A unit-face-value <zero-coupon bond> pays $1$ at maturity. The <martingale deflator> pricing identity is
$$
P_{t,T}=\frac{\mathbb E[Y_T\mid\mathcal F_t]}{Y_t}=\mathbb E\!\left[\exp\!\left(\sum_{j=t+1}^TX_j\right)\middle|\mathcal F_t\right].
$$
Use the <affine process> <Markov property> with respect to the market <filtration>; if that <filtration> contains extra predictive information, the natural <Markov property> alone would not suffice. Part (b), with every future coefficient equal to $1$, gives the exponential-affine form. More explicitly, the <exponential-affine bond pricing> recursion is
$$
\boxed{\alpha(0)=\beta(0)=0,\quad\alpha(n+1)=A(1+\alpha(n)),\quad\beta(n+1)=\beta(n)+B(1+\alpha(n)).}
$$
Indeed, conditioning the first future step in an $(n+1)$-step horizon transforms $e^{(1+\alpha(n))X_{t+1}+\beta(n)}$ into $e^{A(1+\alpha(n))X_t+B(1+\alpha(n))+\beta(n)}$. Therefore
$$
\boxed{P_{t,T}=\exp\bigl(\alpha(T-t)X_t+\beta(T-t)\bigr).}
$$
This uses the true <martingale deflator> pricing identity; a merely local <martingale deflator> would not by itself justify replacing prices by conditional terminal <expectations>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 211 / 3 / d / Solution

{title2=$P_{t,T}=e^{\alpha(T-t)X_t+\beta(T-t)}$}

If a discrete-time <martingale deflator> obeys $Y_t/Y_{t-1}=e^{X_t}$ for an <affine process> $X$, unit <zero-coupon bonds> have exponential-affine prices. For one-step coefficients $A,B$, the recursion is $\alpha(0)=\beta(0)=0$, $\alpha(n+1)=A(1+\alpha(n))$, $\beta(n+1)=\beta(n)+B(1+\alpha(n))$.

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 Incoming links: Exponential-affine bond pricing