5. Let V(S, t) denote the value at time t...

 

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5. Let V(S, t) denote the value at time t <T of a derivative contract when the price of the underlying asset is S where the asset price process S(t) follows the stochastic equation dS = µSdt +oSdW where W = W(t) is a standard Brownian motion, u, o are constants and r is a constant riskless interest rate applicable throughout the life of the option. (a) Explain why the growth factor u does not need to be estimated for the Black- Scholes model. (b) What does the seller of a European call option need to do today to hedge the exposure to the underlying stock? Explain why. (c) [Refer to the formulae at the start of the exam paper] Show that d = d – 2 log(Set/K). Hence, or otherwise, show that the delta of a European call option is Ф(d). as (d) (i) Show that the model for stock prices from the above stochastic equation does not allow stock prices to become negative. (ii) Since some asset classes such as interest rates and commodities can have negative values we propose an alternative model given by the stochastic process: dS = µdt + odW By hedging the delta, construct a partial differential equation that can be used to value derivative contracts in these asset classes. 5. Let V(S, t) denote the value at time t <T of a derivative contract when the price of the underlying asset is S where the asset price process S(t) follows the stochastic equation dS = µSdt +oSdW where W = W(t) is a standard Brownian motion, u, o are constants and r is a constant riskless interest rate applicable throughout the life of the option. (a) Explain why the growth factor u does not need to be estimated for the Black- Scholes model. (b) What does the seller of a European call option need to do today to hedge the exposure to the underlying stock? Explain why. (c) [Refer to the formulae at the start of the exam paper] Show that d = d – 2 log(Set/K). Hence, or otherwise, show that the delta of a European call option is Ф(d). as (d) (i) Show that the model for stock prices from the above stochastic equation does not allow stock prices to become negative. (ii) Since some asset classes such as interest rates and commodities can have negative values we propose an alternative model given by the stochastic process: dS = µdt + odW By hedging the delta, construct a partial differential equation that can be used to value derivative contracts in these asset classes.

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a In the BlackScholes model the growth factor u does not need to be estimated because the model assumes a riskneutral world In a riskneutral world the ... View the full answer