(1

2+m

2− α+

a

2σ,m+ 1,

β2(t, x)

4(λ+K(t))

),

(D.7)

with

m =1

σ

√(a− σ)2 + 4σν, (D.8)

A = b2 + 4µσ, (D.9)

β(t, x) =

√Ax

σ sinh(√

At2

) , (D.10)

K(t) =1

2σ

(√A coth

(√At

2

)+ b

). (D.11)

If

λ < −√b2 + 4µσ + b

2σ, (D.12)

then the transform is well defined for all t < t?, with

t? =1√A

log

(1− 2

√A

b+ 2σλ+√A

). (D.13)

E Figures and Tables

30

0.30.20.10-0.1

b-0.2-0.30Time

5

4000

6000

8000

10000

0

2000

10

Mean

10

Time

5

00.30.20.1

b0-0.1-0.2

#1010

-0.3

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

QV

Figure 1: Simulation of the Radon-Nikodym derivative for the putative risk neutral measure of the i-

th currency denomination Bi(t) = Bi(t)Di(t) given by the SDE (3.9), together with the relative quadratic

variation process. The time horizon is t = 10 years. We consider a one factor specification of the model(i.e. d = 1) and we fix the parameters as κ = 0.49523; θ = 0.53561;σ = 0.67128;V (0) = 1.4338; ρ =−0.89728; a = 0.047360. Parameter b ranges between [−0.4, 0.4]. For b positive the process is a truemartingale, while for b negative the process is a strict local martingale.

Parameter Value Parameter ValueV1(0) 0.8992 V2(0) 1.3011κ1 1.1705 κ2 0.5110θ1 0.6853 θ2 0.5206σ1 0.3980 σ2 0.6786ρ1 -0.8637 ρ2 -0.8925ausd1 0.0729 ausd2 0.0621aeur1 0.0218 aeur2 0.0414

ajpy1 0.1497 ajpy2 0.0687busd1 0.0192 busd2 0.0678beur1 0.1805 beur2 0.0578

bjpy1 -0.0601 bjpy2 -0.0712

Table 1: Parameter values resulting from the calibration procedure of a two factor specification withdeterministic interest rate. Market data as of July 22nd, 2010. Each column corresponds to a volatilityfactor.

Measure Test Value Risk Neutral Pricing Possible

P 2κkθk − σ2k

1.44590.0715

Qusd 2κkθk −(σ2k + 2ρkσkb

usdk

) 1.4591YES

0.1536

Qeur 2κkθk −(σ2k + 2ρkσkb

eurk

) 1.5700YES

0.1415

Qjpy 2κkθk −(σ2k + 2ρkσkb

jpyk

)1.4046

NO-0.0147

Table 2: This table reports the Feller test under different (putative) measures as introduced in Section 3.3.Market data as of July 22nd, 2010. The values of the test for V1 (upper value) and V2 (lower value) areobtained from the calibrated parameters reported in Table 1.

31

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25EURUSD - 1 m

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25EURUSD - 2 m

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25EURUSD - 3 m

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25EURUSD - 6 m

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25EURUSD - 1 y

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25EURUSD - 1.5 y

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25

0.3EURJPY - 1 m

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25

0.3EURJPY - 2 m

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25

0.3EURJPY - 3 m

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25

0.3EURJPY - 6 m

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25

0.3EURJPY - 1 y

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25

0.3EURJPY - 1.5 y

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25USDJPY - 1 m

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25USDJPY - 2 m

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25USDJPY - 3 m

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25USDJPY - 6 m

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25USDJPY - 1 y

15DP 25 DP 35DP ATM 35DC 25DC 15DC0.1

0.15

0.2

0.25USDJPY - 1.5 y

Figure 2: Simultaneous FX calibration. Market data as of July 22nd, 2010. Market volatilities are denotedby crosses, model volatilities are denoted by circles. Moneyness levels follow the standard Delta quotingconvention in the FX option market. DC and DP stand for ”delta call” and ”delta put” respectively.

32

T Risk Neutral Approach Benchmark approach Difference1 week 0.99990 0.99990 1.3922× 10−13

2 weeks 0.99981 0.99981 6.6835× 10−14

3 weeks 0.99971 0.99971 3.0198× 10−14

1 month 0.99959 0.99959 4.6740× 10−14

2 months 0.99917 0.99917 7.3275× 10−15

3 months 0.99879 0.99879 2.3093× 10−14

6 months 0.99740 0.99740 4.6629× 10−15

1 year 0.99377 0.99377 1.5543× 10−15

1.5 years 0.98849 0.98849 4.9960× 10−15

2 years 0.97776 0.97776 6.6613× 10−16

3 years 0.95556 0.95556 1.6653× 10−15

5 years 0.91031 0.91031 5.4401× 10−15

7 years 0.84064 0.84064 1.4211× 10−14

10 years 0.73800 0.73800 1.4100× 10−14

Table 3: Computation of Zero coupon bond prices for the USD market July 22nd, 2010, where risk neutralvaluation is potentially allowed.

T Risk Neutral Approach Benchmark approach Difference1 week 0.99993 0.99993 1.6187× 10−13

2 weeks 0.99987 0.99987 3.2196× 10−14

3 weeks 0.99980 0.99980 9.7811× 10−14

1 month 0.99971 0.99971 8.9928× 10−15

2 months 0.99943 0.99943 1.5543× 10−15

3 months 0.99913 0.99913 8.9928× 10−13

6 months 0.99827 0.99827 1.4083× 10−7

1 year 0.99679 0.99671 7.6955× 10−5

1.5 years 0.99556 0.99482 0.000737262 years 0.99475 0.99229 0.00246173 years 0.99229 0.98319 0.00909515 years 0.98594 0.95580 0.0301407 years 0.96885 0.91430 0.05454410 years 0.92073 0.83195 0.08877715 years 0.83391 0.69949 0.1344220 years 0.74894 0.58296 0.16598

Table 4: Computation of zero coupon bond prices for the JPY market as of July 22nd, 2010, where riskneutral valuation is not possible. The associated price difference increases as the maturity becomes larger.

