closed form approximation of swap exposures - heikki seppala

8/18/2019 Closed Form Approximation of Swap Exposures - Heikki Seppala

1/32

Closed Form Approximation of Swap Exposures

Heikki Seppälä, Ser-Huang Poon and Thomas Schröder∗

March 21, 2013

∗Heikki Seppälä ([email protected]) and Ser-Huang Poon ([email protected]) areboth at the Manchester Business School, Thomas Schröder ([email protected]) is at the European InvestmentBank. Heikki Seppälä is a Marie Curie fellow funded by the European Community’s Seventh Framework Pro-

gramme FP7-PEOPLE-ITN-2008 under grant agreement number PITN-GA-2009-237984 (project name: RISK).The funding is gratefully acknowledged. The opinions expressed in this article are the sole responsibility of theauthors and do not necessarily reflect the views of the European Investment Bank.

1


2/32


Abstract

The collapse of the Lehman Brothers and the 2008 financial crisis have made

Libor risky and increased liquidity cost of some currencies. The basis spreads in

tenor basis swap and cross currency basis swap are substantial post 2008 crisis. No

model exists to date that tackles these new phenomenon. We propose a two-factor

mean reverting Gaussian model for the basis spread, and use it to derive a closed

form expression for the expected exposure of tenor basis swap. An approximation

for the expected exposure of the cross currency basis swaps is derived based on its

upper and lower bounds. We also describe how our model can be calibrated and

show that the calibration of the two-factor Gaussian model to the time series of

term structure of basis spread is quite good. Our analytical solutions can be used to

value swaptions, and as sanity checks for CVA, CVA VaR and Basel risk capital for

counterparty credit risk exposure of OTC derivatives.

JEL Classification: G13, G32, G21

Keywords: Tenor basis swap, cross currency swap, swaption, credit valuation adjust-

ment, expected exposure, Basel

2


3/32


1 Introduction

After the banking crisis of 2007-2008 and the collapse of the Lehman Brothers, companies and

financial institutions have greatly become more risk aware and the new regulations in Basel

III (2011) drastically tightened up the reporting of over-the-counter (OTC) derivatives. Coun-

terparties to the derivative trade made adjustments for the credit worthiness of each other, a

practice now known as credit valuation adjustment (CVA). CVA is waived if the counterparty

chooses to post collateral to reduce the cost of default. The posting of collateral is normally

bilateral unless one of the counterparties is a sovereign or a government agency. Unilateral CVA

is also used in Basel III for determining capital charge against counterparty exposure and the

estimation of CVA VaR (Value at Risk). All such capital risk charge calculations begin with the

expected positive marked to market value of OTC derivatives. In this paper we derive closed

form approximations for expected exposures of tenor basis swaps and cross currency basis swaps.

The presence of basis spread and its volatility have been very prominent during the financial

crisis 2008 (see Figure 1). To date, very little attempt has been made in modeling and estimating

the effect of basis spread on swap risk exposure. We model the tenor basis spread and the cross

currency basis spread using a mean reverting Gaussian two-factor model with deterministic shift,

which allows closed-form solutions and is relatively easy to calibrate. Our model is simpler than

that used in Kenyon and Stamm (2012, 7.2) for pricing fixed-for-floating interest rate swaption,

and our formulation leads to closed form approximations. Our approximations can be used as a

sanity check in trading environment, in cross checking CVA implementation and validation, for

approximating CVA VaR and expected shortfall. Our approximations can also be used to cross-check the price of “stand-alone” options on tenor basis swap and cross-currency swap omitting

3


4/32

the portfolio effect of counterparty default risks.

2006 2008 2010 2012

− 1 0 0

− 5 0

0

Date

C C b

a s i s s p r e a d ( b p )

Figure 1: Time series of cross currency basis swap spreads for EURUSD with maturities 1 year

(black solid), 10 yrs (red dashed), 30 yrs (blue dotted) for the period from 1 February 2006 to30 October 2012.

The paper is organized as follows. Section 2 introduces tenor basis swap and a model for the

swap spread. A closed-form formula was derived for the expected exposure of tenor basis swap.

Section 3 deals with cross currency basis swap. Section 4 gives approximation formulas for the

expected exposure of cross currency basis swap. Section 5 describes the calibration procedures

of our basis spread model and presents some results. Proofs are given in the appendix.

2 Tenor basis swap

In a tenor basis swap (TBS), interest rates of different tenors are exchanged. We consider here

3M vs. 6M Libor tenor basis swap with quarterly payment dates T 1,...,T 2n and fixing dates T k−1,

for k = 1,...2n, on the 3M side, and semi-annual payment dates T 2i and fixing dates T 2(i−1), for

4


5/32

i = 1,...,n, on the 6M side. We denote the notional by N , the x month Libor rates for time

period [t, t+xM ) by LxM t , and the 3M vs. 6M basis spread for period [t, T ] by Πt,T . The spread

Πt,T can be derived from the difference between two fixed-for-floating swaps with identical fixed

legs; 6M vs fixed and 3M vs fixed (see Kenyon, Stamm, 2012, 5.1.1). The spread is by convention

included in the leg with the shorter tenor as the longer (riskier) rate is higher because of credit

and liquidity issues. That is why the tenor basis spread is basically always positive (see figure

2) in contrast to cross currency basis swap (Figure 1). These spreads have existed for a long

time but were negligibly small before the 2008 crisis (see figures 2 and 3).

