Dynamic greeks

Insurance Mathematics and Economics 39 (2006) 123–133 www.elsevier.com/locate/ime

Dynamic greeksI Ragnar Norberg ∗ London School of Economics, London WC2A 2AE, United Kingdom Received February 2005; received in revised form January 2006; accepted 30 January 2006

Abstract The sensitivity of a price (or premium or reserve) to changes in its arguments is given by its derivatives, in finance known as “greeks”. Differential equations for sensitivities are obtained by simply differentiating the differential equation and the side condition that uniquely determine the price function. The device opens up prospects of efficient computation of greeks for a wide range of price functions in parametric models. It is applied here to examples in the Black–Merton–Scholes model and in a Markov chain model. Mathematical issues arising are, firstly, to construct the differential equation for the primary function and, secondly, to prove that the sensitivities actually exist. General resolutions to these problems seem not to be in reach, so only some special situations are discussed here. c 2006 Elsevier B.V. All rights reserved.

JEL classification: C8; C61; C63; G12 MSC: IM10; IM20 Keywords: Sensitivity analysis; Differential equations; Numerical solutions; Black–Merton–Scholes model; Markov chain model

1. Introduction 1.1. Terminology In the finance literature, the derivatives of a price function with respect to its arguments are known as “greeks”. They are so called for the somewhat circumstantial reason that they are denoted by Greek letters (their numerical values, expressed in Arabic numerals, are not called “arabs”). A more descriptive term, commonly used in other quantitative disciplines, is “sensitivities”; see e.g. Saltelli et al. (2000). Sensitivities are useful because they tell which model assumptions are the critical ones, and also because they play a role in the context of hedging; see e.g. Bj¨ork (2004).

I This work was partly supported by the Mathematical Finance Network under the Danish Social Science Research Council, Grant No. 9800335. ∗ Tel.: +44 207 955 6030; fax: +44 207 955 7416.

E-mail address: [email protected]. c 2006 Elsevier B.V. All rights reserved. 0167-6687/$ - see front matter doi:10.1016/j.insmatheco.2006.01.008

124

R. Norberg / Insurance Mathematics and Economics 39 (2006) 123–133

1.2. From static to dynamic greeks Existing finance text-books deal with greeks mainly in situations where the price function admits a closed form expression and the greeks can be obtained simply by differentiating that expression. When a closed form expression does not exist, other methods must be employed. Simulation is widely used. A recent paper by Kalashnikov and Norberg (2003) proposes a “dynamic” method for sensitivity analysis of the reserve in life insurance, which is the solution to a backward differential equation; upon differentiating the differential equation with respect to some parameter in the model, one obtains a differential equation for the sensitivity of the reserve with respect to that parameter. The reserve and its sensitivity are determined by solving the two equations simultaneously, usually by a numerical method. 1.3. Scope and outline of the study The dynamic approach has, of course, not remained unnoticed by financial mathematicians. It was in the air for a while, and was sporadically alluded to by several authors, an early reference being Wilmott (1998). It is outlined in the Black–Merton–Scholes model by Tavella and Randall (2000). However, so far the powers of the method have not been widely recognized, and it is not widely used. The present paper promotes the device, arguing that it works whenever one is able to find a differential equation for the primary function. Issues arising are, firstly, to derive differential equations for non-trivial products, e.g. with path-dependent payoff, and, secondly, to investigate the existence of sensitivities. There is no universal recipe, so all we can do here is to point out these problems and solve them in some special cases. In Section 2, we illustrate the technique in the framework of the basic Black–Merton–Scholes model, in which greeks have been studied extensively. In this simple model, explicit expressions exist for a wide range of price functions, and also for some exotic products with path-dependent pay-off. For a comprehensive account of closed form expressions for functionals of Brownian motion, see Borodin and Salminen (2002). Even when an explicit expression exists, the dynamic approach may provide the superior algorithm for numerical computation. When no explicit expression is at hand, one must resort to numerical methods, either simulation or differential equation numerics. The first step in the latter approach is to derive the differential equation and the side condition that characterize the price, which may be a challenge for complex products. As an example, we consider the down-and-out contract. In Section 3, the programme of dynamic sensitivity analysis is carried out in a market driven by a continuous time Markov chain. Explicit formulas exist only for very simple products, and numerical methods are therefore essential. Numerical results are reported for a simple Poisson model. Proving the existence of sensitivities is usually not easy when closed form expressions do not exist, and the problem is highly dependent on the particulars of the model and the product. In the final Section 4, the problem is discussed in the Markov chain model (which covers the Poisson model), and it is shown that sensitivities with respect to parameters in the transition intensities exist under liberal conditions. 2. The Black–Merton–Scholes market 2.1. The model There are two basic assets, a bank account whose price at time t is Bt = er t ,