33

Parameter 02.23.2015 03.23.2015 04.22.2015 05.22.2015 06.22.2015V1(0) 0.73431 1.1740 1.1719 1.1736 1.1801V2(0) 1.1943 1.4352 1.4338 1.4356 1.4345κ1 0.46330 0.48651 0.48812 0.48968 0.48950κ2 0.52193 0.49237 0.49523 0.49191 0.49282θ1 0.35253 0.29821 0.29999 0.30107 0.30082θ2 0.75203 0.53498 0.53561 0.53464 0.53455σ1 0.49603 0.49110 0.49443 0.49604 0.49561σ2 0.66899 0.66938 0.67128 0.67135 0.67102ρ1 −0.99327 −0.86543 −0.85665 −0.86540 −0.86434ρ2 −0.94111 −0.89690 −0.89728 −0.89706 −0.89558ausd1 0.12445 0.13047 0.12885 0.12399 0.12616ausd2 0.062812 0.036533 0.047360 0.058480 0.058355aeur1 0.045806 0.038959 0.049467 0.053229 0.058580aeur2 0.044107 0.054640 0.051325 0.045735 0.048780

ajpy1 0.16458 0.15589 0.14943 0.14803 0.14984

ajpy2 0.057533 0.080895 0.073395 0.067905 0.073300busd1 −0.10105 −0.041716 −0.057950 −0.059954 −0.055886busd2 −0.11178 −0.094706 −0.12360 −0.12433 −0.12753beur1 −0.065631 −0.041879 −0.063733 −0.072158 −0.074323beur2 0.0076657 −0.037459 −0.039780 −0.044664 −0.041933

bjpy1 −0.059432 −0.047197 −0.053469 −0.054806 −0.055076

bjpy2 −0.065707 −0.051282 −0.058217 −0.062116 −0.056492Res. Norm. 0.0131 0.0121 0.0106 0.0269 0.0141

Table 5: Parameter values resulting from the repeated calibration procedure of a two factor specificationwith constant interest rates.

Measure Test 02.23.2015 03.23.2015 04.22.2015 05.22.2015 06.22.2015

P 2κkθk − σ2k

0.0806 0.0490 0.0484 0.0488 0.04890.3375 0.0787 0.0799 0.0753 0.0766

Qusd 2κkθk −(σ2k + 2ρkσkb

usdk

) −0.0189 0.0135 −0.0007 −0.0027 0.00100.1967 −0.0350 −0.0690 −0.0745 −0.0767

Qeur 2κkθk −(σ2k + 2ρkσkb

eurk

) 0.0159 0.0134 −0.0056 −0.0132 −0.01480.3471 0.0338 0.0320 0.0215 0.0262

Qjpy 2κkθk −(σ2k + 2ρkσkb

jpyk

) 0.0221 0.0089 0.0031 0.0017 0.00170.2547 0.0172 0.0097 0.0005 0.0087

PQusd

Risk neutral pricing?NO NO NO NO NO

Qeur YES YES NO NO NOQjpy YES YES YES YES YES

Table 6: This table reports the Feller test under different (putative) measures as introduced in Section 3.3.The values of the test are obtained from the calibrated parameters reported in Table 5.

