Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2021 / 3

For a long position in the claim, hold $-V_S$ shares and bank value $S{V}_S-V$ so that claim plus hedge has zero initial value. The Black-Scholes equation and Itô's formula show that the total position is self-financing and its Brownian and drift terms cancel. Equivalently, the positive replicating portfolio holds $V_S$ shares and bond value $V-S{V}_S$.

Under the usual growth, integrability and admissibility conditions, no-arbitrage forces the claim price to be $V$: a different price could be traded against the replicating portfolio. The qualification matters because an arbitrary formal PDE solution without a suitable financial class need not be a valid price.

Let ${\tau }_B=\inf \{u\ge t:S_u\le B\}$. While the contract is alive, the down-and-out put pays $(K-S_T{)}^{+}{1}_{\{{\tau }_B>T\}}$ at maturity; a prior touch has already knocked it out.

The truncated terminal payoff can be represented by the ordinary European-price combinationAt maturity its three regions areIndeed it is zero below $B$, equals $K-S$ between $B$ and $K$, and is zero above $K$. This supplies the terminal building block for reflection.

The zero-below-barrier property is a property of the terminal payoff. Before maturity, $F(S,t)$ need not vanish for $S<B$, because the ordinary European contracts can still finish above $B$. The killed barrier option price, rather than this unrestricted European combination, is zero after a touch.

Put $q=1-2r/{\sigma }^2$. The Black-Scholes reflection symmetry maps $F$ to $(S/B{)}^{q}F(B^2/S,t)$, which solves the same PDE. Their difference vanishes at $S=B$. For $S>B$ at maturity the reflected argument is below $B$, where the terminal function in part ii is zero. Thus the boundary and terminal data of the alive down-and-out put are satisfied:The killed-domain Feynman-Kac formula identifies this solution with the discounted payoff conditional on survival. For $S\le B$, or after any prior barrier hit, the actual price is zero. The reflected difference is a formula on the alive domain $S>B$; extending that algebraic expression to $S<B$ generally gives a nonzero, possibly negative function and is not a valid barrier-option price.

A down-and-out digital call with strike at the barrier pays one exactly when the barrier is avoided until expiry. Reflection therefore gives its alive priceThe complementary contract pays one at expiry when the barrier has been hit. Its price isThis is a cash-at-expiry one-touch option, not cash paid at the first touch. After the barrier has already been hit, its price is $e^{-r\tau }$ regardless of the current spot. In particular this branch applies immediately for $S\le B$. Current spot alone does not encode the full path history.

If the barrier is reached, path continuity gives $S_{{\tau }_B}=B$ and the early payment is $K-B$. If it is not reached, the terminal payment is exactly the down-and-out put payoff. When $r=0$, moving a fixed payment from the hitting time to expiry does not change its value. Hence the barrier-triggered put price isHere $G,H$ are evaluated with zero interest rate. If the barrier has already been hit, the exploding put has been exercised and has no remaining value after its payment; the displayed replication concerns an initially alive contract and its cash flows.

The barrier-triggered put exercises at the particular stopping time ${\tau }_B\wedge T$. The American put option optimizes over all allowed exercise times, soIt can be strictly smaller, but can never exceed the American value. Equality is possible if immediate exercise is already triggered and optimal; a fixed barrier is not generally the optimal finite-maturity exercise rule.

The comparison with the European put option can go in either direction. By the early exercise premium for a fixed barrier,Thus an early payment gains interest but sacrifices remaining put time value; the expression inside braces need not have a fixed sign for arbitrary barriers.

For a concrete general construction take current spot $S\ge K$, $r>0$, $\sigma >0$, and $t<T$. Set $\gamma =2r/{\sigma }^2$ and $b_{\infty }=K\gamma /(1+\gamma )$, the perpetual put option exercise boundary. For $0<B<b_{\infty }$, the perpetual value at $B$ equals $K-B$ and strictly dominates the finite-maturity European value. The premium formula and positive probability of hitting $B$ therefore give $E_B>P$. Strictness follows from the perpetual verification argument: before a fixed positive expiry, its discounted value loses drift $rK$ whenever the stock is below $b_{\infty }$, with positive expected occupation time when started at $B<b_{\infty }$.

Conversely, for $B<K$ sufficiently close to $K$, every possible payment of the exploding put is at most $K-B$, whether paid at touch or on surviving to expiry. Hence $0\le E_B\le K-B\to 0$, while $P(S,t;K)>0$. This gives $E_B<P$ for such barriers, and also strict inequality below the American value.

These examples establish both possibilities in general, not a claim that every fixed initial state admits both. If $S<b_{\infty }$ and only alive barriers $B<S$ are allowed, all those barriers are in the same perpetual stopping region and the premium comparison is nonnegative.