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 ...
- 21.3k
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 ...
- 15k
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 ...
- 8,229
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
2
votes
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 ...
- 21.3k
2
votes
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 ...
- 121
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 $...
- 21.3k
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
Kou model — solving PIDE for European and American options in Python
The issue I described in my initial question is linked to the integral term. In the paper, this term is multiply by $ \theta \Delta \text{t} $ but this is only the "implicit" part of the ...
- 96
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-...
- 707
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-...
- 181
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 ...
- 6,094
1
vote
Hawkes process intensity solution
Set $\tilde{\lambda}_t = e^{\kappa t} \lambda_t$ and solve for $\tilde{\lambda}_t$
- 5,702
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
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