(2.1)

and a stock whose price at time t is St = eαt+σ Wt .

(2.2)

The interest rate r , the drift parameter α, and the volatility σ 2 are constants, and W is a standard Brownian motion. In units of the bank account, the discounted prices are B˜ t = Bt /Bt = 1 and S˜t = St /Bt = e(α−r )t+σ Wt . The former is certainly a martingale under any measure, while the latter is a martingale with respect to the equivalent measure P˜ under which the process   σ α −r ˜ + t + Wt Wt = σ 2

R. Norberg / Insurance Mathematics and Economics 39 (2006) 123–133

125

is a standard Brownian motion. The stock price can be recast as St = e(r −

σ2 ˜ 2 )t+σ Wt

,

(2.3)

with dynamics dSt = St (r dt + σ dW˜ t ).

(2.4)

2.2. Option prices A European style option is a contingent claim of the form h(ST )

(2.5)

due at some fixed exercise time T . Its unique arbitrage-free price at time t is ˜ −(T −t)r h(ST )|Ft ], pt = E[e

(2.6)

˜ denotes expectation with respect to P, ˜ and Ft = σ {Wτ ; 0 ≤ τ ≤ t}. Using (2.3) to write where E ST = St e

  2 r − σ2 (T −t)+σ (W˜ T −W˜ t )

,

and noting that W˜ has independent increments, we conclude that the price pt must be of the form pt = v(St , t),

(2.7)

where (r − ˜ v(s, t) = e−(T −t)r E[h(se

σ2 ˜ ˜ 2 )(T −t)+σ ( WT − Wt )

)],

(2.8)

(s, t) ∈ (0, ∞) × [0, T ]. Moreover, since W˜ T − W˜ t is normally distributed with mean zero and variance (T − t) under ˜ we obtain the integral expression P, Z √ σ2 e−(T −t)r ∞ 2 v(s, t) = √ (2.9) h(se(r − 2 )(T −t)+σ T −tw )e−w /2 dw. 2π −∞ Here, and in what follows, the dependence of the price on the model parameters r and σ is suppressed in the notation v(s, t), which rather should be written v(s, t; r, σ ). The price function v is also the solution to the differential equation 1 vt (s, t) = v(s, t)r − vs (s, t)r s − vss (s, t)σ 2 s 2 , 2 subject to the ultimo condition v(s, T ) = h(s),

s > 0.

(2.10)

(2.11)

To save notation, we have used subscripts to signify derivatives: ∂2 ∂ ∂ v, vss = 2 v, (2.12) vt = v. ∂s ∂t ∂s The differential equation can be obtained in several ways. We sketch here a technique that carries over to more complex situations encountered later. The starting point is the martingale vs =

˜ −r T h(ST )|Ft ] = e−r t v(St , t), Mt = E[e the last expression being due to (2.6)–(2.8). By Ito’s formula, using (2.4), the dynamics of M are 1 dMt = e−r t (−r dt)v(St , t) + e−r t (vs (St , t)St (rt dt + σ dW˜ t ) + vt (St , t)dt) + e−r t vss (St , t)σ 2 St2 dt. 2

