Question

(i) Use formulas (4.5.23)-(4.5.25), (4.5.26), and (4.5.29) to determine the delta $p_x(t, x)$, the gamma $p_{x x}(t, x)$, and the theta $p_t(t, x)$ of a European put. (ii) Show that an agent hedging a short position in the put should have a short position in the underlying stock and a long position in the money market account. (iii) Show that $f(t, x)$ of (4.5.26) and $p(t, x)$ satisfy the same Black-ScholesMerton partial differential equation (4.5.14) satisfied by $c(t, x)$.

     (i) Use formulas (4.5.23)-(4.5.25), (4.5.26), and (4.5.29) to determine the delta $p_x(t, x)$, the gamma $p_{x x}(t, x)$, and the theta $p_t(t, x)$ of a European put.
(ii) Show that an agent hedging a short position in the put should have a short position in the underlying stock and a long position in the money market account.
(iii) Show that $f(t, x)$ of (4.5.26) and $p(t, x)$ satisfy the same Black-ScholesMerton partial differential equation (4.5.14) satisfied by $c(t, x)$.

Stochastic Calculus for Finance II : Continuous-Time Models

Steven E. Shreve 1st Edition

Chapter 4, Problem 12

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Step 1

The formulas for these Greeks are derived from the partial derivatives of the put option price with respect to the underlying asset price and time. Assuming the Black-Scholes model, the price of a European put option $p(t, x)$ is given by: \[ p(t, x) = Show more…

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(i) Use formulas (4.5.23)-(4.5.25), (4.5.26), and (4.5.29) to determine the delta $p_x(t, x)$, the gamma $p_{x x}(t, x)$, and the theta $p_t(t, x)$ of a European put. (ii) Show that an agent hedging a short position in the put should have a short position in the underlying stock and a long position in the money market account. (iii) Show that $f(t, x)$ of (4.5.26) and $p(t, x)$ satisfy the same Black-ScholesMerton partial differential equation (4.5.14) satisfied by $c(t, x)$.

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