Dynamic greeks
Ragnar Norberg1
London School of Economics
Abstract: The sensitivity of a price function to changes in its arguments is
given by its derivatives, in finance known as ’greeks’. Differential equations
for greeks 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 virtually any
price function in any parametric model. It is applied here to examples in the
Black-Merton-Scholes model and in a Markov chain model. Mathematical
issues arising are, firstly, to derive the differential equation for the primary
function and, secondly, to prove that the greeks actually exist. General resolutions to these problems seem not to be in reach, so only some special
situations will be discussed here.
Key-words: Derivative prices, sensitivity analysis, differential equations, numerical solutions, Black-Merton-Scholes model, Markov chain model.
JEL classification: C8, C61, C63, G12
1
Introduction
A. Terminology. In the finance literature the derivatives of a price function with respect to its arguments are known as ’greeks’. Presumably, they
are called so simply because they are denoted by Greek letters. (Their numerical values, expressed in Arabic numerals, are not called ’arabs’ though.)
An alternative term, commonly used in other quantitative disciplines, is ’sensitivities’, see e.g. Saltelli et al. (2000). The greeks 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. Björk (2004).
B. 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
1
This work was partly supported by the Mathematical Finance Network under the
Danish Social Science Research Council, Grant No. 9800335
1
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 derivative (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.
C. 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. There
is reason to promote 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 greeks.
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 extensively studied.
In this model explicit expressions exist for a wide range of price functions,
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 on 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 greeks is usually not easy when closed form
2
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
A. The model.
at time t is
There are two basic assets, a bank account whose price
Bt = er t ,
(2.1)
and and a stock whose price at time t is
St = eα t + σWt .
(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 B̃t = Bt /Bt = 1 and S̃t = St /Bt = e(α−r) t+σWt .
The former is certainly a martingale under any measure, while the latter
is a martingale with respect to the equivalent measure P̃ under which the
process
α−r
σ
W̃t =
+
t + Wt
σ
2
is a standard Brownian motion. The stock price can be recast as
2
r− σ2 t+σ W̃t
St = e
,
(2.3)
with dynamics
dSt = St r dt + σdW̃t .
B. Option prices.
form
(2.4)
A European style option is a contingent claim of the
h(ST )
(2.5)
due at some fixed exercise time T . Its unique arbitrage-free price at time t
is
i
h
pt = Ẽ e−(T −t) r h(ST ) Ft ,
3
where Ẽ denotes expectation with respect to P̃, and Ft = σ{Wτ ; 0 ≤ τ ≤ t}.
Using (2.3) to write
2
r− σ2 (T −t)+σ(W̃T −W̃t )
ST = St e
,
and noting that W̃ has independent increments, we conclude that the price
pt must be of the form
pt = v(St , t) ,
(2.6)
where
−(T −t) r
v(s, t) = e
2
r− σ2 (T −t)+σ (W̃T −W̃t )
Ẽ h s e
.
(2.7)
Moreover, since W̃T −W̃t is normally distributed with mean zero and variance
(T − t) under P̃, we obtain the integral expression
e−(T −t) r
v(s, t) = √
2π
Z
∞
√
2
r− σ2 (T −t)+σ T −t w
−w2 /2
h se
e
dw.
(2.8)
−∞
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) rs − vss (s, t) σ 2 s2 ,
2
(2.9)
subject to the ultimo condition
v(s, T ) = h(s) ,
s > 0.
(2.10)
To save notation we have used subscripts to signify derivatives:
vs =
∂2
∂
∂
v , vss = 2 v , vt = v .
∂s
∂s
∂t
(2.11)
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
Mt = Ẽ e−rT h(ST ) Ft = e−rt v(St , t) ,
4
the last expression being due to (2.6) and (2.7). By Itǒ’s formula, the
dynamics of M is
dMt = e−rt (−r dt) v(St , t)
+ e−rt vs (St , t)St (rt dt + σdW̃t ) + vt (St , t) dt
1
+ e−rt vss (St , t) σ 2 St2 dt .
2
The term involving dW̃t 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 the differential
equation (2.9).
C. Greeks.
By tradition, the greeks in (2.11) are denoted
∆ = vs , Γ = vss , Θ = vt .
(2.12)
Two more greeks are standard in the BMS model:
ρ = vr , V = vσ .