34

USD

EU

RJP

YT

RN

AB

AD

iffere

nce

TR

NA

BA

Diff

ere

nce

TR

NA

BA

Diff

ere

nce

0.2

5008

0.9

9935

0.9

9935

6.0

851×

10−

13

0.5

0267

0.9

9940

0.9

9940

1.1

990×

10−

13

0.5

0214

0.9

9931

0.9

9932

1.6

497×

10−

7

0.3

1088

0.9

9915

0.9

9915

7.8

564×

10−

11

0.5

8291

0.9

9930

0.9

9931

1.7

908×

10−

13

0.5

8058

0.9

9919

0.9

9919

1.0

711×

10−

7

0.3

8847

0.9

9889

0.9

9889

1.6

583×

10−

90.6

6872

0.9

9920

0.9

9920

2.4

092×

10−

13

0.6

6567

0.9

9907

0.9

9907

2.6

713×

10−

8

0.4

8311

0.9

9851

0.9

9851

1.0

495×

10−

70.7

4971

0.9

9910

0.9

9910

1.2

557×

10−

13

0.7

4738

0.9

9895

0.9

9895

6.0

677×

10−

9

0.5

5772

0.9

9814

0.9

9814

6.5

352×

10−

70.8

4145

0.9

9899

0.9

9899

1.3

323×

10−

15

0.8

4052

0.9

9882

0.9

9882

4.4

317×

10−

11

0.6

5500

0.9

9766

0.9

9765

3.6

805×

10−

60.9

1573

0.9

9890

0.9

9890

1.5

266×

10−

13

0.9

1445

0.9

9871

0.9

9871

1.7

248×

10−

11

0.7

2984

0.9

9720

0.9

9719

1.0

367×

10−

50.9

9918

0.9

9881

0.9

9881

1.7

464×

10−

13

0.9

9979

0.9

9859

0.9

9859

1.8

563×

10−

13

0.8

0747

0.9

9669

0.9

9666

2.4

915×

10−

51.0

940

0.9

9870

0.9

9870

1.8

863×

10−

13

1.4

984

0.9

9789

0.9

9788

4.0

501×

10−

12

1.0

571

0.9

9467

0.9

9449

0.0

0018107

1.1

660

0.9

9862

0.9

9862

4.8

850×

10−

15

2.0

078

0.9

9714

0.9

9714

6.4

399×

10−

11

1.3

036

0.9

9210

0.9

9146

0.0

0063221

1.2

490

0.9

9852

0.9

9852

7.3

275×

10−

15

3.0

043

0.9

9531

0.9

9531

1.7

337×

10−

11

1.5

694

0.9

8872

0.9

8711

0.0

016107

1.3

381

0.9

9842

0.9

9842

2.6

534×

10−

14

4.0

007

0.9

9248

0.9

9249

5.6

222×

10−

13

1.8

185

0.9

8507

0.9

8200

0.0

030724

1.4

160

0.9

9833

0.9

9833

4.1

855×

10−

14

5.0

000

0.9

8843

0.9

8844

2.2

760×

10−

14

2.0

508

0.9

8128

0.9

7635

0.0

049275

1.4

995

0.9

9823

0.9

9823

9.4

147×

10−

14

6.0

010

0.9

8272

0.9

8272

4.7

500×

10−

12

2.3

162

0.9

7646

0.9

6886

0.0

076017

2.0

046

0.9

9756

0.9

9756

7.1

276×

10−

14

6.9

994

0.9

7557

0.9

7557

3.0

261×

10−

12

2.5

623

0.9

7165

0.9

6109

0.0

10564

3.0

006

0.9

9524

0.9

9524

1.3

289×

10−

13

8.0

024

0.9

6682

0.9

6682

5.0

325×

10−

11

2.8

109

0.9

6659

0.9

5261

0.0

13975

3.9

955

0.9

9131

0.9

9131

6.1

114×

10−

12

8.9

971

0.9

5673

0.9

5673

9.7

525×

10−

11

3.0

623

0.9

6131

0.9

4352

0.0

17795

4.9

924

0.9

8537

0.9

8537

6.8

156×

10−

11

9.9

940

0.9

4504

0.9

4504

5.5

121×

10−

10

3.9

848

0.9

4113

0.9

0701

0.0

34118

5.9

886

0.9

7758

0.9

7758

1.7

274×

10−

10

11.9

87

0.9

1834

0.9

1835

4.0

101×

10−

9

4.9

787

0.9

1801

0.8

6411

0.0

53900

6.9

837

0.9

6781

0.9

6781

3.5

236×

10−

10

14.9

76

0.8

6795

0.8

6795

2.3

196×

10−

8

5.9

721

0.8

9434

0.8

2012

0.0

74226

7.9

833

0.9

5645

0.9

5645

6.0

173×

10−

10

19.9

54

0.7

7948

0.7

7948

1.2

711×

10−

7

6.9

654

0.8

7063

0.7

7653

0.0

94102

8.9

739

0.9

4389

0.9

4389

8.7

084×

10−

10

24.9

33

0.7

0159

0.7

0159

3.1

501×

10−

7

7.9

641

0.8

4702

0.7

3393

0.1

1309

9.9

644

0.9

3107

0.9

3107

1.1

074×

10−

929.9

15

0.6

3701

0.6

3701

5.2

597×

10−

7

8.9

545

0.8

2395

0.6

9328

0.1

3067

10.9

57

0.9

1722

0.9

1722

1.3

277×

10−

934.8

87

0.5

7631

0.5

7631

9.1

548×

10−

7

9.9

446

0.8

0111

0.6

5428

0.1

4683

11.9

49

0.9

0298

0.9

0298

1.5

028×

10−

939.8

63

0.5

1867

0.5

1867

1.3

125×

10−

6

10.9

37

0.7

7855

0.6

1696

0.1

6158

14.9

25

0.8

6068

0.8

6068

2.9

209×

10−

8–

––

–

11.9

30

0.7

5639

0.5

8149

0.1

7490

19.8

87

0.7

9260

0.7

9260

4.8

524×

10−

7–

––

–

14.9

09

0.6

9362

0.4

8653

0.2

0709

24.8

50

0.7

3266

0.7

3266

1.3

366×

10−

6–

––

–

19.8

76

0.6

0153

0.3

6187

0.2

3966

29.8

12

0.6

7847

0.6

7847

1.7

194×

10−

6–

––

–

24.8

44

0.5

2388

0.2

7024

0.2

5364

34.7

69

0.6

2967

0.6

2967

2.1

766×

10−

6–

––

–

29.8

10

0.4

5699

0.2

0214

0.2

5484

39.7

35

0.5

8814

0.5

8815

2.6

240×

10−

6–

––

–

39.7

37

0.3

4968

0.1

1375

0.2

3593

44.7

05

0.5

5337

0.5

5337

1.8

836×

10−

7–

––

–

49.6

77

0.2

7553

0.0

65891

0.2

0964

49.6

80

0.5

2631

0.5

2631

4.6

528×

10−

12

––

––

Table 7: Computation of zero coupon bond prices for the USD, EUR and JPY markets as of February23rd, 2015. T refers to the maturity of the bonds. RNA and BA stand for risk neutral and benchmarkapproach respectively.