2007 2008 2009 2010 2011 2012 2013

0

1 0

2 0

3 0

4 0

Date

T e n o r b a s i s

s p r e a d

( b p )

Figure 2: Time series of tenor basis swap spreads with maturities 1 year (black solid), 10 years(red dashed), 30 years (blue dotted) for period 1 January 2007 – 23 October 2012.

Figure 3 shows that the spread decreases when the maturity increases. This is because the

spread can be considered as risk and liquidity premium, which is more prominent over the short

maturity. As the number of payments increases for the longer contract, the premium included

in each payment is lower.

At contract initialization, the value of the TBS , from the 6M payer’s perspective, can be

5


6/32

0 5 10 15 20 25 30

0

1 0

2 0

3 0

4 0

Maturity of swap

T e n o r b a s i s

s p r e a d

( b p )

Figure 3: The spread term structures on 1 January 2007 (black solid), 13 October 2008 (red

dashed) and 23 October 2012 (blue dotted).

6


7/32

written as

TBS 0(0) = N

2n

k=1δ 3M k

F 3M 0,T k−1 + Π0,T

D(0, T k) −

n

i=1δ 6M i F

6M 0,T 2(i−1)

D(0, T 2i)

,

where N is the notional, D is the discount factor, δ 3M k and δ 6M i are year count fractions between

fixing and payment dates, and F 3M t,T k and F 6M t,T i

are the forward Libor rates. The basis spread is

set such that the contract has a zero value at inception, i.e. TBS 0(0) = 0.1

At time t, the value of this contract is given by

TBS 0(t) = N 2n

k= j∗ δ

3M

k F 3M t,T k−1 + Π0,T D(t, T k) −n

i= j δ

6M

i F

6M

t,T 2(i−1)D(t, T 2i) , (1)where

j∗ =

2 j − 1 if t ∈ [T 2( j−1), T 2 j−1)

2 j if t ∈ [T 2 j−1, T 2 j).

At time t, the market value of a new contract can be written as

TBS t(t) = N

δ t,T j∗−1

L3M t + Πt,T

D(t, T j∗−1) +

2nk=1

δ 3M k

F 3M t,T k−1 + Πt,T

D(t, T k)

−N δ t,T 2(j−1)L

6M t D(t, T 2( j−1)) +

ni=1

δ 6M i F 6M t,T 2(i−1)

D(t, T 2i)

(2)

= 0

This means that TBS 0(t) can be valued with a reversal contract by subtracting (2) from (1) as

follows:

TBS 0(t) = (Π0,T − Πt,T )N 2n

k= j∗

δ 3M k D(t, T k). (3)

Equation (3) shows that the value of a tenor basis swap depends entirely on the changes in the

basis spread Πt,T and the discount factor D(t, T k). Note that this formula holds for t ∈ [T j∗−1, T j∗)

and we do not have to treat separately the cases where T j−1 is odd or even.

1In practice the spread and one of the forward curves (depending on the currency) are given by the marketand the other forward curve is built such that TBS 0 (0) = 0 (see Kenyon and Stamm, 2012, 5.2).

7


8/32

Given the 6-month Libor forward curve, we can solve the forward spread at time t from (2)

as follows:

Πt,T = (S 6M t,T − S 3M t,T )

A6M t,T

A3M

t,T

, (4)

where A is the annuity factor and S 3M t,T and S 6M t,T are swap rates of fixed-for-floating swap with

frequency 6M on the fixed side, i.e.

A6M t,T = δ t,T 2(j−1)D(t, T 2( j−1)) +n

i= j

δ 6M i D(t, T 2i),

A3M t,T = δ t,T j∗−1D(t, T j∗−1) +2n

k= j∗ δ 3M k D(t, T k),

S 3M t,T =δ t,T j∗−1L

3M t D(t, T j∗−1) +

2nk= j∗ δ

3M k F

3M t,T k−1

D(t, T k)

A6M t,T ,

S 6M t,T =δ t,T 2(j−1)L

6M t D(t, T 2( j−1)) +

ni= j δ

6M i F

6M t,T 2(i−1)

D(t, T 2i)

A6M t,T .

2.1 Expected exposure of tenor basis swap

Credit exposure becomes a risk only when the value of the OTC derivative is positive. A formal

definition of expected exposure is provided below.

Definition 1 Let V (t) be the value of a contract at time t. Then the exposure at time t is

E[V (t)] = max{V (t), 0} and the expected exposure at time t seen from time s is

EEs[V (t)] = Es[max

{V (t), 0

}] := E[max

{V (t), 0

}|F s],

where F s is the sigma algebra containing all the information up to time s.

8


9/32

We assume the basis spread dynamics is described by

Πt,T = 1

T

−tEt

ˆ T t

πsds (5)

πt = xt + yt + ϕ(t)

dxt = −axtdt + σxdW xtdyt = −bytdt + σydW yt

where ϕ : [0, T ] → R is a deterministic function of time, and both xt and yt mean revert to zero

at the speed a and b, respectively.