(2.13)
(Actually, V is not a Greek letter, but rather an anglo-hellenic hybrid pronounced ’vega’.)
The greeks in (2.12) are qualitatively different from those in (2.13). 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 referred to as “local greeks” and “global greeks”, respectively.
Local greeks are computed as part of any difference scheme for solving
the PDE (2.9) for fixed parameters. Global greeks must be computed by
different methods, and we advocate the following “dynamical” approach:
Differentiating through (2.9) and (2.10) 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) rs − vs (s, t) s − ρss (s, t) σ 2 s2 (2.14)
2
and the side condition
ρ(s, T ) = 0 ,
5
s > 0.
(2.15)
Similarly, differentiating with respect to σ, we obtain
1
Vt (s, t) = V(s, t) r − Vs (s, t) rs − Vss (s, t) σ 2 s2 − vss (s, t) σs2
2
(2.16)
and
V(s, T ) = 0 ,
s > 0.
(2.17)
Now, to determine the price function and its greeks, solve the differential
equations (2.9), (2.14), and (2.16) subject to the side conditions.
D. Computation. Any European option admits an explicit pricing formula (2.8). 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 (2.8) does not admit a
closed algebraic form unless the payoff function h is trivial. If h is piecewise affine, as is the case for e.g. a call option or a put option, then (2.8) can
be expressed as a linear combination of standard normal integrals, which are
tabulated, but none the less are just integrals.) Similar considerations go
for global greeks upon differentiating (2.8) with respect to r and σ.
Alternatively, one may compute prices and greeks by solving numerically
the differential equations (2.9), (2.14), and (2.16). Compared with the former method based on (2.8), 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 way faster if we are interested in the price and its sensitivities, not only
for a given value of the stock at a given time, but for a range of times and
stock prices; Computation based on the explicit formula (2.8) has to be done
separately for each time t, while the dynamic approach delivers numerical
values for each s and each t in one single run.
E. 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 Paragraph C 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
6
(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
where
= Ẽ[e−r (T −t) h(ST ) IT | Ft ] = It v (m) (St , t) ,
h
i
v (m) (s, t) = Ẽ e−r(T −t) h(ST ) IT St = s , It = 1 .
To obtain a differential equation for the price function v (m) (s, t), we introduce the martingale
Mt = Ẽ e−rT h(ST ) IT Ft = e−rt It v (m) (St , t) .
By the general Itǒ’s formula, the dynamics of M is
dMt = e−rt (−r dt) It v (m) (St , t)
(m)
+ e−rt It vs(m) (St , t)St (rt dt + σdW̃t ) + vt (St , t) dt
1 (m)
(St , t) σ 2 St2 dt
+ e−rt It vss
2
+ e−rt It v (m) (St , t) − It− v (m) (St− , t−) .
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 Paragraph B above to arrive at
the differential equation
(m)
vt
(s, t) = v (m) (s, t) r − vs(m) (s, t) rs −
1 (m)
v (s, t) σ 2 s2 .
2 ss
(2.18)
This PDE is the same as (2.9), but it is now to be solved for (s, t) ∈ [m, ∞)×
[0, T ] subject to the conditions
v (m) (s, T ) = h(s), s > m,
v (m) (m, t) = 0, 0 ≤ t ≤ T.
(2.19)
In fact, there exists an explicit expression,
m 2r2 −1 m2 σ
(m)
v (s, t) = v(s, t) −
v
,t ,
s
s
where v(s, t) is the price (2.8) of the clean-cut European option, see Björk
(1998). As argued above, the price function can be straightforwardly computed by solving the differential equation (2.18) subject to (2.19), and so can
(m)
(m)
greeks by the dynamical recipe. For instance, ρ(m) = vr and V (m) = vσ
are solutions to (2.14) – (2.17) together with (2.18) – (2.19).
7
3
The Markov chain market
A. 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
finite state space Y = {1, . . . , n}. Denote by λefP
the intensity of transition
from state e to state f (6= e), and set λee = − f ; f 6=e λef (minus the total intensity of transition out of state e). Introduce the indicator processes
Ite = 1[Yt = e], e ∈ Y, and the counting processes Ntef = ]{s; 0 < s ≤
t, Ys− = e, Ys = f }, e 6= f ∈ Y. The compensated counting processes Mtef
defined by
dMtef = dNtef − Ite λef dt
(3.20)
and M0ef = 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 No. 1 is a bank account with
price process
RtP e e
Bt = e 0 e r Iu du .
