Exercise 11 (Forward-Start Option, 5 points) Consider the Black -Scholes model ($2, Shejo,p] and let 0 0 is a fixed strike . What is the limit for large N ? Exercise 13 (Up -and -Out Call in the Heston Model , 7 points ) Consider the risk-neutral Heston model (S!, S. ),cj,pp, where the dynamics ds 9/80 = rat, as 1/8) = rat + Vodw 1.Q do . = A(v - v. )dt + nvv,(pdw ! ." + v1 - p2 dw 2:0), of the safe and risky assets as well as the squared volatility process are modeled directly under a pricing measure Q . Suppose a liquidly traded European option is available as an additional hedging instrument (i) Derive a PDE for the price of an up-and -out call option . (ii) How can this barrier option be hedged by trading in the primary safe and risky assets , as well as the liquidly traded European option ? Exercise 10 (No - Arbitrage and Completeness , 8 points ) Let S) = 1, 82 = 2 and let W 1, W 2, W 3 be three independent standard Brownian motions . Explain which of the following discounted risky asset dynamics correspond to a market with an equivalent martingale measure . If no martingale measure exists , construct an arbitrage ! d$1 /81 = 2dt + dw l, (1) d82/82 = 1dt + 2dw 1, d$1 /81 = 2dt + dw ! - 2dw ?, (2) d$2/87 = 1dt - dw l + dw?. d$1 /8) = 2dt + sin (W 3) dw l - 2dw 2, (3) as; /87 = 1dt - dw / + dw?. Which of these markets are complete? Justify your argument