126

R. Norberg / Insurance Mathematics and Economics 39 (2006) 123–133

The term involving dW˜ t on the right hand side is a martingale increment. It follows that the remaining terms on the right hand side must constitute a martingale increment and, being of order dt (continuous and of bounded variation), they must be null almost surely. This leads to (2.10). The differential equation (2.10) and the side condition (2.11) determine the function v(s, t) uniquely as a mathematical object. For computational purposes, one needs to add side conditions at finite boundaries in the sdimension, and these conditions must be based on auxiliary probabilistic arguments. For instance, for the European call option with maturity T and strike K , h(ST ) = (ST − K )+ , one would set v(0, t) = 0 and v(¯s , t) = s¯ −e−r (T −t) K , 0 ≤ t ≤ T , for some suitably chosen large value s¯ . 2.3. Greeks By tradition, the greeks in (2.12) are denoted ∆ = vs ,

Γ = vss ,

Θ = vt .

(2.13)

Two more greeks are standard in the BMS model: ρ = vr ,

V = vσ .

(2.14)

The greeks in (2.13) are qualitatively different to those in (2.14). The former are sensitivities with respect to time and the state variable within a given point in the model space (i.e. given parameters), whereas the latter are sensitivities with respect to moves across the space of models. They can suitably be called “local greeks” and “global greeks”, respectively. Local greeks are computed as part of any difference scheme for solving the PDE (2.10) for fixed parameters. Global greeks must be computed by different methods, and we advocate the following “dynamical” approach: Differentiating through (2.10) and (2.11) with respect to r , assuming tacitly that this is permitted, we obtain the differential equation 1 ρt (s, t) = ρ(s, t)r + v(s, t) − ρs (s, t)r s − vs (s, t)s − ρss (s, t)σ 2 s 2 2 and the side condition ρ(s, T ) = 0,

s > 0.

(2.15)

(2.16)

Similarly, differentiating with respect to σ , we obtain 1 Vt (s, t) = V(s, t)r − Vs (s, t)r s − Vss (s, t)σ 2 s 2 − vss (s, t)σ s 2 2

(2.17)

V(s, T ) = 0,

(2.18)

and s > 0.

Now, to determine the price function and its greeks, solve the differential equations (2.10), (2.15) and (2.17) subject to the side conditions. 2.4. Computation Any European option admits an explicit pricing formula (2.9). Computation by this formula goes by numerical integration, see e.g. Los (2001), which essentially is the same as solving numerically some ordinary differential equation(s). (The expression (2.9) does not admit a closed algebraic form unless the payoff function h is trivial. If h is piece-wise affine, as is the case for e.g. a call option or a put option, then (2.9) can be expressed as a linear combination of standard normal integrals, which are tabulated, but nonetheless are just integrals.) Similar considerations go for global greeks upon differentiating (2.9) with respect to r and σ . Alternatively, one may compute prices and greeks by solving numerically the differential equations (2.10), (2.15) and (2.17). Compared with the former method based on (2.9), the latter may be just as easy to implement, may be just a little bit slower (if you cannot spare a second), and may be just a little bit less accurate (if you cannot spare a change). Actually, it may be much faster if we are interested in the price and its sensitivities, not only for a given

R. Norberg / Insurance Mathematics and Economics 39 (2006) 123–133

127

value of the stock at a given time, but for a range of times and stock prices; computation based on the explicit formula (2.9) has to be done separately for each s and t in which one might be interested, while the dynamic approach delivers numerical values for all s and t in one single run. 2.5. The down-and-out contract The starting point of a dynamic sensitivity analysis is the differential equation for the primary price function. This is straightforwardly obtained for any European option of the form (2.5) by the martingale technique in Section 2.3 above. For more complex claims, it may be an issue to derive a constructive differential equation. As an example, we consider a down-and-out contract, which is a modification of the European option obtained by making the payoff (2.5) contingent on the stock price staying above a certain level m throughout the contract period [0, T ]. Introducing the stopping time Tm = inf{t; St ≤ m} and the (right-continuous) indicator process It = 1[Tm > t], the payoff under the down-and-out contract is h(ST )IT , and its price at time t is (m)

