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Assume a constant risk-free rate $r$ and no dividends. Generalisation is straightforward. To preclude arbitrage opportunities, under the risk-neutral measure $\Bbb{Q}$, the discounted asset price process should be a $\Bbb{Q}$-martingale i.e. $$S_0 = \Bbb{E}^\Bbb{Q}_0 \left[ e^{-rt} S_t \right] \iff \Bbb{E}^\Bbb{Q}_0 \left[ S_t \right] = S_0 \exp(rt) \... 8 Given for all i the mean of \epsilon_i is k and that the \{\epsilon_i\}_i are i.i.d., we have^{\text{(1)}}:$$\begin{align} E\left[\prod_{i=1}^{N_t}(1+\epsilon_i)\right] &=E\left[E\left[\prod_{i=1}^{N_t}(1+\epsilon_i)|N_t\right]\right] \\[6pt] &=E\left[\prod_{i=1}^{N_t}E\left[(1+\epsilon_i)|N_t\right]\right] \\[6pt] &=E\left[\prod_{i=1}...

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We assume that the process $\{J_t, \, t\ge 0\}$ is defined at the jump times of the Poisson process $\{N_t, \, t \ge 0\}$, and all the jump sizes are independent and identically distributed. That is, \begin{align*} Q_t \equiv \int_0^t (J_t-1) dN_t = \sum_{n=1}^{N_t} (J_i-1), \end{align*} where $J_i$, for $i=1, \ldots, \infty$, are independent and $\xi_i = \... 6 Intuitively, cadlag expresses the fact that we know a jump has occurred after the fact, but we never have advance knowledge that the jump is about to occur (i.e no knowledge of the starting point for the jump or that a jump is "under way"). Each jump is a surprise, after which we believe there will be no jumps at least for a little while. I hear it in the ... 6 Answer Assuming the Poisson process$N_t$is independent from the Brownian motions$(W_{1,t},W_{2,t}), you'll have \begin{align} df(X_{1,t},X_{2,t}) &= \frac{\partial f}{\partial X_{1,t}} dX_{1,t}^c + \frac{\partial f}{\partial X_{2,t}} dX_{2,t}^c + \dots \\ &+ \frac{1}{2} \frac{\partial^2 f}{\partial X_{1,t}^2 } d\langle X_{1,t} \rangle_t^c + \... 5 I'll assume $$J_t = \sum_{i=1}^{N_t} Z_i$$ be a compound Poisson process, with(T_n)_{n\geq 1}$being the jump times for Poisson process$(N_t)_{t\geq 0}$and$(Z_i)_{i\geq 1}$sequence of i.i.d. variables independent of$(N_t)_{t\geq 0}$. For SDE $$dP_t = P_{t^-} dJ_t$$ we notice that at jump times we have $$dP_{T_i} = P_{T_i} - P_{T_i^-} = Z_{i} P_{... 4 There is a whole literature on risk-neutral modeling with Levy processes. Consider an arbitrage-free market where asset prices are modeled by a stochastic process (S_t)_{t \in [0,T]}, \mathcal{F}_t represents the history of the asset S and \hat{S}_t=e^{-rt}S_T the stochastic discounted value of the asset. The discounted expectation of the terminal ... 4 The book Stochastic calculus for finance by Steven Shreve gives a good introduction to stochastic calculus applied to finance. A whole chapter is dedicated to the Itô Integral for example. It covers a large spectrum ranging from probability theory to stochastic financial models. I strongly recommend it! 4 Elementary Stochastic Calculus by Thomas Mikosch is an excellent introduction to the topic in a very compact way. Alternatively, Stochastic Calculus for Finance II: Continuous-Time Models by Steven Shreve is a more comprehensive reference which is very much oriented to applications in finance. 4 Let$$ J_t = \sum_{i=1}^{N_t} Z_i$$be a compound Poisson process, with (T_n)_{n\geq 1} being the jump times for Poisson process (N_t)_{t\geq 0} and (Z_i)_{i\geq 1} sequence of i.i.d. variables independent of (N_t)_{t\geq 0}. We need the stochastic integral against dJ_t in order to make sense of dJ_t. For discrete jump size we have$$\delta J_t = ... 3 Jumps are an attempt to solve a math mistake in Modern Portfolio Theory. In the 19502-70s, economists were working on solving the variance-mean tradeoff. Furthermore, they needed to do so with punchcard computing. That radically restricted the set of computable, potential solutions. Both the normal distribution and the log-normal distribution are ... 3 Try this paper (although it's advanced): https://www.sciencedirect.com/science/article/pii/S0377042702009032 The topic you picked is not necessarily an easy one :) 3 For diffusion models (i.e. no jumps): Local volatility models: match vanilla options market prices; give unrealistic volatility dynamic (smile flattens when we move forward in time); Stochastic volatility models: don't match vanilla options market prices (not enough skew for short dated expiries); give realistic volatility dynamics. You might want to ... 3 Commonly, we employ the Euler scheme for$\Delta\ln(S_t)$, not for$\Delta S_t$. Let us specify the jump part as $$S_{t+}=S_{t}J\Rightarrow dS_t=S_t(J-1)$$ where$J$is a strictly positive random variable. (NB: Under Merton we would have$\ln(J)\sim N(\mu_J,\sigma_J^2)$and$\mathrm{E}(J)=e^{\mu_J+\frac{1}{2}\sigma_J^2}$) And for the solution scheme we ... 3 $$X_t = B_t 1_{t<0.5} + (x+ B_t) 1_{t\geq 0.5} = B_t + x1_{t\geq 0.5}$$ $$[X, X]_t = [B, B]_t + x^2 1_{t\geq 0.5} = t+ x^2 1_{t\geq 0.5}$$ (the author probably intended to use$0.5$as jump size too) 2 Consider “Paul Wilmott Introduces Quantitative Finance” if you look for an enjoyable read, good intuition and a not too academic approach 2 For anyone else searching for good Merton Jump Diffusion examples, found a much better notated reference here: https://www.codearmo.com/python-tutorial/merton-jump-diffusion-model-python 2 Let$P(t, r_t, T)$be the bond price at time$t$, where$0 \leq t \leq T. Then, by Ito's formula, \begin{align*} &\ P(t, r_t, T) \\ =& P(0, r_0, T) + \int_0^t\partial_s P(s, r_s, T) ds + \int_0^t\partial_r P(s, r_{s-}, T) dr_s + \frac{1}{2}\sigma^2 \int_0^t r_s\partial_{rr} P(s, r_s, T)ds\\ & \quad +\sum_{s \leq t}\big[P(s, r_s, T) - P(s, r_{s-}... 2dJ_{t}$can be understood as a Steljes measure , when you want to define jumps using bounded variation function , but you can simply understand it as$J_{t}-J_{t-}Those processes belong to a more general class of process called Levy processes through Lévy–Khintchine representation where you can define clearly the jump part, you can find better expanding ... 2 First, we need to be careful about putting the condition at the right place: \begin{align} e^{kt}\mathbb{E}[\mu_t] -\mu_0 &= \mathbb{E}\bigg[\sum_{m=1}^{N_t} e^{k\tau_m}\eta_m\bigg]\\ &= \mathbb{E}\bigg[\sum_{m=1}^{N_t} \mathbb{E}\bigg[e^{k\tau_m}\eta_m|N_t\bigg]\bigg]\\ &=\mathbb{E}\Big[\eta_1\Big] \cdot \mathbb{E}\bigg[\sum_{m=1}^{N_t} \mathbb{... 2 No arbitrage means that you can't have a portfolio with a positive expectation without risk. let's suppose that the value of option with jumps is lower thanC_{BS}(0,S_0)$Please consider the following portfolio at time 0: Sell BS hedge on option with jumps with price$C_{BS}(0,S_0)$. Buy option with jumps with price$P_J(0,S_0)$. Keep the rest of money$...

