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10

Let $$dS_t = \mu S_t dt + \sigma S_t dW_t + S_{t^-} dJ_t$$ where $$J_t = \sum_{j=1}^{N_t} (V_j - 1)$$ is a compound Poisson process, with $V_j$ i.i.d. jump sizes (positive random variables) whose statistical properties are not relevant for what needs to be proven and $N_t$ a standard Poisson process of intensity $\lambda$. The processes $W_t$, $N_t$ and ...

4

Let $\{P_t \mid t \geq 0\}$ be a compound Poisson process, where \begin{align*} P_t = \sum_{i=1}^{N_t} (V_i -1), \end{align*} and $N_t$ is a Poisson process with intensity $\lambda$ and jump times $\tau_i$, $i = 1, \ldots, \infty$. Let $Y_i=\ln V_i$ and $f(x)$ be the density function. Then \begin{align*} P_t - \lambda t E(V_1) &= P_t - \lambda t \int_{\...

3

The error is, you are not storing the random numbers for the same path at the end: xbefore = x + c*tau + sigma*sqrt(tau)*randn() A = muA + sigmaA*randn(); xafter = xbefore + A; But then at end you set a different path here by creating a new random number: xT = log(S0)+(c+muA*lambda)*T+sqrt((sigma^2+(muA^2+sigmaA^2)*lambda)*T)*randn(); randn() ...

2

The Feller condition applies without modification. That is under the assumption that $v$ is square-root process with poisson-arrival jumps (as you wrote), and assuming the jump distribution is strictly positive and initial level $v_0>0$. The reason is, conditional on no jumps occuring, the process is just a square root process, for which the references ...

2

Diffusion brings about a standard deviation which increases with the square root of time (just like in Brownian motion), while jumps add variability proportional to time (since the jump times are a Poisson process). So they are quite different. Experience shows that sharp stock market moves do occur (in connection with big news events for example), so ...

2

The problem is that what some mean when they say "volatility" is BS implied vol from an option price. What some others mean when they say "volatility" is some diffusion parameter from a drift diffusion model (with or without jumps). These are the same value in the log normal model of stock prices but different for many other models including those with jumps....

2

Could it be that your problem is only due to the $t^-$ notation convention? Think of it that way, it is only worth distinguishing $S_{t^-}$ from $S_t$ at a jump time. Elsewhere, knowing that Brownian motion paths are continuous, you'll always have $S_t = S_{t^-}$. Thus you could also write the SDE: $$\frac {dS_t}{S_{t^-}} = \alpha dt+\sigma dW_t+ d\... 2 Use Ito for jumps$$ dS_t = \frac{\partial S_t}{\partial t} dt + \frac{\partial S_t}{\partial W_t}dW_t + \frac{1}{2}\frac{\partial^2 S_t}{\partial W_t^2} dt + \frac{\partial S_t}{\partial N_t}d N_t $$The first part is pretty straight forward$$ \frac{\partial S_t}{\partial t} dt = S_t(\mu - \frac{1}{2}\sigma^2)  \frac{\partial S_t}{\partial W_t}dW_t ...

1

There is nothing to model in the payoff. A payoff is a collection of cash flows. A cash flow is a function of market observables. Your function just happens to be discontinuous. From a risk point of view this means that you are exposed to the volatility skew. So any model used for valuation should be calibrated to the volatility smile (you cannot simply ...

1

Jumps are totally different from volatility. Imagine a stock whose price has jumps but has no volatility. The asset pricing implications for options on that stock are totally different than from a stock with volatility. Below I simulated 3 stock paths: (i) Jumps and volatility, (2) Only Jumps and (3) No jumps but higher volatility. As you can imagine the ...

1

Let's start with the main idea, I hope you can finish the computations yourself. Whenever you want to derive a pricing equation, try the following approach: discounted value of portfolio/option/derivative must be a martingale for non-arbitrage reasons. Since you have a Markovian dynamics in variables $t$ and $S$, you assume that the price is some function \$V(...

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Do these work for you? P34 of http://web.mit.edu/junpan/www/SVJ.pdf P1360 of http://www.darrellduffie.com/uploads/pubs/DuffiePanSingleton2000.pdf P2045 of http://www.math.ku.dk/~rolf/bakshi.pdf

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Look at Gilli & Schumann's paper. They provide a Bates' model estimates set, the way to improve such estimates calibrating those ones using an Heuristic model and, lastly, the relative codes in matlab, in order to be able to replicate the model. Unfortunately, there are not available the relative call prices estimated time series; I think that noone ...

1

What you do is: You simulate a Brownian path - with the correct standard deviation. Then you simulate the Compound Poisson process. In each time step you sample a jump or no jump and the jump size if there was one. In each time step you draw from a Bernoulli distribution - which is to my knowledge just an approximation. For the compound Poisson process I ...

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