pt

˜ −r (T −t) h(ST )IT |Ft ] = It v (m) (St , t), = E[e

where ˜ −r (T −t) h(ST )IT |St = s, It = 1], v (m) (s, t) = E[e (s, t) ∈ (m, ∞)×[0, T ]. To obtain a differential equation for the price function v (m) (s, t), we introduce the martingale ˜ −r T h(ST )IT |Ft ] = e−r t It v (m) (St , t). Mt = E[e By the general Ito’s formula, the dynamics of M are (m) dMt = e−r t (−r dt)It v (m) (St , t) + e−r t It (vs(m) (St , t)St (rt dt + σ dW˜ t ) + vt (St , t)dt) 1 (m) (St , t)σ 2 St2 dt + e−r t (It v (m) (St , t) − It− v (m) (St− , t−)). + e−r t It vss 2 The last term on the right hand side, which is the jump part, is null: if t < Tm , then It− = It = 1 and v (m) (St− , t−) = v (m) (St , t); if t > Tm , then It− = It = 0; if t = Tm , then, due to the diffuse nature of S, v (m) (St− , t−) = v (m) (St , t) = 0. We can now proceed as in Section 2.2 above to arrive at the differential equation

1 (m) (s, t) = v (m) (s, t)r − vs(m) (s, t)r s − vss (s, t)σ 2 s 2 . (2.19) 2 This PDE is the same as (2.10), but it is now to be solved for (s, t) ∈ (m, ∞) × [0, T ] subject to the conditions (m)

vt

v (m) (s, T ) = h(s),

s > m,

v (m) (m, t) = 0,

0 ≤ t ≤ T.

(2.20)

In fact, there exists an explicit expression,  m  2r −1  m 2  σ2 v (m) (s, t) = v(s, t) − v ,t , s s where v(s, t) is the price (2.9) of the clean-cut European option; see Bj¨ork (2004). As argued above, the price function can be computed straightforwardly by solving the differential equation (2.19) subject to (2.20), and so can greeks by the dynamical recipe. 3. The Markov chain market 3.1. The model A Markov chain driven market was introduced in a recent paper by the author (Norberg, 2003), from which we fetch some basic definitions and results. Let {Yt }t≥0 be a homogeneous Markov chain on a finitePstate space Y = {1, . . . , n}. Denote by λe f the intensity of transition from state e to state f (6= e), and set λee = − f ; f 6=e λe f (minus the total intensity of transition out of state e). Introduce the indicator processes Ite = 1[Yt = e], e ∈ Y, and the counting

128

R. Norberg / Insurance Mathematics and Economics 39 (2006) 123–133 ef

processes Nt by

ef

= ]{s; 0 < s ≤ t, Ys− = e, Ys = f }, e 6= f ∈ Y. The compensated counting processes Mt defined

ef

ef

dMt = dNt − Ite λe f dt

(3.21)

ef

and M0 = 0 are square integrable, orthogonal martingales. Taking Yt to represent the state of the economy at time t, we introduce a market with n basic tradeable assets: asset number 1 is a bank account with the price process Bt = e

Rt P

er

0

e I e du u

.

The remaining n − 1 assets are stocks, and the price process of stock number i is Sti = e

Rt P 0

e (α

ie I e du+P u f ; f 6=e

ef

β ie f dNu )

,

(3.22)

i = 1, . . . , n − 1. The r e , α ie , and β ie f are constants with the following interpretation: r e is the interest rate in economy state e; α ie is the rate of return on stock number i during sojourns in economy state e (of the same nature ie f as r e ); eβ is the factor with which the stock price changes instantaneously upon a market transition from state e to state f . Again, the discounted bank account price is trivially a martingale. The discounted stock prices S˜ti = Sti /Bt have dynamics   X X ie f ef i (α ie − r e )Ite dt + d S˜ti = S˜t− (eβ − 1)dNt  , (3.23) e

f ; f 6=e

and would be martingales with respect to an equivalent martingale measure P˜ under which the terms within the parentheses on the right hand side are martingale increments. With a view to (3.21), this means that (3.23) should be of the form X X ie f ef i d S˜ti = S˜t− (eβ − 1)d M˜ t , (3.24) e

f ; f 6=e

ef ˜ More specifically, where the M˜ t are the compensated counting processes under P. ef

ef

d M˜ t = dNt − Ite λ˜ e f dt,

(3.25)

