Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 29K (Stochastic Financial Models)

PutThe call payoff is $(S_T-K{)}^{+}$, and it is positive whenWriting $\varphi$ for the standard normal density, the given risk-neutral pricing formula becomesCompleting the square givesso the two tail integrals are $\Phi (d_{+})$ and $\Phi (d_{-})$, where $\Phi$ is the standard normal distribution function. Therefore the Black-Scholes formula is

The payoff identityreplicates the call-minus-put position by one stock and borrowing the present value of $K$. Hence put-call parity givesand therefore

For $0\le t<T$, set $\tau =T-t$ andUnder the equivalent martingale measure, conditional log-normality givesThe risk-neutral pricing value of the digital call option is consequentlyAt $t=T$ this converges to the stated payoff away from $S_T=K$, with the payoff convention specifying the boundary value.

The delta hedge holds the derivative of the claim value with respect to the current stock price. Sincethe number of risky-asset units for $t<T$ isThis is the Black-Scholes digital option formula. The hedge becomes singular close to maturity near the strike, reflecting the discontinuity of the payoff.

The digital call option and digital put option indicators partition the possible terminal stock prices:Thus the digital put-call parity isAt time zero, part b gives $D_{\mathrm{call}}(0,S_0)=e^{-rT}\Phi (d_{-})$, so