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Some European options

For each of these options, give the analytical formula of the payo at maturity and draw it on a coordinate system (payo value in function of stock price). Provide also a possible nancial application for such options (i.e. what kind of risk it hedges or on what moves it speculates). 1. Straddle option : you hold a Call option and a Put option with the same strike price K and maturity T . 2. Strangle option : you hold a Call option and a Put option with the same maturity T but a dierent strike. 3. Strip option : you hold a Call option and two Put options with the same strike K and maturity T . 4. Strap option : you hold two Call options and one Put option with the same strike K and maturity T . 5. Bull spread option : you hold a Call option of strike K1 and have sold a Call option with strike K2 > K1 . Both options have maturity T . 6. Bear spread option : you hold a Put option of strike K1 and have sold a Put option with strike K2 > K1 . Both options have maturity T . 7. Buttery option : you hold a Call option of strike K + δK and a Call option of strike K − δK , and you have sold two Call options of strike K . All options have maturity T . 8. Condor option : you hold a Call option of strike K1 and another of strike K4 , and you have sold two Call options of strike K2 and K3 respectively. The strikes verify K1 < K2 < K3 < K4 . All options have maturity T . 9. You have written a forward option of strike K for some holder, and you hold two Call options of same strike K . All contracts have delivery at T . How would you name that option ? 2

Arbitrage bounds for Call and Put options

Suppose that there is no arbitrage opportunity on the market (Assumption NA holds). Let us denote B(, 0, T ) the zero coupon price in time 0 which delivers 1$ at T . We also denote 1

C0 and P0 the prices at time 0 of a Call option and a Put option respectively, on the same stock S , with the same strike K and maturity T .

1. Show, using arbitrage reasonning, that (S0 − KB(0, T ))+ ≤ C0 ≤ S0 . You can also use the Propositions of the course if you want. 2. Deduce from the above inequality that (KB(0, T ) − S0 )+ ≤ P0 ≤ KB(0, T ). 3. Show that the Call option price is a decreasing function of the strike K . 4. Assume that there exists a zero coupon at any date for any maturity : B(t, T ) ≤ 1 for all t ≤ T arbitrarily chosen. By using answers to questions 1 and 3, show that the Call option price is an increasing function of the maturity T . 5. Repeat questions 3 and 4 for the Put option. 3

One period Binomial model

Consider a one period nancial model with two traded securities : a stock and a forward contract on the stock. Assume that the initial price of the stock equals S0 = 120 and that the terminal price of the stock can take two values, 110 and 150. Assume also that the forward price equals K = 135. Compute the arbitrage-free price of the zero-coupon bond which pays 10000$ at maturity, by making the arbitrage reasoning (so don't use the course proposition 1). 4

Hedging a long position - one period

Do exercise 1.6 in Steven E. Schreve Textbook (p.20). 5

Path-dependent Barrier Option

Consider the two-period binomial model with S0 , u = 1, 2, d = 0.8 and take r = 0.1. Consider a derivative security, hich expires at time t = 2, and which payo equals ( V2 =

100 if max(S1 , S2 ) ≥ K 0 if max(S1 , S2 ) < K

where the threshold K (the barrier) equals 110 and Si denotes the price of the stock at time i ∈ {1, 2}. Compute 1. the arbitrage-free price of the option at time 0. 2. the number of stocks ∆0 in the replicating strategy of the option at time 0. 2

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