Consider the Black-Scholes model with volatility and interest rate r. Let G = (ST, AT) be...

 

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Consider the Black-Scholes model with volatility and interest rate r. Let G = (ST, AT) be an Asian- style option, where A = Sdt is the arithmetic average of the stock price over [0,T]. Denote by II the unique arbitrage-free price process of G. (i) Given the ansatz II+ F = F(t, St, At), where At = Sudu, derive a PDE for the pricing function (ii) Now consider the mean payoff G = Stdt. Compute the conditional expectation II = er(T-t) Tt) EQ[G|Ft] and write it in the form H(t, St, At) for an (explicit) function H. Show that this function H satisfies the PDE derived in part (i). What is the vega of this contract? Consider the Black-Scholes model with volatility and interest rate r. Let G = (ST, AT) be an Asian- style option, where A = Sdt is the arithmetic average of the stock price over [0,T]. Denote by II the unique arbitrage-free price process of G. (i) Given the ansatz II+ F = F(t, St, At), where At = Sudu, derive a PDE for the pricing function (ii) Now consider the mean payoff G = Stdt. Compute the conditional expectation II = er(T-t) Tt) EQ[G|Ft] and write it in the form H(t, St, At) for an (explicit) function H. Show that this function H satisfies the PDE derived in part (i). What is the vega of this contract?