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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 ...

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The call for the stock that can jump downward will be more valuable due to put-call parity. Suppose you have two stocks, both with a price of $100 and the same diffusive volatility. Stock A does not jump, whereas stock B can at some random time jump (for example) to zero. Clearly a put on stock B will be worth more, but the call must therefore also be worth ... 5 Let$\{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*} andN_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_{\... 4 The Feller condition applies without modification. That is under the assumption thatv$is square-root process with poisson-arrival jumps (as you wrote), and assuming the jump distribution is strictly positive and initial level$v_0>0$. The reason is, conditional on no jumps occuring, the process is just a square root process, for which the references ... 4 Let us define the auxiliary process$\Lambda_t=e^{\kappa t}\lambda_t$. Note that: $$\Lambda_t = \kappa e^{\kappa t} \int_0^t(\rho_s-\lambda_s)ds+\delta e^{\kappa t}\int_0^tdN_t$$ Hence after a jump occurs at$t: $$\Lambda_t=\Lambda_{t-}+\delta e^{\kappa t}$$ Therefore by Ito's lemma for jump-diffusion processes: \begin{align} d\Lambda_t & = \... 4 There's a lot left unspecified in this question, since it is stated without precision, but the effective idea of the answer given here is that those jumps introduce extra variation into the forward distribution of the underlying. And such variation is the bread-and-butter of option value. That said, the ambiguity in the question leaves room for other ... 3 Assume that the value of the sample path of the geometric Brownian motion equals 10 at time t_0 and equals 100 at time t_0 + \Delta t. For the value to change from 10 to 100, the path should necessarily go over all the values between 10 and 100 (possibly with fluctuations) during the intermediate time \Delta t; it cannot jump directly from ... 3 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 ... 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() generates ... 2 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 2 Try $$\mathbb{E}\frac{1}{T} \int_0^T V_t dt = \frac{1}{T} \int_0^T \mathbb{E} V_t dt$$ and use $$\frac{1}{dt}\mathbb{E} V_t = \kappa\theta - \kappa \mathbb{E}V_t + (\lambda_0 + \lambda_1\mathbb{E} V_t)\mu_V,$$ which is in fact a simple ODE. 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 You can check out those discussions in Merton paper when introducing jumps "Option pricing when underlying stock returns are discontinuous". In the very last part he discusses the influence of considering jumps compared to the usual Black Sholes model. From what i remember it'sall about considering your option is ATM or not , that will usually make ... 2 Could it be that your problem is only due to thet^-$notation convention? Think of it that way, it is only worth distinguishing$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\... 1 Set$\tilde{\lambda}_t = e^{\kappa t} \lambda_t$and solve for$\tilde{\lambda}_t$1 Alternatively, let$\{\tau_i\}_{i=1}^{\infty}$be the jump time of the Poisson process$N. Moreover, let \begin{align*} X_t = \int_0^t (J_s-1)dN_s. \end{align*} Here, we assume that the jump sizesJ(\tau_i)$, for$i=1, \ldots, \infty, are independent identically distributed. Then, \begin{align*} X_t &= \sum_{0<s\le t}(J_s-1)1_{\Delta N(s) >0}.... 1 The closest contract to this is gap risk which does trade, either as OTC swap (client looking for a hedge) or embedded inside a structured note (bank looking to recycle risk). Basic starting point is daily close-to- close observations against a 80-90% putspread, cancel upon payment, equity payer receives a spread for being short the risk (major equity ... 1 Y in the CGMY model is not defined for negative integer values due to divergence of the gamma function at those values, and implicitly the characteristic function. However, in the case of negative non-integerx$we extend the gamma function in the sense that whenever$x \in (-\infty,0) \setminus \mathbb{Z_{-}}$, we define the value of$\Gamma(x)$via the ... 1 Defining$\tilde{S}_n = S_n/S_{n-1}$(which is well defined, assuming$S_n > 0$for all$n$), the problem becomes that of barrier option pricing. In particular, you're looking to price a down-and-out barrier option. I wrote my dissertation on barrier options a couple of years ago. You might be able to find some inspiration there. You can find it on ... 1 I don't think you can correlate discrete processes in the traditional sense. Instead, I would make the two Poisson intensities time-varying through which a degree of "jump similarity" can be injected Say each jump intensity is a positive mean reverting process, such as an exponential OU, where the increments are jointly distributed (I.e. with correlation) ... 1 TLDR: The jump frequency depends on how you specify the jump size distribution. If you want the$\lambda$to actually represent the jump frequency under a certain jump-diffusion model, then you should jointly estimate all model parameters, e.g. using maximum likelihood estimation (MLE) or generalized method of moments (GMM). Example: Consider a general ... 1 You should take a look at the BENCHOP project. There we benchmarked around 15 different numerical methods against 6 option pricing problems. One of the problems was the Merton model. The methods were split into 4 families: Monte Carlo, Fourier, Finite Difference, and Radial Basis Function methods. This is the paper containing the results: http://dx.doi.... 1 There is nothing to model in the payoff. A payoff is a collection of cash flows. A cash flow is a function of market observables. Your function just happens to be discontinuous. From a risk point of view this means that you are exposed to the volatility skew. So any model used for valuation should be calibrated to the volatility smile (you cannot simply ... 1 Jumps are totally different from volatility. Imagine a stock whose price has jumps but has no volatility. The asset pricing implications for options on that stock are totally different than from a stock with volatility. Below I simulated 3 stock paths: (i) Jumps and volatility, (2) Only Jumps and (3) No jumps but higher volatility. As you can imagine the ... 1 Let's start with the main idea, I hope you can finish the computations yourself. Whenever you want to derive a pricing equation, try the following approach: discounted value of portfolio/option/derivative must be a martingale for non-arbitrage reasons. Since you have a Markovian dynamics in variables$t$and$S$, you assume that the price is some function$V(...

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One of these two books may help you: A Guide to Quantitative Finance Option Pricing via Quadrature They are both from the same author. Both price European vanilla options under various stochastic processes.

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