LIFE INSURANCE MATHEMATICS 2002

Ragnar NorbergLondon School of Economics

Abstract

Since the pioneering days of Black, Merton and Scholes financial mathematicshas developed rapidly into a flourishing area of science. Its impacts on insuranceare great by any calculation: applications are virtually countless and even thebasic paradigms are being rethought. This talk focuses on life insurance andshows how the mathematics of finance and of insurance dovetail into a consistent,model-based approach to measurement and management of combined insurancerisk and finance risk.

1 Introduction

Finance was always an essential part of insurance. Trivially, one might say,because any business has to attend to its money affairs. However, for at leasttwo reasons insurance is not just any business.In the first place, its products are not physical goods or services; they are

financial contracts with provisions related to uncertain future events. Therefore,pricing of insurance products is not just an accounting exercise involving thefour basic arithmetical operations; it is a matter of risk assessment based onstochastic models and methods.In the second place, insurance policies are more or less long term contracts

(in life insurance up to several tens of years) under which the customers pays inadvance for benefits to come later, hence the term premium (= first) for theprice. Therefore, the insurance industry is a major accumulator of capital intodays society, and the insurance companies are major institutional investors.It follows that the financial operations of an insurance company may be as

decisive of its revenues as its insurance operations and that the financial risk (orasset risk) may be as severe as, or even more severe than, the insurance risk (orliability risk). Insurance risk, which is due to the random nature of the cash-flowof premiums less insurance claims, is diversifiable in the sense that gains andlosses on individual policies will average out in a sufficiently large portfolio ofindependent risks: the law of large numbers is at work. Financial risk, whichis due to the uncertain yields on the financial investments, is not diversifiablein the same simple sense. Any investment portfolio is affected by booms andrecessions of the market in large, interest rate variations, and movements inprices of individual stocks. The composition of an investment portfolio may bemore or less risky (e.g. stocks are more risky than government bonds), but thevolume of the portfolio is of course of no relevance to its riskiness; losses andgains on investments do not subject themselves to the law of large numbers.

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On this background one may ask why insurance mathematics traditionallycenters on measurement and control of the insurance risk. The answer maypartly be found in institutional circumstances: The insurance industry usedto be heavily regulated, solvency being the primary concern of the regulatoryauthority. Possible adverse developments of economic factors (inflation, weakreturns on investment, low interest rates, etc.) would be safeguarded againstby placing premiums on the safe side. The surpluses, which would typicallyaccumulate under this regime, were redistributed as bonuses (dividends) to thepolicyholders only in arrears, after interest and other financial parameters hadbeen observed. Furthermore, the insurance industry used to be separated fromother forms of business and protected from competition within itself, and severerestrictions were placed on its investment operations. In these circumstancesfinancial matters appeared to be something the traditional actuary did not needto worry about. Another reason why insurance mathematics used to be void offinancial considerations was, of course, the absence of a well developed theoryfor description and control of financial risk.All this has changed. National and institutional borders have been downsized

or eliminated and regulations have been liberalized: Mergers between insurancecompanies and banks are now commonplace, new insurance products are beingcreated and put on the market virtually every day, by insurance companies andother financial institutions as well, and without prior licencing by the super-visory authority. The insurance companies of today find themselves placed ona fiercely competitive market. Many new products are directly linked to eco-nomic indices, like unit-linked life insurance and catastrophe derivatives. Byso-called securitization also insurance risk can be put on the market and thusopen new possibilities of inviting investors from outside to participate in riskthat previously had to be shared solely between the participants in the insuranceinsurance schemes. These developments in practical insurance coincide with theadvent of modern financial mathematics, which has equipped the actuaries witha well developed theory within which financial risk and insurance risk can beanalyzed, quantified and controlled.A new order of the day is thus set for the actuarial profession. The purpose

of this chapter is to give a glimpse into some basic ideas and results in modernfinancial mathematics and to indicate by examples how they may be applied toactuarial problems involving management of financial risk.

2 Classical life insurance mathematics

A. The multi-state life insurance policy.We consider an insurance policy issued at time 0 and terminating at a fixedfinite time T . There is a finite set of mutually exclusive states of the policy,Z = {1, . . . , JZ}. By convention, 1 is the initial state at time 0. Let Z(t) denotethe state of the policy at time t [0, T ]. Taking Z to be a stochastic processwith right-continuous paths and at most a finite number of jumps, the sameholds also for the associated indicator processes Ij and counting processes Njk

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defined, respectively, by IZj (t) = 1[Z(t) = j] (1 or 0 according as the policy is in

the state j or not at time t) and NZjk(t) = ]{ ; Z() = j, Z() = k, (0, t]}(the number of transitions from state j to state k (6= j) during the time interval(0, t]). The history of the policy up to and including time t is represented bythe sigma-algebra FZt = {Z() ; [0, t]}. The development of the policy isgiven by the filtration (increasing family of sigma-algebras) FZ = {FZt }t[0,T ].Let B(t) denote the total amount of contractual benefits less premiums

payable during the time interval [0, t]. We assume that it develops in accor-dance with the dynamics

dB(t) =

j

IZj (t) dBj(t) +j 6=k

bjk(t) dNZjk(t) , (2.1)

where each Bj is a deterministic payment function specifying payments due dur-ing sojourns in state j (a general life annuity), and each bjk is a deterministicfunction specifying payments due upon transitions from state j to state k (a gen-eral life assurance). We assume that each Bj is finite-valued, right-continuous,and decomposes into an absolutely continuous part and a discrete part with atmost a finite number of jumps in [0, T ]. Thus,

dBj(t) = bj(t)dt + Bj(t) ,

where Bj(t) = Bj(t) Bj(t), when different from 0, is a jump representinga lump sum payable at time t if the policy is then in state j. The functions bjand bjk are assumed to be finite-valued and piecewise continuous.

B. The Markov chain description of the policy.The process Z is assumed to be a time-continuous Markov chain on the statespace Z . We denote its transition probabilities by

pZjk(t, u) = P[Z(u) = k |Z(t) = j] ,

t u, and the corresponding intensities of transition by

jk(t) = limh0

pZjk(t, t+ h)/h ,

j 6= k, implying that these exist for all t [0, n). We assume, moreover, thatthe intensities are piece-wise continuous. The total intensity of transition fromstate j is j =

k; k 6=j jk. Here and elsewhere a dot in the place of a subscript

signifies summation over that subscript. We shall need Kolmogorovs forwarddifferential equations for s < t:

dt pZij(s, t) =

g; g 6=j

pZig(s, t)gj(t) dt pZij(s, t)j(t) dt . (2.2)

We remind of the fact that the compensated counting processesMZjk , j 6= k,defined by

dMZjk(t) = dNZjk(t) I

Zj (t)jk(t) dt , (2.3)

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are square integrable, mutually orthogonal, zero mean FZ -martingales.Let Hjk , j, k Z , j 6= k, be predictable processes satisfying

Ej 6=k

(0,t]

H2jk()jk()d

at time t is defined presicely as the conditional expected present value of thefuture benefits less premiums, given the past history of the policy:

V (t) = E

[ nt

e

trdB()

Ft].

By the Markov assumption,

V (t) = VZ(t)(t) =

j

IZj (t)Vj(t) , (2.5)

where the Vj are the statewise reserves,

Vj(t) = E

[ nt

e

trdB()

Z(t) = j]

=

nt

e

tr

g

pjg(t, )

dBg() +

h;h6=g

bgh()gh() d

.(2.6)(2.7)

We need the backward (so-called Thieles) differential equations

dVj(t) = r(t)Vj(t) dt dBj(t)

k; k 6=j

Rjk(t)jk(t) dt , (2.8)

where

Rjk(t) = bjk(t) + Vk(t) Vj(t) (2.9)

is the (for evident reasons so-called) sum at risk associated with a possible tran-sition from state j to state k at time t. The differential equation is easily derivedby differentiating the defining expression (2.6) and using the Kolmogorov equa-tion (2.2).

E. The principle of equivalence.The contractual payments (or rather the premiums for given benefits) are con-strained by the principle of equivalence, which lays down that

E

[ T0

e

0r dB()

]= 0 , (2.10)

that is,

V1(0) = B1(0) . (2.11)

The rationale of this principle, which is the basic paradigm of classical lifeinsurance, is that it will establish balance on the average in a large portfolio ofindependent policies.

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The reserve defined in the previous paragraph is motivated the same way:upon providing currently a reserve equal to the conditional expected value ofthe future net liability on each individual reserve, the company will meet itsliabilities on the average.

F. A martingale proof of Thieles differential equation.We take here the opportunity to demonstrate a martingale technique that will beomnipresent throughout the text. Thieles differential equation could be provedby elementary calculations as explained above, but the following argument givesalso an insight into the dynamics of the total cash-flow.Define the martingale

M(t) = E

[ n0

e

0rdB()

FZt]

=

t0

e

0rdB() + e

t

0rE

[ nt

e

trdB()

FZt]

=

t0

e

0rdB() + e

t

0r

j

IZj (t)Vj(t) . (2.12)

Now apply Itos formula to (2.12):

dM(t) = e

t

0rdB(t) + e

t

0r(r(t) dt)

j

IZj (t)Vj(t)

+ e

t

0r

j

IZj (t) dVj(t) + e

t

0rj 6=k

dNZjk(t) (Vk(t) Vj(t)) .

The last term on the right takes care of the jumps of the Markov process: upona jump from state j to state k the last term in (2.12) changes immediatelyfrom the discounted value of the reserve in state j just before the jump to thevalue of the reserve in state k at the time of the jump. Since the state-wisereserves are deterministic functions with finite variation, they have at most acountable number of discontinuities at fixed times. The probability that theMarkov process jumps at any such time is 0. Therefore, we need not worryabout possible common points of discontinuity of the Vj(t) and the I

Zj (t). For

the same reason we can also disregard the left limit in Vj(t) in the last term.We proceed by inserting (2.1) for dB(t) and the expression dNZjk(t) = dM

Zjk(t)

IZj (t)jk(t) dt obtained from (2.3), and gather

dM(t) = e

t

0r

j

IZj (t)

dBj(t) r(t)Vj(t) dt + dVj(t) +

k; k 6=j

jk dtRjk(t)

+ e

t

0rj 6=k

Rjk(t) dMZjk(t) , (2.13)

Since the last term on the right of (2.13) is the increment of a martingale, the firstterm of the right is the difference between the increments of two martingales

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and is thus itself the increment of a martingale. This martingale has finitevariation and, as will be explained below, is also continuous, and must thereforebe constant. For this to be true for all realizations of the indicator functionsIZj , we must have

dBj(t) rj Vj(t) dt + dVj(t) +

k; k 6=j

jk dtRjk(t) = 0 , (2.14)

which is Thieles differential equation. Furhtermore we obtain that

dM(t) = e

t

0rj 6=k

Rjk(t) dMZjk(t) ,

which displays the dynamics of the martingale M .Finally, we explain why the (2.14) is the increment at t of a continuous

function. The dt terms are continuous increments, of course. Outside jumptimes of the Bj both the Bj themselves and the Vj are continuous. At any timet where there is a jump in some Bj the reserve Vj jumps by the same amount inthe opposite direction since Vj(t) = Bj(t) + Vj(t). Thus, Bj + Vj is indeedcontinuous.

