(3.68) where is a Wiener process in the domestic measure. This is the funda- mental relationship that connects the two Wiener processes.
complementarity problem with finite differences is a straightforward modification (quite trivial in most cases) of the finite difference formu- lation for European pricing.
European Derivatives
Although we often refer to the pricing equation of European derivatives as a partial differential equation (PDE), in general the equation is a partial- integro differential equation (PIDE). The difference is significant mathe- matically, but it is not a substantial complication numerically.
The derivation of a PDE as the pricing equation for a European deriva- tive is possible if the underlying processes are Markovian. Difficulties arise if the underlying process is not Markovian; that is, if the underlying process depends on the path of the Wiener processes involved. The best-known example is the case of the short rate process in the Heath-Jarrow-Morton model (Heath, Jarrow, and Morton, 1992). Some of the well-known mod- els for the short rate are ways of getting around this problem. Such models are designed with features that make the short rate Markovian (Bhar and Chiarella, 1997).
The typical pricing equation looks as follows:
(3.71)
where S and r are Ito processes, a, b, c, d, e, and f are functions of S,r, and t,I is a coordinate that does not have a corresponding diffusion term, h is a Poisson jump intensity (this could also be a stochastic process), and (.) is the jump density.
The classical Black-Scholes equation lacks the I coordinate and the jump term. The convolution integral does not add significant difficulties to the numerical task of solving this equation. We now describe the two main approaches for deriving the pricing equation.
In the absence of jumps in the underlying processes, there are two funda- mental approaches for deriving the pricing equations for European derivatives.
The first approach consists in constructing a hedging portfolio whose value
∂V∂t ---
Convection a∂V
---∂S b∂V ---∂r c∂V
---∂I
+ +
Diffusion
d∂2V
∂S2
--- e∂S---∂2∂rV f∂2V
∂r2 ---
+ +
+ +
Source
rV – h
∫
( )V( ) dConvolution V
–
Jump
=
tracks the value of the derivative as a function of time. The second approach is based on the Feynman-Kac theorem, which states that the conditional expec- tation of a stochastic process obeys a partial differential equation. If there are jumps in the underlying processes, we need to introduce additional assump- tions about the effect of the jumps. We devote a separate section to the deri- vation of the pricing equation in the presence of jumps.
Hedging Portfolio Approach
Consider an option on an underlying process S(t), t [0, T], with a payoff V(S(T), T) g(S(T)). We want to obtain the equation describing the value of the option at time t [0, T], V(S(t), t). Assume that the underlying pro- cess is as follows, where (S(t), t) and (S(t), t) are known given the infor- mation available at time t:
(3.72) For notational convenience, we will from now on omit the arguments inV and its derivatives, ,, and W unless there is need for additional clar- ity. Applying Ito’s lemma to V(S(t), t), we get
(3.73)
Assume now a hedging portfolio, Y(t), designed to track the evolu- tion of the value of the derivative from time 0 to T, as we did earlier in the chapter. The portfolio will consist of(t) units of S(t), plus borrowing or lending. Over an infinitesimal interval of time, the portfolio value will change as described by Equation 3.6. Denoting the instantaneous bor- rowing or lending rate by r(t), the change in the replicating portfolio value is
(3.74) Substituting dS from Equation 3.72 into Equation 3.74, the replicating portfolio process is
(3.75) In order for the portfolio to hedge the option, V(S(t), t) and Y(t) must have the same drift and the same volatility. Equating the drifts in Equations 3.73 and 3.75, we find the following equation for the value of the option:
∈
=
∈
dS t( ) S t( )
--- = (S t( ),t)dt+(S t( ),t)dW t( )
dV S t( ( ),t) ∂V
∂t
--- S∂V---∂S 12---2S2∂2V
∂S2 ---
+ +
dt+S∂V---∂SdW
=
dY = ( )dSt +(Y t( )–( )S tt ( ))r t( )dt
dY = [Y t( )r t( )+( )S tt ( )(( )t –r t( ))]dt+( )S tt ( )dW
(3.76) Equating the volatilities, we find the number of units of S must satisfy the relationship
(3.77) Since we require that the hedging portfolio should replicate the option for any t [0, T], we set Y(t) V(S(t), t) in Equation 3.76. Replacing for n in Equation 3.76, we get the following pricing equation,
(3.78)
with the end condition
(3.79) This is the Black-Scholes partial differential equation of option pricing.
Notice that the drift (S(t), t) of the underlying process does not appear in this expression.
Equation 3.78 applies if r is a deterministic function of time. If r itself were governed by a stochastic process, this should be reflected in the deri- vation and additional terms would appear. Notice also that if we set r and Y V in Equation 3.76, we obtain the Black and Scholes equation.
