Consider a European call option 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. Suppose there are traded discount bonds maturing at T and u. Assume the volatility of the process Zt =

[St − xPt(T)]/Pt(u) if t < T St/Pt(u) if T ≤ t ≤ u is a constant σ during [0,u].

(a) Show that the value at date 0 of the call option is

[S0 − xP0(T)]N(d1)− e−yuK N(d2), where yis the yield at date 0 of the discount bond maturing at u and d1 = log[(S0 − xP0(T))/K] +

y + 1 2σ2



u

σ

√u , d2 = d1 −σ

√u.

(b) Interpret the process Z.