3 Insurance risk

A. Differential equations for moments of present values.We want to determine higher order moments of future net liabilities. By theMarkov property, we need only the state-wise conditional moments

V(q)j (t) = E

[( Tt

e

trdB()

)qZ(t) = j],

q = 1, 2, . . .

The functions V(q)j are determined by the differential equations

d

dtV

(q)j (t) = (qr(t) + j(t))V

(q)j (t) qbj(t)V

(q1)j (t)

k; k 6=j

jk(t)

qp=0

(q

p

)jk

(t))pV(qp)k (t) ,

valid on (0, T )\D and subject to the conditions

V(q)j (t) =

qp=0

(q

p

)(Bj(t)Bj(t))

pV(qp)j (t) , (3.1)

t D. A rigorous proof is given in [25]. The computation goes as follows. Firstsolve the differential equations in the upper interval (tm1, n), where the sideconditions (3.1) are just

V(q)j (n) = (Bj(n) Bj(n))

q (3.2)

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since V(q)j (n) = q0 (the Kronecker delta). Then, if m > 1, solve the differential

equations in the interval (tm2, tm1) subject to (3.1) with t = tm1, andproceed in this manner downwards.

Letting m(q)j (t) denote the q-th central moment corresponding to the non-

central V(q)j (t), we have

m(1)j (t) = V

(1)j (t) , (3.3)

m(q)j (t) =

qp=0

(1)qp(q

p

)V

(p)j (t)

(V

(1)j (t)

)qp. (3.4)

B. Numerical examples. Referring to disability model, consider a maleinsured at age 30 for a period of 30 years. We assume that the intensities oftransition at time t, when the insured is 30 + t years old, are

ad(t) = id(t) = 0.0005 + 0.000075858 100.038(30+t) ,

ai(t) = 0.0004 + 0.0000034674 100.06(30+t) ,

ia(t) = 0.005 ,

and that the interest rate is constant, r = 0.05.

The central moments m(q)j (0) defined in (3.3) (3.4) have been computed

for the states a and i (state d is uninteresting) and are shown in Table 1 for(1) a term insurance with sum 1 (= b02 = b12);(2) an annuity payable in active state with level intensity 1 (= b0);(3) an annuity payable in disabled state with level intensity 1 (= b1);(4) for a combined policy providing a term insurance with sum 1 (= b02 = b12)and a disability annuity with level intensity 0.5 (= b1) against level net premium0.0125 (= b0) payable in active state.

Table 1: Central moments m(q)j (0), q = 1, 2, 3, of the present value of the four

payments streams (1) (4) listed above.

m(1)a (0) m

(1)i (0) m

(2)a (0) m

(2)i (0) m

(3)a (0) m

(3)i (0)

(1) 0.063 0.063 0.027 0.027 0.012 0.012(2) 14.748 0.790 4.816 6.510 39.540 61.399(3) 0.245 13.573 1.477 8.675 12.061 69.004(4) 0.000 7.158 0.397 2.234 1.597 9.379

C. Solvency margins.Let W be the present value of all future net liabilities in respect of an insuranceportfolio. Denote the q-th central moment of W by m(q). The so-called normal

8

power approximation of the upper -fractile of the distribution of W , which wedenote by w1, is based on the first three moments and is

w1 m(1) + c1

m(2) +

c21 1

6

m(3)

m(2),

where c1 is the upper -fractile of the standard normal distribution. Adoptingthe so-called break-up criterion in solvency control, w1 can be taken as aminimum requirement on the technical reserve at the time of consideration.It decomposes into the premium reserve, m(1), and what can be termed thefluctuation reserve, w1 m(1). A possible measure of the riskiness of theportfolio is the ratio R =

(w1 m(1)

)/P , where P is some suitable measure

of the size of the portfolio. By way of illustration, consider a portfolio of Nindependent policies, all identical to the combined policy (4) in Table 1. Takingas P the total premium income per year, the value of R at the time of issue is48.61 for N = 10, 12.00 for N = 100, 3.46 for N = 1000, 1.06 for N = 10000,and 0.332 for N = 100000.

4 Stochastic interest and financial risk.

A. A Markov chain interest model.The economy (or rather the part of the economy that governs the interest) ismodeled as a homogeneous time-continuous Markov chain Y on a finite statespace Y = {1, . . . , JY }, with intensities of transition ef , e, f J Y , e 6= f .The associated indicator and counting processes are denoted by IYe and N

Yef ,

respectively, and the filtration generated by Y is denoted by FY = {FYt }t[0,T ].We assume that the force of interest takes a fixed value re when the economy

is in state e, that is,

r(t) = rY (t) =

e

IYe (t)re . (4.1)

Figure 2 outlines a simple Markov chain interest rate model with three states{1, 2, 3} and state-wise rates of interest r1 = 0.02 (low), r2 = 0.05 (medium),and r3 = 0.08 (high). Direct transition can only be made to a neighbouringstate, and the total intensity of transition out of any state is 0.5, that is, theinterest rate changes every two years on the average. By symmetry, the station-ary (long run average) interest rate is 0.05.

B. The full Markov model.We assume that the processes Y and Z are defiined on the same probabilityspace and that they are stochastically independent. Then (Y, Z) is a Markovchain on Y Z with intensities

ej,fk(t) =

ef (t) , e 6= f, j = k ,jk(t) , e = f, j 6= k ,0 , e 6= f, j 6= k .

9

r1 = 0.02 r2 = 0.05 r3 = 0.08-12 = 0.5

21 = 0.25

-23 = 0.2532 = 0.5

Figure 2: Sketch of a simple Markov chain interest model.

For the purpose of assessing the contractual liability we are interested inaspects of its conditional distribution, given the available information at timet. We focus here on determining the conditional moments. By the Markovassumption, the functions in quest are the state-wise conditional moments

V(q)ej (t) = E

[( Tt

e

tr dB()

)q Y (t) = e, Z(t) = j].

The differential equations extend straightforwardly to

d

dtV

(q)ej (t) = (qre + j(t) + e)V

(q)ej (t) qbj(t)V

(q1)ej (t)

k;k 6=j

jk(t)

qp=0

(q

p

)(bjk(t))

pV(qp)ek (t)

f ;f 6=e

efV(q)fj (t) . (4.2)

Denote by m(q)ej (t) the q-th central moment corresponding to V

(q)ej (t).

C. Numerical results for a combined insurance policy.Consider the combined life insurance and disability pension policy and theMarkov chain interest model in Figure 2 modified such that the infinitesimalmatrix is

=

1 1 00.5 1 0.5

0 1 1

. (4.3)

The scalar can be interpreted as the expected number of interest changes pertime unit.Table 2 displays the first three central moments of the present value at time

0. The level premium rate (= b1) is the net premium rate in state (2,a) (i.e.the rate that establishes expected balance between discounted premiums andbenefits when the insured is active and the interest is at medium level 0.05 attime 0).The first three rows in the body of the table form a benchmark; = 0

means no interest fluctuation, and we therefore obtain the results for threecases of fixed interest. It is seen that the second and third order moments of the

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Table 2: Central moments m(q)ej (0) of orders q = 1, 2, 3 of the present value of

future benefits less premiums for the combined policy in interest state e andpolicy state j at time 0, for some different values of the rate of interest changes,. Second column gives the net premium pi of a policy starting from intereststate 2 (medium) and policy state 1 (active).

e, j : 1, 1 1, 2 2, 1 2, 2 3, 1 3, 2

pi q

1 0.072 10.206 0.000 7.158 0.030 5.3180 .0125 2 1.156 6.024 0.397 2.234 0.159 0.938

3 6.845 36.093 1.597 9.379 0.465 2.947

1 0.030 9.166 0.000 7.254 0.018 5.843.05 .0127 2 0.838 5.622 0.438 3.151 0.236 1.793

3 4.652 21.814 1.895 7.923 0.806 3.110

1 0.001 7.601 0.000 7.212 0.001 6.854.5 .0126 2 0.467 3.104 0.416 2.767 0.370 2.467

3 2.070 10.546 1.740 8.808 1.462 7.357

1 0.000 7.207 0.000 7.165 0.000 7.1235 .0125 2 0.404 2.330 0.399 2.302 0.394 2.274

3 1.643 9.491 1.613 9.322 1.585 9.155

1 0.000 7.158 0.000 7.158 0.000 7.158 .0125 2 0.397 2.234 0.397 2.234 0.397 2.234

3 1.597 9.379 1.597 9.379 1.597 9.379

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present value are strongly dependent on the (fixed) force of interest and, in fact,their absolute values decrease when the force of interest increases (as could beexpected since increasing interest means decreasing discount factors and, hence,decreasing present values of future amounts).It is seen that, as increases, the differences across the three pairs of columns

get smaller and in the end they vanish completely. The obvious interpretationis that the initial interest level is of little importance if the interest changesrapidly.The overall impression from the two central columns corresponding to medium

interest is that, as increases from 0, the variance of the present value will firstincrease to a maximum and then decrease again and stabilize. This observationsupports the following piece of intuition: the introduction of moderate interestfluctuation adds uncertainty to the final result of the contract, but if the interestchanges sufficiently rapidly, it will behave like fixed interest at the mean level.Presumably, the values of the net premium in the second column reflect thesame effect.

