Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 211 / 2

A T-forward measure is a probability measure $Q^T$, equivalent to the physical measure, under which prices expressed in units of the positive maturity-$T$ bond are martingales. Equivalently, every attainable payoff $X_T$ has time-$t$ priceThe zero-coupon bond is the numéraire.

The forward contract initiated at $t$ has payoff $S_T-F_t^T$ and zero value. Pricing under the T-forward measure givesBecause $F_t^T$ is ${\mathcal{F}}_t$-measurable and $B_t^T>0$,The tower property of conditional expectation therefore makes $(F_t^T{)}_{t<T}$ a $Q^T$-martingale.

If $K_1\le K_2$, then pointwiseThe positive pricing formula under the T-forward measure givesTherefore $C_t^{T,K_1}\ge C_t^{T,K_2}$, so the call price is non-increasing in strike.

Define the piecewise-linear functionOn $[K_j,K_{j+1})$, its slope is $g^{\prime }(K_j)$. Since $g$ is convex, $g^{\prime }$ is nondecreasing and hence $h^{\prime }(s)\le g^{\prime }(s)$ wherever the derivatives exist. As $h(0)=g(0)$, integration gives $h(s)\le g(s)$ for every $s\ge 0$.

Positive no-arbitrage pricing, the forward identity ${\mathbb{E}}_{Q^T}[S_T\mid {\mathcal{F}}_t]=F_t^T$, and the call-price formula now give