35

USD

EU

RJP

YT

RN

AB

AD

iffere

nce

TR

NA

BA

Diff

ere

nce

TR

NA

BA

Diff

ere

nce

0.2

5533

0.9

9932

0.9

9932

9.5

861×

10−

11

0.5

1135

0.9

9955

0.9

9955

9.0

594×

10−

14

0.5

1103

0.9

9928

0.9

9928

1.8

545×

10−

7

0.3

0542

0.9

9915

0.9

9915

8.5

724×

10−

11

0.5

8555

0.9

9949

0.9

9949

1.5

099×

10−

14

0.5

8844

0.9

9915

0.9

9915

1.1

805×

10−

7

0.3

9958

0.9

9880

0.9

9880

7.7

737×

10−

90.6

6625

0.9

9942

0.9

9942

1.8

086×

10−

13

0.6

7126

0.9

9903

0.9

9903

3.0

184×

10−

8

0.4

7454

0.9

9848

0.9

9848

4.1

679×

10−

80.7

5840

0.9

9934

0.9

9934

1.3

567×

10−

13

0.7

6384

0.9

9889

0.9

9889

1.0

159×

10−

8

0.5

7174

0.9

9805

0.9

9805

5.2

282×

10−

70.8

3249

0.9

9927

0.9

9927

3.6

859×

10−

14

0.8

3801

0.9

9877

0.9

9878

1.9

202×

10−

11

0.6

4658

0.9

9761

0.9

9761

2.1

247×

10−

60.9

1610

0.9

9920

0.9

9920

1.5

021×

10−

13

0.9

2271

0.9

9865

0.9

9865

1.9

128×

10−

11

0.7

2439

0.9

9714

0.9

9713

6.8

585×

10−

61.0

107

0.9

9912

0.9

9912

3.2

863×

10−

14

1.0

133

0.9

9850

0.9

9851

1.8

328×

10−

10

0.9

7381

0.9

9533

0.9

9524

8.5

524×

10−

51.0

833

0.9

9906

0.9

9906

4.4

187×

10−

14

1.5

090

0.9

9774

0.9

9774

8.3

891×

10−

11

1.2

204

0.9

9304

0.9

9265

0.0

0039463

1.1

657

0.9

9899

0.9

9899

2.2

204×

10−

16

2.0

076

0.9

9688

0.9

9688

7.0

679×

10−

11

1.4

862

0.9

9008

0.9

8888

0.0

011957

1.2

551

0.9

9890

0.9

9890

1.9

984×

10−

15

3.0

039

0.9

9483

0.9

9483

2.0

172×

10−

11

1.7

353

0.9

8687

0.9

8434

0.0

025306

1.3

330

0.9

9883

0.9

9883

1.8

985×

10−

14

4.0

007

0.9

9188

0.9

9188

1.7

073×

10−

12

1.9

680

0.9

8352

0.9

7917

0.0

043494

1.4

165

0.9

9875

0.9

9875

1.8

796×

10−

13

5.0

023

0.9

8762

0.9

8762

1.5

022×

10−

11

2.2

335

0.9

7928

0.9

7218

0.0

071035

1.5

022

0.9

9866

0.9

9866

4.3

410×

10−

14

6.0

010

0.9

8222

0.9

8222

4.6

875×

10−

12

2.4

797

0.9

7506

0.9

6479

0.0

10271

2.0

040

0.9

9804

0.9

9804

4.8

406×

10−

14

6.9

994

0.9

7557

0.9

7557

2.1

447×

10−

12

2.7

283

0.9

7060

0.9

5658

0.0

14016

3.0

007

0.9

9579

0.9

9579

2.5

235×

10−

13

8.0

028

0.9

6771

0.9

6771

6.6

605×

10−

11

2.9

798

0.9

6593

0.9

4763

0.0

18296

3.9

959

0.9

9172

0.9

9172

5.4

496×

10−

12

8.9

976

0.9

5872

0.9

5872

6.0

607×

10−

11

3.2

258

0.9

6110

0.9

3820

0.0

22900

4.9

930

0.9

8605

0.9

8605

6.3

637×

10−

11

9.9

952

0.9

4905

0.9

4905

3.0

073×

10−

10

3.9

863

0.9

4654

0.9

0743

0.0

39111

5.9

895

0.9

7950

0.9

7950

1.1

563×

10−

10

11.9

89

0.9

2458

0.9

2458

2.5

580×

10−

9

4.9

806

0.9

2532

0.8

6252

0.0

62801

6.9

858

0.9

7205

0.9

7205

1.9

374×

10−

10

14.9

79

0.8

7882

0.8

7882

1.4

759×

10−

8

5.9

746

0.9

0349

0.8

1622

0.0

87274

7.9

870

0.9

6366

0.9

6366

2.8

133×

10−

10

19.9

62

0.7

9462

0.7

9462

7.7

188×

10−

8

6.9

686

0.8

8158

0.7

7036

0.1

1122

8.9

767

0.9

5431

0.9

5431

3.6

996×

10−

10

24.9

40

0.7

2118

0.7

2118

2.2

061×

10−

7

7.9

677

0.8

5950

0.7

2547

0.1

3404

9.9

715

0.9

4464

0.9

4464

4.4

520×

10−

10

29.9

23

0.6

5910

0.6

5910

3.7

710×

10−

7

8.9

558

0.8

3790

0.6

8288

0.1

5502

10.9

66

0.9

3435

0.9

3435

5.1

396×

10−

10

34.8

97

0.6

0089

0.6

0089

6.7

433×

10−

7

9.9

494

0.8

1655

0.6

4214

0.1

7441

11.9

60

0.9

2407

0.9

2407

5.5

639×

10−

10

39.8

76

0.5

4584

0.5

4584

9.6

345×

10−

7

10.9

42

0.7

9239

0.6

0114

0.1

9125

14.9

44

0.8

9313

0.8

9313

1.0

177×

10−

8–

––

–

11.9

36

0.7

7470

0.5

6682

0.2

0788

19.9

21

0.8

4496

0.8

4496

1.6

587×

10−

7–

––

–

14.9

16

0.7

1570

0.4

6956

0.2

4614

24.8

92

0.8

0163

0.8

0164

4.3

668×

10−

7–

––

–

19.8

89

0.6

2827

0.3

4345

0.2

8482

29.8

69

0.7

6029

0.7

6029

5.1

555×

10−

7–

––

–

24.8

55

0.5

5373

0.2

5226

0.3

0148

34.8

38

0.7

2270

0.7

2270

6.0

885×

10−

7–

––

–

29.8

27

0.4

8890

0.1

8556

0.3

0334

39.8

13

0.6

8800

0.6

8800

7.0

764×

10−

7–

––

–

39.7

60

0.3

8294

0.1

0093

0.2

8201

44.7

94

0.6

6212

0.6

6212

5.0

278×

10−

8–

––

–

49.7

04

0.3

0658

0.0

56088

0.2

5049

49.7

78

0.6

4076

0.6

4076

2.5

646×

10−

13

––

––

Table 8: Computation of Zero coupon bond prices for the USD EUR and JPY markets as of March23rd, 2015. T refers to the maturity of the bonds. RNA and BA stand for risk neutral and benchmarkapproach respectively.