5 Getting rid of financial risk

A. Non-diversifiable risk.The principle of equivalence rests on the implicit assumption that the experi-ence basis, that is the transition intensities, interest, and administration coststhroughout the contract period, are known at the time of inception of the con-tract. In reality, however, the experience basis may undergo significant andunforeseeable changes within the time horizon of the contract, thus exposingthe insurer to a risk that is non-diversifiable. In the present paper, which fo-cuses on the interplay between insurance and finance, we will concentrate onthe interest rate and assume that all other elements of the experience basis areknown and fixed throughtout the term of the contract.The risk stemming from the uncertain development of the interest rate can,

under certain ideal market conditions, be eliminated by letting the contractualpayments depend on the returns on the companys investments. Products ofthis type, known as unit-linked insurances, have been gaining increasing marketshares ever since they emerged some few decades ago, and today they are alsotheoretically well understood, see Aase and Persson (1994), Mller (1998), andreferences therein.Unlike the unit-linked concept, a standard life insurance policy specifies con-

tractual payments in nominal amounts, binding to both parties throughout theentire term of the contract. Thus, an adverse development of the interest ratescan not be countered by raising premiums or reducing benefits and also not bycancelling the contract (the right of withdrawal remains one-sidedly with theinsured). The only way the insurer can prevent the non-diversifiable financialrisk is to charge premiums to the safe side. In practice this is done by calcu-lating premiums on a conservative so-called technical basis or first order basis,which represents a provisional worst-case scenario for the future development

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of the experience basis. In our simplified set-up, with interest as the only un-certain element in the experience basis, this means that premiums and reservesare calculated under the assumption that the interest rate is r, typically lowerthan expected. Denote the corresponding first order state-wise reserve by V j .Premiums are set in accordance with the principle of equivalence,

E

[ n0

e

0r dB()

]= 0 , (5.1)

or, equivalently,V 1 (0) = B1(0) . (5.2)

B. Definition of the surplus.With premiums based on prudent first order assumptions, the portfolio willcreate a systematic technical surplus if everything goes well. We define thesurplus at time t as

S(t) =

t0

e

t

r d(B)() V Z(t)(t) (5.3)

= e

t

0r

t0

e

0 dB() V Z(t)(t) , (5.4)

which is past net income (premiums less benefits), compounded with the factualsecond order interest, minus expected discounted future liabilities valuated onthe conservative first order basis. This definition complies with practical ac-countancy regulations in insurance since S(t) is precisely the difference betweenthe current cash balance and the first order reserve that by statute has to beprovided to meet future liabilities. Notice that S(0) = 0, a consequence of (5.1),

and S(T ) = T0e

T

dB(), as it ought to be.

Differentiating (5.4) gives

dS(t) = e

t

0rr(t) dt

t0

e

0r dB() dB(t) dV Z(t)(t)

= r(t) dt S(t) + r(t) dt V Z(t)(t) dB(t) dVZ(t)(t) .

Upon substituting dB(t) from (2.1) and V Z(t)(t) from (2.5), using the generalIt formula to write

dV Z(t)(t) =

j

IZj (t) dV

j (t) +j 6=k

{V k (t) Vj (t)} dN

Zjk(t)

(there are almost surely no common jumps of the deterministic state-wise re-serves and the counting processes), and picking dV j (t) from (2.8), we find

dS(t) = (t) dt S(t) + dC(t) + dM(t) , (5.5)

where

dC(t) =

j

IZj (t) cj(t) dt ,

13

with

cj(t) = (r(t) r)V j (t) (5.6)

anddM(t) =

j 6=k

Rjk(t) dMZjk(t) ,

with the MZjk defined in (2.3).The processM is a zero mean H-martingale in the conditional model, given

Gn, that is,E[M(t) | Hs Gn] = M(s)

for s t, and M(0) = 0. Then it is also a zero mean F-martingale in the fullmodel since E[M(t) | Fs] = E [ E[M(t) | Hs Gn] | Fs] = E[M(s) | Fs] = M(s).The term dM(t) in (5.5) is the purely accidental part of the surplus increment.The two first terms on the right of (5.5) are the systematic parts, which make thesurplus drift to something with expected value different from 0. The first termis the earned interest on the surplus itself, and what remains is quite naturallythe policy-holders contribution to the technical surplus.To put it another way, let us switch the first term on the right of (5.5) over

to the left and multiply the equation with e

t

0 to form a complete differential

on the left hand side. Integrating from 0 to t and using the fact that S(0) =C(0) = M(0) = 0, we arrive at

e

t

0S(t) =

t0

e

0dC() +

t0

e

0dM() , (5.7)

showing that the discounted surplus at time t is the discounted total contribu-tions plus a martingale representing noise. We can rewrite (5.7) as

S(t) =

t0

e

t

dC() +

t0

e

t

dM() , (5.8)

displaying the surplus at time t as the compounded total of contributions andaccidental martingale increments. (Beware that the last term on the right of(5.8) is not a martingale although its expected value is 0 for all t.)

C. Redistribution of surplus Bonus.The technical surplus belongs to the insured and has to paid back as bonus.There are many possible schemes. The simplest is cash bonus : The rate atwhich bonus will be paid at some fixed future time u, provided the insured isthen alive, is

W = (r(u) r)V (u) .

Adopting the Markov chain interest model, we can make model-based predictionof this quantity. At time t < u, given r(t) = re,W is predicted by its conditionalexpected value

We(t) = E[W | r(t) = re] .

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It is easy to show that the functions We(t) are the solution to the differentialequations

d

dtWe(t) =

`;f 6=e

ef (We(t)Wf (t)) ,

subject to the conditions

We(u) = (re r)V u ,

e = 1, . . . , JY .By terminal bonus the surpluses are accumulated and paid back as a lump

sum at the term of the contract T , provided the insured is then alive. Theamount paid is

W =

T0

e

T

r(r() r)V () d

= W (t)

t0

e

t

r(r() r)V () d +W (t) ,

where

W (t) = e

T

tr ,

W (t) =

Tt

e

T

r(r() r)V () d .

The random variables W (t) and W (t), which are unknown at time t, arepredicted by

W e(t) = E[W(t) | r(t) = re] ,

W e (t) = E[W(t) | r(t) = re] .

Writing

W (t) = er(t)dtW (t+ dt) ,

W (t) = W (t) (r(t) r)V (t) dt + W (t+ dt) ,

we easily show by a backward argument that the functions W e(t) and We (t)

are the solution to the differential equations

d

dtW e(t) = re W

e(t) +

f ;f 6=e

ef (We(t)W

f (t)) ,

d

dtW e (t) = W

e(t)(re r

)V (t) +

f ;f 6=e

ef (We (t)W

f (t)) ,

subject to the conditions

W e(T) = 1 ,

W e (T) = 0 ,

e = 1, . . . , JY .

15

6 Reintroducing financial risk, and eliminating

it again

A. Guaranteed interest.Recall the basic rules of the with profit insurance contract: On the one hand,any surplus is to be redistributed to the insured. On the other hand, benefits andpremiums set out in the contract cannot be altered to the insureds disadvantage.This means that negative surplus, should it occur, cannot result in negativebonus. Thus, the with profit policy comes with an interest rate guarantee tothe effect that bonus is to be paid as if factual interest were no less than firstorder interest, roughly speaking. For instance, cash bonus is to be paid at rate

(r(t) r)+V(t)

per survivor at time t, hence the insurer has to cover

(r r(t))+V(t) . (6.1)

Similarly, terminal bonus (typical for e.g. a pure endowment benefit) is to bepaid as a lump sum ( T

0

e

T

r(r() r)V () d

)+

per survivor at time n, hence the insurer has to cover( T0

e

T

r(r r())V () d

)+

. (6.2)

(We write a+ = max(a, 0) = a 0.)An interest guarantee of this kind represents a liability on the part of in-

surer. It cannot be offered for free, of course, but has to be compensated by apremium. This can certainly be done without violating the rules of game forthe participating policy, which lay down that premiums and benefits be set outin the contract at time 0. Thus, for simplicity, suppose a single premium is tobe collected at time 0 for the guarantee. The question is, how much should itbe?Being brought up with the principle of equivalence, we might think that the

expected discounted value of the liability is an agreeable candidate for the pre-mium. However, the rationale of the principle of equivalence, which was to makepremiums and benefits balance on the average in an infinitely large portfolio,does not apply to financial risk. Interest rate variations cannot be eliminatedby increasing the size of the portfolio; all policy-holders are faring together inone and the same boat on their once-in-a-lifetime voyage through the troubledwaters of their chapter of economic history. This risk cannot be averaged outin the same way as the risk associated with the lengths of the individual lives.

16

None the less, in lack of anything better, let us find the expected discountedvalue of the interest guarantee, and just anticipate here that this actually wouldbe the correct premium in an extended model specifying a so-called completefinancial market. Those who are familiar with basic arbitrage theory know whatthis means. Those who are not should at this stage just imagine that, in ad-dition to the bank account with the interest rate r(t), there are some otherinvestment opportunities, and that any future financial claim can be duplicatedperfectly by investing a certain amount at time 0 and thereafter just sellingand buying available assets without any further infusion of capital. The initialamount required to perform this duplicating investment strategy is, quite natu-rally, the price of the claim. It turns out that this price is precisely the expecteddiscounted value of the claim, only under a different probability measure thanthe one we have specified in our physical model. With these reassuring phrases,let us proceed to find the expected discounted value of the interest guarantee.

B. Pricing guaranteed interest.Consider cash bonus with gurantee given by (6.1): Given that r(0) = re (say),the price of the total claims under the guarantee, averaged over an infinitelylarge portfolio, is

E

[ T0

e

0r(r r())+V

() px d

r(0) = re]. (6.3)

A natural starting point for creating some useful differential equations by thebackward construction is the price of future claims under the guarantee in statee at time t,

We(t) = E

[ Tt

e

tr(r r())+V

() px d

r(t) = re], (6.4)

e = 1, . . . , JY , 0 t T . The price in (6.3) is precisely We(0).Conditioning on what happens in the time interval (t, t+ dt] and neglecting

terms of order o(dt) that will disappear in the end anyway, we find

We(t) = (1edt)((r re)+V

(t) tpx dt+ eredtWe(t+ dt)

)+

f ; f 6=e

efdtWf (t) .

From here we easily arrive at the differential equations

d

dtWe(t) = (r

re)+V(t) tpx+reWe(t)

f ; f 6=e

efdt (Wf (t)We(t)) , (6.5)

which are to be solved subject to the conditions

We(T) = 0 . (6.6)

17

Next, consider terminal bonus at time n given by (6.2): Given r(0) = re,the price of the claim under the guarantee, averaged over an infinitely largeportfolio, is

E

[e

n

0r

( T0

e

T

r(r r())V () d

)+

T px

r(0) = re]

= E

[( T0

e

0r(r r())V () d

)+

r(0) = re]

T px . (6.7)

The price of the claim at time t should be the conditional expected dis-counted value of the claim, given what we know at the time:

E

[e

T

tr

( T0

e

T

r(r r())V () d

)+

T px

r(); 0 t]

= E

[(U(t) +

nt

e

tr(r r())V () d

)+

r(); 0 t]

T px .

