Suppose that X t follows a GBM with volatility σ Recall that a forward contract

Suppose that x t follows a gbm with volatility σ

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being the respective risk-free rates in the US and Britain. Suppose that X t follows a GBM with volatility σ . Recall that a forward contract for purchasing a pound at time T has payoff X T - F , where X T is the prevailing exchange rate at maturity T , and F is the price associated with the forward contract at time t = 0. Determining F through an arbitrage argument consists of creating two portfolios: (A) longing a forward and x US bonds at a cost of x US dollars, and (B) longing y pounds at a cost of yX 0 US dollars (as X 0 is the current exchange rate). At maturity, portfolio (A) has a payoff X T - F + xe rT US dollars and portfolio (B) has a payoff yX T e r f T US dollars. If x = Fe - rT and y = e - r f T , then both portfolios are worth X T US dollars at time t = T . To ensure no arbitrate is possible, the costs of each portfolio at time t = 0 must be equal, i.e. x = yX 0 or Fe - rT = e - r f T X 0 , meaning F = X 0 e ( r - r f ) T . Assuming X t follows a GBM with drift μ , the risk-free rate r f paid for holding British pounds implies d X t = ( μ - r f ) X t d t + σX t d W t or, in the risk-neutral world, d X t = ( r - r f ) X t d t + σX t d W Q t . This happens because the price F of the forward contract is chosen so that E Q { X T - F } = 0 implying F = E Q { X T } = X 0 e ( r - r f ) T , where the latter equality follows since ( r - r f ) X t is the drift of d X t in the risk-neutral world. The price of a call option on the pound is then C 0 = e - r f T C BS (0 , X 0 ; K, T, r - r f , σ ) . 64
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  • Fall '11
  • COULON
  • Dividend, Mathematical finance, Black–Scholes

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