Options
American options allow early exercise at any time, therefore the value of an American call option V(S, t) can be written as
V(S, t) = sup
t≤τ≤T
e−rτE£
(S(τ)−K)+¤
It can be proved16 that for each t there exists a stock priceS∗(t) such that
if S(t)≥S∗(t) the value of the American call option is the payoff which can
be obtained by immediately exercising the option, that is [S(t)−K], while if S(t) < S∗(t) the call value exceeds this payoff. The curve S∗(t) is called
exercise boundary as it defines over time the level at which after which it is best to exercise the option. On the exercise boundary the value of the non exercised option is the same as the value of the exercised option. The region
{(S, t)|S < S∗(t)}
is defined the continuation region of an American call option. Its complement is called exercise region. Symmetric results apply to the American put option: its value can be expressed as
P(S, t) = sup
t≤τ≤T
e−rτE£
(K−S(τ))+¤
and for each t there exists a stock price S∗(t) such that if S(t)≤ S∗(t), the
value of the American put option is the payoff [K−S(t)], while if S(t) > S∗(t) the put value exceeds this payoff. The continuation region for the American put option is given by
{(S, t)|S > S∗(t)} the exercise region being its complement.
It can be proved17 that the discounted price of the option is a martingale in the continuation region and hence the PIDE (4.6) holds. However in the
16Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus,
Springer Verlag, second edition, 1991.
17Ioannis Karatzas and Steven E. Shreve,Methods of Mathematical Finance, Application
exercise region the PIDE does not hold anymore. Being the exercise and the continuation region being switched for calls and puts with respect to the exercise boundary it is a good idea to treat the two cases distinctly.
The Variance Gamma PIDE for American Vanilla Call Options
We know that the PIDE (4.6) does not apply to the case where the option is exercised. Let’s define the exercise region for call options in terms of our variable x , ln(S) as the area where x > x(τ). Hence x(τ) is the exercise barrier expressed in terms of ln(S), as a function of time, in the same as
S∗(t) defined the barrier before. Let’s write an equivalent expression for the
PIDE in the exercise region; in particular we want to write the value of the infinitesimal generator for the Markov process x correspondent to the PIDE we wrote. Let’s define the operator L(f) applied to the generical function f(x, t) as the infinitesimal generator
L(f), ∂f(x, t) ∂t + (r−q+ω)· ∂f(x, t) ∂x + + Z +∞ −∞ [f(x+y, t)−f(x, t)]k(y)dy−rf(x, t) (4.17) When we are in the exercise region, the option is worth its payoff that is
W(x, t) = ex −K
and hence we have
∂W(x, t)
∂t = 0
∂W(x, t)
∂x =e
x
We can now substitute this values in the operator L(f) to obtain its value
in the exercise region. We get
L(W) = (r−q+ω)ex+
+
Z +∞ −∞
Because values of the underlying overx(τ) correspond to the exercise region, we have that after a jump of sizey, we are in the exercise region ifx+y > x(τ), that is if the size of the jump is bigger than x(τ)−x. In this area
W(x+y, t) =ex+y −K
We can therefore divide the integral in equation (4.18) in two pieces dividing them at the level x(τ)−x. Moreover remembering that
Z +∞ −∞
(ey −1)k(y)dy=−ω (4.19) and dividing also this integral in two pieces we can rewrite equation (4.18) as L(W) = (r−q)ex−r(ex−K) + + Z x(τ)−x −∞ [W(x+y, t)−(ex−K) + (1−ey)ex]k(y)dy+ + Z +∞ x(τ)−x £ ex+y −K−(ex −K) + (1−ey)ex¤ k(y)dy
The second integral is equal to zero and we remain with
L(W) =rK −qex+ Z x(τ)−x −∞ £ W(x+y, t) +K −ex+y¤ k(y)dy (4.20) We found that for American call options, the PIDE (4.6) holds in the con- tinuation region therefore L(W) = 0 for x < x(τ); moreover for x > x(τ) L(W) is given by equation (4.20). Therefore we can incorporate both the
behavior in the continuation and in the exercise region by writing
∂W(x, t) ∂t + (r−q+ω)· ∂W(x, t) ∂x + + Z +∞ −∞ [W(x+y, t)−W(x, t)]k(y)dy−rW(x, t) + −1x>x(τ) ( rK −qex+ Z x(τ)−x −∞ £ W(x+y, t) +K −ex+y¤ k(y)dy ) = 0 (4.21) where the indicator function is defined as
1A ½
1 if A
It is interesting to compare the European and American PIDE: we can see that the difference is given by the part multiplied by the indicator function. To obtain the European PIDE it is necessary to extract from the American option the value of early exercise which is expressed by the dividend yield minus the interest on the strike times the time the stock spends in the ex- ercise region18. In the case of a pure jump process, this amount has to be further modified by adding the expected shortfall the strategy may experi- ence because of the jumping back of the process in the continuation region after having reached the exercised region19. This last correction is realized with the integral in the last term.