˜ Inspection of (3.23)–(3.25) shows that the requested where the λ˜ e f would be the transition intensities of Y under P. martingale measure P˜ exists if the equations X ie f α ie − r e + (eβ − 1)λ˜ e f = 0, (3.26) f ; f 6=e

i = 1, . . . , n − 1, e = 1, . . . , n, have non-negative solutions λ˜ e f such that λ˜ e f = 0 if and only if λe f = 0. The existence of an equivalent martingale ensures that the market is free of arbitrage. Moreover, if the equation (3.26) admits one and only one solution, then the market is complete. 3.2. Option prices A general European style stock option pays an amount of the form h YT (ST` ) at time T > 0; the payoff depends on ˜ the price of the the state of the economy and the price of stock number ` at the term T . Under the pricing measure P, claim at time t < T is ˜ − pt = E[e

RT t

ru du YT

h

(ST` )|Ft ],

129

R. Norberg / Insurance Mathematics and Economics 39 (2006) 123–133

where Ft = σ {Yτ ; 0 ≤ τ ≤ t}. Due to the structure (3.22) of the stock price and the Markov property of Y , the price process must be of the form X pt = v Yt (St` , t) = Ite v e (St` , t), e

where the state-wise price functions are # " R T P `g g P `gh gh ! β dNu ) RT g (α Iu du+ t e − r du Y h;h6=g ˜ e t u h T se v (s, t) = E Yt = e . Explicit formulas exist for claims of the simple form h(Yt ), e.g. zero coupon bonds, caplets, and other interest derivatives; see Norberg (2003). (They involve the exponential function of a matrix, which is an infinite sum, and is in this sense just as “explicit” as the exponential function of a real.) For stock derivatives, one usually has to resort to simulation or numerical solution of differential equations. Aiming at dynamical sensitivity analysis, we take the latter approach. Rt The discounted price e− 0 ru du v Yt (St` , t) is a martingale under the equivalent measure. Operating on it with Ito’s formula and identifying the drift term that must vanish, we arrive at the following system of first order partial differential equations for the state-wise price functions: X `e f vte (s, t) = r e v e (s, t) − vse (s, t)sα `e − (v f (seβ , t) − v e (s, t))λ˜ e f , (3.27) f ; f 6=e

e = 1, . . . , n. These are to be solved subject to the conditions v e (s, T ) = h e (s),

(3.28)

e = 1, . . . , n. 3.3. Greeks Consider the stock option price discussed in the previous paragraph, which (except in some very simple special cases) has to be determined as the solution to the boundary-value PDE problem (3.27) and (3.28). Greeks are straightforwardly obtained as solutions to the differential equations and side conditions obtained upon differentiating (3.27) and (3.28). For an example, take the sensitivities of the state-wise price functions v e (s, t) with respect to the rate of return α `h of the stock in state h. Denoting these sensitivities ad hoc by the Greek letter ν, we obtain the PDE X `e f νte (s, t) = r e ν e (s, t) − νse (s, t)sα `e − vse (s, t)sδeh − (ν f (seβ , t) − ν e (s, t))λ˜ e f f ; f 6=e

−

X

(v f (seβ

`e f

, t) − v e (s, t))

f ; f 6=e

∂ ef λ˜ ∂α `h

(3.29)

(δeh is the Kronecker delta) and the side condition ν e (s, T ) = 0,

(3.30)

e = 1, . . . , n. The derivatives of the λ˜ e f with respect to α `h are obtained upon differentiating (3.26) and solving X ∂ ie f δie,`h + (eβ − 1) `h λ˜ e f = 0, ∂α f ; f 6=e i = 1, . . . , n − 1, e = 1, . . . , n. 3.4. Computation Numerical solution of the differential equations (3.27) and (3.29) goes by the Lax–Wendroff difference scheme, modified to account for the non-standard feature that the differential equations are shifted (i.e. involve function-values at different values of the s-argument).