The remaining n − 1 assets are stocks, and the price process of stock No. i
is
R P P
Sti = e
t
0
e
αie Iue du+
f ; f 6=e
β ief dNuef
,
(3.21)
i = 1, . . . , n − 1. The re , αie , and β ief are constants with the following
interpretation: re is the interest rate in economy state e; αie is the rate of
return on stock No. i during sojourns in economy state e (of the same nature
ief
as re ); 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 S̃ti = Sti /Bt have dynamics


X
X ief
i
(αie − re ) Ite dt +
dS̃ti = S̃t−
eβ − 1 dNtef  ,
(3.22)
e
f ; f 6=e
and would be martingales with respect to an equivalent martingale measure
P̃ under which the terms within the parentheses on the right hand side are
martingale increments. With a view to (3.20) this means that (3.22) should
be of the form
X X ief
i
dS̃ti = S̃t−
eβ − 1 dM̃tef ,
(3.23)
e f ; f 6=e
8
where the M̃tef are the compensated counting processes under P̃. More
specifically,
dM̃tef = dNtef − Ite λ̃ef dt
(3.24)
where the λ̃ef would be the transition intensities of Y under P̃. Inspection
of (3.22), (3.23), and (3.24) shows that the requested martingale measure P̃
exists if the equations
X ief
eβ − 1 λ̃ef = 0 ,
(3.25)
αie − re +
f ; f 6=e
i = 1, . . . , n − 1, e = 1, . . . , n, have non-negative solutions λ̃ef such that
λ̃ef = 0 if and only if λef = 0.
The existence of an equavalent martingale ensures that the market is
free of arbitrage. Moreover, if the equations (3.25) admits one and only one
solution, then the market is complete.
B. Option prices. A general European style stock option pays an amount
of the form hYT (ST` ) at time T > 0; the payoff depends on the state of the
economy and the price of stock No. ` at the term T . Under the pricing
measure P̃ the price of the claim at time t < T is
i
h RT
pt = Ẽ e− t ru du hYT (ST` ) Ft ,
where Ft = σ{Yτ ; 0 ≤ τ ≤ t}. Due to the structure (3.21) of the stock price
and the Markov property of Y , the price process must be of the form
X
Ite v e (St` , t),
pt = v Yt (St` , t) =
e
where the state-wise price functions are
R
RT P P
`g g
`gh dN gh
u
e
− tT ru du YT
g α Iu du+ h; h6=g β
t
v (s, t) = Ẽ e
h
se
Yt = e .
Explicit formulas exist for zero coupon bonds and for claims of the simple
form h(Yt ), e.g. 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 will take the latter approach.
9
Rt
The discounted price e− 0 ru du v Yt (St` , t) is a martingale under the equivalent measure. Operating on it with Itǒ’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 vte (s, t) = re v e (s, t) − vse (s, t) s α`e −
v f (sβ `ef , t) − v e (s, t) λ̃ef ,
f ; f 6=e
(3.26)
e = 1, . . . , n. These are to be solved subject to the conditions
v e (s, T ) = he (s),
(3.27)
e = 1, . . . , n.
C. 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 (3.26) – (3.27). Greeks
are straightforwardly obtained as solutions to the differential equations and
side conditions obtained upon differentiating (3.26) and (3.27).
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
νte (s, t) = re ν e (s, t) − νse (s, t) s α`e − vse (s, t) s δeh
X `ef
−
ν f (s eβ , t) − ν e (s, t) λ̃ef
(3.28)
f ; f 6=e
−
X v f (s eβ
`ef
, t) − v e (s, t)
f ; f 6=e
∂ ef
λ̃
∂α`h
(δeh is the Kronecker delta) and the side condition
ν e (s, T ) = 0 ,
(3.29)
e = 1, . . . , n. The derivatives of the λ̃ef with respect to α`h are obtained
upon differentiating (3.25) and solving
δie,`h +
X eβ
ief
−1
f ; f 6=e
i = 1, . . . , n − 1, e = 1, . . . , n.
10
∂ ef
λ̃ = 0 ,
∂α`h
D. Computation. Numerical solution of the differential equations (3.26)
and (3.28) goes by the Lax-Wendroff difference scheme, modified to account
of the non-standard feature that the differential equations are shifted (i.e.
involve function-values at different values of the s-argument).