However, Equation 3.76 only enforces the drifts of dV and dY to be the same, not their unanticipated changes. In other words, assuming that r has the same effect as requiring that the volatility component of the option price process and the volatility component of the replicating portfolio pro- cess should be the same. As we saw earlier, if r(t) is the instantaneous risk- free rate, a situation where we assume that the rate of growth of any non- dividend paying asset is equal to r(t) is equivalent to expressing the process for the asset in the risk neutral measure. As a result, the invocation of risk neutrality allows us, in the absence of cash flows associated with the under- lying processes, to determine the pricing equation by simply enforcing the equality Et[dV(S(t), t)] r(t)V(S(t), t), where dV(S(t), t) is the increment of V obtained through the application of Ito’s lemma. Invoking risk neutrality means that the drifts of the underlying processes must be consistent with risk neutral returns of traded assets. If the underlying process is the price of a stock that does not pay dividends, this consistency with risk neutrality
∂V∂t
--- S∂V---∂S 12---2S2∂2V
∂S2 ---
+ + = rY+n(–r)S
( )t ∂V
∂S ---
=
∈ =
∂V∂t
--- rS∂V
∂S
--- 1
2---2S2∂2V
∂S2 ---
+ + = rV
V T S T( , ( )) = g S T( ( ))
=
=
=
=
simply means that Et[dS(t)] r(t)S(t). In more complex cases, obtaining the appropriate risk neutral drift may require significant elaboration, as we dis- cussed earlier in the chapter.
Feynman-Kac Approach
The Feynman-Kac theorem establishes a relationship between stochastic differential equations and partial differential equations. Given the stochas- tic differential equation
(3.80) the Feynman-Kac theorem states that the expectation
(3.81) is the solution to the following partial differential equation,
(3.82) subject to the end condition
(3.83) In the case of several underlying processes, y1(t), y2(t),…, yn(t), follow- ing the stochastic differential equations,
(3.84) the function
(3.85) is given by the solution of the differential equation:
(3.86)
subject to
(3.87) where ij cov(dWi,dWj)/dt.
=
dy t( ) = (y t( ),t)dt+(y t( ),t)dW t( )
f y,( t) = Ey,t[g y T( ( ))]
∂f
∂t
--- ∂y---∂f 1 2---2∂2f
∂y2 ---
+ + = 0
f y,( T) = g y( )
dyi = i(y1, y2,…,yn,t)dt+i(y1, y2,…,yn,t)dWi
f y( 1,y2,…,yn,t) Ey
1,y2,…,yn,t[g y( 1( ),T y2( ),T …,yn( )T )]
=
∂f
∂t--- i∂y∂f
i --- i=0
i=n
∑
12--- ijij∂y∂fiyj --- i j=0,
i j=n,
∑
+ + = 0
f y( 1,y2,…,yn,T) = g y( 1, y2,…,yn)
=
As an example of the Feynman-Kac approach, consider the derivation of the pricing equation for a claim in the case where the interest rate is described by the Hull and White model (Equation 2.102). In the risk neu- tral measure, the price of a claim on S(t) paying g(S(T)) at maturity is given by
(3.88)
Assume that the processes for S(t) and r(t) are given by
(3.89) and
(3.90) where S and r may be functions of the state variables r and S and time.
Notice that the argument in the expectation in Equation 3.88 is not simply a function of T, but it depends on the trajectories of r. In order to apply the Feynman-Kac formula, we need to transform the argument of the expecta- tion, so that it depends on the values of processes at time T only. We can accomplish this at the expense of temporarily increasing the dimensionality of the problem by defining an auxiliary process of the form
(3.91) With this, the value of the claim is (notice the conditions required of the expectation)
(3.92) We now define
(3.93) and apply the three-dimensional Feynman-Kac formula to U(r, S, z, t):
(3.94) V r t( ( ),S t( ),t) Er,S,t g T( )e–
∫
Ttr( )d
=
dS = rSdt+sSdWs
dr = (a t( )–br)dt+rdWr
dz = –r t( )dt
V r S t( , , ) = e–zEr,S,z,t[g T( )ez T( )]
V r S t( , , ) = e–zU r,( S,z,t)
∂U∂t
--- rS∂U
∂S
--- (a t( )–br)∂U
---∂r r∂U ---∂z
– 1
2---s2 S2∂2U
∂S2 ---
+ + +
Ssrs,r∂---∂S2U∂r 1 2---r2∂2U
∂r2 ---
+ = 0
+
We can now eliminate the additional dimension, z, by replacing U(r,S, z,t) ezV(r, S, t) in Equation 3.95. This gives the two-dimensional pricing equation
(3.95)
with the end condition
(3.96)