Proposition 2 From the basis spread dynamic defined in (5), we have

πt = e−atx0 + σx

ˆ t0

e−a(t−u)dW xu + e−bty0 + σy

ˆ t0

e−b(t−u)dW yu + ϕ(t)

Πt,T = 1

T − tβ (t ,T ,a)e−atx0 + β (t ,T ,b)e−bty0 +

ˆ T t

ϕ(s)ds

+

1

T − t β (t ,T ,a)σx ˆ t

0 e−a(t

−u)

dW x

u + β (t ,T ,b)σyˆ

t

0 e−b(t

−u)

dW y

u .In particular, xt, yt, πt and Πt,T are Gaussian with means

µx(t) := E[xt] = e−atx0

µy(t) := E[yt] = e−bty0

µπ(t) := E[πt] = e−atx0 + e

−bty0 + ϕ(t)

µΠ(t) := E[Πt,T ] = 1

T − tβ (t ,T ,a)e−atx0 + β (t ,T ,b)e

−bty0 +ˆ T t

ϕ(u)du

,

(6)

9


10/32

and variances

ν x(t) := var[xt] = σ2xβ (0, t, 2a)

ν y(t) := var[yt] = σ2yβ (0, t, 2b)

ν π(t) := var[πt] = σ2xβ (0, t, 2a) + σ

2yβ (0, t, 2b)

ν Π(t) := var[Πt,T ] = 1

(T − t)2β (t ,T ,a)2var[xt] +

1

(T − t)2β (t ,T ,b)2var[yt],

(7)

where β (t ,T ,a) = 1−e−at

a .

The proof of this proposition follows from the results in Brigo and Mercurio (2006, 4.2). Notethat this model allows the basis spread to be negative.2

Theorem 3 Under the Gaussian 2-factor model and assuming independence between the dis-

count rate and the basis spread for t ∈ [T j−1, T j),

EE[TBS 0(t)] = [√ ν ΠN

(d) + (µΠ − Π0,T )(1 − Φ(d))]2n

k= jδ 3M k E[D(t, T i)],

where µΠ and ν Π are given in equations (6) and (7), Φ is cumulative distribution function of

standard normal distribution and

d = Π0,T − µΠ√

ν Π.

As an example, suppose we have a 5-year Euribor 3M for 6M tenor basis swap initiated on 2

July 2012. The expected exposure profile at initialization is shown in figure 4. Our basis spread

model was calibrated to the basis spread term structure on 2 July 2012. As shown in figure 4,

the expected exposure drops on each 3M leg payment date.

The assumption that the discount rate and the tenor basis spread are independent could be

unrealistic, because higher Libor rate could be due to increased riskiness or liquidity shortage

of the Libor rate, and the basis spread should widen. The higher Libor rate would have an

2According to Brigo and Mercurio (2006, 4.2.1) the two-factor model in (5) is equivalent to the Hull-Whitetwo-factor model, with πt following a mean reverting Gaussian process with a stochastic long-run mean that isitself mean reverting. The second specification is more intuitive but harder to calibrate.

10


11/32

0 1 2 3 4 5

0 . 0

0

0 . 0

2

0 . 0

4

0 . 0

6

0 . 0

8

0 . 1

0

Time (yrs)

E x p e c t e d e x p o s u r e ( % )

Figure 4: Expected exposure profile of a 5-year tenor basis swap starting on 28 September 2012.

immediate negative impact on the discount factor. So the assumption of independence between

basis spread and discount factor could lead to an over-estimation of the changes in the value of

tenor basis swap, and an overestimation of the expected exposure.

3 Cross Currency Basis Swap

The valuation of cross currency basis swap (CCBS) is more complicated than tenor basis swap

since, apart from interest rate and tenor basis, the value of CCBS also depends on exchange

rate and cross currency basis. For ease of exposition, we consider below a cross currency swap

with the same tenor on both legs thereby allowing us to omit the confounding effects of tenor

basis spread. Before the financial crisis, such a CCBS predominantly contained exchange rate

risk. Sometimes, the swap quotes included a small basis spread to reflect the currency relative

shortage. However, due to the severe dollar shortage during and since the 2008 crisis, there is

11


12/32

now a significant and fluctuating spread involved in such a basis swap quoted as “3M USD Libor

flat vs 3M Euribor + spread”. In the case of EURUSD CCBS, the spread is negative since 2008,

with the 5Y spread reaching the peaks of over -60bp in 2009 and in 2011 (see figure 1).

Suppose we have a EURUSD T -period CCBS initiated at time 0 with notional N e0 and N $0 ,

exchange rate S 0 = N $0N e0

, and a basis spread Π0,T . At maturity T , the notional are to be returned

in the exact amount thereby preserving the exchange rate at inception. The FX spot rate S t,

expressed as USD per EUR, is time varying. The contract has payments at dates (T i)ni=0, with

T 0 = 0, T n = T . At time 0, the value of the contract in euros (from the e–payer’s perspective)

can be written as

CCBS e0 (S 0) = −De(0, T )N e0 −n

i=1

δ T i−1,T iDe(0, T i)(r

e

T i−1,T i + Π0,T )N

e

0

+ D$(0, T )N $0S 0

+n

i=1

δ T i−1,T iD$(0, T i)r

$T i−1,T i

N $0S 0

= 0

(8)

where reT i−1,T i (r$T i−1,T i) is the Euribor ($–Libor ) at time T i−1 for the period [T i−1, T i), and δ T i−1,T i

is the year fraction between T i−1 and T i, D$(0, T ) is from the 3M $-Libor curve (typically the

US OIS, Overnight Index Swap rates) and De(0, T i) is the basis adjusted discount factor such

that the value of the swap is 0 at inception (see Kenyon and Stamm, 2012, 6.1).

At time t ∈ [T j−1, T j) the value of this CCBS for the e–payer is

CCBS et (S 0) = −De(t, T )N e0 −n

i= j

δ T i−1,T iDe(t, T i)(reT i−1,T i + Π0,T )N e0

+ D$(t, T )S 0N

e0

S t+

ni= j

δ T i−1,T iD$(t, T i)r

$T i−1,T i

S 0N e0

S t.