E. 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+βNt ,
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 α11 = α12 = α, β 112 = β 121 = β, and put Nt = Nt12 + Nt21 . The statewise 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
r−α
.
λ̃ = β
e −1
(The physical intensity of N does not matter since it is not a path property
of the process.) The price function of the call option is
n
∞ λ̃(T − t)
X
v(s, t) = e−r(T −t)
seα(T −t)+βn − K
e−λ̃(T −t) . (3.30)
n!
+
n=0
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 (3.26) and the side condition (3.27) reduce to
vt (s, t) = r v(s, t) − vs (s, t) s α − v(s eβ , t) − v(s, t) λ̃ , (3.31)
v(s, T ) = (s − K)+ .
The sensitivity ν(s, t) =
∂
∂α v(s, t)
(3.32)
is the solution to
νt (s, t) = r ν(s, t) − νs (s, t) s α − vs (s, t) s − ν(s eβ , t) − ν(s, t) λ̃
1
+ v(s eβ , t) − v(s, t) β
,
(3.33)
e −1
11
ν(s, T ) = 0 .
(3.34)
We interpose here that the Poisson model is haunted by non-smoothness
problems. Indeed, upon inspecting (3.30) one realizes that the derivatives
involved in (3.33) 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 (3.30). A recent paper by the
author (Norberg, 2002) shows how to locate the 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 = 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 = eβ − 1. (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 {Zt }t≥0
be a continuous time Markov with finite state space Z = {1, . . . , n} and
intensities that are parametric functions, µjk
θ (t). Denote the infinitesimal
matrix and the matrix of transition probabilities over the time interval from
t to u by
k∈Z
(t)
Mθ (t) = µjk
,
θ
j∈Z
k∈Z
(t,
u)
Pθ (t, u) = pjk
,
θ
j∈Z
respectively. Being mainly interested in the first order derivative in one
direction at a time, we can as well assume that θ is real-valued and assumes
its values in an open interval.
12
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 .
(4.35)
∂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
jk
continuous: for fixed j, k, and t either µjk
θ (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 Lθ,η (T )]
where Lθ, 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
Lθ, η (t)
RtP
=e
0
j6=k
jk
jk
jk
j
jk
(ln µjk
θ+η (τ )−ln µθ (τ ))dN (τ )−(µθ+η (τ )−µθ (τ ))I (τ ) dτ
jk
t
X µjk
θ+η (τ ) − µθ (τ )
Lθ, η (τ −)
µjk
0
θ (τ )
j6=k
Z
=1 +
dMθjk (τ ) ,
(4.36)
(4.37)
where the Mθjk are the compensated counting processes given by
dMθjk (t) = dN jk (t) − I j (t)µjk
θ (t) dt .
Under Pθ the Mθjk are martingales with respect to the natural filtration
Ft = σ{Zτ ; 0 ≤ τ ≤ t}, and they are square integrable and, moreover,
mutually orthogonal:
h
i
dhMθgh , Mθjk i(t) = Eθ dMθgh (t) dMθjk (t) | Ft− = δgh,jk I j (t) µjk
θ (t) dt .
(4.38)
Here δgh,jk is the Kronecker delta, which is 1 or 0 according as (g, h) is equal
to (j, k) or not.
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Under Pθ the random variable X has the martingale representation
Z TX
X = Eθ [X] +
ξθjk (τ ) dMθjk (τ ) ,
(4.39)
0
j6=k
where the ξθjk are predictable processes.
