Answer Happy

(Asset-or-Nothing Option - Probabilistic Ap- proach) Let {W, :t> 0} be a P-standard Wiener process on the probability space (12,F,P) and let the stock price S, follow a GBM with the following SDE ds= S(x - D}dt+ SodW.. where he is the drift parameter, D is the continuous dividend yield, o > is the volatility parameter, and let r > 0 denote the risk-free interest rate. An asset-or-nothing call option is a contract that pays Sy at expiry time T if the spot price Sy > K and nothing if Sy SK. In contrast, an asset-or-nothing put pays Sy >0 at expiry time I if the spot price S; <K and nothing if S > K. (0) (15p) By denoting Ca(So, t, K,T) and P.(S, tK.T) as the asset-or-nothing call and put option prices, respectively, at time t, t < T show using the risk-neutral valuation approach that C.(S.,t;K,T) = See-D{T-$(dt) and P.(S.,t;K,T) = See-DT)+(-d+) where 1 d. log(S:/K) + (r - D+ 40?)(T-1) OVT-t and (I) = - L etud du. (ii) (10p) Verify that the put-call parity for an asset-or-nothing option is C.(Sc,t;K,T)+ P. (Sc,t;K,T) = Se-D(T-) Answer: