Exercise (Arbitrage 5 points) On July 1 you notice the following quotes in the option market (supposedly perfect otherwise!). the underlying (spot) asset being the (not dividend paying) FTSE 100 index: Call 7.200 December: 377 Call 7.400 December: 278 Put 7.200 December: 276 Put 7.400 December: 384 The FTSE 100 index quotes 7.230 and the 6-month (annualized) interest rate in the money market is 2 (simple, or proportional, rate). You can take 6 months as 0.5 year. a) What arbitrage involving these 4 options should you undertake? b) Establish the precise sequence of cash-flows (ie the cash-flow at date 0 (July 1") and the cash-flow at date T (December 31") induced by this arbitrage if you do not cancel by either lending or borrowing, the initial (at date 0) cash-inflow or outflow c) What is the quarterly IRR (Internal Rate of Retur) of this sequence? Check that indeed it reflects an arbitrage. Exercise 2 (Vanilla and barrier options in discrete time) (7 points) a) Stock BCD currently quotes 505) Compute the current value of the European Call (with 2 decimals) written on BCH strike K = 510, using the binomial model (all intermediary variables you will use should have at least 5 decimals). You also know that The option maturity is T=0.25 (91.25 days) The yield curye.is flat at 2% (annualized, discrete rate) A dividend et 16 is paid out at date t = 50 days (between time nodes #1 and 12 in a 3. period tree): Use the proportional dividend approach. The tree has 3 periods (4 dates), the true probability of a positive jump is 0.57 and the expected volatility of BCD is 42% (annual). b) Compute the delta. gamma, theta and (approximate) vega (each with 4 decimals). c) Let a European Up-und-In Call be also written on BCD. Its strike K is 510, its maturity is T=0.25 and the barrier is H = 565. Such a call becomes a vanilla European call if and only if BCD's price hits the 565 barrier (even though it may fall afterwards), but expires worthless if the 565 barrier has never been hit (so. beware of the path followed by BCD's price). Compute the current value of this barrier call using the tree you built for BCD's price in question a). I 20. Exercise 3 (Dynamic strategies) (2.5 points) a) Suppose there are 256 business days in a year. You buy today an ATMF put written on the CAC40 price index, of maturity 1 year. You and the market anticipate that the average annualized volatility of the CACAO price index will be 16% next year. You manage your gamma" so that your portfolio is always delta-neutral. What is the average realized daily volatility of the CACAO price index such that your P&L will be approximately) zero at the end of the year? Explain precisely why. Under what circumstances will your final P&L be positive? Negative? b) You buy a bull spread involving 2 calls with strikes K, and K (> Ki), such that (K,+Kx)/2 = S(O), where S(O) is the underlying asset price today. What should you do to be delta-neutral? If you do that, what will be the approximate values of your gamma, theta, and vega? Exercise 4 ("Best of" option in a simple case) 18 points) I-14 Let a European "Best of" option B be defined as the right to receive, at maturity date T. one unit of asset #1 or one unit of asset #2, whichever has the highest market price: B(T) = Max[S.(T); S;(7)] These two assets are spot assets not paying dividends. The risk-neutral, dynamics of their respective prices are given by: and ds,/S;=rdt +0,dw, (t) + 0W2(t) dS:/S, Erdt +20W200) wherer is the constant, instantaneous interest rate, w, and W (t) are independent Wiener processes, and o, and 2 are constant parameters. a) Express the values of S (1) (as a function, in particular, of S2(0)) and of Sys260 (as a function, in particular of S, (O)/S2(0); use Ito's Lemma). Then show that these two processes are independent. b) Compute, under the risk-neutral probability. ECS (7)) and E(S(T)/S2(T)) at date r = 0. (Hint: think of the Laplace transform of a Gaussian variable.) c) Write the payoff at date T of option B as a portfolio of 2 risky assets, one of which is a European call exchange option, denoted by CE. CE is the right, but not the obligation to buy at date T, one unit of asset 1 at the price of one unit of asset #2. d) Rewrite CEPs payoff at date T so that a product of two terms appears. e) Compute the value of option CE at date r = 0. (Hint: use b) and a) for t = T and try to recover a Black- Scholes-like formula, which is assumed to be known without proof). ) Give the value of the "Best of" option B at date 1 = 