(9)

The basis spread, FX rate and OIS rates are likely to have changed since inception, thus the

contract value is not zero at time t. We can value the CCBS in (9) by a reversal contract with

a notional value N et

= N e0 , and N $

t = S

tN e

0 = S t

S 0N $

0. Then by making use of the fact that

N $tS t

= N $0S 0

, the cash flows and the value of the new contract can be written as

12


13/32

CCBS et (S t) = −De(t, T )N e0 −n

i= jδ T i−1,T iD

e(t, T i)(re

T i−1,T i + Πt,T )N

e

0

+ D$(t, T )N $0S 0

+n

i= j

δ T i−1,T iD$(t, T i)r

$T i−1,T i

N $0S 0

= 0.

(10)

Subtracting (10) from (9), we get, by setting N e0 = $1,

CCBS e

t

(S 0) = (Πt,T −

Π0,T )n

i= j

δ T i−1,T iDe(t, T i)

A(t)

+ D$(t, T )S 0S t − 1 C (t)

(11)

A(t)

C (t) ≈ (Πt,T − Π0,T )(T n − t)

S 0S t− 1

.

where A(t) and C (t) denote the spread and the FX components of the CCBS respectively.

As an illustration, we consider a 4-year CCBS with quarterly payments starting on 1 April

2008 with notional 10Me. The unreported spreads are approximated by appropriate weighted

averages of the reported market spreads, e.g., a 3 year 9 month spread would be approximated

by Πt,3 912= 0.75Πt,4 + 0.25Πt,3. figure (5) plots the CCBS value and its two subcomponents as

they evolve through time up to maturity. As we see in Figure 5 the FX-part, which was the value

of the CCBS before crisis, still approximates the CCBS fairly well, but there is a slight deviation

especially in when the time to maturity is 2 – 3.5 years. For a longer contract the deviation

would potentially be larger. In this example, the FX component approximation is higher than

the CCBS value at every time point, but this needs not always be the case. It is obvious that

the CCBS could contain very large exposure at maturity due to the FX component, which is not

the case in tenor basis swap, whose exposure converges to at contract maturity. Comparison of

the evolution of the value of CCBS in Figure (5) to the expected exposure profile of the tenor

basis swap shown in Figure 4 illustrates this fact.

Figure 6 plots the cross currency basis swap spread term structures on three different dates

13


14/32

2009 2010 2011 2012

0 . 0

0

0 . 0

5

0 . 1

0

0 . 1

5

0 . 2

0

0 . 2

5

Date

V a l u e

Figure 5: Value of a 4-year CCBS initiated on 1 April 2008 (black solid), the approximation of FX component,C , (red dashed) and the approximation of spread component, A, (blue dotted)during the life of the contract.

and the average over our sample period. The steepness of the term structure is very different on

the three dates with the short term spread being a lot more volatile than the long term spread.

As such, the term (Πt,T − Π0,T ) needs not always be negative. Nevertheless, in our data period,

the spread component tends to decrease instead of increase the value of the CCBS. However,

since the EURUSD basis spread was almost non-existent before 2008, we do not have sufficient

data to examine closely on how the spread component affects the value of CCBS over the long

term.

The value CCBS et (S 0) is dominated by the FX component, C (t), especially at short maturity.

Figure 5 shows a clear dominance of the FX component on the CCBS valuation. The spread

component reached its peak at 300bp just after 2009, but in general has a value of much less than

100bp in absolute term. In contrast, the changes in the FX component

S 0S t− 1

can exceed

25%. When the time to maturity is long, the spread component, A, has many terms in the

summation. But, as shown in figure 6, the changes in the long term spreads are very small.This, combined with the discounting effect, means that the spread component, A, is always

14


15/32

0 5 10 15 20 25 30 − 1

5 0

− 1 0 0

− 5 0

0

Time (yrs)

S p r e a d

( b p )

Figure 6: Spread term structures on dates 1 April 2008 (blue dotted), 25 November 2008 (reddashed), 2 May 2012 (green dashdotted) and average over the period 1 April 2008 to 2 May2012 (black solid).

relatively small. The FX component, C , gets smaller on average as time to maturity increases

because of the discounting effect, but the FX rate changes can be very volatile. As such, the FX

component dominates completely when the CCBS is closed to maturity.

4 Expected exposure of CCBS

From the analysis in the previous section, it is clear that a closed form formula for the expected

exposure for the CCBS is difficult to obtain. Our objective here is to find a simple closed form

approximation for the expected exposure of CCBS. By Definition 1, the exposure and expected

exposure of CCBS, at time t, are

E[CCBS et (S 0)] = max{A(t) + C (t), 0}

EE0[CCBS e

t (S 0)] = E0[max{A(t) + C (t), 0}].

15


16/32

According to (11) the value of the CCBS depends on the evolution of the 3M Libor rates on

both currencies, the instantaneous cross currency basis swap spread πt and the FX rate S t. From

exchange rate parity, the FX rate also depends on the risk-free (OIS) rates of both currencies

and the OIS spread. An examination of the data post 2008 crisis suggests that all the interest

rates are correlated but the spreads and interest rates do not appear to be strongly correlated.

Hence, we assume in the following section that the cross currency basis spread and the interest

rates are not correlated. This assumption is essential in order to achieve our simple closed form

approximation.

4.1 Bounds for expected exposure

The following proposition gives approximations for the lower and the upper bounds of the ex-

pected exposure.

