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-$t$ price
$$
\pi_t=B_t^T\mathbb E_{Q^T}[X_T\mid\mathcal F_t].
$$
The zero-coupon bond is the <numéraire>.

Solution

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

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

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

If $K_1\leq K_2$, then pointwise
$$
(S_T-K_1)^+\geq(S_T-K_2)^+.
$$
The positive pricing formula under the <T-forward measure> gives
$$
C_t^{T,K}=B_t^T\mathbb E_{Q^T}[(S_T-K)^+\mid\mathcal F_t].
$$
Therefore $C_t^{T,K_1}\geq C_t^{T,K_2}$, so the call price is non-increasing in strike.

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

{c}

For a maturity $T$, the T-forward measure uses the zero-coupon bond maturing at $T$ as <numéraire>. If $B_t^T$ is its time-$t$ price, an attainable payoff $X_T$ has value $B_t^T\mathbb E_{Q^T}[X_T\mid\mathcal F_t]$.

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