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

Assume the innovations are independent of the previous history and have finite exponential moments at every real argument, as required by the finite-valued affine definition. Conditioning on $X_{t-1}=x$ givesHence the autoregressive process of order one is an affine process, withHere and in the remaining parts, an affine process uses a time-homogeneous Markov process: the one-step transform is the same at every time. The displayed definition at time $1$ alone would not determine later transitions of an arbitrary time-inhomogeneous Markov process.

Use the time-homogeneous Markov property and define backward effective parameters byConditioning the last exponential factor on ${\mathcal{F}}_{t-1}$ replaces ${\theta }_t{X}_t$ by $A(q_t)X_{t-1}+B(q_t)$. Repeating the tower property of conditional expectation combines this with the preceding exponent, then with each earlier exponent. The last remaining conditional transform is at time $1$, soThese are finite because the one-step affine process transforms are finite at every real parameter. The argument establishes the entire joint affine transform, including when the coefficients ${\theta }_j$ have different signs.

Iterating the autoregressive process of order one givesFor $q_r={\sum }_{j=r}^t{\theta }_j{a}^{j-r}$, regrouping the sum of the exponents givesThe independent innovations factorize the moment-generating function. ConsequentlyFinite sums avoid a division by $a-1$, so these formulas include $a=0$ and $a=1$; zeroth powers in these sums are $1$.

A unit-face-value zero-coupon bond pays $1$ at maturity. The martingale deflator pricing identity isUse 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 isIndeed, conditioning the first future step in an $(n+1)$-step horizon transforms $e^{(1+\alpha (n))X_{t+1}+\beta (n)}$ into $e^{A(1+\alpha (n))X_t+B(1+\alpha (n))+\beta (n)}$. ThereforeThis uses the true martingale deflator pricing identity; a merely local martingale deflator would not by itself justify replacing prices by conditional terminal expectations.