Proposition 4 We have that

1

2 [EC (t) + EA(t) + |E|C (t)| − E|A(t)||] ≤ EE0[A(t) + C (t)] ≤ EE0[C (t)] + EE0[A(t)] (12)

The lower bound of CCBS (t) is closed to the FX part, C (t), of the expected exposure, which

is convenient since most of the time A(t) and C (t) are of different sign and negatively correlated.

This suggests that EE0[CCBS et (S 0)] ≤ EE0[C (t)]. However, we should be very careful with this

inference since we do not want to under estimate the exposure. On the other hand, the upper

bound in (12) is likely to be over pessimistic as it assumes that A(t) and C (t) always have the

same signs.

16


17/32

4.2 Spread component of CCBS

To apply Proposition 4, we need to calculate E[A(t)], E[|A(t)|] and EE0[A(t)] for the spread

component, A. From (11), we rewrite the spread component, A(t), as

A(t) = (Πt,T − Π0,T )n

i= j

δ T i−1,T iDe(t, T i) =

ni= j

Ai(t)

Ai(t) := (Πt,T − Π0,T )δ T i−1,T iDe(t, T i)

De(t, T ) = e−´ T t (res+Πt,T )ds

where re

s is the 3M Euribor spot rate.

Using this discount factor, we have the following theorem.

Theorem 5 Under the Gaussian 2-factor model and assuming independence between the spot

interest rate r and the spot spread π we have, for t ∈ [T j−1, T j),

E[A(t)] =n

i= j E[Ai(t)]E[A(t)+] =

ni= j

E[Ai(t)+]

E[|A(t)|] =n

i= j

E[|Ai(t)|]

where

E[Ai(t)] = Bt,T iC Π,i [µΠ − (T i − t)ν Π −Π0,T ] (13)

E[Ai(t)+] = Bt,T iC Π,i [

√ ν ΠN

(d) + (µΠ − (T i − t)ν Π − Π0,T )(1 −N (d))] (14)

E[|Ai(t)|] = Bt,T iC Π,i [2√ ν ΠN

(d) + (µΠ − (T i − t)ν Π −Π0,T )(1 − 2N (d))] (15)

17


18/32

with µΠ and ν Π given in equations (6) and (7),

Bt,T i = δ T i−1,T iE

e−

´ T t re s ds

,

C Π,i = e−(T i−t)µΠ+(T i−t)2ν

Π2 and

d = Π0,T − (µΠ − (T i − t)ν Π)√

ν Π.

4.3 FX component of CCBS

In order to use Proposition 4, we also need to find E[C (t)], E[|C (t)|] and EE0[C (t)]. Each of

these components can be separated into E[C (t)+] and E[C (t)−] as follows:

E[C (t)] = E[C (t)+] − E[C (t)−],

E[|C (t)|] = E[C (t)+] + E[C (t)−] and

EE0[C (t)] = E[C (t)+].

Notice that C (t)+ = max{C (t), 0} resemblance to the payoff of a call option to buy 1S 0

euros for

one dollar at time t;

max{C (t), 0} = maxD$(t, T )

S 0S t− 1

, 0

= S 0D$(t, T )max

1S t− 1

S 0, 0,

whereas C (t)− resemblance to the payoff of a put option to sell 1S 0

euros for one dollar at time t.

If we assume independence between the FX rate and the 3M $-Libor rate (which is not entirely

true), these call and put option values can be obtained directly from market data on valuation

date.

In fact, according to the example in the previous section, the simplest approximation for the

expected exposure of CCBS at time t is the expected exposure of the FX component, C (t), i.e.,

EE0[CCBS e

t (S 0)]

≈E D$(t, T )Emax

S 0

S t −

1, 0 . (16)

18


19/32

The data post 2008 suggests that the spread component, A, and the FX component, C , have

different signs in most cases. So it would be tempting to use equation (16) as an upper bound for

the expected exposure of CCBS . However, one should be careful with this since the data period

is relatively short and we do not have any economic theory to justify this negative relationship.

5 Calibration

The closed form approximation we derived for the exposure of tenor basis swap and cross currency

basis swap in the previous sections depends on how well the two-factor Gaussian model we

proposed for the basis spread fit the market observed spread. In this section, we will show that

this empirical fit is quite good. Our calibration approach follows the calibration of credit spreads

in Brigo and Alfonsi (2005). We consider the integrated process and define

Π0,T = 1

T

ˆ T 0

πsds,

Z 0,T = 1

T ˆ

T

0

xs + ysds,

Φ(0, T ) = 1

T

ˆ T 0

ϕ(s)ds.

(17)

The procedures involve first fitting Z 0,T i to the market observed basis spread term structure

{Π̂0,T 1, ..., Π̂0,T n} as closely as possible. After this, Φ is chosen such that

Π̂0,T i =EΠ0,T i = EZ 0,T i + Φ(0, T i)

for all T i.

Since (17) does not contain any volatility information, we adopt the “bond price” calculation

in Brigo and Mercurio (2005, Theorem 4.2.1) and reproduced below:

Theorem 6 (Brigo and Mercurio, 2005, Theorem 4.2.1, Zero Coupon Bond Price) Supposed

19


20/32

the dynamic of the instantaneous short rate is given by a two-factor Gaussian model

r (t) = x (t) + y (t) + ϕ (t) ,

then the price at time t of a zero coupon bond maturing at time T is 3

Ee−Φ(0,T )−Z 0,T = e−Φ(0,T )− 1T β(a,T )x0− 1T β(b,T )y0+ 12T 2 V (T ),

where

β (a, T ) = (1 − e−aT )

a

V (T ) = σ2xa2

T +

2e−aT

a − e

−2aT

2a − 3

2a

+σ2y

b2

T +

2e−bT

b − e

−2bT

2b − 3

2b

+ 2ρxyσxσy

ab (T − β (a, T ) − β (b, T ) + β (a + b, T )) .

