Consider an American call option with strike K on an asset that pays a single known discrete dividend x at a known date T < u, where u is the date the option expires. Assume the asset price S drops by x when it goes ex-dividend at date T (i.e., ST = limt↑T St −x) and otherwise is an Itô process. Assume there is a constant risk-free rate r.

(a) Show that if x < 

1 −e−r(u−T)



K, then the call should not be exercised early.

For the remainder of the exercise, assume x > 

1− e−r(u−T)



K. Assume the volatility of the process Zt =

St −e−r(T−t)

x if t < T St if T ≤ t ≤ u is constant over[0,u]. Let V(t,St) denote the value of a European call on the asset with strikeK maturing at u. Let S∗ denotethe value ofthe stock price just beforeT suchthatthe holder ofthe American option would be indifferent about exercising just before the stock goes ex-dividend. This value is given by S∗ −K = V(T,S∗ −

x). Exercise is optimal just before T if limt↑T St > S∗, and equivalently, if ST >

S∗ − x. Let A denote the event ST > S∗ − x and let C denote the set of states of the world such that ST ≤ S∗ − x and Su > K. The cash flows to a holder of the option who exercises optimally are (ST + x − K)1A at (or, rather, “just before”)

date T and (Su − K)1C at date u.

(b) Show that the value at date 0 of receiving (ST + x− K)1A at date T is



S0 − e

−rTx



N(d1) − e−rT(K −x)N(d2), where d1 = log(S0 −e−rTx) − log(S∗ − x) + 

r + 1 2σ2



T

σ

√T , d2 = d1 − σ

√

T .

(c) Show that the value at date 0 of receiving (Su −K)1C at date u is 
S0 − e−rTx 
M(−d1,d
1,−

T/u) −e−ruK M(−d2,d
2,−

T/u), where M

(a, b,ρ) denotes the probability that ξ1 < a and ξ2 < b when ξ1 and ξ2 are standard normal random variables with correlation ρ, and where d
1 = log(S0 −e−rTx) − logK + 
r + 1 2σ2 
u σ
√u , d
2 = d
1 − σ
√u.