130

R. Norberg / Insurance Mathematics and Economics 39 (2006) 123–133

3.5. An example As an illustration, we consider a European call option with maturity T and strike K , h(ST ) = (ST − K )+ , in a Poisson analogue to the Brownian motion scenario in Section 2. The bank account is given by (2.1) and the price process of the stock is St = eαt+β Nt , where α and β are constants and N is a Poisson process with intensity λ. A more general Poisson market was investigated by Gerber and Shiu (1996). It is a special case of the Markov chain market, and the present simple situation can be constructed as follows. Let the Markov chain have two states, Y = {1, 2}, let there be one stock with price process S 1 = S given by α 11 = α 12 = α, β 112 = β 121 = β, and put Nt = Nt12 + Nt21 . The state-wise price function reduces to a function of s and t only, and the equivalent martingale measure is seen to be the one under which N is a homogeneous Poisson process with intensity λ˜ =

r −α . eβ − 1

(The physical intensity λ does not matter, since it is not a path property of the process.) The price function of the call option is v(s, t) = e−r (T −t)

∞ X

(seα(T −t)+βn − K )+

n=0

(λ˜ (T − t))n −λ˜ (T −t) e . n!

(3.31)

Being an infinite sum, this formula is only “semi-explicit” and not particularly convenient for numerical computation of the price and its greeks. The differential equation (3.27) and the side condition (3.28) reduce to vt (s, t) = r v(s, t) − vs (s, t)sα − (v(seβ , t) − v(s, t))λ˜ ,

(3.32)

v(s, T ) = (s − K )+ .

(3.33)

The sensitivity ν(s, t) =

∂ ∂α v(s, t)

is the solution to

νt (s, t) = r ν(s, t) − νs (s, t)sα − vs (s, t)s − (ν(seβ , t) − ν(s, t))λ˜ + (v(seβ , t) − v(s, t)) ν(s, T ) = 0.

1 , eβ − 1

(3.34) (3.35)

We interpose here that the Poisson model is haunted by non-smoothness problems. Indeed, upon inspecting (3.31) one realizes that the derivatives involved in (3.34) do not exist on those curves in the positive quadrant of the (s, t)plane where seα(T −t)+βn = K for some integer n because, at such points, an additional term enters into the sum (3.31). A recent paper by the author (Norberg, 2005) shows how to determine the locations of points of non-smoothness in a more general Markov chain setting and how to get about the problems they create in numerical computations. Let it suffice here to say that, in the present situation, the difference scheme works well, since it essentially only requires continuity and piece-wise differentiability. The following numerical results were obtained for r = ln(1.05), α = 0.045, γ = eβ − 1 = 0.02, T = 1, K = 1.05, and S0 = 1: the sensitivities of the price at time 0 of the European call option are 0.85 with respect to r , −0.70 with respect to α, and 0.040 with respect to γ . (One should contemplate these findings: increasing α, and hence the performance of the stock, makes the option worth less; increasing β, and hence the performance of the stock, makes the option worth more.) 4. Do greeks exist? The answer to this question is very much dependent on the features of the model and of the claim. We shall be content to discuss the problem only for a general Markov chain model (which includes the Poisson model), and will focus on global greeks with respect to parameters in transition intensities, which is the hard part. Consider a family of probability measures {Pθ ; θ ∈ Θ} indexed by a parameter θ in some open finite-dimensional Euclidean set. Let {Z t }t≥0 be a continuous time Markov with finite state space Z = {1, . . . , n} and intensities that are