From Theorem 6, we can write

e−Π̂0,T = Ee−Φ(0,T )−Z 0,T = e−Φ(0,T )−EZ 0,T +12var(Z 0,T )=e−Φ(0,T )−f (x0,y0,a,b,σx,σy,ρ;T )

f (x0, y0,a ,b ,σx, σy, ρ; T ) = EZ 0,T − 12var (Z 0,T ) ≈ Π̂0,T (18)

where c is a constant representing the error of the fit. From Theorem 6,

f (x0, y0,a ,b ,σx, σy, ρ; T ) = 1

T β (a, T )x0 +

1

T β (b, T )y0 − 1

2T 2V (T ) (19)

Equation (18) will be calibrated to the market observed term structure of the basis spread for

all relevant T .

We can impose some restrictions for the minimisation. In our case the parameters a, b, σx

3

Function β is positive and decreasing, and V is positive. In the limit, we have limT →01T β (a, T ) =

1, limT →∞1T β (a, T ) = 0 and limT →0

V (T )T 2

= 0, limT →∞V (T )T 2

= 0.

20


21/32

and σy must be positive so that we calibrate the model by minimizing the quadratic difference

between Π̂ and f , i.e., solve arguments from

minx0,y0,a,b,σx,σy,ρ

ni=1

Π̂0,T i − f (x0, y0,a ,b ,σx, σy, ρ; T )2 + γ A , (20)where

A = {(x0, y0,a ,b ,σx, σy, ρ) ∈ R7 : a,b,σx, σy > 0}.

The last term on the right-hand side of (20) imposes a penalty determined by parameter γ if

the positivity conditions on parameters a, b, σ

x and σ

y are not satisfied. If we want to have morerestrictions or fix some parameters, we can add these restrictions to A. Our calibration results

suggest that for cross currency basis spread, x0 and y0 should be of different sign, whereas for

tenor basis spread, x0 and y0 are both positive.

When all the parameters are fitted, we solve the values of the mean integral of the determin-

istic drift from

Φ̂(0, T i) = Π̂0,T i − EZ 0,T i = − 12T 2V (T i) + [Π̂0,T i − f (x0, y0,a ,b ,σx, σy, ρ; T i)] (21)

for all i. We then fit a function to these points to get fitted values of Φ̂. The values of ϕ depend

on the choice of the functional form of Φ. Here, we do not extract the values of ϕ because we are

not modelling the ”spot spreads”. Nevertheless, to recover ϕ, Φ must be differentiable4. Then

from (17)4One could try to find a smooth approximation for the error function (T i) := Π̂0,T i − f (·) in equation (21).

For example Nelson-Siegel-Svensson method (Svensson, 1994) could be used to recover . The derivative of V can be obtained using Mathematica. For the calibration of Nelson-Siegel-Svensson method, see Gilli, Große, andSchumann (2010).

21


22/32

ˆ T 0

ϕ(t)dt = T Φ(0, T )

ϕ(t) = d

dt [tΦ(0, t)] = Φ(0, t) + t

d

dtΦ(0, t).

5.1 Calibration results

The term structure of basis spread does not contain information about volatility, which is the

first major problem. We use the “bond equivalent” to get the volatility term artificially appear

and set

e−Π̂0,T = Ee−Φ(0,T )−Z 0,T = e−Φ(0,T )−EZ 0,T +12var[Z 0,T ]

The volatility term should really be treated as input parameter here.

In the calibration, we first fit the dynamical part as well as possible to the term structure.

That is we obtain x0, y0,a ,b ,σx, σy and ρxy by minimising

M i=1

Π̂0,T i − EZ 0,T i +

1

2var[Z 0,T i]

2× T 1.5i ,

where M is the number of observations in the term structure, and T 1.5i is our choice of a weighting

scheme to give a greater emphasis on calibrating the long maturity swap spread well .

The deterministic part, Φ(0, t), is a function such that for each i = 1,...,M

Φ(0, T i) = Π̂0,T i − EZ 0,T i ,

where Z holds the calibrated parameters.

Figures 7, 8 and 9 present our calibration results. Figure 7 shows the fit to 1 year, 5 year,

10 years and 30 years basis spread for EURUSD cross currency basis swap. Figure 8 shows

22


23/32

2007 2009 2011 2013

− 1 0

0

1 0

2 0

3 0

4 0

Date

B a s i s p o i n t s

2007 2009 2011 2013

0

5

1 0

1 5

2 0

Date


2007 2009 2011 2013

0

5

1 0

1 5

Date


2007 2009 2011 2013

0

2

4

6

8

Date


Figure 7: Calibration performance of dynamic part for 1Y (top left), 5Y (top right), 10Y (bottom

left) and 30Y spreads (bottom right). The parameters a = 0.05, σx = 0 are fixed and the restof the parameters are calibrated. Black solid line is the actual term structure, red dashed lineis the dynamic part of daily calibrated model and blue dotted line is the deterministic part.

