HWASSGNINE_CHERO - = 1/√2π S σ √dt ≈ 0.4 x S 0...

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HWASSGN #9 EXERCISE HW 9-1 A company’s cash position, measured in millions of dollars, follows a generalized Brownian motion with a drift rate a=0.1 per month and a volatility rate b = 0.4 per month. The initial cash position is 2.0. a)What are the probability distributions of the cash position after 1 month, 6 month and 1 year? b) What are the probabilities of a negative cash position after 6 months and 1 year? c) At what time in the future in the probability of a negative cash position greatest? EXERCISE HW 9-2. Consider a stock S currently trading at price S 0 . The future evolution of the stock price (S t ) t≥0 is modeled as the driftless geometric Brownian motion with instant increments: dS t = σS t dW t , where µ and σ > 0 are constant parameters and W is a standard Brownian motion. a) Assuming zero interest rates, argue that the fair value of an instant at-the-money call expiring at time dt is c 0 = σS 0 E(max(0, W dt )). b)Let X be a random variable following a normal distribution with mean 0 and standard deviation s. Show that: E(max(0,X)) = s/√2π. c) Combining results from questions (a) and (b) show that c
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Unformatted text preview: = 1/√2π S σ √dt ≈ 0.4 x S 0 σ√dt d) Estimate the fair value of a 1-week at-the-money call on the S&P 500 index using the following data: S&P 500 spot price: 2000 S&P 500 volatility: 20% HWASSGN #9 EXERCISE HW 9-3. Consider a call opton on ABC Inc. wiTh The following informaton: STrike price $1 MaTuriTy: 1 yearVolatliTy: σ InTeresT raTe: 0% No dividends LeT c(S) be The Black-Sholes call value as a functon of ABC Inc’s spoT price S. a) show ThaT SN’ (d 1 ) = N’ (d 2 ) where N’ (x) = e –x2/2 / √2π is The sTandard normal partal disTributon densiTy functon and d 1 = ln S + σ 2 /2, d 2 = d 1 – σ. /σ b) Carefully show ThaT c’ (S) = N (d 1 ) where N(.) is The sTandard normal cumulatve disTributon functon. c) Assuming σ = 30% calculaTe c’ (1) and inTerpreT This number using a FrsT-order ±aylor series expansion, wiTh emphasis on Fnancial inTerpreTaton. d) Assuming σ = 30% produce a graph of c’ (S) for 0 < S ≤ 2....
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