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As a complement to @ir7’s comprehensive derivation, in the case of Burgard and Kjaer’s the jump process $J_t$ models the default of the issuer. You specialize the process by setting $Z_1=-1$, while the values of $\{Z_i:i\geq2\}$ are irrelevant. You then notice that as soon as the process jumps once, the product of jump sizes becomes null. We therefore have: $... 2 If you're interested in learning about stochastic calculus outside of the context of quant finance (which I think is a better approach than learning about it solely in the context of finance), check out Stochastic Integration and Differential Equations by Protter. 2 I suppose the expectation could be used to get at some time-weighted average price (TWAP) where we assume each instant of observation has infinitesimal and equal weight$\frac{dt}{T}$:$\bar{S}_T := \frac{1}{T} \int_{t=0}^T S_t dt$. One problem with this is we don't often care that much about an average stock price. When we do care, we often look at: an ... 1 The bellwether Indices for testing, are NASDAQ, Technology sector, S & P 500 Big 500 capital weighted Stocks, Russell 2000, MID sector stocks and some small stocks. It is better to use the data fro their relative, ETF's eg. QQQ, SPY, IWM. The Dow is covered by the Nasdaq and the S&P 500, it is the 20 biggest stocks on the market and is not useful. ... 1 Jumps do not imply fat tails. See the simulation in R. Note that the excess kurtosis of [normal variable + jump] is negative. > set.seed(1) > Normal_Variable <- rnorm(1e8) > kurtosis(Normal_Variable)  -0.000628316 > Jump <- 2 * ((runif(1e8) < 0.5) * 2 - 1) > kurtosis(Normal_Variable + Jump)  -1.280009 1 Actually, I do not think it's true. Jumps, when added to the Black-Scholes (BS) dynamics, do modify the volatility surface. However, the volatility skew may get inverted: the implied BS volatility may be higher when the strike is closer to the current value$S(0)$of the underlying asset$S$. Consider an idealized example:$$\log(S(t+dt) / S(t)) ={\rm[... 1 Most of the work you will find on jump diffusion models will be in derivative pricing or related work on insurance. In essence, they tend to be interesting ways to think about future distributions. If your performance metric is the mean squared error, we can easily show that what you should be trying to estimate is the conditional expected value of 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 Using a pre-defined fixed time grid is not exactly accurate since you don't know when in an interval the jump occurred. The correct way is to first simulate the time points of the jumps and then the values of the process$S(t)_{before}$and$S(t)_{after}\$ by a Brownian increment. The latter then depends on the time between the jumps. This way you can have ...

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