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Unformatted text preview: Introduction to derivatives, Fall 2011 BUSI 588, Homework 2 solutions Homework 2 solutions 1. (*) (a) Following our class discussion, the only forward price consistent with no-arbitrage is given by F t = S t (1 + r $ ) T- t (1 + r e ) T- t = 0 . 90 1 . 05 1 . 15 = 0 . 8217 (b) Venis Telroy is selling the euro forward at too high a price. Thereby we would like to sell forward and buy a synthetic forward contract by trading on the underlying asset and cash. CF at t CF at T Sell euros forward- ( S T- . 86) Borrow PV(0.86) in US +0 . 8190- . 86 Buy 1/1.15 euros and put in euro bonds- . 7826 S T Net . 0364 2. (*) (a) The risk-neutral probabilities in this problem were ˆ p u = 1 . 05- . 90 1 . 20- . 90 = 0 . 5; ˆ p d = 1 . 20- 1 . 05 1 . 20- . 90 = 0 . 5 The value of the call option is thereby given by . 5(1) 1 . 05 = 0 . 4762; since the call is only in the money in the u-state (giving its owner $1). Note that one can manufacture a “synthetic option” by trading in the stock and bonds. In particular, let Δ denote the number of units of the stock in our replicating portfolio, and let B be the dollar amount in the riskfree bonds. Then we can try to find Δ and B such that Δ12 + B (1 . 05) = 1; Δ9 + B (1 . 05) = 0 . My calculations suggest that Δ = 1 / 3 and B =- 2 . 8571 do the job. The cost of this replicating portfolio is, as expected, equal to 1 3 10- 2 . 8571 = 0 . 4762 (b) If the value of call option were $0.32, it would be cheap. The arbitrage opportunity would be to buy the option and sell the portfolio that replicates the payoffs from the call. The following table sketches the details of the arbitrage opportunity. c Diego Garc´ ıa, Kenan-Flagler Business School Page 1 of 5 Introduction to derivatives, Fall 2011 BUSI 588, Homework 2 solutions Today d state u state Buy call option- . 32 +1 Sell 1 / 3 units of stock +3 . 33- 3- 4 Lend $2.8571- 2 . 86 3 +3 Net +0 . 1562 (c) If the value of call option in this case is too high. The arbitrage opportunity would be to sell the option and buy the portfolio that replicates the payoffs from the call. The following table sketches the details of the arbitrage opportunity. Today d state u state Sell call option +0 . 61- 1 Buy 1 / 3 units of stock- 3 . 33 +3 +4 Borrow $2.8571 +2 . 86- 3- 3 Net +0 . 1338 3. (**) Looking at the ratios of forward prices for the different commodities we immediately idenfity stark differences. Gold has a steady, clock-wise increase in its futures prices. Natural gas is all over the place, with prices being higher during winter months. Coffee is a bit strange as well, with prices first rising then falling. According to our forward pricing theories, the forward prices ought to be given by F t = S (1 + r f ) t , where S is the current asset price. This is so since neither of the assets under consideration pays any dividends. In particular, our theories imply that F t F s = (1 + r f ) t- s ; ⇒ r f = F t F s 1 t- s- 1; where F t and F s denote future prices with different maturities.denote future prices with different maturities....
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