In the following discussion we find the Δ t that yields such a portfolio If P t

In the following discussion we find the δ t that

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). In the following discussion we find the Δ t that yields such a portfolio. If P ( t, S ) is a function such that P t = P ( t, S t ), Itˆo’s formula gives us t = parenleftbigg ∂P ∂t + 1 2 σ 2 S 2 t 2 P ∂S 2 parenrightbigg d t + ∂P ∂S d S t - Δ t d S t - Δ t D 0 S t d t = parenleftbigg ∂P ∂t + 1 2 σ 2 S 2 t 2 P ∂S 2 - Δ t D 0 S t parenrightbigg d t + parenleftbigg ∂P ∂S - Δ t parenrightbigg d S t . 62
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with ∂P ∂t , 2 P ∂S 2 , and ∂P ∂S each evaluated at t and S = S t . Choosing Δ t = ∂P ∂S ( t, S t ) yields t = parenleftbigg ∂P ∂t + 1 2 σ 2 S 2 t 2 P ∂S 2 - Δ t D 0 S t parenrightbigg d t. (43) For the portfolio Π t to be hedged with respect to the underlying S t t is unaffected by changes in S t ), it must provide the risk-free rate of return r in the risk-neutral world. And thus it should satisfy: t = r Π t d t = r ( P t - Δ t )d t = r ( P t - ∂P ∂S S t )d t. (44) The equations (43) and (44) will be equivalent only when the drift terms are equal. Thus, matching the drift terms in those equations and rearranging implies that the function P ( t, S ) satisfies the PDE: ∂P ∂t + 1 2 σ 2 S 2 2 P ∂S 2 + ( r - D 0 ) S ∂P ∂S - rP = 0 , for all t T , and S > 0, with a known boundary condition P ( T, S T ). Thus, in the risk neutral world, the dynamics of S t are given by: d S t = ( r - D 0 ) S t d t + σS t d W Q t , where W Q t a Brownian motion under the risk neutral measure Q . European call option on dividend paying stock Consider a stock S t that moves like a Geometric Brownian Motion (GBM) and pays constant dividend rate D 0 , and suppose C 0 is the price of a European call option on the stock, with maturity T , and strike K . We showed above that GBM associated with dividend-paying stock S t in the risk-neutral world has a drift parameter r - D 0 and volatility σ . Pricing a call option on S t requires evaluating the expectation: C 0 := e - rT E Q { ( S T - K ) + } . Recall that the Black-Scholes formula for a European call option, assuming no dividend payments, is just is given by C BS (0 , S 0 ; K, T, r, σ ) = E Q { e - rT ( S T - K ) + } , where S t is a GBM with drift r and volatility σ in the risk-neutral world. Hence, we can express the price of our call option on a dividend-paying stock as: C 0 := E Q { e - rT ( S T - K ) + } = E Q { e - ( r - D 0 + D 0 ) T ( S T - K ) + } = e - D 0 T E Q { e - ( r - D 0 ) T ( S T - K ) + } = e - D 0 T C BS (0 , S 0 ; K, T, r - D 0 , σ ) . Simply put, we replace r by r - D 0 in the Black-Scholes formula but add a multiplicative term of e - D 0 T to ensure we discount the expectation (in risk-neutral world) the option’s payout at maturity using only the risk-free rate r . 63
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Foreign Exchange Options Every major currency has an associated prevailing interest rate that corresponds to the return on a “risk-free” asset denominated in that currency (in quotes because, in practice, US Treasury securities are the closest to being risk-free). Speculation in foreign exchange markets entails purchasing a foreign currency X t and placing it in that currency’s risk-free asset paying an interest rate r f . Such a scenario is akin to buying a stock with constant dividend rate r f . For example, X t can represent the number of US dollars to buy one British pound, with r and r f being the respective risk-free rates in the US and Britain. Suppose
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  • Fall '11
  • Dividend, Mathematical finance, Black–Scholes

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