This PIDE has to be solved by imposing the following final condition:
W(x, T) = max(ex
−K,0)
Moreover, being the option American we need to impose an early exercise condition:
W(x, t)>max(ex−K,0)∀t < T
Finally we have the usual boundary conditions for call options
W(−∞, t) = 0 ∀t W(+∞, t) =ex ∀t
The Variance Gamma PIDE for American Vanilla Put Options
The approached used for call options can be replicated for put options. We define here the exercise region as the area wherex < x(τ). In this region the following equations are true
W(x, t) = K−ex ∂W(x, t) ∂t = 0 ∂W(x, t) ∂x =−e x
18See on this Peter Carr, R. A. Jarrow and R. Myneni, “Alternative Characterization
of American Put Options”,Mathematical Finance, 2, 1992, pages 87-106.
19See more on this in C. R. Gukhal, “Analytical Valuation of American Options on
Therefore the operator L(f) as defined in equation (4.17), in the exercise region, is equal to L(W) = −(r−q+ω)ex+ + Z +∞ −∞ [W(x+y, t)−(K−ex)]k(y)dy−r(K−ex)
Using again equation (4.19) and remembering that this time the process jumps in the exercise region if y < x(τ)−x, we can write
L(W) = (q−r)ex−r(K−ex) + + Z x(τ)−x −∞ £ K−ex+y −(K−ex) + (ey −1)ex¤ k(y)dy+ + Z +∞ x(τ)−x [W(x+y, t)−(K−ex) + (ey −1)ex]k(y)dy that is equal to L(W) =qex−rK+ Z +∞ x(τ)−x £ W(x+y, t)−K+ex+y¤ k(y)dy (4.22) For American put option we can say that in the continuation regionL(W) =
0 and in the exercise region L(W) is given by equation (4.22). Everything
can be written in the same equation as
∂W(x, t) ∂t + (r−q+ω)· ∂W(x, t) ∂x + + Z +∞ −∞ [W(x+y, t)−W(x, t)]k(y)dy−rW(x, t) + −1x<x(τ) ½ qex−rK+ Z +∞ x(τ)−x £ W(x+y, t)−K+ex+y¤ k(y)dy ¾ = 0 (4.23) The difference between European and American PIDE for put option can be interpreted in the same way as we did for the call options as subtraction of the early exercise value from the American option. However here the European case is obtained by subtracting the interest on the strike minus the dividend yield for the time spent in the exercise region and adding the expected shortfall the strategy would have if the stock jumps back in the continuation region after having reached the exercise region.
The PIDE for American put options will be solved imposing the following final condition:
W(ex, T) = max(K
−ex,0)
Moreover since early exercise is allowed for American option, we need to impose
W(ex, t)>max(K
−ex,0) ∀t < T
And finally we have boundary conditions for put options
W(−∞, t) =K ∀t W(+∞, t) = 0 ∀t