Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 28J (Stochastic Financial Models)

For positive spot, strike, volatility and maturity, the Black-Scholes formula iswhere $d_1=[\log (S_0/K)+(r+{\sigma }^2/2)T]/(\sigma \sqrt{T})$ and $d_2=d_1-\sigma \sqrt{T}$. Degenerate zero-volatility or zero-maturity cases are obtained by limits.

At zero interest, interchanging spot and strike gives $d_1(K,S_0)=-d_2(S_0,K)$ and $d_2(K,S_0)=-d_1(S_0,K)$. Substitution into the put formula therefore givesScaling both positive arguments by $\alpha$ leaves $\log (S_0/K)$ and both $d$ values unchanged, while multiplying every price prefactor by $\alpha$. ThusThese are Put-call symmetry in the Black-Scholes model and degree-one homogeneity.

Assume a positive barrier $B\le K<S_0$. Under the zero-rate risk-neutral Black-Scholes model, $X_t=\log (S_t/B)$ starts at $x=\log (S_0/B)>0$ and is Brownian motion with variance rate ${\sigma }^2$ and drift $\mu =-{\sigma }^2/2$. The drifted transition density killed at zero isTo see the factor, multiply the zero-drift reflected Gaussian difference by $\exp [\mu (y-x)/{\sigma }^2-{\mu }^{2}t/(2{\sigma }^2)]$; its second term is the ordinary drifted Gaussian from starting point $-x$ times $e^{-2\mu x/{\sigma }^2}$.

Integrate the payoff $(B{e}^y-K{)}_{+}$ against this density. The first term is $C(S_0,K)$. Since $K\ge B$, the payoff is zero for $y\le 0$, so the second integral may be extended to all real $y$ and equals the ordinary call price from reflected spot $B{e}^{-x}=B^2/S_0$. The reflection weight is $e^x=S_0/B$. Thus the down-and-out European call value isThis assumes continuous monitoring and knock-out on first hitting, with no rebate.

At $B=K$, homogeneity givesThe price is therefore $C(S_0,K)-P(S_0,K)$. Zero-rate put-call parity yields

The zero-rate barrier-strike call hedge givesHold one share and borrow $80$. Stop and liquidate when the price first hits $80$: continuity makes the portfolio value exactly zero then, so it meets the knocked-out payoff. If no hit occurs before maturity, the stock remains above $80$ and the portfolio pays $S_T-80$, exactly the call payoff. There is no dependence on the stated volatility or maturity in this special contract.

The hedge is pathwise, so it works for other continuously evolving stock models, including stochastic or time-dependent volatility, as long as the interest rate and dividends are zero and trading/monitoring are continuous and frictionless. It does not require lognormal terminal prices.

A downward jump can overshoot the barrier: liquidation then leaves $S_{\tau }-K<0$ instead of the required zero, creating gap risk. Discrete monitoring or delayed execution can likewise destroy the stopping replication. Nonzero interest, dividends, transaction costs, funding differences and trading constraints require a changed portfolio or pricing model. Thus continuous barrier hitting and zero carry are the key assumptions, not constant volatility.