(6.8)

where

U(t) =

t0

e

t

r(r r())V () d .

The quantity in (6.8) is more involved than the one in (6.4) since it dependseffectively on the past history of interest rate through U(t). We can, therefore,not hope to end up with the same simple type of problem as above and in allother situations encountered so far, where we essentially had to determine theconditional expected value of some function depending only on the future courseof the interest rate. Which was easy since, by the Markov property, we couldlook at state-wise conditional expected values We(t), e = 1, . . . , J

Y , say. Theseare deterministic functions of the time t only and can be determined by solvingordinary differential equations.Let us proceed and see what happens. Due to the Markov property (condi-

tional independence between past and future, given the present) the expressionin (6.8) is a function of t, r(t) and U(t). Dropping the uninteresting factor T px,consider its value for given U(t) = u and r(t) = re,

We(t, u) = E

[(u+

Tt

e

tr(r r())V () d

)+

r(t) = re].

Use the backward construction:

We(t, u) =

(1 edt)E

[(u+ (r re)V (t) dt+ ere dt

Tt+dt

e

t+dtr(r r())V () d

)+

r(t + dt) = re]

18

+

f ; f 6=e

efdtWf (t, u) =

(1 edt)ere dtWe(t+ dt, e

re dtu+ (r re)Vt dt) +

f ; f 6=e

efdtWf (t, u) .

Insert here ere dt = 1 redt+ o(dt),

We(t+ dt, ere dtu+ (r re)V

(t) dt) =

We(t, u) +

tWe(t, u) dt+

uWe(t, u)(u re + (r

re)V(t)) dt+ o(dt) ,

and proceed in the usual manner to arrive at the partial differential equations

tWe(t, u)+(ure+(r

re)V(t))

uWe(t, u)r

eWe(t, u)+

f ; f 6=j

ef (Wf (t, u)We(t, u)) = 0 .

These are to be solved subject to the conditions

We(T, u) = u+ ,

e = 1, . . . , JY .Since the functions we are interested in involved both t and U(t), we are lead

to state-wise functions in two arguments and, therefore, quite naturally end upwith partial differential equations for those.

7 A Markov chain financial market

A. Motivation.The theory of diffusion processes, with its wealth of powerful theorems andmodel variations, is an indispensable toolkit in modern financial mathematics.The seminal papers of Black and Scholes [6] and Merton [21] were crafted withBrownian motion, and so were most of the almost countless papers on arbitragepricing theory and its bifurcations that followed over the past quarter of acentury.A main course of current research, initiated by the martingale approach to

arbitrage pricing ([15] and [16]), aims at generalization and unification. Todaythe core of the matter is well understood in a general semimartingale setting,see e.g. [9]. Another course of research investigates special models, in particu-lar various Levy motion alternatives to the Brownian driving process, see e.g.[10] and [27]. Pure jump processes have been widely used in finance, rangingfrom plain Poisson processes introduced in [22] to quite general marked pointprocesses, see e.g. [4]. And, as a pedagogical exercise, the market driven by abinomial process has been intensively studied since it was launched in [8].We will here present a model where the financial market is driven by a con-

tinuous time homogeneous Markov chain. The idea was launched in [26] and

19

reappeared in [11], the context being limited to modelling of the spot rate ofinterest. This will allow us to synthesize insurance and finance within the math-ematical model framework already familiar to us. Some additional notation andresults are presented in Appendix H.

B. The Markov chain marketWe are going to extend the interest model in 4. Thus let {Yt}t0 be a contin-uous time Markov chain with finite state space Y = {1, . . . , JY }. Recall thatthe associated indicator and counting processes are denoted by IYe and N

Yef .

The FY = {FYt }t0, is taken to satisfy the usual conditions of right-continuity(Ft = u>tFu) and completeness (F0 contains all subsets of P-nullsets), andF0 is assumed to be the trivial (,). This means, essentially, that Y is right-continuous (hence the same goes for the IYe and the N

Yef ) and that Y0 deter-

ministic.We assume that Y is time homogeneous so that the transition probabilities

pYef (s, t) = P[Yt = f | Ys = e]

depend only on the length of the transition period, ts. Henceforth we thereforewrite pYef (s, t) = p

Yef (t s). This implies that the transition intensities

ef = limt0

pYef (t)

t, (7.1)

e 6= f , exist and are constant. To avoid repetitious reminders of the typee, f Y, we reserve the indices e and f for states in Y throughout. We willfrequently refer to

Ye = {f ;ef > 0} ,

the set of states that are directly accessible from state e, and denote the numberof such states by

JYe = |Ye| .

Putee = e =

f ;fYe

ef

(minus the total intensity of transition out of state e). We assume that all statesintercommunicate so that pYef (t) > 0 for all e, f (and t > 0). This implies that

JYe > 0 for all e (no absorbing states). The matrix of transition probabilities,

PY (t) = (pef (t)) ,

and the infinitesimal matrix, = (ef ) ,

are related by (7.1), which in matrix form reads = limt01t (P

Y (t) I), andby the backward and forward Kolmogorov differential equations,

d

dtPY (t) = PY (t) = PY (t) . (7.2)

20

Under the side condition PY (0) = I, (7.2) integrates to

PY (t) = exp(t) . (7.3)

In the representation (H.2),

PY (t) = De=1,...,JY (eet)1 =

ne=1

eetjj , (7.4)

the first (say) eigenvalue is 1 = 0, and corresponding eigenvectors are 1 = 1and 1 = (p1, . . . , pn) = limt(pe1(t), . . . , peJY (t)), the stationary distribu-tion of Y . The remaining eigenvalues, 2, . . . , n, are all strictly negative sothat, by (7.4), the transition probabilities converge exponentially to the sta-tionary distribution as t increases.We now turn to the subject matter of our study and, referring to introduc-

tory texts like [5] and [32], take basic notions and results from arbitrage pricingtheory as prerequisites.

C. The continuous time Markov chain market.We consider a financial market driven by the Markov chain described above.Thus, Yt represents the state of the economy at time t, FYt represents theinformation available about the economic history by time t, and FY representsthe flow of such information over time.In the market there are m+ 1 basic assets, which can be traded freely and

frictionlessly (short sales are allowed, and there are no transaction costs). Aspecial role is played by asset No. 0, which is a locally risk-free bank accountwith state-dependent interest rate

r(t) = rYt =

e

Ie(t)re ,

where the state-wise interest rates re, e = 1, . . . , JY , are constants. Thus, its

price process is

S0(t) = exp

( t0

r(s) ds

)= exp

(e

re

t0

Ie(s) ds

),

with dynamics

dS0(t) = S0(t) r(t) dt = S0(t)

e

reIe(t) dt .

(Setting S0(0) = 1 a just a matter of convention.)The remaining m assets, henceforth referred to as stocks, are risky, with

price processes of the form

Si(t) = exp

e

ie

t0

Ie(s) ds+

e

fYe

iefNef (t)

, (7.5)

21

i = 1, . . . ,m, where the ie and ief are constants and, for each i, at least oneof the ief is non-null. Thus, in addition to yielding state-dependent returns ofthe same form as the bank account, stock No. i makes a price jump of relativesize

ief = exp (ief ) 1

upon any transition of the economy from state j to state k. By the general Itsformula, its dynamics is given by

dSi(t) = Si(t)

e

ieIe(t) dt+

e

fYe

iefdNef (t)

. (7.6)

Taking the bank account as numeraire, we introduce the discounted stockprices Si(t) = Si(t)/S0(t), i = 0, . . . ,m. (The discounted price of the bankaccount is Bt 1, which is certainly a martingale under any measure). Thediscounted stock prices are

Si(t) = exp

e

(ie re)

t0

Ie(s) ds+

e

fYe

iefNef (t)

, (7.7)

with dynamics

dSi(t) = Si(t)

e

(ie re)Ie(t) dt +

e

fYe

iefdNef (t)

, (7.8)

i = 1, . . . ,m.We stress that the theory we are going to develop does not aim at explain-

ing how the prices of the basic assets emerge from supply and demand, businesscycles, investment climate, or whatever; they are exogenously given basic enti-ties. (And God said let there be light, and there was light, and he said letthere also be these prices.) The purpose of the theory is to derive principles forconsistent pricing of financial contracts, derivatives, or claims in a given market.

D. Portfolios.A dynamic portfolio or investment strategy is an m + 1-dimensional stochasticprocess

(t) = ((t), (t)) ,

where (t) represents the number of units of the bank account held at time t,and the i-th entry in

(t) = (1(t), . . . , m(t))

represents the number of units of stock No. i held at time t. As it will turnout, the bank account and the stocks will appear to play different parts in theshow, the latter being the more visible. It is, therefore, convenient to costume

22

the two types of assets and their corresponding portfolio entries accordingly. Tosave notation, however, it is useful also to work with double notation

(t) = (0(t), . . . , m(t)) ,

with 0(t) = (t), i(t) = i(t), i = 1, . . . ,m, and work with

St = (S1(t), . . . , Sm(t)) .

andSt = (S0(t), . . . , Sm(t))

.

The portfolio is adapted to FY (the investor cannot see into the future), andthe shares of stocks, , must also be FY -predictable (the investor cannot, e.g.upon a sudden crash of the stock market, escape losses by selling stocks atprices quoted just before and hurry the money over to the locally risk-free bankaccount.)The value of the portfolio at time t is

V (t) = (t)S0(t) +m

i=1

i(t)Si(t) = (t)S0(t) + (t)S(t) = (t)S(t)

Henceforth we will mainly work with discounted prices and values and, inaccordance with (7.7), equip their symbols with a tilde. The discounted valueof the portfolio at time t is

V (t) = (t) + (t) S(t) = (t) S(t) . (7.9)

The strategy is self-financing (SF) if dV (t) = (t) dS(t) or, equivalently,

dV (t) = (t) dS(t) =

mi=1

i(t) dSi(t) . (7.10)

We explain the last step: Put D(t) = S0(t)1, a continuous process. The dy-

namics of the discounted prices S(t) = D(t)S(t) is then dS(t) = dD(t)S(t)+D(t) dS(t). Thus, for V (t) = D(t)V (t), we have

dV (t) = dD(t)V (t) +D(t) dV (t) = dD(t)(t)S(t) +D(t)(t) dS(t)

= (t) (dD(t)S(t) +D(t) dS(t)) = (t) dS(t) ,

hence the property of being self-financing is preserved under discounting.The SF property says that, after the initial investment of V 0 , no further

investment inflow or dividend outflow is allowed. In integral form:

V (t) = V 0 +

t0

(s) dS(s) = V 0 +

t0

(s) dS(s) . (7.11)

Obviously, a constant portfolio is SF; its discounted value process isV (t) = S(t), hence (7.10) is satisfied. More generally, for a continuous port-folio we would have dVt() = d

(t) S(t)+(t) dS(t), and the self-financing

23

condition would be equivalent to the a budget constraint d(t) S(t) = 0, whichsays that any purchase of assets must be financed by a sale of some other assets.We urge to say that we shall typically be dealing with portfolios that are notcontinuous and, in fact, not even right-continuous so that d(t) is meaningless(integrals with respect to the process are not well defined).

