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Jump-Diffusion Model

Consider an up-and-in call option and assume that the price of the underlying asset follows a jump-diffusion model, then under certain conditions the price of an up-and-in call with barrier \(H > S_0 = x\), strike \(K\) and maturity \(T\) is given by the following expectation

\(\mathbb{E}_x^*\left[e^{-rT}(S_T - K)^+ \mathbb{I}_{\{\max_{0\leq t\leq T} S_t \geq H\}}\right]\).

where the superscript (*) denotes the expectation under the risk neutral measure. In general this expectation is difficult to compute analytically but it can be approximated by numerical methods such as Monte Carlo simulations. Specifically, let \(\mathbb{G} \subset [0,\infty)\) be a finite set of grid points. We wish to construct a Markov chain with state space \(\mathbb{G}\) which approximates the price process and evaluate the option by taking samples of the Markov chain.

Approximation method

To approximate the price process with a Markov chain the first step is to calculate the transition rate Q-matrix \(\Lambda\) from the generator of the process. In the case when the price process is modelled by a jump-diffusion process we can approximate the Q-matrix via the decomposition \(\Lambda\) = \(\Lambda_J + \Lambda_D\) where \(\Lambda_J\) and \(\Lambda_D\) corresponds to the jump component and the diffusion component of the process respectively. Once we have the generator we sample the trajectories of the chain as follows:

  1. Sample a waiting time \(s\) from the distribution \(s\sim\exp(-\Lambda(x,x))\) where \(x\) is the current state of the Markov chain.
  2. Sample the next state with \(\mathbb{P}[X_s = y] = -\frac{\Lambda(x,y)}{\Lambda(x,x)}\)

We then repeat this process untill we reach the time of maturity \((T)\). This gives us a sample trajectory of the price process. We sample a large number of trajectories and for each sample we calculate the discounted payoff. The price of the option is then esitimated by the sample mean of the discounted payoffs.

The model

As an example we chose Double-exponential jump diffusion process to model price process. We model the price process of the underlying asset \(S_t\) by the following SDE \(M\)

\(\frac{\textrm{d}S_t}{S_{t^-}}= \left(r-d-\lambda\zeta\left(\frac{S_{t^-}}{S_0}\right)^\beta\right)\textrm{d}t + \left(\frac{S_{t^-}}{S_0}\right)^\beta\textrm{d}\Bigg(\sigma W_t + \sum_{i=1}^{N_t} (e^{K_i}-1)\Bigg),\)

where \(\beta\in\mathbb{R}\), \(r\) is the risk-free interest rate and \(d\) is the dividend yield. Also \(W_t\) is a standard Brownian motion, \(N_t\) is a Poisson process with intensity \(\lambda\) and \(K_1, K_2,\ldots\) are i.i.d. with the double exponential density

\( f_K(k)=p\eta_1e^{-\eta_1k}\mathbb{I}_{\{k\geq 0\}} +(1-p)\eta_2 e^{-\eta_2k}\mathbb{I}_{\{k<0\}},\)

where \(\eta_1>1\), \(\eta_2>0\) and \(p\in[0,1]\) and \(\zeta\) is given by the expectation

\( \zeta = \mathbb{E}[e^{K_1} - 1] = \frac{p\eta_1}{\eta_1 -1} + \frac{(1-p)\eta_2}{\eta_2 +1} - 1. \)

following [1] the jump measure \(\nu\) can be given by

\( \nu(x,\textrm{d}y)= \lambda\left(\frac{x}{S_0}\right)^\beta\Big(p\eta_1(y+1)^{-1-\eta_1}\mathbb{I}_{\{y\geq 0\}}+(1-p)\eta_2(y+1)^{\eta_2-1}\mathbb{I}_{\{y<0\}}\Big)\;\textrm{d}y. \)

Since \(y\) is relative jump size we can write above application as

\Lambda_J(x,x(1+y_i)) & = \lambda\left(\frac{x}{S_0}\right)^\beta\begin{cases} p\left((\alpha_x(y_{i-1})+1)^{-\eta_1}-(\alpha_x(y_i)+1)^{-\eta_1}\right) & 0\leq\alpha_x(y_{i-1}) \\
(1-p)\left((\alpha_x(y_i)+1)^{\eta_2}-(\alpha_x(y_{i-1})+1)^{-\eta_2}\right) & 0\geq\alpha_x(y_{i}) \\
p\left((\alpha_x(y_{i-1})+1)^{-\eta_1}-1\right) + (1-p)\left((\alpha_x(y_i)+1)^{\eta_2}-1\right) & \alpha_x(y_{i-1})<0<\alpha_x(y_i).