11
votes
Risk neutral measure for jump processes
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 ...
- 14.5k
8
votes
Accepted
Merton's jump diffusion
Given for all $i$ the mean of $\epsilon_i$ is $k$ and that the $\{\epsilon_i\}_i$ are i.i.d., we have$^{\text{(1)}}$:
$$\begin{align}
E\left[\prod_{i=1}^{N_t}(1+\epsilon_i)\right] &=E\left[E\left[...
- 7,999
7
votes
Accepted
Black-Scholes formula for Poisson jumps
We assume that the process $\{J_t, \, t\ge 0\}$ is defined at the jump times of the Poisson process $\{N_t, \, t \ge 0\}$, and all the jump sizes are independent and identically distributed. That is,
\...
- 21k
6
votes
Accepted
Cadlag Property of Jump Proccesses
Intuitively, cadlag expresses the fact that we know a jump has occurred after the fact, but we never have advance knowledge that the jump is about to occur (i.e no knowledge of the starting point for ...
- 10.9k
6
votes
Accepted
exercise on multivariate Ito's lemma + jumps (Poisson)
Answer
Assuming the Poisson process $N_t$ is independent from the Brownian motions $(W_{1,t},W_{2,t})$, you'll have
\begin{align}
df(X_{1,t},X_{2,t}) &= \frac{\partial f}{\partial X_{1,t}} dX_{1,...
- 14.5k
5
votes
Accepted
Solution for a SDE for a Bond found in Bugard & Kjaer
I'll assume
$$ J_t = \sum_{i=1}^{N_t} Z_i$$ be a compound Poisson process, with $(T_n)_{n\geq 1}$ being the jump times for Poisson process $(N_t)_{t\geq 0}$ and $(Z_i)_{i\geq 1}$ sequence of i.i.d. ...
- 5,008
4
votes
Levy process and random measure
There is a whole literature on risk-neutral modeling with Levy processes.
Consider an arbitrage-free market where asset prices are modeled by a stochastic process $(S_t)_{t \in [0,T]}, \mathcal{F}_t$ ...
- 2,906
4
votes
Basic book on stochastic calculus, Itô and jump processes and Brownian Motion
The book Stochastic calculus for finance by Steven Shreve gives a good introduction to stochastic calculus applied to finance. A whole chapter is dedicated to the Itô Integral for example. It covers a ...
- 111
4
votes
Basic book on stochastic calculus, Itô and jump processes and Brownian Motion
Elementary Stochastic Calculus by Thomas Mikosch is an excellent introduction to the topic in a very compact way. Alternatively, Stochastic Calculus for Finance II: Continuous-Time Models by Steven ...
- 7,999
4
votes
SDE Jump-Diffusion
Let
$$ J_t = \sum_{i=1}^{N_t} Z_i$$ be a compound Poisson process, with $(T_n)_{n\geq 1}$ being the jump times for Poisson process $(N_t)_{t\geq 0}$ and $(Z_i)_{i\geq 1}$ sequence of i.i.d. variables ...
- 5,008
4
votes
What is the intuition behind "jumps" causing volatility skew?
Jumps are an attempt to solve a math mistake in Modern Portfolio Theory. In the 19502-70s, economists were working on solving the variance-mean tradeoff. Furthermore, they needed to do so with ...
- 4,359
4
votes
Accepted
Variance of the log returns in jump diffusion with time-varying jump sizes
Note: Time-dependent parameters can be introduced quite easily into affine jump diffusion models. Even if the corresponding (time) integrals cannot be solved in closed form, option pricing and moment ...
- 6,470
3
votes
What is the intuition behind "jumps" causing volatility skew?
Actually, I do not think it's true. Jumps, when added to the Black-Scholes (BS) dynamics, do modify the volatility surface. However, the volatility skew may get inverted: the implied BS volatility may ...
- 282
3
votes
Accepted
Binomial tree with jumps
Try this paper (although it's advanced): https://www.sciencedirect.com/science/article/pii/S0377042702009032
The topic you picked is not necessarily an easy one :)
- 5,998
3
votes
stochastic vol modelling not enough for smile
For diffusion models (i.e. no jumps):
Local volatility models:
match vanilla options market prices;
give unrealistic volatility dynamic (smile flattens when we move forward in time);
Stochastic ...
- 2,200
3
votes
What can the area under a GBM jump curve tell you
I suppose the expectation could be used to get at some time-weighted average price (TWAP) where we assume each instant of observation has infinitesimal and equal weight $\frac{dt}{T}$: $\bar{S}_T := \...
- 2,880
3
votes
Accepted
Euler Scheme for Jump-Diffusion models
Commonly, we employ the Euler scheme for $\Delta\ln(S_t)$, not for $\Delta S_t$.
Let us specify the jump part as
$$
S_{t+}=S_{t}J\Rightarrow dS_t=S_t(J-1)
$$
where $J$ is a strictly positive random ...
- 6,470
3
votes
Accepted
Second variation of a Brownian motion under jump-diffusion process
$$ 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 ...
- 5,008
2
votes
Basic book on stochastic calculus, Itô and jump processes and Brownian Motion
Consider “Paul Wilmott Introduces Quantitative Finance” if you look for an enjoyable read, good intuition and a not too academic approach
- 1,651
2
votes
Understanding and simulating the jumps in Merton's Jump-Diffusion SDE?
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
votes
Accepted
Bond PDE under an Affine Jump Diffusion model
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 + \...
- 21k
2
votes
Solution for a SDE for a Bond found in Bugard & Kjaer
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 ...
- 7,999
2
votes
Accepted
SDE Jump-Diffusion
$dJ_{t}$ 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 ...
- 98
2
votes
Accepted
Expected Value of Mean-Reverting Jump Process
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]\\
&= \...
- 136
2
votes
Accepted
Vanilla Call Option Priced Using Jump Diffusion Model
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 than $C_{BS}(0,S_0)$ Please consider the ...
- 852
2
votes
Predicting time series using Jump Diffusion model and Neural Networks
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.
...
- 2,436
2
votes
Basic book on stochastic calculus, Itô and jump processes and Brownian Motion
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 ...
- 1,172
2
votes
Does discretizing a diffusion model make it look like a jump diffusion model?
This not possible. The compensated Poisson process $N_t-\lambda t$ converges in the limit of large intensity $\lambda$ to a Brownian motion with variance rate $\lambda\,.$ Therefore, the pure jump ...
- 2,003
1
vote
What is the intuition behind "jumps" causing volatility skew?
Jumps do not imply fat tails. See the simulation in R. Note that the excess kurtosis of [normal variable + jump] is negative.
...
- 282
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