E. Absence of arbitrage.An SF portfolio is called an arbitrage if, for some t > 0,

V 0 < 0 and V(t) 0 a.s. P ,

or, equivalently,V 0 < 0 and V

(t) 0 a.s. P .

A basic requirement on a well-functioning market is the absence of arbitrage.The assumption of no arbitrage, which appears very modest, has surprisinglyfar-reaching consequences as we shall see.A martingale measure is any probability measure P that is equivalent to P

and such that the discounted asset prices S(t) are martingales (F, P). Thefundamental theorem of arbitrage pricing says: If there exists a martingalemeasure, then there is no arbitrage. This result follows from easy calculationsstarting from (7.11): Forming expectation E under P and using the martingaleproperty of S under P, we find

E[V (t)] = V 0 + E[

t0

(s) dS(s)] = V 0

(the stochastic integral is a martingale). It is seen that arbitrage is impossible.We return now to our special Markov chain driven market. Let

= (ef )

be an infinitesimal matrix that is equivalent to in the sense that ef = 0 if and

only if ef = 0. By Girsanovs theorem, there exists a measure P, equivalent to

P, under which Y is a Markov chain with infinitesimal matrix . Consequently,the processes MYef , j = 1, . . . , J

Y , f Ye, defined by

dMYef (t) = dNef (t) Ie(t)ef dt , (7.12)

and Mef (0) = 0, are zero mean, mutually orthogonal martingales w.r.t. (FY , P).

Rewrite (7.8) as

dSi(t) = Sit

e

ie re +

fYe

ief ef

Ie(t) dt+

e

fYe

iefdMYef (t)

,(7.13)

i = 1, . . . ,m. The discounted stock prices are martingales w.r.t. (FY , P) if andonly if the drift terms on the right vanish, that is,

ie re +

fYe

ief ef = 0 , (7.14)

24

j = 1, . . . , JY , i = 1, . . . ,m. From general theory it is known that the existenceof such an equivalent martingale measure P implies absence of arbitrage.The relation (7.14) can be cast in matrix form as

re1e = ee , (7.15)

j = 1, . . . , JY , where 1 is m 1 and

e = (ie)i=1,...,m , e = (ief )fYei=1,...,m , e =

(ef

)fYe

.

The existence of an equivalent martingale measure is equivalent to the existenceof a solution e to (7.15) with all entries strictly positive. Thus, the market isarbitrage-free if (and we can show only if) for each j, re1e is in the interiorof the convex cone of the columns of e.Assume henceforth that the market is arbitrage-free so that (7.13) reduces

to

dSi(t) = Si(t)

e

fYe

iefdMYef (t) , (7.16)

where the Mef defined by (7.12) are martingales w.r.t. (FY , P) for some measure

P that is equivalent to P.Inserting (7.16) into (7.10), we find that is SF if and only if

dV (t) =

e

fYe

mi=1

i(t)Si(t)iefdMYef (t) , (7.17)

implying that V is a martingale w.r.t. (FY , P) and, in particular,

V (t) = E[V (t) | Ft] . (7.18)

Here E denotes expectation under P. (Note that the tilde, which in the firstplace was introduced to distinguish discounted values from the nominal ones, isalso attached to the equivalent martingale measure and certain related entities.This usage is motivated by the fact that the martingale measure arises from thediscounted basic price processes, roughly speaking.)

F. Attainability.A T -claim is a contractual payment due at time T . Formally, it is an FYT -measurable random variable H with finite expected value. The claim is attain-able if it can be perfectly duplicated by some SF portfolio , that is,

V T = H . (7.19)

If an attainable claim should be traded in the market, then its price mustat any time be equal to the value of the duplicating portfolio in order to avoid

25

arbitrage. Thus, denoting the price process by pi(t) and, recalling (7.18) and(7.19), we have

pi(t) = V (t) = E[H | Ft] , (7.20)

or

pi(t) = E[e

T

trH Ft] . (7.21)

By (7.20) and (7.17), the dynamics of the discounted price process of anattainable claim is

dpi(t) =

e

fYe

mi=1

i(t)Si(t)iefdMYef (t) . (7.22)

G. Completeness.Any T -claim H as defined above can be represented as

H = E[H ] +

T0

e

fYe

ef (t)dMYef (t) , (7.23)

where the ef (t) are FY -predictable and integrable processes. Conversely, any

random variable of the form (7.23) is, of course, a T -claim. By virtue of (7.19),and (7.17), attainability of H means that

H = V 0 +

T0

dV (t)

= V 0 +

T0

e

fYe

i

i(t)Si(t)iefdMYef (t) . (7.24)

Comparing (7.23) and (7.24), we see that H is attainable iff there exist pre-dictable processes 1(t), . . . , m(t) such that

mi=1

i(t)Si(t)ief = ef (t) ,

for all j and f Ye. This means that the JYe -vector

e(t) = (ef (t))fYe

is in R(e).

The market is complete if every T -claim is attainable, that is, if every nj-vector is in R(e

). This is the case if and only if rank(e) = JYe , which can be

fulfilled for each e only if m maxe JYe .

26

8 Arbitrage-pricing of derivatives in a complete

market

A. Differential equations for the arbitrage-free price.Assume that the market is arbitrage-free and complete so that prices of T -claimsare uniquely given by (7.20) or (7.21).Let us for the time being consider a T -claim of the form

H = h(Y (T ), S`(T )) . (8.1)

Examples are a European call option on stock No. ` defined by H = (S`(T )K)+, a caplet defined by H = (r(T ) g)+ = (rYT g)

+, and a zero couponT -bond defined by H = 1.For any claim of the form (8.1) the relevant state variables involved in the

conditional expectation (7.21) are t, Y (t), S`(t), hence pi(t) is of the form

pi(t) =

JYe=1

Ie(t)fe(t, S`(t)) , (8.2)

where the

fe(t, s) = E[e

T

trH Y (t) = e, S`(t) = s] (8.3)

are the state-wise price functions.The discounted price (7.20) is a martingale w.r.t. (FY , P). Assume that the

functions fe(t, s) are continuously diferentiable. Using It on

pi(t) = e

t

0r

JYe=1

Ie(t)fe(t, S`(t)) , (8.4)

we find

dpi(t) = e

t

0r

e

Ie(t)

(re fe(t, S`(t)) +

tfe(t, S`(t)) +

sfe(t, S`(t))S`(t)`j

)dt

+e

t

0r

e

fYe

(ff (t, S`(t)(1 + `ef )) fe(t, S`(t))) dNef (t)

= e

t

0r

e

Ie(t) (re fe(t, S`(t)) +

tfe(t, S`(t)) +

sfe(t, S`(t))S`(t)`e

+

fYe

{ff (t, S`t(1 + `ef )) fe(t, S`(t))}ef

)dt

+e

t

0r

e

fYe

(ff (t, S`(t)(1 + `ef )) fe(t, S`(t))) dMYef (t) . (8.5)

27

By the martingale property, the drift term must vanish, and we arrive at thenon-stochastic partial differential equations

re fe(t, s) +

tfe(t, s) +

sfe(t, s)s`e

+

fYe

(ff (t, s(1 + `ef )) fe(t, s)) ef = 0 , (8.6)

e = 1, . . . , JY , which are to be solved subject to the side conditions

fe(T, s) = h(e, s) , (8.7)

e = 1, . . . , JY .In matrix form, with

R = Dj=1,...,JY (re) , A` = Dj=1,...,JY (`e) ,

and other symbols (hopefully) self-explaining, the differential equations and theside conditions are

Rf(t, s) +

tf(t, s) + sA`

sf(t, s) + f(t, s(1 + )) = 0 , (8.8)

f(T, s) = h(s) . (8.9)

B. Identifying the strategy.Once we have determined the solution fe(t, s), e = 1, . . . , J

Y , the price processis known and given by (8.2).The duplicating SF strategy can be obtained as follows. Setting the drift

term to 0 in (8.5), we find the dynamics of the discounted price;

dpi(t) = e

t

0r

e

fYe

(ff (t, S`(t)(1 + `ef )) fe(t, S`(t))) dMYef (t) .(8.10)

Identifying the coefficients in (8.10) with those in (7.22), we obtain, for eachstate j, the equations

mi=1

i(t)Si(t)ief = ff (t, S`(t)(1 + `ef )) fe(t, S`(t)) , (8.11)

f Ye. The solution e(t) = (i,e(t))i=1,...,m (say) certainly exists since

rank(e) m, and it is unique iff rank(e) = m. Furthermore, it is a function oft and S(t) and is thus predictable. This simplistic argument works on the openintervals between the jumps of the process Y , where dMYef (t) = Ie(t)ef dt.For the dynamics (8.10) and (7.22) to be the same also at jump times, thecoefficients must clearly be left-continuous. We conclude that

(t) =JYe=1

Ie(t)(t) ,

28

which is predictable.Finally, is determined upon combining (7.9), (7.20), and (8.4):

(t) = e

t

0r

JYe=1

(Ie(t)fe(t, S`(t)) Ie(t)

mi=1

i,e(t)Si(t)

).

C. The Asian option.As an example of a path-dependent claim, let us consider an Asian option, which

essentially is a T -claim of the form H =( T

0S`() d K

)+, where K 0.

The price process is

pi(t) = E

e Tt r

( T0

S`() d K

)+FYt

=

e

Ie(t)fe

(t, S`(t),

t0

S`() d

),

where

fe(t, s, u) = E

e Tt r

( Tt

S`() + uK

)+Y (t) = j, S`(t) = s .(8.12)

The discounted price process is

pi(t) = e

t

0r

nj=1

Ie(t) fe

(t, S`(t),

t0

S`(s)

).

