Jump diffusion and incomplete hedging

Definition Under a chosen pricing measure $\mathbb{Q}$, let $W^{\mathbb{Q}}$ be a Brownian motion, $N$ a Poisson process with constant intensity $\lambda \ge 0$, and $J_1, J_2, \dots$ independent, identically distributed marks with $J_k > 0$ and $\mathbb{E}^{\mathbb{Q}}[J_k] < \infty$, where $W^{\mathbb{Q}}$, $N$ and the marks are independent. Let $r$, $q$ and $\sigma \ge 0$ be constants and write $\kappa_J = \mathbb{E}^{\mathbb{Q}}[J - 1]$ for the mean proportional jump. The jump diffusion with proportional jumps is

Between jumps the price follows a geometric Brownian motion with drift $r - q - \lambda\kappa_J$, and at the $k$th jump time $T_k$ it is multiplied by the $k$th mark, $S_{T_k} = S_{T_k-} J_k$. The left limit $S_{t-}$ appears because the size of a jump is proportional to the price just before it. The solution is

with the empty product equal to $1$ when $N_t = 0$. It is positive because every mark is positive. A mark $J \le 0$ lies outside this multiplicative model.

Definition A Poisson process with intensity $\lambda \ge 0$ is a counting process $N$ with $N_0 = 0$, right-continuous paths that increase only by jumps of size one, and independent increments with $N_t - N_s \sim \operatorname{Poisson}(\lambda(t - s))$ for $s < t$, that is

for $n = 0, 1, 2, \dots$. The jump times are the times at which $N$ increases. Every bounded interval contains finitely many of them almost surely, which is what finite activity means. The probability generating function is

for every real $z$. Because the marks are independent of $N$, conditioning on $N_t$ gives

Definition For a Poisson process with intensity $\lambda$, the compensated process

is a martingale for any filtration to which $N$ is adapted and whose past is independent of the future increments of $N$, because $\mathbb{E}^{\mathbb{Q}}[N_t - N_s \mid \mathcal{F}_s] = \lambda(t - s)$. Its variance is $\operatorname{Var}(\widetilde{N}_t) = \lambda t$. For marks with $\mathbb{E}^{\mathbb{Q}}\lvert J - 1 \rvert < \infty$, the compensated jump sum

is a martingale too, since the sum has independent increments and mean $\lambda\kappa_J t$. The compensator $\lambda\kappa_J t$ is a deterministic drift that removes the mean of the jump sum. It does not replace the realized marks by their mean, and the jump times and the law of the marks stay as they were. In the language of random measures, the compensator of the jumps is $\lambda \, dt \, F(dy)$, where $F$ is the law of a mark.

Definition Drift compensation is the choice of the drift between jumps that makes the discounted stock with reinvested dividends, $e^{-(r - q)t} S_t$, a $\mathbb{Q}$-martingale. In the jump diffusion that drift is $r - q - \lambda\kappa_J$. From the explicit solution, with $\tau = T - t$,

because the Brownian factor $e^{\sigma(W^{\mathbb{Q}}_T - W^{\mathbb{Q}}_t) - \sigma^2\tau/2}$ has mean one and the product of the marks over $(t, T]$ has mean $e^{\lambda\kappa_J\tau}$. Written with the compensated jump sum $M$, the stock equation is

The condition fixes only the product $\lambda\kappa_J$, not $\lambda$ and the law of $J$ separately. Suppose the physical model has the same form with drift $\mu^{\mathbb{P}}$ between jumps, $\sigma > 0$, intensity $\lambda^{\mathbb{P}} > 0$ and the same law of the marks. Then every intensity $\lambda^{\mathbb{Q}} > 0$ gives an equivalent martingale measure, with density

where $W^{\mathbb{Q}}_t = W^{\mathbb{P}}_t + \eta t$ is the $\mathbb{Q}$-Brownian motion and $\eta = (\mu^{\mathbb{P}} - r + q + \lambda^{\mathbb{Q}}\kappa_J)/\sigma$. This construction is a standard Girsanov result for jump diffusions, stated here without proof. The choice $\lambda^{\mathbb{Q}} = 0$ is excluded, since a measure under which no jump occurs is not equivalent to one under which jumps have positive probability.

Definition For a twice continuously differentiable function $f$ on $(0, \infty)$ for which the expectation below is finite, the generator of the jump diffusion under $\mathbb{Q}$ is

Expanding the last term gives the equivalent form $(r - q - \lambda\kappa_J) s f' + \tfrac12\sigma^2 s^2 f'' + \lambda\,\mathbb{E}^{\mathbb{Q}}[f(sJ) - f(s)]$ with the raw drift between jumps. The generator comes from Itô’s formula for jump processes, in which $f(S_t) - f(S_0)$ is the sum of the usual diffusion terms and of the jump changes $f(S_{T_k-} J_k) - f(S_{T_k-})$, and the jump changes have the compensator $\lambda \int_0^t \mathbb{E}^{\mathbb{Q}}[f(S_{u-} J) - f(S_{u-})] \, du$, where the expectation is over $J$ with $S_{u-}$ held fixed. The jump term evaluates $f$ at the point $sJ$, away from $s$, which is what nonlocal means. For a European claim with payoff $g(S_T)$ the pricing equation is the partial integro-differential equation

with $\mathcal{L}$ acting on the variable $s$. If it has a solution that is continuously differentiable in $t$ and twice in $s$ with suitable growth, and the payoff and the marks are integrable enough, then $V(t, s) = \mathbb{E}^{\mathbb{Q}}[e^{-r(T - t)} g(S_T) \mid S_t = s]$, the Feynman-Kac representation for jump diffusions, stated here without proof. The price depends on the chosen $\mathbb{Q}$ through $\lambda$ and the law of $J$.

Definition Let $V$ be an option value at a fixed time, differentiable at the current price $s$, and hold one option together with $-V'(s)$ shares, the delta hedge chosen just before a jump. If the price jumps from $s$ to $sJ$, the jump residual is the change in value of this hedged position,

It is the exact change, not an approximation. If $V$ is twice continuously differentiable near $s$, Taylor’s theorem gives $R_J(s) \approx \tfrac12 V''(s) s^2 (J - 1)^2$ for small jumps, and for a convex $V$ the residual is nonnegative for every $J$. The residual vanishes for every mark in the support of the jump law only if all the jumped values $V(sJ)$ lie on the tangent line of $V$ at $s$. With $\sigma > 0$ the hedge ratio is already pinned to $V'(s)$ by the Brownian term, so unless every jumped value lies on that tangent line, the stock and the money-market account cannot hedge both risks, and the market made of these two assets is incomplete. Further traded options can span some of the jump risk, which changes the market. A degenerate model can differ. With $\sigma = 0$ and a single jump factor, the chord slope $(V(sJ) - V(s))/(s(J - 1))$ removes the jump risk exactly.