Change of Numéraire and a Jump-Diffusion Option Pricing Formula

Abstract

We note that a general change in numéraire formula in Geman, El Karoui and Rochet (1995) also applies to the case where the stock price and the bond price dynamics are driven by semimartingales with discontinuities. It is observed in the markets that both stock and bond prices generally exhibit discontinuities. We thus consider the case where both the stock price and bond price processes contain a compound Poisson component, in addition to a continuous log-normally distributed component, and we extend the option pricing formula in Geman et al. to this situation.

Appendix

In this appendix, we state a result involving the distribution of the arrival times $(T_1, T_2,\dots , T_n)$ of a non-homogeneous Poisson process $N_T$ conditioned on $n$ arrivals $N_T = n$ over the interval $(0, T]$.

Lemma A.1

Consider a non-homogeneous Poisson process $N_t$ with intensity $\lambda _t$ under some probability measure $\mathbb {P}$. Let the time of the $i$th arrival be $T_i = t_i$ where $0 < t_1 < t_2 < \cdots < t_n \le T$. Conditioned on $N_T = n$, the joint density of the times of the arrivals is

$$\begin{aligned} g_{{\mathbb P}} (t_1,\cdots ,t_n) = \left\{ \begin{array}{ll} \frac{n! \prod _{i=1}^n \lambda _{t_i}}{\left[ \int _0^T \lambda _t \mathrm{{d}}t\right] ^n} &{} {f\!or} \; 0 < t_1 < t_2 < \cdots < t_n \le T,\\ 0 &{} { otherwise.}\end{array} \right. \end{aligned}$$

(A.1)

The proof can be found in standard texts on point processes, e.g. Daley and Vere-Jones (1988). In the context of Theorem 3 in this paper, for $0 < t_1 < t_2 < \cdots < t_n \le T$, the joint density $g_{\mathbb Q_S} (t_1,\ldots ,t_n) = n!\frac{ \prod _{i=1}^n \hat{\lambda }_{t_i}}{\left[ \int _0^T \hat{\lambda }_t \mathrm{{d}}t\right] ^n}$ and $g_{\mathbb Q_P} (t_1,\ldots ,t_n) = n!\frac{\prod _{i=1}^n \tilde{\lambda }_{t_i}}{\left[ \int _0^T \tilde{\lambda }_t \mathrm{{d}}t\right] ^n}$.