We obtain partial differential equations in three variables.The special case K = 0 is simpler, with only two state variables.

D. Interest rate derivatives.A particularly simple, but still important, class of claims are those of the formH = h(YT ). Interest rate derivatives of the form H = h(rT ) are included sincer(T ) = rYT . For such claims the only relevant state variables are t and Y (t), sothat the function in (8.3) depends only on t and e. The equation (8.6) reducesto

d

dtfe(t) = refe(t)

fYe

(ff (t) fe(t))ef , (8.13)

and the side condition is (put h(e) = he)

fe(T ) = he . (8.14)

In matrix form,d

dtf(t) = (R )f(t) ,

29

subject tof(T ) = h .

The solution is

f(t) = exp{(R)(T t)}h . (8.15)

It depends on t and T only through T t.In particular, the zero coupon bond with maturity T corresponds to h = 1.

We will henceforth refer to it as the T -bond in short and denote its price processby p(t, T ) and its state-wise price functions by p(t, T ) = (pe(t, T ))e=1,...,JY ;

p(t, T ) = exp{(R)(T t)}1 . (8.16)

For a call option on a U -bond, exercised at time T (< U) with price K, hhas entries he = (pe(T, U)K)+.In (8.15) (8.16) it may be useful to employ the representation shown in

(H.2),

exp{(R)(T t)} = Dj=1,...,JY (ej (Tt)) 1 , (8.17)

say.

9 Numerical procedures

A. Simulation.The homogeneous Markov process {Y (t)}t[0,T ] is simulated as follows: Let Kbe the number of transitions between states in [0, T ], and let T1, . . . , TK be thesuccessive times of transition. The sequence {(Tn, Y (Tn))}n=0,...,K is generatedrecursively, starting from the initial state Y (0) at time T0 = 0, as follows. Hav-ing arrived at Tn and Y (Tn), generate the next waiting time Tn+1 Tn as anexponential variate with parameter Y (n) (e.g. ln(Un)/Y (n), where Un has auniform distribution over [0, 1]), and let the new state Y (Tn+1) be k with proba-bility Y (n)k/Y (n). Continue in this mannerK+1 times until TK < T TK+1.

B. Numerical solution of differential equations.Alternatively, the differential equations must be solved numerically. For interestrate derivatives, which involve only ordinary first order differential equations,a Runge Kutta will do. For stock derivatives, which involve partial first orderdifferential equations, one must employ a suitable finite difference method, seee.g. [35].

10 Risk minimization in incomplete markets

A. Incompleteness.The notion of incompleteness pertains to situations where a contingent claim

30

cannot be duplicated by an SF portfolio and, consequently, does not receive aunique price from the no arbitrage postulate alone.In Paragraph ??F we were dealing implicitly with incompleteness arising

from a scarcity of traded assets, that is, the discounted basic price processesare incapable of spanning the space of all martingales w.r.t. (FY , P) and, inparticular, reproducing the value (7.23) of every financial derivative (functionof the basic asset prices).Incompleteness also arises when the contingent claim is not a purely finan-

cial derivative, that is, its value depends also on circumstances external to thefinancial market. We have in mind insurance claims that are caused by eventslike death or fire and whose claim amounts are e.g. inflation adjusted or linkedto the value of some investment portfolio.In the latter case we need to work in an extended model specifying a basic

probability space with a filtration F = {Ft}t0 containing FY and satisfying theusual conditions. Typically it will be the natural filtration of Y and some otherprocess that generates the insurance events. The definitions and conditions laiddown in Paragraphs ??C-E are modified accordingly, so that adaptedness of and predictability of are taken to be w.r.t. (F,P) (keeping the symbol P forthe basic probability measure), a T -claim H is FT measurable, etc.

B. Risk minimization.Throughout the remainder of the paper we will mainly be working with dis-counted prices and values without any other mention than the notational tilde.The reason is that the theory of risk minimization rests on certain martingalerepresentation results that apply to discounted prices under a martingale mea-sure. We will be content to give just a sketchy review of some main conceptsand results from the seminal paper of Fllmer and Sondermann [12].Let H be a T -claim that is not attainable. This means that an admissible

portfolio satisfyingV (T ) = H

cannot be SF. The cost, C(t), of the portfolio by time t is defined as that partof the value that has not been gained from trading:

C(t) = V (t)

t0

()dS() .

The risk at time t is defined as the mean squared outstanding cost,

R(t) = E[(C(T ) C(t))2

Ft] . (10.1)By definition, the risk of an admissible portfolio is

R(t) = E

[(H V (t)

Tt

()dS())2

Ft],

which is a measure of how well the current value of the portfolio plus futuretrading gains approximates the claim. The theory of risk minimization takes

31

this entity as its object function and proves the existence of an optimal admis-sible portfolio that minimizes the risk (10.1) for all t [0, T ]. The proof isconstructive and provides a recipe for how to actually determine the optimalportfolio.One sets out by defining the intrinsic value of H at time t as

V H(t) = E[H | Ft

].

Thus, the intrinsic value process is the martingale that represents the naturalcurrent forecast of the claim under the chosen martingale measure. By theGalchouk-Kunita-Watanabe representation, it decomposes uniquely as

V H(t) = E[H ] +

t0

H

(t)dS(t) + LH(t) ,

where LH is a martingale w.r.t. (F, P) which is orthogonal to S. The port-folio H defined by this decomposition minimizes the risk process among alladmissible strategies. The minimum risk is

RH(t) = E

[ Tt

dLH()

Ft].

C. Unit-linked insurance.As the name suggests, a life insurance product is said to be unit-linked if thebenefit is a certain predetermined number of units of an asset (or portfolio)into which the premiums are currently invested. If the contract stipulates aminimum value of the benefit, disconnected from the asset price, then one speaksof unit-linked insurance with guarantee. A risk minimization approach to pricingand hedging of unit-linked insurance claims was first taken by Mller [23], whoworked with the Black-Scholes-Merton financial market. We will here sketchhow the analysis goes in our Markov chain market, which conforms well withthe life history process in that they both are intensity-driven.Let Tx be the remaining life time of an x years old who purchases an insur-

ance at time 0, say. The conditional probability of survival to age x+ u, givensurvival to age x+ t (0 t < u), is

utpx+t = P[Tx > u |Tx > t] = e

u

tx+s ds , (10.2)

where y is the mortality intensity at age y. We have

d utpx+t = utpx+t x+t dt . (10.3)

Introduce the indicator of survival to age x+ t,

I(t) = 1[Tx > t] ,

and the indicator of death before time t,

N(t) = 1[Tx t] = 1 I(t) .

32

The process N(t) is a (very simple) counting process with intensity I(t)x+t,that is, M given by

dM(t) = dN(t) I(t)x+t dt (10.4)

is a martingale w.r.t. (F,P). Assume that the life time Tx is independent of theeconomy Y . We will work with the martingale measure P obtained by replacingthe intensity matrix of Y with the martingalizing and leaving the rest ofthe model unaltered.Consider a unit-linked pure endowment benefit payable at a fixed time T ,

contingent on survival of the insured, with sum insured equal to one unit ofstock No. `, but guaranteed no less than a fixed amount g. This benefit is acontingent T -claim,

H = (S`(T ) g) I(T ) .

The single premium payable as a lump sum at time 0 is to be determined.Let us assume that the financial market is complete so that every purely

financial derivative has a unique price process. Then the intrinsic value of H attime t is

V H(t) = pi(t) I(t) Ttpx+t ,

where pi(t) is the discounted price process of the derivative S`(T ) g.Using It and inserting (10.4), we find

dV H(t) = dpi(t) I(t) Ttpx+t + pi(t) I(t) Ttpx+t x+t dt+ (0 pit Ttpx+t) dN(t)

= dpi(t) I(t) Ttpx+t pit Ttpx+t dM(t) .

It is seen that the optimal trading strategy is that of the price process of thesum insured multiplied with the conditional probability that the sum will bepaid out, and that

dLH(t) = Ttpx+t pit dM(t) .

Consequently,

RH(t) =

Tt

Tsp2x+s E

[pi(s)2

Ft] stpx+t x+s ds= Ttpx+t

Tt

E[pi(s)2

Ft] Tspx+s x+s ds . (10.5)11 Trading with bonds: How much can be hedged?

A. A finite zero coupon bond market.Suppose an agent faces a contingent T -claim and is allowed to invest only inthe bank account and a finite number m of zero coupon bonds with maturitiesTi, i = 1, . . . ,m, all post time T . For instance, regulatory constraints may beimposed on the investment strategies of an insurance company. The questionis, to what extent can the claim be hedged by self-financed trading in theseavailable assets?

33

An allowed SF portfolio has discounted value process V (t) of the form

dV (t) =

mi=1

i(t)

e

fYe

(pf (t, Ti)pe(t, Ti))dMYef (t) =

e

d(Me(t))Fe(t)(t) ,

where is predictable, MY

e (t) = (MYef (t))

fYe is the ne-dimensional row vector

comprising the non-null entries in the j-th row of MY (t) = (MYef (t)), and

Fe(t) = YeFt

where

Ft = (pe(t, Ti))i=1,...,me=1,...,JY

= (p(t, T1), , p(t, Tm)) , (11.1)

and Ye is the JYe J

Y matrix which maps Ft to (pf (t, Ti) pe(t, Ti))i=1,...,mfYe

.

If e.g. Yn = {1, . . . , p}, then YJY

= (Ipp , 0p(np1) , 1p1).The sub-market consisting of the bank account and them zero coupon bonds

is complete in respect of T -claims iff the discounted bond prices span the spaceof all martingales w.r.t. (FY , P) over the time interval [0, T ]. This is the caseiff, for each e, rank(Fe(t)) = J

Ye . Now, since Ye obviously has full rank J

Ye ,

the rank of Fe(t) is determined by the rank of Ft in (11.1). We will argue that,typically, Ft has full rank. Thus, suppose c = (c1, . . . , cm)

is such that

Ftc = 0JY 1 .

Recalling (8.16), this is the same as

mi=1

ci exp{(R)Ti}1 = 0 ,

or, by (8.17) and since has full rank,

Dj=1,...,JY (

mi=1

cieeTi)11 = 0 . (11.2)

Since 1 has full rank, the entries of the vector 11 cannot be all null.Typically all entries are non-null, and we assume this is the case. Then (11.2)is equivalent to

mi=1

cieeTi = 0 , j = 1, . . . , JY . (11.3)

Using the fact that the generalized Vandermonde matrix has full rank, we knowthat (11.3) has a non-null solution c if and only if the number of distinct eigen-values e is less than m.

