Find the parameter $d$ of the Affine Option Pricing Model in Duffie, Pan and Singleton (2000)

Question

According to Duffie, Pan and Singleton (2000) for any real number $y$ and any $a$ and $b \in \mathbb{R}^n$, the price of a security that pays $\exp(aX_t)$ at time $T$ in the event that $bX_t \leq y$ is given by:

$$G_{a,b}(y;X_{0},T,\chi)=\frac{\psi^{\chi}(a, X_{0}, 0, T)}{2}-\frac{1}{\pi} \int_{0}^{\infty} \frac{\operatorname{Im}\left [ \psi^{\chi} (a + \operatorname{i}vb,X_{0}, 0, T)e^{-ivy} \right ]}{v}dv$$

Where:

• $X$ is a $n$-dimensional Affine Jump Diffusion process;
• $\psi^{\chi}(\cdot)$ is the characteristic function of $X_T$ conditional on $X_t=x$;
• $\operatorname{Im}(\theta)$ determines the imaginary part of $\theta \in \mathbb{C}$;
• $v \in \mathbb{R}$

As an example they show that the price $p$ at date $0$ of a call option with payoff $[\exp(dX_T)-c]^{+}$ at date $T$, for given $d \in \mathbb{R}$ and strike $c$ is given by:

$$p=G_{d,-d}(-\ln c)-cG_{0,-d}(-\ln c)$$

My question is: how can one identify the parameter $d$ in practice?

For example: suppose we are in a Black-Scholes world where the interest rate $r$ is set to zero. This means that:

$$\ln S_{T} \sim N \left ( \ln S_t - \frac{1}{2}\sigma^2(T-t), \sigma^2(T-t) \right )$$

If we define the price of the underlying of a call option as $S_t = \exp(dX_t)$ and we know $\ln S_t, K, \sigma$ and $T-t$, is it possible to recover the value of $d$ to price the call option applying the model specified above?

Answer 1

$d$ is a vector that collapses the $n$-dimensional vector into a real number. In the BS case $d=1$. There is nothing to be estimated. Also not that in practice affine pricing is done through FFT (and variants) rather than the direct transform you quote.