In Section 5.2.3, we have seen that the European call price is already uniquely described by the parabolic initial–value problem (5.3.3). For a numerical determination of the European call price function c(ˆσ(·, ·); x, T ), however, we can only consider a finite dimensional number of
discretization points regarding the state variable x. Thus, we have to choose a lower bound xminand
an upper bound xmax for the state variable x so that the resulting interval [xmin, xmax] contains
the region relevant for the construction of the instantaneous volatility function, i.e. xmin and xmax
have to be chosen sufficiently small and large, respectively. Hence, we are approximating
IR × [0 , ¯T ] by the bounded domain [xmin, xmax] × [0 , ¯T ] and the pure initial–value problem
becomes a initial–boundary–value problem. Therefore, we need to derive wise values of
c(ˆσ(·, ·); x, T ) at the lateral boundary of [xmin, xmax] × [0 , ¯T ], i.e. we need information about the
boundary values c(ˆσ(·, ·); xmin, T ) and c(ˆσ(·, ·); xmax, T ) for T ∈ [0 , ¯T ].
We first examine the value of c(ˆσ(·, ·); x, T ) at the left boundary, i.e. when x is small. Suppose x tends to −∞. Because of the change of variables (5.2.9), this is equivalent to assuming that the
strike price K is very small, i.e. K is tending to 0. In this case, the holder of the European call option gets the underlying asset almost for free at the end of the period T . If we suppose that the volatility is constant, then with (5.1.5) it follows that in this case the European call price for K = 0 is given by C(σ(·, ·); 0, T ) = S exp(− T Z 0 d(τ ) dτ ) , T ∈ [0 , ¯T ] , (5.4.1)
with d(T ) , T ∈ [0 , ¯T ] is the known dividend yield. This equality also holds for the nonconstant
volatility case. We will derive the value of C(σ(·, ·); 0, T ) by building an equivalent portfolio that generates the same payout at the expiration date T as the European call option. Suppose first that
5.4 Determination of the Discrete Instantaneous Volatility Function 93
no dividends are paid on the asset underlying the option, i.e. d(T ) ≡ 0 for T ∈ [0 , ¯T ]. Since the
exercise price K is equal to zero, the payout of the option at the expiration date is equivalent to the value of the underlying asset at the expiration date. Hence, the option can be replicated by using a portfolio which consists of one share of the underlying asset and is held constant over the life of the option. Thus, to avoid possibilities of arbitrage the value of the option and the value of this portfolio at the current time must be the same if no payments on either of these assets are made during the life of the option, i.e. the value of the option is equal to the current price of the asset underlying the option. If this relationship is not true, an arbitrageur can easily make a riskless profit by buying the option and selling the stock or vice versa.
Now, suppose that dividends are paid on the underlying asset. The payment of a dividend causes the stock to drop by an amount equal to the dividend. Supposing that a continuous dividend yield of rate d is paid and the stock price rises from S today to ST at time T , it follows that without
dividend payments the stock would have risen from S exp(−d(T − t)) today up to ST at time T .
Thus, if the underlying asset of a European call option pays a continuous dividend yield of rate d then the European call option has the same value than the corresponding European call option with the same underlying asset paying no dividends and having the value S exp(−d(T − t)) today since the final value of the asset is the same in both cases. Therefore, for a valuation of a European call option with an underlying asset paying a continuous dividend yield of d we can base the calculations on the current value of the asset of S exp(−d(T − t)), see for instance Hull [58]. Hence, assuming a time varying dividend yield d(T ), T ∈ [0 , ¯T ], (5.4.1) presents the value of a European call option
with strike price K = 0 and an underlying asset with value S that pays a dividend yield of rate
d(T ) , T ∈ [0 , ¯T ].
Since we carry out the numerical computation of the value of the European call option on a bounded domain with respect to the space variable x, we are interested in the value of the Euro- pean call option for a sufficiently small, but finite, left boundary xmin = ln(KminS ). In this case, the
holder of the European call option still has to pay a small strike price Kminat maturity. Considering
the time value of this payment, the value of the European call option at the left boundary
Kmin = S exp(xmin) becomes
C(σ(·, ·); Kmin, T ) = S exp(− T Z 0 d(τ ) dτ ) − Kmin exp(− T Z 0 r(τ ) dτ ) , (5.4.2)
for T ∈ [0 , ¯T ] and with r(T ) , T ∈ [0 , ¯T ], being the riskless interest rate.
For the right boundary of the domain [xmin, xmax], the following observation can be made for
the value of a European call option. As the strike price increases without bound, it becomes ever more likely that the option will not be exercised, i.e. it becomes ever more likely that at the expiration date the payoff is zero. Thus, the call option is worthless for K → ∞ even if there is a long time to expiration. Hence,
lim
K→∞C(σ(·, ·); K, T ) = 0 , T ∈ [0 , ¯T ] , (5.4.3)
such that the condition for the right boundary Kmax= S exp(xmax) is given by
94 5 Constructing the Instantaneous Volatility Function
Note that the boundary conditions (5.4.2) and (5.4.4) ensure the compatibility of the initial and boundary condition of the initial–boundary–value problem.
The transformation (5.2.10) from C(K, T ) to c(x, T ) and the boundary conditions (5.4.2) and (5.4.4) for the original parabolic differential equation (5.2.7) lead to the boundary condition for the transformed parabolic differential equation (5.2.12):
c(ˆσ(·, ·); xmin, T ) = exp(−
T
Z
0
d(τ ) dτ ) − exp(xmin) exp(−
T Z 0 r(τ ) dτ ) , c(ˆσ(·, ·); xmax, T ) = 0 , (5.4.5) for T ∈ [0 , ¯T ].