34

In the case where rank(Fe(t)) < nj for some j we would like to determine the

Galchouk-Kunita-Watanabe decomposition for a given FYT -claim. The intrinsicvalue process has dynamics

dHt =

e

fYe

ef (t)dMYef (t) =

e

d(Me(t))e(t) . (11.4)

We seek a decomposition of the form

dV (t) =

i

i(t) dp(t, Ti) +

e

fYe

ef (t) dMYef (t)

=

e

jYe

i

i(t) (pf (t, Ti) pe(t, Ti)) dMYef (t) +

e

fYe

ef (t)dMYef (t)

=

e

d(Me(t))Fe(t)e(t) +

e

d(Me(t))e(t) ,

such that the two martingales on the right hand side are orthogonal, that is,e

Ijt

fYe

(Fe(t)e(t))ee(t) = 0 ,

where e = D(e). This means that, for each e, the vector e(t) in (11.4) is tobe decomposed into its ,

eprojections onto R(Fe(t)) and its orthocomple-

ment. From (H.3) and (H.4) we obtain

Fe(t)e(t) = Pe(t)e(t) ,

wherePe(t) = Fe(t)(Fe(t)

eFe(t))

1Fe(t)e ,

hence

e(t) = (Fe(t)eFe(t))

1Fe(t)ee(t) . (11.5)

Furthermore,

e(t) = (IPe(t))e(t) , (11.6)

and the risk is Tt

e

pY (t)e(s t)

fYe

ef (ef (s))2 ds . (11.7)

The computation goes as follows: The coefficients ef involved in the in-trinsic value process (11.4) and the state-wise prices pj(t, Ti) of the Ti-bondsare obtained by simultaneously solving (8.6) and (8.13), starting from (8.9) and(8.13), respectively, and at each step computing the optimal trading strategy by (11.5) and the from (11.6), and adding the step-wise contribution to the

35

variance (11.7) (the step-length times the current value of the integrand).

B. First example: The floorlet.For a simple example, consider a floorlet H = (r rT )+, where T < mini Ti.The motivation could be that at time T the insurance company will ascribeinterest to the insureds account at current interest rate, but not less than aprefixed guaranteed rate r. Then H is the amount that must be provided perunit on deposit and per time unit at time T .Computation goes by the scheme described above, with the ef (t) = ff (t)

fe(t) obtained from (8.13) subject to (8.14) with he = (r re)+.

C. Second example: The interest guarantee in insurance.A more practically relevant example is an interest rate guarantee on a life in-surance policy. Premiums and reserves are calculated on the basis of a prudentso-called first order assumption, stating that the interest rate will be at somefixed (low) level r throughout the term of the insurance contract. Denote thecorresponding first order reserve at time t by V (t). The (portfolio-wide) meansurplus created by the first order assumption in the time interval [t, t + dt) is(r r(t))+tpxV

(t) dt. This surplus is currently credited to the account of theinsured as dividend, and the total amount of dividends is paid out to the insuredat the term of the contracts at time T . Negative dividends are not permitted,however, so at time T the insurer must cover

H =

T0

e

T

sr(r r(s))+sp

xV

(s) ds .

The intrinsic value of this claim is

Ht = E

[ T0

e

s

0r(r r(s))+ sp

xV

(s) ds

Ft]

=

t0

e

s

0r(r r(s))+ sp

xV

(s) ds+ e

t

0r

e

Ie(t)fe(t) ,

where the fe(t) are the state-wise expected values of future guarantees, dis-counted at time t,

fe(t) = E

[ Tt

e

s

tr(r r(s))+ sp

xV

(s) ds

Y (t) = e].

Working along the lines of Section 8, we determine the fe(t) by solving

d

dtfe(t) = (r

re)+

tpxV

(t) + refe(t)

fYe

(ff (t) fe(t))ef ,

subject to

fe(T ) = 0 . (11.8)

36

The intrinsic value has dynamics (11.4) with ef (t) = ff (t) fe(t).From here we proceed as described in Paragraph A.

D. Computing the risk.Constructive differential equations may be put up for the risk. As a simpleexample, for an interest rate derivative the state-wise risk is

Re(t) =

Tt

g

peg( t)

f ;f 6=g

gf (gf ())2d .

Differentiating this equation, we find

d

dtRe(t) =

f ;f 6=e

ef (ef (t))2+

Tt

g

d

dtpeg( t)

f ;f 6=g

(gf ())2d ,

and, using the backward version of (7.2),

d

dtpeg(s t) =

h;h6=e

ehphg(s t) + epeg(s t) ,

we arrive at

d

dtRe(t) =

f ;f 6=e

ef (ef (t))2

f ;f 6=e

ef Rf (t) + eRe(t) .

References

[1] Aase, K.K. and Persson, S.-A. (1994). Pricing of unit-linked life insurancepolicies. Scand. Actuarial J., 1994, 26-52.

[2] Andersen, P.K., Borgan, ., Gill, R.D., Keiding, N. (1993). Statistical Mod-els Based on Counting Processes. Springer-Verlag, New York, Berlin, Heidel-berg.

[3] Bibby J.M., Mardia, K.V., and Kent J.T. Multivariate Analysis. AcademicPress, 1979.

[4] Bjrk, T., Kabanov, Y., Runggaldier, W. (1997): Bond market structures inthe presence of marked point processes. Mathematical Finance, 7, 211-239.

[5] Bjrk, T. (1998): Arbitrage Theory in Continuous Time, Oxford UniversityPress.

[6] Black, F., Scholes, M. (1973): The pricing of options and corporate liabili-ties. J. Polit. Economy, 81, 637-654.

[7] Bremaud, P. (1981). Point processes and queues. Springer-Verlag, New York,Heidelberg, Berlin.

37

[8] Cox, J., Ross, S., Rubinstein, M. (1979): Option pricing: A simplified ap-proach. J. of Financial Economics, 7, 229-263.

[9] Delbaen, F., Schachermayer, W. (1994): A general version of the fundamen-tal theorem on asset pricing. Mathematische Annalen, 300, 463-520.

[10] Eberlein, E., Raible, S. (1999): Term structure models driven by generalLvy processes. Mathematical Finance, 9, 31-53.

[11] Elliott, R.J., Kopp, P.E. (1998): Mathematics of financial markets,Springer-Verlag.

[12] Fllmer, H., Sondermann, D. (1986): Hedging of non-redundant claims. InContributions to Mathematical Economics in Honor of Gerard Debreu, 205-223, eds. Hildebrand, W., Mas-Collel, A., North-Holland.

[13] Gantmacher, F.R. (1959): Matrizenrechnung II, VEB Deutscher Verlag derWissenschaften, Berlin.

[14] Gerber, H.U. (1995). Life Insurance Mathematics, 2nd edn. Springer-Verlag.

[15] Harrison, J.M., Kreps, D.M. (1979): Martingales and arbitrage in multi-period securities markets. J. Economic Theory, 20, 1979, 381-408.

[16] Harrison, J.M., Pliska, S. (1981): Martingales and stochastic integrals inthe theory of continuous trading. J. Stoch. Proc. and Appl., 11, 215-260.

[17] Hoem, J.M. (1969): Markov chain models in life insurance. Bltter Deutsch.Gesellschaft Vers.math., 9, 91107.

[18] Hoem, J.M. and Aalen, O.O. (1978). Actuarial values of payment streams.Scand. Actuarial J. 1978, 38-47.

[19] Jordan, C.W. (1967). Life Contingencies. The Society of Actuaries,Chicago.

[20] Karlin, S., Taylor, H. (1975): A first Course in Stochastic Processes, 2nd.ed., Academic Press.

[21] Merton, R.C. (1973): The theory of rational option pricing. Bell Journalof Economics and Management Science, 4, 141-183.

[22] Merton, R.C. (1976): Option pricing when underlying stock returns arediscontinuous. J. Financial Economics, 3, 125-144.

[23] Mller, T. (1998): Risk minimizing hedging strategies for unit-linked lifeinsurance. ASTIN Bull., 28, 17-47.

[24] Norberg, R. (1991). Reserves in life and pension insurance. Scand. ActuarialJ. 1991, 1-22.

38

[25] Norberg, R. (1995): Differential equations for moments of present valuesin life insurance. Insurance: Math. & Econ., 17, 171-180.

[26] Norberg, R. (1995): A time-continuous Markov chain interest model withapplications to insurance. J. Appl. Stoch. Models and Data Anal., 245-256.

[27] Norberg, R. (1998): Vasiek beyond the normal. Working paper No. 152,Laboratory of Actuarial Math., Univ. Copenhagen.

[28] Norberg, R. (1999): A theory of bonus in life insurance. Finance andStochastics., 3, 373-390.

[29] Norberg, R. (2001): On bonus and bonus prognoses in life insurance. Scand.Actuarial J. 1991, 1-22.

[30] Papatriandafylou, A. and Waters, H.R. (1984). Martingales in life insur-ance. Scand. Actuarial J. 1984, 210-230.

[31] Protter, P. (1990). Stochastic Integration and Differential Equations.Springer-Verlag, Berlin, New York.

[32] Pliska, S.R. (1997): Introduction to Mathematical Finance, Blackwell Pub-lishers.

[33] Ramlau-Hansen, H. (1991): Distribution of surplus in life insurance.ASTINBull., 21, 57-71.

[34] Sverdrup, E. (1969): Noen forsikringsmatematiske emner. Stat. Memo. No.1, Inst. of Math., Univ. of Oslo. (In Norwegian.)

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[36] Width, E. (1986): A note on bonus theory. Scand. Actuarial J. 1986, 121-126.

Appendices

A Calculus

A. Piecewise differentiable functions.Being concerned with operations in time, commencing at some initial date, wewill consider functions defined on the positive real line [0,). Thus, let usconsider a generic function X = {Xt}t0 and think of Xt as the state or valueof some process at time t. For the time being we take X to be real-valued.

