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

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?

$d$ 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.
• Thank you for your answer! Can I ask you how can I see (or show) that $d=1$? I mean, I was trying to find $d$ in different ways starting from the fact that if $S_t = \exp (dX_t)$ then $d = \ln(S_t) / X_t$ and that under Black-Scholes assets dynamics are determined by a GBM, but I was not getting anywhere. Actually if $d=1$ then $\ln S_t = X_t$, but what $X_t$ is under Black-Scholes? A GBM? Jun 6, 2016 at 8:12
• $X_t$, which is the logarithm of the spot price, is a BM. $S_t=\exp (X_t)$ is GBM. BM is affine. Jun 6, 2016 at 20:12