23


24/32

2007 2008 2009 2010 2011 2012 2013

0 . 0

0 0 0

0 . 0

0 1 0

Date

x 0

2007 2008 2009 2010 2011 2012 2013

0 . 0

0 0

0

. 0 0 2

0 . 0

0 4

Date

y 0

2007 2008 2009 2010 2011 2012 2013

1 . 0 9 9 7

1 . 1

0 0 0

1 . 1

0 0 3

Date

b

2007 2008 2009 2010 2011 2012 2013

0 . 0

0 0 1 6

0 . 0

0 0 1 8

Date

s i g m a y

Figure 8: Calibrated parameters x0, y0, b and σy.

0 5 10 15 20 25 30

0 . 0

0 . 2

0

. 4

0 . 6

time(yrs)

S p r e a d ( b p )

0 5 10 15 20 25 30

0

5

1 0

1 5

2 0

time(yrs)

S p r e a d ( b p )

0 5 10 15 20 25 30

− 5

0

5

1 0

1 5

2 0

time(yrs)

S p r e a d

( b p )

0 5 10 15 20 25 30

− 1 0

0

1 0

2 0

3 0

4 0

time(yrs)

S p r e a d

( b p )

Figure 9: Basis term structure (black), calibrated dynamical part of the model (red dashed) and

deterministic part of the model (blue dotted) on dates 1 January 2007, 29 September 2008, 29October 2010 and 20 July 2012.

24


25/32

the calibrated values for x0, y0, b and σy; note that x0 and y0 vary a lot through time, but b

and σy are practically constant. Figure 9 shows the fit of the model to the basis spread for

different dates. Apart from the very short maturity spread, our two factor Gaussian model fit

the basis spread term structure reasonably well. Finally, one should note that the two mean

reversion rates, a and b, and the two volatilities, σx and σy, should be considered as inputs in

the calibration as they are basically unaffected by calibration. However, the optimal choice of

their (initial) values is not clear. A more flexible parameter setting could fit the dynamical part

capture the the term structure better, but the stability of the calibrated parameters values could

suffer.

We are interested in the evolution of Πt,T with respect to t. We try to model this evolution

making Πt,T time homogeneous,

Πt,T = 1

T − tEtˆ T

t

πsds

,

which is quite intuitive, but it does not seem to be good as the π is zero mean reverting and hence

Πt,T is decreasing on expectation. This is not realistic according to the market observed data;

the shorter maturity spreads are generally a lot higher higher than the long maturity spreads.

Another choice is to model the evolution of Πt,T by setting

Πt,T = 1

T − tˆ T −t0

πsds,

which may be less intuitive, but should give more realistic values for shorter spreads.

25


26/32

References

Basel Committee on Banking Supervision, Basel III: A global regulatory framework

for more resilient banks and banking systems , December 2010 (rev June 2011),

www.bis.org/publ/bcbs189.pdf.

Brigo, D., Alfonsi, A. (2005): Credit Default Swaps and Option Pricing with SSRD Stochastic

Intensity and Interest-Rate Model, Finance & Stochastics, Vol. IX(1), 2005.

Brigo, D., Mercurio, F. (2006): Interest Rate Models - Theory and Practice , Second edition,

Springer.

Gilli, M., Große, S., Schumann, E., (2010), Calibrating the Nelson-Siegel-Svensson model,

COMISEF working paper series, WPS-031.

Kenyon, C., Stamm, E., (2012), Discounting, Libor, CVA and Funding, Interest Rate and Credit

Pricing , (Applied Quantitive Finance) Palgrave Macmillan.

Siegel, J., (1972), Risk, Interest Rates and the Forward Exchange, The Quarterly Journal of

Economics, Vol. 86, No. 2 (May, 1972), pp. 303-309. 5.1

Svensson, L., (1994), Estimating and Interpreting Forward Interest Rates: Sweden 1992–1994.

IMF Working Paper 94/114, 1994.

26


27/32

Appendix

Proof of Theorem 3

We have that

EE0[TBS 0(t)] = E

(Πt,T − Π0,T )

ni=2 j

δ 3M i D(t, T i)

+

= E

(Πt,T − Π0,T )+ ni=2 j

δ 3M i E[D(t, T i)]

and

E

(Πt,T − Π0,T )+

=

ˆ ∞Π0,T

(x− Π0,T )e−(x−µΠ)

2

2νΠdx√ 2πν Π

=

ˆ ∞d

(√ ν Πz + µΠ −Π0,T ) e−z

2

2dz √ 2π

=√ ν ΠN

(d) + (µΠ −Π0,T ) (1 −N (d)),

where d =

Π0,T

−µΠ

√ ν Π . Here we made the change of variable z = x

−µΠ

√ ν Π

Proof of Theorem 4

The fact that X + = 12

[X + |X |], implies

(A(t) + B(t))+ = max{A(t) + C (t), 0}

= 12

[A(t) + C (t) + |A(t) + C (t)|].

and since

||A(t)| − |C (t)| | ≤ |A(t) + C (t)| ≤ |A(t)| + |C (t)|

27


28/32

we see that

A(t) + C (t) + ||A(t)| − |C (t)|| ≤ A(t) + C (t) + |A(t) + C (t)|

≤ A(t) + C (t) + |A(t)| + |C (t)|(22)

Moreover,

E [||A(t)| − |C (t)||] = E (|A(t)| − |C (t)|) {|A|≥|C |} + E (|C (t)| − |A(t)|) {|C |>|A|}≥ |E[|A(t)|] − E[|C (t)|]|

and according to this taking expectations in equation (22) gives

EA(t) + EC (t) + |E|A(t)| − E|C (t)|| ≤ 2E[(A(t) + B(t))+]

≤ 2([E[A(t)+] + E[C (t)+])

and the result follows.