36

USD

EU

RJP

YT

RN

AB

AD

iffere

nce

TR

NA

BA

Diff

ere

nce

TR

NA

BA

Diff

ere

nce

0.2

5270

0.9

9930

0.9

9930

2.9

560×

10−

11

0.5

1332

0.9

9965

0.9

9965

9.3

880×

10−

11

0.5

1337

0.9

9928

0.9

9929

1.8

200×

10−

7

0.3

1924

0.9

9907

0.9

9907

1.9

363×

10−

10

0.5

8321

0.9

9959

0.9

9959

9.9

334×

10−

10

0.5

8574

0.9

9918

0.9

9918

1.0

922×

10−

7

0.3

9414

0.9

9882

0.9

9882

2.1

562×

10−

90.6

6615

0.9

9954

0.9

9954

8.7

244×

10−

90.6

6845

0.9

9907

0.9

9907

2.7

860×

10−

8

0.4

9117

0.9

9843

0.9

9843

1.8

228×

10−

70.7

5313

0.9

9948

0.9

9948

5.1

639×

10−

80.7

5611

0.9

9896

0.9

9896

7.4

406×

10−

9

0.5

6612

0.9

9807

0.9

9807

1.1

914×

10−

60.8

3296

0.9

9943

0.9

9943

1.9

228×

10−

70.8

3837

0.9

9885

0.9

9885

4.0

214×

10−

11

0.6

4376

0.9

9770

0.9

9769

5.1

237×

10−

60.9

1576

0.9

9937

0.9

9937

5.9

486×

10−

70.9

1738

0.9

9875

0.9

9875

2.0

599×

10−

12

0.7

3806

0.9

9715

0.9

9713

2.0

281×

10−

51.0

029

0.9

9932

0.9

9932

1.6

072×

10−

61.0

051

0.9

9863

0.9

9863

1.7

454×

10−

12

0.8

9345

0.9

9613

0.9

9603

0.0

0010597

1.0

826

0.9

9927

0.9

9927

3.4

880×

10−

61.5

034

0.9

9794

0.9

9794

1.5

695×

10−

11

1.1

401

0.9

9412

0.9

9352

0.0

0060114

1.1

658

0.9

9921

0.9

9920

7.0

272×

10−

62.0

021

0.9

9717

0.9

9717

6.4

672×

10−

12

1.4

061

0.9

9147

0.9

8944

0.0

020368

1.2

524

0.9

9915

0.9

9913

1.3

257×

10−

53.0

016

0.9

9536

0.9

9536

3.3

628×

10−

12

1.6

554

0.9

8855

0.9

8399

0.0

045639

1.3

322

0.9

9909

0.9

9907

2.2

195×

10−

54.0

007

0.9

9262

0.9

9262

8.3

755×

10−

13

1.8

880

0.9

8546

0.9

7736

0.0

080942

1.4

222

0.9

9902

0.9

9898

3.7

160×

10−

55.0

026

0.9

8855

0.9

8855

1.9

938×

10−

11

2.1

536

0.9

8148

0.9

6796

0.0

13519

1.4

991

0.9

9896

0.9

9890

5.5

059×

10−

56.0

068

0.9

8315

0.9

8315

1.8

158×

10−

10

2.3

998

0.9

7743

0.9

5764

0.0

19794

1.9

994

0.9

9848

0.9

9813

0.0

0035315

7.0

025

0.9

7659

0.9

7659

3.5

968×

10−

11

2.6

486

0.9

7311

0.9

4589

0.0

27224

2.9

979

0.9

9656

0.9

9409

0.0

024724

7.9

977

0.9

6880

0.9

6880

3.9

513×

10−

11

2.9

001

0.9

6855

0.9

3285

0.0

35694

3.9

964

0.9

9321

0.9

8626

0.0

069478

8.9

978

0.9

5970

0.9

5970

4.8

851×

10−

11

3.1

462

0.9

6380

0.9

1903

0.0

44765

4.9

942

0.9

8839

0.9

7506

0.0

13324

9.9

955

0.9

5025

0.9

5025

2.4

617×

10−

10

3.9

863

0.9

4677

0.8

6705

0.0

79724

5.9

969

0.9

8263

0.9

6163

0.0

21001

11.9

95

0.9

2638

0.9

2638

5.5

962×

10−

10

4.9

805

0.9

2505

0.8

0044

0.1

2461

6.9

907

0.9

7564

0.9

4631

0.0

29328

14.9

79

0.8

8141

0.8

8141

1.3

357×

10−

8

5.9

799

0.9

0237

0.7

3310

0.1

6927

7.9

833

0.9

6708

0.9

2912

0.0

37961

19.9

58

0.8

0015

0.8

0015

8.8

003×

10−

8

6.9

707

0.8

7959

0.6

6886

0.2

1073

8.9

786

0.9

5803

0.9

1132

0.0

46714

24.9

40

0.7

2892

0.7

2892

2.0

908×

10−

7

7.9

615

0.8

5695

0.6

0858

0.2

4836

9.9

744

0.9

5001

0.8

9452

0.0

55496

29.9

22

0.6

6975

0.6

6975

3.6

738×

10−

7

8.9

547

0.8

3428

0.5

5248

0.2

8180

10.9

69

0.9

3920

0.8

7524

0.0

63966

34.9

05

0.6

1216

0.6

1216

5.4

720×

10−

7

9.9

480

0.8

1206

0.5

0103

0.3

1104

11.9

69

0.9

2950

0.8

5714

0.0

72361

39.8

87

0.5

5796

0.5

5796

7.5

849×

10−

7

10.9

41

0.7

9023

0.4

5400

0.3

3623

14.9

48

0.9

0062

0.8

0468

0.0

95936

––

––

11.9

40

0.7

6854

0.4

1086

0.3

5769

19.9

24

0.8

5925

0.7

2804

0.1

3121

––

––

14.9

14

0.7

0797

0.3

0511

0.4

0286

24.9

00

0.8

1887

0.6

5793

0.1

6094

––

––

19.8

80

0.6

1796

0.1

8565

0.4

3231

29.8

78

0.7

8306

0.5

9659

0.1

8647

––

––

24.8

47

0.5

4144

0.1

1336

0.4

2808

34.8

58

0.7

4921

0.5

4124

0.2

0797

––

––

29.8

14

0.4

7533

0.0

69355

0.4

0598

39.8

41

0.7

1994

0.4

9315

0.2

2679

––

––

39.7

56

0.3

6929

0.0

26153

0.3

4314

44.8

26

0.6

9814

0.4

5343

0.2

4471

––

––

49.6

93

0.2

9391

0.0

10107

0.2

8380

49.8

07

0.6

7931

0.4

1835

0.2

6096

––

––

Table 9: Computation of zero coupon bond prices for the USD, EUR and JPY markets as of April 22nd,2015. T refers to the maturity of the bonds. RNA and BA stand for risk neutral and benchmark approachrespectively.