Using the device
Eθ+η [X | Ft ] =
Eθ [XLθ, η (T ) | Ft ]
Eθ [Lθ, η (T ) | Ft ]
together with (4.37), (4.39), and (4.38), we have
1
(Eθ+η [X | Ft ] − Eθ [X | Ft ])
η
1 Eθ [XLθ, η (T ) | Ft ]
=
− Eθ [X | Ft ]
η
Eθ [Lθ,η (T ) | Ft ]
1 1
=
Covθ [X, Lθ, η (T ) | Ft ]
η Lθ, η (t)


Z T
jk
X jk
µjk
(τ
)
−
µ
(τ
)
1
θ+η
θ
jk
jk
=
Eθ 
Lθ, η (τ −)
ξθ (τ )
dhMθ , Mθ i(τ ) Ft 
jk
Lθ, η (t)
η µθ (τ )
t
j6=k


Z T
X jk
µjk
(τ ) − µjk
(τ ) j
1
θ+η
θ
=
I (τ ) dτ Ft  . (4.40)
Eθ 
Lθ, η (τ )
ξθ (τ )
Lθ, η (t)
η
t
j6=k
(The left-limit is annihilated by dτ .) We can now formulate the following
result:
Lemma If the intensities µjk
θ (τ ) are differentiable functions of θ and the
process
jk
X jk
µjk
θ+η (τ ) − µθ (τ ) j
Lθ, η (τ )
ξθ (τ )
I (τ )
(4.41)
η
j6=k
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
1
d
jk
j
Eθ [X | Ft ] =
Eθ 
Lθ, η (τ )
ξθ (τ ) µθ (τ ) I (τ ) dτ Ft  .
dθ
Lθ, η (t)
dθ
t
j6=k
(4.42)
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The existence of the global greek is what is important here; the integral
expression in (4.42) is not necessarily useful for computations. The lemma
is only a preparatory result that needs to be supplemented with 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 of our discussion of this
issue by indicating some more specific sufficient conditions.
d
Corollary The global greek dθ
Eθ [X | Ft ] exists under the following conditions:
d jk
(i) The derivatives dθ
µθ (τ ), 0 ≤ τ ≤ T , j 6= k ∈ Z, exist, are continuous in
t (at least piece-wise), and constitute an equicontinuous family of functions
of θ.
(ii) For any given θ the functions |ξθjk (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
µjk
θ+η (τ ) − µθ (τ )
η
=
d jk
µ ∗ (τ )
dθ θ
for some θ∗ between θ and θ + η (it may depend on τ and, of course, on
j and k). By the equicontinuity condition, we can choose η small enough
d jk
d jk
that dθ
µθ∗ (τ ) < dθ
µθ (τ ) + b(θ) for some positive number b(θ). By the
d jk
assumed 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) gives that the function in (4.41) is bounded
in absolute value by Lθ, η (τ ) n(n − 1) a(θ) (b(θ) + c(θ)), which is integrable
since
Lθ, η (τ ) Eθ
Ft = 1 .
Lθ, η (t) By dominated convergence, we conclude that (4.40) tends to the expression
on the right of (4.42) as η goes to 0.
The conditions (i) and (ii) are easy to check and are usually satisfied. It
may be helpful to know that ξθjk (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.
15
Z(t)`
In particular, for X = Iu` , we have Eθ [X | Ft ] = pθ (t, u), t ≤ u and
jk
j`
ξθ (τ ) = pk`
θ (τ, u) − pθ (τ, u), t ≤ τ ≤ u. Inserting this into (4.42), one
obtains (4.35).
References
Andersen, P.K., Borgan, Ø., Gill, R.D., and Keiding, N. (1993): Statistical
Models Based on Counting Processes, Springer-Verlag.
Björk, T. (2004): Arbitrage Theory in Continuous Time 2nd. ed., Oxford
University Press.
Borodin, A.N. and Salminen, P. (2002): Handbook of Brownian Motion 2nd.
ed., Birkhäuser.
Gerber, H.U. and Shiu, E. (1996): Actuarial bridges to dynamic hedging
and option pricing. Insurance: Mathematics & Economics 18, 183-218.
Kalashnikov, V.V. and Norberg, R. (2003): On the sensitivity of premiums
and reserves to changes in valuation elements. Scand. Actuarial J, 2003,
238-256.
Los, C.A. (2001): Computational Finance, World Scientific.
Norberg, R. (2002): Anomalous PDE-s in Markov chains; domains of validity and numerical solutions. Working paper 89, Department of Statistics,
London School of Economics. (To appear in Finance & Stochastics.)
Norberg, R. (2003): The Markov chain market. ASTIN Bull. 33, 265-287.
Saltelli, A., Chan, K., and Scott, E.M. (eds.) (2000): Sensitivity Analysis,
Wiley.
Tavella, D. and Randall, C. (2000): Pricing Financial Instruments, Wiley.
Wilmott, P. (1998): Derivatives, Wiley.
Address for correspondence:
Ragnar Norberg
Department of Statistics
London School of Economics
Houghton Street
London WC2A 2AE
e-mail: [email protected]
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