0. Exercise (Arbitrage 5 points) On July 1 you notice the following quotes in the option market (supposedly perfect otherwise!). the underlying (spot) asset being the (not dividend paying) FTSE 100 index: Call 7.200 December: 377 Call 7.400 December: 278 Put 7.200 December: 276 Put 7.400 December: 384 The FTSE 100 index quotes 7.230 and the 6-month (annualized) interest rate in the money market is 2 (simple, or proportional, rate). You can take 6 months as 0.5 year. a) What arbitrage involving these 4 options should you undertake? b) Establish the precise sequence of cash-flows (ie the cash-flow at date 0 (July 1") and the cash-flow at date T (December 31") induced by this arbitrage if you do not cancel by either lending or borrowing, the initial (at date 0) cash-inflow or outflow c) What is the quarterly IRR (Internal Rate of Retur) of this sequence? Check that indeed it reflects an arbitrage. Exercise 2 (Vanilla and barrier options in discrete time) (7 points) a) Stock BCD currently quotes 505) Compute the current value of the European Call (with 2 decimals) written on BCH strike K = 510, using the binomial model (all intermediary variables you will use should have at least 5 decimals). You also know that The option maturity is T=0.25 (91.25 days) The yield curye.is flat at 2% (annualized, discrete rate) A dividend et 16 is paid out at date t = 50 days (between time nodes #1 and 12 in a 3. period tree): Use the proportional dividend approach. The tree has 3 periods (4 dates), the true probability of a positive jump is 0.57 and the expected volatility of BCD is 42% (annual). b) Compute the delta. gamma, theta and (approximate) vega (each with 4 decimals). c) Let a European Up-und-In Call be also written on BCD. Its strike K is 510, its maturity is T=0.25 and the barrier is H = 565. Such a call becomes a vanilla European call if and only if BCD's price hits the 565 barrier (even though it may fall afterwards), but expires worthless if the 565 barrier has never been hit (so. beware of the path followed by BCD's price). Compute the current value of this barrier call using the tree you built for BCD's price in question a). I 20. Exercise 3 (Dynamic strategies) (2.5 points) a) Suppose there are 256 business days in a year. You buy today an ATMF put written on the CAC40 price index, of maturity 1 year. You and the market anticipate that the average annualized volatility of the CACAO price index will be 16% next year. You manage your gamma" so that your portfolio is always delta-neutral. What is the average realized daily volatility of the CACAO price index such that your P&L will be approximately) zero at the end of the year? Explain precisely why. Under what circumstances will your final P&L be positive? Negative? b) You buy a bull spread involving 2 calls with strikes K, and K (> Ki), such that (K,+Kx)/2 = S(O), where S(O) is the underlying asset price today. What should you do to be delta-neutral? If you do that, what will be the approximate values of your gamma, theta, and vega? Exercise 4 ("Best of" option in a simple case) 18 points) I-14 Let a European "Best of" option B be defined as the right to receive, at maturity date T. one unit of asset #1 or one unit of asset #2, whichever has the highest market price: B(T) = Max[S.(T); S;(7)] These two assets are spot assets not paying dividends. The risk-neutral, dynamics of their respective prices are given by: and ds,/S;=rdt +0,dw, (t) + 0W2(t) dS:/S, Erdt +20W200) wherer is the constant, instantaneous interest rate, w, and W (t) are independent Wiener processes, and o, and 2 are constant parameters. a) Express the values of S (1) (as a function, in particular, of S2(0)) and of Sys260 (as a function, in particular of S, (O)/S2(0); use Ito's Lemma). Then show that these two processes are independent. b) Compute, under the risk-neutral probability. ECS (7)) and E(S(T)/S2(T)) at date r = 0. (Hint: think of the Laplace transform of a Gaussian variable.) c) Write the payoff at date T of option B as a portfolio of 2 risky assets, one of which is a European call exchange option, denoted by CE. CE is the right, but not the obligation to buy at date T, one unit of asset 1 at the price of one unit of asset #2. d) Rewrite CEPs payoff at date T so that a product of two terms appears. e) Compute the value of option CE at date r = 0. (Hint: use b) and a) for t = T and try to recover a Black- Scholes-like formula, which is assumed to be known without proof). ) Give the value of the "Best of" option B at date 1 = 0