37

USD

EU

RJP

YT

RN

AB

AD

iffere

nce

TR

NA

BA

Diff

ere

nce

TR

NA

BA

Diff

ere

nce

0.2

5563

0.9

9927

0.9

9927

1.2

972×

10−

10

0.5

1118

0.9

9972

0.9

9972

2.4

469×

10−

10

0.5

1072

0.9

9930

0.9

9930

1.7

574×

10−

7

0.3

0270

0.9

9909

0.9

9909

2.2

813×

10−

10

0.5

8823

0.9

9967

0.9

9967

3.2

072×

10−

90.5

9164

0.9

9917

0.9

9917

1.1

215×

10−

7

0.3

9981

0.9

9873

0.9

9873

3.3

414×

10−

90.6

6611

0.9

9961

0.9

9961

2.3

930×

10−

80.6

7107

0.9

9906

0.9

9906

2.8

733×

10−

8

0.4

7462

0.9

9839

0.9

9839

1.2

277×

10−

70.7

4977

0.9

9954

0.9

9954

1.3

172×

10−

70.7

6137

0.9

9894

0.9

9894

8.5

841×

10−

9

0.5

5216

0.9

9803

0.9

9803

9.3

915×

10−

70.8

4147

0.9

9946

0.9

9946

5.8

497×

10−

70.8

4044

0.9

9883

0.9

9883

6.2

951×

10−

11

0.6

4661

0.9

9752

0.9

9752

5.7

385×

10−

60.9

1575

0.9

9939

0.9

9939

1.5

835×

10−

60.9

1989

0.9

9871

0.9

9871

2.0

474×

10−

11

0.7

2154

0.9

9704

0.9

9702

1.7

472×

10−

50.9

9921

0.9

9930

0.9

9930

4.0

873×

10−

61.0

021

0.9

9860

0.9

9860

7.9

620×

10−

12

0.8

0203

0.9

9653

0.9

9648

4.6

130×

10−

51.0

858

0.9

9922

0.9

9921

9.4

268×

10−

61.5

090

0.9

9785

0.9

9785

4.5

244×

10−

11

1.0

486

0.9

9455

0.9

9418

0.0

0036928

1.1

661

0.9

9914

0.9

9912

1.8

388×

10−

52.0

104

0.9

9701

0.9

9701

9.6

194×

10−

11

1.3

146

0.9

9187

0.9

9036

0.0

015073

1.2

496

0.9

9904

0.9

9901

3.3

775×

10−

53.0

069

0.9

9483

0.9

9483

6.4

644×

10−

11

1.5

638

0.9

8885

0.9

8512

0.0

037304

1.3

324

0.9

9894

0.9

9888

5.7

406×

10−

54.0

032

0.9

9149

0.9

9149

2.1

211×

10−

11

1.7

964

0.9

8558

0.9

7855

0.0

070238

1.4

163

0.9

9884

0.9

9875

9.2

592×

10−

55.0

020

0.9

8657

0.9

8657

1.2

929×

10−

11

2.0

620

0.9

8132

0.9

6903

0.0

12290

1.5

054

0.9

9871

0.9

9857

0.0

0014544

6.0

004

0.9

8003

0.9

8003

8.1

002×

10−

13

2.3

080

0.9

7698

0.9

5844

0.0

18546

1.9

989

0.9

9782

0.9

9694

0.0

0088618

6.9

983

0.9

7171

0.9

7171

2.4

473×

10−

11

2.5

569

0.9

7232

0.9

4622

0.0

26100

3.0

030

0.9

9467

0.9

8858

0.0

060964

8.0

040

0.9

6233

0.9

6233

1.7

680×

10−

10

2.8

083

0.9

6736

0.9

3253

0.0

34826

3.9

973

0.9

8886

0.9

7216

0.0

16701

8.9

986

0.9

5198

0.9

5198

2.6

855×

10−

11

3.0

543

0.9

6217

0.9

1792

0.0

44254

4.9

901

0.9

8026

0.9

4887

0.0

31388

9.9

929

0.9

4066

0.9

4066

8.7

440×

10−

10

3.9

879

0.9

4210

0.8

5656

0.0

85549

5.9

843

0.9

6919

0.9

2068

0.0

48509

11.9

86

0.9

1316

0.9

1316

5.4

188×

10−

9

4.9

787

0.9

1811

0.7

8486

0.1

3325

6.9

776

0.9

5599

0.8

8931

0.0

66678

14.9

77

0.8

6306

0.8

6306

2.2

262×

10−

8

5.9

718

0.8

9314

0.7

1296

0.1

8019

7.9

698

0.9

4126

0.8

5627

0.0

84983

19.9

55

0.7

7348

0.7

7348

1.2

978×

10−

7

6.9

647

0.8

6791

0.6

4427

0.2

2365

8.9

641

0.9

2550

0.8

2257

0.1

0293

24.9

34

0.6

9601

0.6

9601

3.1

878×

10−

7

7.9

656

0.8

4271

0.5

7986

0.2

6285

9.9

525

0.9

0943

0.7

8936

0.1

2007

29.9

15

0.6

2979

0.6

2979

5.4

640×

10−

7

8.9

526

0.8

1802

0.5

2146

0.2

9655

10.9

43

0.8

9252

0.7

5623

0.1

3629

34.8

83

0.5

6855

0.5

6855

1.0

099×

10−

6

9.9

422

0.7

9335

0.4

6797

0.3

2538

11.9

34

0.8

7604

0.7

2442

0.1

5162

39.8

60

0.5

1170

0.5

1170

1.4

134×

10−

6

10.9

35

0.7

6994

0.4

1988

0.3

5006

14.9

09

0.8

2897

0.6

3676

0.1

9221

––

––

11.9

27

0.7

4631

0.3

7613

0.3

7018

19.8

69

0.7

6070

0.5

1638

0.2

4432

––

––

14.9

07

0.6

8031

0.2

7023

0.4

1008

24.8

31

0.7

0451

0.4

2255

0.2

8196

––

––

19.8

71

0.5

8288

0.1

5553

0.4

2735

29.7

88

0.6

5375

0.3

4649

0.3

0726

––

––

24.8

34

0.5

0311

0.0

90162

0.4

1295

34.7

49

0.6

0524

0.2

8342

0.3

2181

––

––

29.8

00

0.4

3411

0.0

52238

0.3

8188

39.7

16

0.5

6625

0.2

3426

0.3

3199

––

––

39.7

21

0.3

2779

0.0

17799

0.3

0999

44.6

88

0.5

3578

0.1

9579

0.3

3999

––

––

49.6

57

0.2

5451

0.0

062285

0.2

4828

49.6

62

0.5

0802

0.1

6398

0.3

4404

––

––

Table 10: Computation of zero coupon bond prices for the USD, EUR and JPY markets as of May 22nd,2015. T refers to the maturity of the bonds. RNA and BA stand for risk neutral and benchmark approachrespectively.

