as3_sol - Columbia University Instructor Rama CONT M.S in...

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Columbia University M.S. in Financial Engineering IEOR 4706: Foundations of Financial Engineering Instructor: Rama CONT Summer 2011. Solution for Assignment 3. Arbitrage relations. Part I: Consider an arbitrage-free market in which investors can trade in stocks with prices S i ( t ) , i = 1 ..d . Investors can lend or borrow in the interbank market at the prevailing LIBOR rates. Denote by C i t ( T, K i ) (resp. P i t ( T, K i )) the values, at t T , of a European call option (resp. a put option) with maturity T and strike K i on the underlying asset S i . D ( t, T ) denotes the discount factor and ( x - K ) + = max( x - K, 0). We assume the stocks do not pay dividends in the next 3 months. 1. Show that if T 2 > T 1 , C i t ( T 2 , K ) C i t ( T 1 , K ). (Answer) Suppose C i t ( T 2 , K ) < C i t ( T 1 , K ). Then consider the following strategy: At t T 1 : sell the call with maturity T 1 , buy the call with maturity T 2 > T 1 (this is called a calendar spread ) and use the net profit to buy a zero coupon bond with notional [ C i t ( T 1 , K ) - C i t ( T 2 , K )] /D (0 , T 2 ) > 0 and maturity T 2 . The net cash flow is zero from these operations is zero. At T 1 : (a) if S ( T 1 ) K : do nothing. (b) if S ( T 1 ) > K : short one share of stock (gets you S ( T 1 )), buy a zero coupon bond with maturity T 2 , notional K/D ( T 1 , T 2 ) (costs you K ), and pay off the holder of the call option with maturity T 1 . Net cash flow: S ( T 1 ) - K - ( S ( T 1 ) - K ) = 0. In both cases the net cash flow is zero: the strategy is self-financing and does not require us to inject capital at T 1 . At T 2 : (a) if S ( T 1 ) K : your portfolio is now worth V T = max( S ( T 2 ) - K, 0) + C i t ( T 1 , K ) - C i t ( T 1 , K ) D (0 , T 2 ) > 0 (b) if S ( T 1 ) > K : your portfolio is now worth V T = K |{z} zero - coupon - S ( T 2 ) | {z } short pos .
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