39

In the present text we will work exclusively in the space of so-called piecewisedifferentiable functions. From a mathematical point of view this space is tinysince only elementary calculus is needed to move about in it. From a practicalpoint of view it is huge since it comfortably accommodates any idea, howeversophisticated, that an actuary may wish to express and analyse. It is convenientto enter this space from the outside, starting from a wider class of functions.We first take X to be of finite variation (FV), which means that it is the

difference between two non-decreasing, finite-valued functions. Then the left-limit Xt = limstXs and the right-limit Xt+ = limst Xs exist for all t, andthey differ on at most a countable set D(X) of discontinuity points of X .We are particularly interested in FV functions X that are right-continuous

(RC), that is, Xt = limst Xs for all t. Any probability distribution functionis of this type, and any stream of payments accounted as incomes or outgoes,can reasonably be taken to be FV and, as a convention, RC. If X is RC, thenXt = Xt Xt, when different from 0, is the jump made by X at time t.For our purposes it suffices to let X be of the form

Xt = X0 +

t0

x d +

0

provided that the individual terms on the right and also their sum are welldefined. Considered as a function of t the integral is itself PD and RC withcontinuous increments Ytxt dt and jumps Yt(Xt Xt). One may think of theintegral as the weighted sum of the Y -values, with the increments of X asweights, or vice versa. In particular, (A.1) can be written simply as

Xt = Xs +

ts

dX , (A.4)

saying that the value of X at time t is its value at time s plus all its incrementsin (s, t].By definition, t

s

Y dX = limrt

ts

Y dX =

ts

Y dX Yt(Xt Xt) =

(s,t)

Y dX ,

a left-continuous function of t. Likewise, ts

Y dX = limrs

tr

Y dX =

ts

Y dX + Ys(Xs Xs) =

[s,t]

Y dX ,

a left-continuous function of s.

C. The chain rule (Its formula).Let Xt = (X

1t , . . . , X

mt ) be an m-variate function with PD and RC components

given by dX it = xit dt + (X

it X

it). Let f : R

m 7 R have continuous partialderivatives, and form the composed function f(Xt). On the open intervals wherethere are neither discontinuities in the xi nor jumps of the X i, the functionf(Xt) develops in accordance with the well-known chain rule for scalar fieldsalong rectifiable curves. At the exceptional points f(Xt) may change (only) dueto jumps of the X i, and at any such point t it jumps by f(Xt) f(Xt). Thus,we gather the so-called change of variable rule or Its formula, which in oursimple function space reads

df(Xt) =

mi=1

f

xi(Xt) x

it dt+ f(Xt) f(Xt) , (A.5)

or, in integral form,

f(Xt) = f(Xs) +

ts

mi=1

xif(X )x

i d +

s

If X and Y have no common jumps, as is certainly the case if one of them iscontinuous, then (A.7) reduces to the familiar

d(XtYt) = Xt dYt + Yt dXt . (A.8)

The integral form of (A.7) is the so-called rule of integration by parts: ts

YdX = YtXt YsXs

ts

X dY . (A.9)

B Probability and expectation

Taking basic measure theoretic probability as a prerequisite, we represent therelevant part of the world and its uncertainties by a probability space (,F ,P).Here is the set of possible outcomes , F is a sigmaalgebra of subsets of representing the events to which we want to assign probabilities, and P : F 7[0, 1] is a probability measure.A set A F such that P[A] = 0 is called a nullset, and a property that

takes place in all of , except possibly on a nullset, is said to hold almost surely(a.s.). If more than one probability measure are in play, we write nullset (P)and a.s. (P) whenever emphasis is needed. Two probability measures P andP are said to be equivalent, written P P, if they are defined on the same Fand have the same nullsets.Let G be some sub-sigmaalgebra of F . We denote the restriction of P to G

by PG ; PG [A] = P[A], A G. Note that also (,G,PG) is a probability space.A G-measurable random variable (r.v.) is a function X : 7 R such that

X1(B) G for all B R, the Borel sets in R. We write X G in short.The expected value of a r.v. X is the probability-weighted average E[X ] =

X dP =X() dP(), provided this integral is well defined.

The conditional expected value of X , given G, is the r.v. E[X |G] G satis-fying

E{E[X |G]Y } = E[XY ] (B.1)

for each Y G such that the expected value on the right exists. It is uniqueup to nullsets (P). To motivate (B.1), consider the special case when G ={B1, B2, . . .}, the sigma-algebra generated by the F-measurable setsB1, B2, . . .,which form a partition of . Being G-measurable, E[X |G] must be of the form

k bk1Bk . Putting this together with Y = 1Bj into the relationship (B.1) wearrive at

E[X |G] =

j

1Bj

BjX dP

P[Bj ],

as it ought to be. In particular, taking X = IA, we find the conditional proba-bility P[A|B] = P[A B]/P[B].One easily verifies the rule of iterated expectations, which states that, for

H G F ,

E {E[X |G]| H} = E[X |H] . (B.2)

42

C Change of measure

If L is a r.v. such that L 0 a.s. (P) and E[L] = 1, we can define a probabilitymeasure P on F by

P[A] =

A

LdP = E[1AL] . (C.1)

If L > 0 a.s. (P), then P P.The expected value of X w.r.t. P is

E[X ] = E[XL] (C.2)

if this integral exists; by the definition (C.1), the relation (C.2) is true forindicators, hence for simple functions and, by passing to limits, it holds formeasurable functions. Spelling out (C.2) as

X dP =

XLdP suggests the

notation dP = LdP or

dP

dP= L . (C.3)

The function L is called the Radon-Nikodym derivative of P w.r.t. P.Conditional expectation under P is formed by the rule

E[X |G] =E[XL|G]

E[L|G]. (C.4)

To see this, observe that, by definition,

E{E[X |G]Y } = E[XY ] (C.5)

for all Y G. The expression on the left of (C.5) can be reshaped as

E{E[X |G]Y L} = E{E[X |G] E[L|G]Y } .

The expression on the right of (C.5) is

E[XY L] = E{E[XL|G]Y } .

It follows that (C.5) is true for all Y G if and only if

E[X |G] E[L|G] = E[XL|G] ,

which is the same as (C.4).For X G we have

EG [X ] = E[X ] = E[XL] = E {X E[L|G]} = EG {X E[L|G]} , (C.6)

showing that

dPGdPG

= E[L|G] . (C.7)

43

D Stochastic processes: general concepts

To describe the evolution of random phenomena over some time interval [0, T ],we introduce a family F = {Ft}0tT of sub-sigmaalgebras of F , where Ftrepresents the information available at time t. More precisely, Ft is the set ofevents whose occurrence or non-occurrence can be ascertained by time t. If noinformation is ever sacrificed, we have Fs Ft for s < t. We then say that F isa filtration, and (,F ,F,P) is called a filtered probability space.A stochastic process is a family of r.v.-s, {Xt}0tT . It is said to be adapted

to the filtration F if Xt Ft for each t [0, T ], that is, at any time thecurrent state (and also the past history) of the process is fully known if weare currently provided with the information F. An adapted process is said tobe predictable if its value at any time is entirely determined by its history inthe strict past, loosely speaking. For our purposes it is sufficient to think ofpredictable processes as being either left-continuous or deterministic.

E Martingales

An adapted process X with finite expectation is a martingale if

E[Xt|Fs] = Xs

for s < t. The martingale property depends both on the filtration and on theprobability measure, and when these need emphasis, we shall say that X ismartingale (F,P). The definition says that, on the average, a martingaleis always expected to remain on its current level. One easily verifies that,conditional on the present information, a martingale has uncorrelated futureincrements. Here are some useful general results:Abbreviate Pt = PFt , introduce

Lt =dPtdPt

,

and put L = LT . By (C.7) we have

Lt = E[L|Ft] , (E.1)

which is a martingale (F,P).Let X be a real-valued random variable such that E|X | < . Then the

process M defined byM(t) = E[X |Ft]

is a martingale. This follows by the rule of iterated expectation and the filtrationproperty, Fs Ft for s < t :

E[M(t) | Fs] = E{E[X |Ft] | Fs} = E[X | Fs] = M(s) .

A martingaleM with paths that are (almost surely) continuous and of finitevariation in every finite interval is constant as a function of time; M(t) = M(0)

44

for all t. This is seen as follows. Since M has finite variation, it obeys the rulesof ordinary calculus and, in particular,

M2(t) = M2(0) + 2

t0

M(s) dM(s) .

SinceM is continuous, it is also predictable so that the integral t0 2M(s) dM(s)

is a martingale. It follows that

E[M2(t)

]= M2(0) .

Since E[M(t)] = M(0), we conclude that

Var[M(t)] = 0 ,

hence M is constant.

F Counting processes

As the name suggests, a counting process is a stochastic processN = {Nt}0tTthat commences from zero (N0 = 0) and thereafter increases by isolated jumpsof size 1 only. The natural filtration of N is FN = {FNt }0tT , where F

Nt =

{Ns; s t} is the history of N by time t. This is the smallest filtration towhich N is adapted. The strict past history of N at time t is denoted by FNt.An FN -predictable process {t}0tT is called a compensator of N if the

process M defined by

Mt = Nt t (F.1)

is a zero mean FN -martingale. If is absolutely continuous, that is, of the form

t =

t0

s ds ,

then the process is called the intensity of N . We may also define the intensityinformally by

t dt = P [dNt = 1 | Ft] = E [dNt | Ft] ,

and we sometimes write the associated martingale (F.1) in differential form,

dMt = dNt t dt . (F.2)

A stochastic integral w.r.t. the martingale M is an FN -adapted process Hof the form

Ht = H0 +

t0

hs dMs , (F.3)

where H0 is FN0 -measurable and h is an F

N -predictable process such that H isintegrable. The stochastic integral is also a martingale.

45

A fundamental representation result states that every FN martingale is astochastic integral w.r.t. M . It follows that every integrable FNt measurabler.v. is of the form (F.3).

If H(1)t = H

(1)0 +

t0h

(1)s dMs and H

(2)t = H

(2)0 +

t0h

(2)s dMs are stochastic

integrals with finite variance, then an easy heuristic calculation shows that

Cov[H(1)T , H

(2)T |Ft] = E

[ Tt

h(1)s h(2)s s ds | Ft

], (F.4)

and, in particular,

Var[HT |Ft] = E

[ Tt

h2ss ds | Ft

].

H(1) and H(2) are said to be orthogonal if they have conditionally uncorrelatedincrements, that is, the covariance in (F.4) is null. This is equivalent to sayingthat H(1)H(2) is a martingale.The intensity is also called the infinitesimal characteristic if the counting

process since it entirely determines it probabilistic properties. If t is deter-ministic, then Nt is a Poisson process. If depends only on Nt, then Nt is aMarkov process.The change of variable rule (A.6) becomes particularly simple when the