PDEs for Arithmetic Asian Options a) Use the higher-dimensional Ito-formula ( Appendix B2) to show that the value function V (S, A, t) of an

PDEs for Arithmetic Asian Options

a) Use the higher-dimensional Itˆo-formula (→ Appendix B2) to show that the value function V (S, A, t) of an Asian option satisfies dV = ∂V ∂t + S ∂V ∂A + μS ∂V ∂S + 1 2 σ2S2 ∂2V ∂S2  dt + σS ∂V ∂S dW , where S is the price of the asset and A its average.

b) Construct a suitable riskless portfolio and derive the Black–Scholes equation ∂V ∂t + S ∂V ∂A + 1 2 σ2S2 ∂2V ∂S2 + rS ∂V ∂S − rV = 0 .

c) Use the transformation V (S, A, t) = V(S, R, t) = SH(R, t), with R = A S and transform the Black–Scholes equation (6.5) to ∂H ∂t + 1 2 σ2R2 ∂2H ∂R2 + (1 − rR) ∂H ∂R = 0 .

d) From Rt+ dt = Rt + dRt , dSt = μSt dt + σSt dWt derive the SDE dRt = (1 + (σ2 − μ)Rt) dt − σRt dWt

e) For At := 1 t t 0 Sθ dθ show dA = 1 t (S − A) dt and derive the PDE ∂V ∂t + 1 2 σ2S2 ∂2V ∂S2 + rS ∂V ∂S + 1 t (S − A) ∂V ∂A − rV = 0 .