R. Norberg / Insurance Mathematics and Economics 39 (2006) 123–133

131

jk

parametric functions, µθ (t). Denote the infinitesimal matrix and the matrix of transition probabilities over the time interval from t to u by k∈Z

jk

Mθ (t) = (µθ (t)) j∈Z ,

jk

k∈Z

Pθ (t, u) = ( pθ (t, u)) j∈Z ,

respectively. Being mainly interested in the first order derivative in one direction at a time, we can also assume that θ is real-valued and assumes its values in an open interval. Using classical techniques, Kalashnikov and Norberg (2003) proved that, if Mθ (t) is sufficiently smooth, then Pθ (t, u) is differentiable with respect to θ and Z u ∂ ∂ Pθ (t, u) = Pθ (t, τ ) Mθ (τ )Pθ (τ, u)dτ. (4.36) ∂θ ∂θ t We will sketch the proof of a more general result. Generality is gained at the expense of introducing the additional assumption that the probability measures generated by varying the parameter are mutually absolutely continuous: for jk jk fixed j, k, and t, either µθ (t) > 0 for all θ or µθ (t) = 0 for all θ. The price function of a contingent claim is in general an expected value Eθ [X ], where X is an integrable FT measurable random variable. The problem is to prove the existence of the derivative of this function with respect to θ. By Girsanov’s theorem for counting processes (see e.g. Andersen et al. (1993)), Eθ+η [X ] = Eθ [X L θ,η (T )] where L θ,η (T ) is the Radon–Nikodym derivative, or likelihood, of Pθ+η with respect to Pθ . It is the value at time T of the likelihood process L θ,η (t) = e

Rt P 0

jk jk jk jk j jk j6=k ((ln µθ+η (τ )−ln µθ (τ ))dN (τ )−(µθ+η (τ )−µθ (τ ))I (τ )dτ )

Z = 1+

jk jk t X µθ+η (τ ) − µθ (τ ) jk dMθ (τ ), L θ,η (τ −) jk 0 µθ (τ ) j6=k

(4.37) (4.38)

jk

where the Mθ are the compensated counting processes given by jk

jk

dMθ (t) = dN jk (t) − I j (t)µθ (t)dt. jk

Under Pθ , the Mθ are martingales with respect to the natural filtration Ft = σ {Z τ ; 0 ≤ τ ≤ t}, and they are square integrable and, moreover, mutually orthogonal: gh

jk

gh

jk

jk

dhMθ , Mθ i(t) = Eθ [dMθ (t)dMθ (t)|Ft− ] = δgh, jk I j (t)µθ (t)dt. Here, δgh, jk is the Kronecker delta, which is 1 or 0 according as (g, h) is equal to ( j, k) or not. Under Pθ , the random variable X has the martingale representation Z TX jk jk X = Eθ [X ] + ξθ (τ )dMθ (τ ), 0

j6=k

jk

where the ξθ are predictable processes. Using the device Eθ +η [X |Ft ] =

Eθ [X L θ,η (T )|Ft ] Eθ [L θ,η (T )|Ft ]

together with (4.38)–(4.40), we have
1 Eθ+η [X |Ft ] − Eθ [X |Ft ] η   1 Eθ [X L θ,η (T )|Ft ] − Eθ [X |Ft ] = η Eθ [L θ,η (T )|Ft ]

(4.39)

(4.40)

132

R. Norberg / Insurance Mathematics and Economics 39 (2006) 123–133

1 1 Covθ [X, L θ,η (T )|Ft ] η L θ,η (t) # "Z jk jk T X jk µθ +η (τ ) − µθ (τ ) 1 jk jk = ξθ (τ ) Eθ L θ,η (τ −) dhM , M i(τ ) Ft θ θ jk L θ,η (t) t ηµθ (τ ) j6=k # "Z jk jk T X jk µθ +η (τ ) − µθ (τ ) j 1 ξθ (τ ) Eθ L θ,η (τ ) I (τ )dτ Ft . = L θ,η (t) η t j6=k =

(4.41)

(The left-limit is annihilated by dτ .) We can now formulate the following result: jk

Lemma. If the intensities µθ (τ ) are differentiable functions of θ and the process L θ,η (τ )

X

jk jk µθ +η (τ ) − µθ (τ ) j jk I (τ ) ξθ (τ )

