The price function of a European call option under stochastic interest rates can also be solved using the partial differential equation approach. Let the asset value process S_t and the short rate process r_t be governed by

where dZ_S dZ_r = ρdt. Let λ_r denote the market price of risk of the short rate. Let c(S,r,t) denote the time-t value of a European call option with strike price X and maturity date T.

(a) Show that the governing equation c(S,r,t) is given by 

(b) Let B(r,t; T) denote the discount bond price, which satisfies 

The bond price function admits a solution of the form 

Defining ξ = S/B and u = c/B, show that u(ξ,t) satisfies 

(c) Show that the value of the European equity call option under stochastic interest rates is given by 

Transcribed Image Text:

d St St =rt dt+os (t) dZs drt = µr(t) dt + or (t) dZr,