38

USD

EU

RJP

YT

RN

AB

AD

iffere

nce

TR

NA

BA

Diff

ere

nce

TR

NA

BA

Diff

ere

nce

0.2

5554

0.9

9928

0.9

9928

1.0

829×

10−

10

0.5

0840

0.9

9976

0.9

9976

2.2

031×

10−

10

0.5

0820

0.9

9931

0.9

9931

1.6

587×

10−

7

0.3

2460

0.9

9903

0.9

9903

6.2

311×

10−

10

0.5

8660

0.9

9968

0.9

9968

3.1

197×

10−

90.5

8865

0.9

9920

0.9

9920

1.0

811×

10−

7

0.3

9953

0.9

9872

0.9

9872

3.4

141×

10−

90.6

6657

0.9

9961

0.9

9961

2.4

997×

10−

80.6

7119

0.9

9908

0.9

9908

2.6

756×

10−

8

0.4

7725

0.9

9838

0.9

9838

1.3

855×

10−

70.7

4954

0.9

9953

0.9

9953

1.3

705×

10−

70.7

5075

0.9

9897

0.9

9897

6.3

637×

10−

9

0.5

7165

0.9

9791

0.9

9791

1.4

954×

10−

60.8

3597

0.9

9945

0.9

9945

5.6

901×

10−

70.8

3837

0.9

9885

0.9

9885

1.4

196×

10−

11

0.6

4663

0.9

9747

0.9

9747

5.9

978×

10−

60.9

1688

0.9

9937

0.9

9937

1.7

041×

10−

60.9

1767

0.9

9874

0.9

9874

1.5

913×

10−

12

0.7

2707

0.9

9698

0.9

9696

1.9

628×

10−

50.9

9976

0.9

9931

0.9

9931

4.3

878×

10−

61.0

022

0.9

9862

0.9

9862

7.8

938×

10−

12

0.9

7378

0.9

9511

0.9

9489

0.0

0022707

1.0

851

0.9

9922

0.9

9921

1.0

046×

10−

51.5

147

0.9

9786

0.9

9786

2.7

632×

10−

10

1.2

397

0.9

9255

0.9

9143

0.0

011145

1.1

662

0.9

9911

0.9

9909

1.9

817×

10−

52.0

077

0.9

9702

0.9

9702

4.8

335×

10−

11

1.4

890

0.9

8962

0.9

8659

0.0

030258

1.2

545

0.9

9898

0.9

9894

3.7

722×

10−

53.0

041

0.9

9476

0.9

9476

2.4

972×

10−

11

1.7

213

0.9

8641

0.9

8041

0.0

060010

1.3

331

0.9

9887

0.9

9880

6.2

478×

10−

54.0

003

0.9

9123

0.9

9123

6.3

061×

10−

14

1.9

870

0.9

8220

0.9

7128

0.0

10919

1.4

162

0.9

9872

0.9

9862

0.0

0010050

5.0

021

0.9

8608

0.9

8608

1.9

075×

10−

11

2.2

331

0.9

7788

0.9

6100

0.0

16881

1.5

073

0.9

9856

0.9

9840

0.0

0015985

6.0

002

0.9

7940

0.9

7940

4.3

776×

10−

13

2.4

819

0.9

7320

0.9

4904

0.0

24159

2.0

046

0.9

9737

0.9

9638

0.0

0098958

6.9

981

0.9

7137

0.9

7137

2.8

516×

10−

11

2.7

332

0.9

6819

0.9

3556

0.0

32629

2.9

990

0.9

9278

0.9

8607

0.0

067067

8.0

013

0.9

6216

0.9

6216

1.9

938×

10−

11

2.9

790

0.9

6292

0.9

2110

0.0

41819

3.9

921

0.9

8453

0.9

6612

0.0

18405

8.9

960

0.9

5177

0.9

5177

2.2

599×

10−

10

3.2

249

0.9

5739

0.9

0564

0.0

51755

4.9

861

0.9

7255

0.9

3798

0.0

34571

9.9

931

0.9

4151

0.9

4151

8.1

510×

10−

10

3.9

844

0.9

3952

0.8

5396

0.0

85560

5.9

785

0.9

5766

0.9

0445

0.0

53210

11.9

86

0.9

1443

0.9

1443

5.0

668×

10−

9

4.9

774

0.9

1356

0.7

8121

0.1

3235

6.9

693

0.9

4047

0.8

6769

0.0

72777

14.9

75

0.8

6546

0.8

6546

2.5

412×

10−

8

5.9

699

0.8

8650

0.7

0875

0.1

7775

7.9

648

0.9

2159

0.8

2921

0.0

92371

19.9

54

0.7

7896

0.7

7896

1.2

903×

10−

7

6.9

621

0.8

5913

0.6

3974

0.2

1939

8.9

487

0.9

0216

0.7

9108

0.1

1109

24.9

34

0.7

0246

0.7

0246

3.0

852×

10−

7

7.9

596

0.8

3194

0.5

7542

0.2

5652

9.9

375

0.8

8237

0.7

5344

0.1

2893

29.9

15

0.6

3652

0.6

3652

5.2

877×

10−

7

8.9

460

0.8

0541

0.5

1711

0.2

8831

10.9

26

0.8

6257

0.7

1694

0.1

4562

34.8

87

0.5

7692

0.5

7692

9.1

025×

10−

7

9.9

378

0.7

7937

0.4

6388

0.3

1549

11.9

15

0.8

4301

0.6

8188

0.1

6113

39.8

64

0.5

2026

0.5

2026

1.2

892×

10−

6

10.9

29

0.7

5401

0.4

1581

0.3

3820

14.8

81

0.7

8758

0.5

8665

0.2

0093

––

––

11.9

21

0.7

2944

0.3

7257

0.3

5687

19.8

32

0.7

0937

0.4

6013

0.2

4924

––

––

14.8

96

0.6

6049

0.2

6774

0.3

9275

24.7

86

0.6

4764

0.3

6572

0.2

8192

––

––

19.8

58

0.5

6111

0.1

5453

0.4

0657

29.7

44

0.5

9288

0.2

9143

0.3

0145

––

––

24.8

19

0.4

7879

0.0

89580

0.3

8921

34.6

94

0.5

4186

0.2

3191

0.3

0994

––

––

29.7

82

0.4

0948

0.0

52033

0.3

5745

39.6

52

0.4

9796

0.1

8552

0.3

1244

––

––

39.7

01

0.3

0345

0.0

17796

0.2

8565

44.6

17

0.4

6386

0.1

5040

0.3

1346

––

––

49.6

32

0.2

3032

0.0

062287

0.2

2409

49.5

82

0.4

3275

0.1

2212

0.3

1064

––

––

Table 11: Computation of zero coupon bond prices for the USD, EUR and JPY markets as of June22nd, 2015. T refers to the maturity of the bonds. RNA and BA stand for risk neutral and benchmarkapproach respectively.