η

j6=k

(4.42)

can be dominated by an integrable process, then the derivative of Eθ [X |Ft ] with respect to θ exists and is given by # "Z T X jk d jk d 1 j L θ,η (τ ) Eθ ξθ (τ ) µθ (τ )I (τ )dτ Ft . (4.43) Eθ [X |Ft ] = dθ L θ,η (t) dθ t j6=k The existence of the global greek is what is important here; the integral expression in (4.43) is not necessarily useful for computations. The lemma is only a preparatory result that needs to be supplemented with further examination of the intensity functions and the random variable X case by case. It is realized that the existence of greeks is not something that one can make very general statements about. We round off our discussion of this issue by indicating some more specific sufficient conditions. d dθ Eθ [X |Ft ] exists under the following conditions: jk d dθ µθ (τ ), 0 ≤ τ ≤ T , j 6= k ∈ Z, exist, are continuous in t

Corollary. The global greek

(i) The derivatives (at least piece-wise), and constitute an equicontinuous family of functions of θ. jk (ii) For any given θ, the functions |ξθ (τ )|, 0 ≤ τ ≤ T , j 6= k ∈ Z, are uniformly bounded by some constant a(θ ). We render a few words of explanation to the corollary and further clues to verification of the conditions. If condition (i) is satisfied, then the intensities are continuously differentiable with respect to θ and so jk

jk

µθ +η (τ ) − µθ (τ ) η

=

d jk µ ∗ (τ ) dθ θ

θ∗

for some between θ and θ + η (it may depend on τ and, of course, on j and k). By the equicontinuity condition, d jk d jk µθ ∗ (τ ) < dθ µθ (τ ) + b(θ ) for some positive number b(θ ). By the assumed we can choose η small enough that dθ d jk continuity with respect to t, the functions dθ µθ (τ ) are bounded on the closed interval [0, T ] by some positive number c(θ). These things, together with assumption (ii), give that the function in (4.42) is bounded in absolute value by L θ,η (τ )n(n − 1)a(θ )(b(θ ) + c(θ )), which is integrable since   L θ,η (τ ) Eθ Ft = 1. L θ,η (t) By dominated convergence, we conclude that (4.41) tends to the expression on the right of (4.43) as η goes to 0. jk The conditions (i) and (ii) are easy to check and are usually satisfied. It may be helpful to know that ξθ (τ ) is the change in the conditional expected value of X upon a jump from state j to state k at time τ , typically a bounded function. jk j` Z (t)` In particular, for X = Iu` , we have Eθ [X |Ft ] = pθ (t, u), t ≤ u, and ξθ (τ ) = pθk` (τ, u)− pθ (τ, u), t ≤ τ ≤ u. Inserting this into (4.43), one obtains (4.36).

R. Norberg / Insurance Mathematics and Economics 39 (2006) 123–133

133

References Andersen, P.K., Borgan, Ø., Gill, R.D., Keiding, N., 1993. Statistical Models Based on Counting Processes. Springer-Verlag. Bj¨ork, T., 2004. Arbitrage Theory in Continuous Time, second ed. Oxford University Press. Borodin, A.N., Salminen, P., 2002. Handbook of Brownian Motion, second ed. Birkh¨auser. Gerber, H.U., Shiu, E., 1996. Actuarial bridges to dynamic hedging and option pricing. Insurance: Mathematics & Economics 18, 183–218. Kalashnikov, V.V., Norberg, R., 2003. On the sensitivity of premiums and reserves to changes in valuation elements. Scandinavian Actuarial Journal 2003, 238–256. Los, C.A., 2001. Computational Finance. World Scientific. Norberg, R., 2003. The Markov chain market. ASTIN Bulletin 33, 265–287. Norberg, R., 2005. Anomalous PDE-s in Markov chains; domains of validity and numerical solutions. Finance and Stochastics 9, 519–537. Saltelli, A., Chan, K., Scott, E.M. (Eds.), 2000. Sensitivity Analysis. Wiley. Tavella, D., Randall, C., 2000. Pricing Financial Instruments. Wiley. Wilmott, P., 1998. Derivatives. Wiley.