Proof of Theorem 5

Assuming independence between interest rate and spread, we have.

D(t, T i) = Et

e−

´ T it r

es+Πt,T ds

= Et

e−

´ T it r

es dsEt

e−

´ T it Πt,T ds

= B

e

(t, T i)Et e−(T i−t)Πt,T ,(23)

where Be(t, T i) = Et

e−

´ T it r

es ds

.

28


29/32

Because δ T i−1,T i and D(t, T i) are non-negative, we have

E[A(t)] =n

i= jE[Ai(t)]

E[A(t)+] =n

i= j

E[Ai(t)+]

E[A(t)−] =n

i= j

E[Ai(t)−]

E[|A(t)|] =n

i= j

E[|Ai(t)|]

Furthermore, under the assumption that r and π are independent, we have

E[Ai(t)] = Bt,T,iE

(Πt,T − Π0,T )Ete−(T i−t)Πt,T

, (24)

E[Ai(t)+] = Bt,T,iE

(Πt,T − Π0,T )+Et

e−(T i−t)Πt,T

(25)

E[Ai(t)−] = Bt,T,iE

(Πt,T − Π0,T )−Et

e−(T i−t)Πt,T

(26)

where Bt,T,i = δ T i−1,T iEe−

´ T it r

es ds

. We see this simply by calculating

E[Ai(t)+] = E

(Πt,T − Π0,T )δ T i−1,T iD(t, T i)

+

= δ T i−1,T iE e− ´ T it res dsE (Πt,T − Π0,T )+Et e− ´ T it Πt,T ds .(27)

Formulas for (24) and (26) essentially follow from this. Furthermore

E[|A(t)|] = E[A(t)+] + E[A(t)−]. (28)

29


30/32

Equation (23) and Lemma 7 with change of variable z = x−(µΠ−(T i−t)ν Π2√ ν Π

yields

E

(Πt,T − Π0,T )+D(t, T i)

= B(t, T i)E (Πt,T − Π0,T )+e−(T i−t)Πt,T

= B(t, T i)ˆ ∞Π0,T

(Πt,T − Π0,T )e−(T i−t)Πt,T dP(Πt,T )

= B(t, T i)

ˆ ∞Π0,T

(x− Π0,T )e−(T i−t)xe−(x−µΠ)

2

2νΠdx√ 2πν Π

= B(t, T i)e−(T i−t)µΠ+ (T i−t)

2νΠ2 [

√ ν ΠN

(d) + (µΠ − (T i − t)ν Π − Π0,T )(1 −N (d))] ,

where d = Π0,T −(µΠ−(T i−t)ν Π)

2√ ν Π

. This concludes the proof of (25). Changing the integration limits

in the obvious way we get (24) and we also see that

E[Ai(t)−] = Bt,T,iC Π,i [

√ ν ΠN

(d) − (µΠ − (T i − t)ν Π −Π0,T )N (d)] .

Finally, the result follows from (28).

Lemma 7 Suppose X ∼ N (µ, σ) and A ⊂ Ω. Then for all α, β,γ we have that

ˆ Ω

X γ eβX +α AdPX = e

βµ+β

2σ2

2 +α

ˆ ∞−∞

(σz + µ + βσ2)γ e−z2

2 Â

dz √ 2π

. (29)

Proof. We have that

ˆ Ω

X γ eβX +α AdPX =

ˆ ∞−∞

xγ eβx+αe−(x−µ)2

2σ2 Ã

dx√ 2πσ

=ˆ ∞−∞

xγ e− (x−µ)2

2σ2 −βx−α Ã

dx√ 2πσ

30


31/32

and since

(x− µ)22σ2

− βx − α = 12σ2

x2 − 2µx + µ2 − 2σ2βx

− α

= 12σ2x2 − 2(µ + σ2β )x + µ2− α

= 1

2σ2x2 − 2(µ + σ2β )x + (µ− σ2β )2 − 2µσ2β − σ4β 2− α

= −µβ − σ2β 2

2 − α + 1

2σ

x2 − 2(µ + σ2β )x + (µ + σ2β )2

we have

ˆ Ω

X γ e−βX +α AdPX = eµβ+

σ2β2

2 +α ˆ ∞

−∞xγ e− (

x−(µ+σ2β

))

2

2σ2 Ã dx√

2πσ.

Now change of variable z = x−(µ+βσ2)

σ gives the result.

FX Call

We show how to get values for E [max{C (t), 0}] from volatility data. Suppose we know theBlack-Scholes volatility σBS . Then

S t = S 0e

r$−(re+π)−σ

2BS2

t+σBSW t

,

where r$, re, π and σBS are constants. Now

C (t) = S 0

S t− 1 =

e−(r$−(re+π)t−σ

2BS2 t+σBSW t

− 1

= e(re+π)t

e−r

$teσ2BS2 t−σBS

√ tZ − e−(re+π)t

where Z follows the standard normal distribution. Clearly C (t) is positive whenever

Z ≤ d2 := (re + π +

σ2BS2

−r$)√ t

σBS

31


32/32

and using Lemma 7 we obtain

E [max{C (t), 0}] = e(re+π)tˆ d1−∞

e−r

$teσ2BS2 t−σBS

√ tZ − e(−re+π)t

dPZ

= e(re+π)t e(−r$+σ2BS)tN (d1) − e−(re+π)tN (d2) ,

where d1 = d2 + σBS √ t. Notice that this differs from the traditional Black-Scholes price a little

due to the Siegel’s paradox [6].

closed form approximation of swap exposures - heikki seppala

Documents