15
votes
Accepted
Solution of Merton's Jump-Diffusion SDE
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 ...
- 14.5k
6
votes
Accepted
Price of Call Option with or without jumps
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 $100 and the same diffusive volatility. Stock A does not ...
- 176
4
votes
Accepted
Stochastic integral involving Poisson Process
Let
\begin{align*}
X_t=\left(\int_0^t f(s,N_{s-}) d\tilde{N}_s\right)^2
\end{align*}
and
\begin{align*}
Y_t = \int_0^t f(s,N_{s-}) d\tilde{N}_s.
\end{align*}
By It$\hat{\text{o}}$'s product rule for ...
- 21k
4
votes
Price of Call Option with or without jumps
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 ...
- 14.7k
4
votes
Accepted
Hawkes process intensity solution
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 ...
- 7,999
3
votes
Accepted
Geometric Brownian Motion unable to model / predict jumps
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 ...
- 165
3
votes
Solution of Merton's Jump-Diffusion SDE
Use Ito for jumps
$$ dS_t = \frac{\partial S_t}{\partial t} dt + \frac{\partial S_t}{\partial W_t}dW_t + \frac{1}{2}\frac{\partial^2 S_t}{\partial W_t^2} dt + \frac{\partial S_t}{\partial N_t}d N_t $$...
- 624
2
votes
Realized variance in SVJJ (Heston with jumps) model
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} ...
- 624
2
votes
Where to find pricing formulas for affine stochastic volatility jump-diffusion models?
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
- 4,307
2
votes
How to compute the conditional variance of this jump process?
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 sizes $...
- 21k
2
votes
Accepted
Simple question on jump-diffusion
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 ...
- 14.5k
2
votes
Price of Call Option with or without jumps
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 ...
- 98
1
vote
Modelling considerations for a jump model
Note that,
\begin{align*}
d\big( e^{-\mu t}S_t \big) &= -\mu e^{-\mu t}S_t dt + e^{-\mu t}S_{t-}(\mu dt + Y_t dN_t)\\
&=e^{-\mu t}S_{t-}Y_t dN_t.
\end{align*}
From the Doleans-Dade exponential ...
- 21k
1
vote
What is the purest way to get exposure to Jump risk premia, is there a jump swap
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 ...
1
vote
Characteristic function of CGMY model
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-...
- 717
1
vote
Crash cliquet price
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-...
- 171
1
vote
Simulate double exponential process with correlated jumps?
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 ...
1
vote
Accepted
How to estimate lambda for Jump-Diffusion Process from Empirical data?
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 ...
- 5,959
1
vote
Marked poisson process vs compounded
The compound Poisson process isn't technically a marked process because we formulate the process with respect to $\sum_i D_i$ instead of $(\tau_i, D_i)$. However the compound process is constructed ...
- 111
1
vote
Hawkes process intensity solution
Set $\tilde{\lambda}_t = e^{\kappa t} \lambda_t$ and solve for $\tilde{\lambda}_t$
- 5,632
1
vote
Accepted
Numerical Methods for Merton Model
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 ...
- 116
1
vote
Accepted
How to price jumps in payoffs
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 ...
- 3,936
1
vote
Where to find pricing formulas for affine stochastic volatility jump-diffusion models?
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 ...
- 526
Only top scored, non community-wiki answers of a minimum length are eligible