39

T Risk Neutral Approach Benchmark approach Difference1 week 0.99995 0.99995 5.2625× 10−14

2 weeks 0.99989 0.99989 7.1054× 10−14

3 weeks 0.99984 0.99984 3.4195× 10−14

1 month 0.99977 0.99977 3.6859× 10−14

2 months 0.99953 0.99953 4.8850× 10−14

3 months 0.99930 0.99930 4.5419× 10−13

6 months 0.99832 0.99832 2.7218× 10−7

1 year 0.99502 0.99474 0.000275691.5 years 0.98966 0.98652 0.00314112 years 0.98224 0.97103 0.0112053 years 0.96295 0.92028 0.0426705 years 0.91380 0.78027 0.133537 years 0.85910 0.63795 0.2211510 years 0.77870 0.46128 0.31742

Table 12: Computation of zero coupon bond prices for the USD market as of June 22nd, 2015, whererisk neutral valuation is not possible. The associated price difference increases as the maturity becomeslarger.

40

References

Abramowitz, M. and Stegun, I. A. (1965). Handbook of Mathematical Functions with Formulas, Graphs,and Mathematical Tables. Dover Books on Mathematics. Dover Publications, New York, first edition.

Baldeaux, J., Fung, M. C., Ignatieva, K., and Platen, E. (2015a). A hybrid model for pricing and hedgingof long dated bonds. Applied Mathematical Finance, to appear.

Baldeaux, J., Grasselli, M., and Platen, E. (2015b). Pricing currency derivatives under the benchmarkapproach. Journal of Banking and Finance, 53(0):34–48.

Baldeaux, J. and Platen (2013). Functionals of Multidimensional Diffusions with Applications to Finance.Bocconi & Springer Series, Vol. 5.

Biagini, F., Cretarola, A., and Platen, E. (2014). Local risk-minimization via the benchmark approach.Mathematics and Financial Economics, 8(2):109–134.

Bouleau, N. and Lamberton, D. (1989). Residual risks and hedging strategies in Markovian markets.Stochastic Processes and their Applications, 33(1):131–150.

Carr, P. and Madan, D. B. (1999). Option valuation using the fast Fourier transform. Journal ofComputational Finance, 2:61–73.

Craddock, M. and Lennox, K. (2009). The calculation of expectations for classes of diffusion processesby Lie symmetry methods. Annals of Applied Probability, 19(1):127–157.

De Col, A., Gnoatto, A., and Grasselli, M. (2013). Smiles all around: FX joint calibration in a Multi-Heston model. Journal of Banking and Finance, 37(10):3799–3818.

Du, K. and Platen, E. (2016). Benchmarked risk minimization. Mathematical Finance, 26(3):617–637.

Duffee, G. R. (2002). Term premia and interest rate forecasts in affine models. Journal of Finance,57:405–443.

Duffie, D., Filipovic, D., and Schachermayer, W. (2003). Affine processes and applications in finance.Annals of Applied Probability, 13:984–1053.

Duffie, D. and Richardson, H. (1991). Mean-variance hedging in continuous time. The Annals of AppliedProbability, 1(1):1–15.

Follmer, H. and Sondermann, D. (1986). Hedging of non-redundant contingent claims. In Hildenbrand,W. and Mas-Colell, A., editors, Contributions to Financial Economics, pages 205–223. North-Holland.

Gnoatto, A. and Grasselli, M. (2014). An affine multi-currency model with stochastic volatility andstochastic interest rates. SIAM Journal on Financial Mathematics, 5(1):493–531.

Gourinchas, P. and Parker, J. A. (2002). Consumption over the life cycle. Econometrica, 70(1):47–89.

Grasselli, M. (2016). The 4/2 stochastic volatility model. Mathematical Finance, to appear.

Heath, D. and Platen, E. (2006). Currency derivatives under a minimal market model. The ICFAIJournal of Derivatives Markets, 3:68–86.

Heston, S. (1997). A simple new formula for options with stochastic volatility. Technical report, Wash-ington University of St. Louis.

Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications tobond and currency options. Review of Financial Studies, 6:327–343.

41

Hulley, H. and Ruf, J. (2015). Weak tail conditions for local martingales. Working Paper, available athttp://arxiv.org/abs/1508.07564.

Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer-Verlag.

Lee, R. (2004). Option pricing by transform methods: Extensions, unification and error control. Journalof Computational Finance, 7(3):51–86.

Lewis, A. (2001). A simple option formula for general jump-diffusion and other exponential Levy pro-cesses. Envision Financial Systems and OptionCity.net.

Loewenstein, M. and Willard, G. A. (2000). Local martingales, arbitrage, and viability: Free snacks andcheap thrills. Economic Theory, 2(16):135–161.

Long, J. (1990). The numeraire portfolio. Journal of Financial Economics, 26(1):29–69.

Lukacs, B. (1970). Characteristic Functions. Griffin, London, 2nd edition.

Platen, E. (1997). A non-linear stochastic volatility model. Financial Mathematics Research Report No.FMRR 005-97, Center for Financial Mathematics, Australian National University, Canberra.

Platen, E. and Heath, D. (2010). A Benchmark Approach to Quantitative Finance. Springer-Verlag.

Schweizer, M. (1991). Option hedging for semimartingales. Stochastic Processes and Their Applications,37(2):339–363.

Schweizer, M. (1994). Approximating random variables by stochastic integrals. The Annals of Probability,22(3):1536–1575.

Schweizer, M. (2001). A guided tour through quadratic hedging approaches. In Jouini, E., Cvitanic,J., and Musiela, M., editors, Option Pricing, Interest Rates and Risk Management, pages 538–574.Cambridge University Press.

42