11
votes
Accepted
Hyperbolic and Elliptic PDEs in Quant Finance
PDE Classification (Background)
Linear second-order PDEs can be classified as either elliptic, parabolic or hyperbolic. A general PDE in two dimensions for $u=u(x,y)$ would look like
$$Au_{xx}+2Bu_{xy}...
- 15.2k
5
votes
Accepted
Proving $\mathbb{P}(S_t<0|S_0=s_0)=0$ for Geometric BM
I think the easiest way to derive the solution to the GBM is via Ito's Lemma.
The GBM: $dS_t = \mu S_t dt + \sigma S_t dW_t$ is a short hand for:
$$ S_t = S_0 + \int_{h=0}^{h=t}\left(\mu S_h\right)dh ...
- 5,998
5
votes
An example of Feynman-Kac
Hope it is okay to attempt an answer to this slightly old question. The PDE
$$f_t +\frac12 \sigma^2 s^2 f_{ss}=0$$
with terminal condition $f(T,s)=s^4$, is solved by
$$f(t,s)=\mathbb{E}(S_T^4|S_t=s),$$...
- 225
4
votes
Accepted
The derivation of vega/gamma relationship
Just want to add the observation that the pricing PDE solution can be formally written as
$$
C(\tau) = e^{\tau \mathcal H} C(0) \quad (*)
$$
where $\tau$ is time to maturity and $\mathcal H$ is a ...
- 1,739
2
votes
The derivation of vega/gamma relationship
Even though it is true that the volatility is constant in this setting, the relationship is valid for all terminal condition or pay-off function -- beyond the typical $(\pm(S-K))_+$ -- so long as the ...
- 2,746
1
vote
Deriving the Heston-Hull-White PDE
(Just what I think is the right start)
The pricing PDE comes out of the dynamics of a self-financing portfolio, $\Pi$, hedged against the movements of stock, $S$, its volatility, $v$, and interest ...
- 5,008
1
vote
Accepted
Feynman-Kac representation of Black-Cox model
I'm not sure if this answers your question, but what you call the 'pde solution' does come directly from your probabilistic setup.
With $t=0$, we have:
$$ E \left[ e^{- rT}p 1_{x_T\geq p, \tau_b\geq ...
- 5,008
1
vote
Proving $\mathbb{P}(S_t<0|S_0=s_0)=0$ for Geometric BM
Another sketch of proof:
If you move to the equivalent PDE (using Feynman-Kac), you can assume that S is positive, find the solution by log-transfomation.
Then as the solution is unique given initial ...
- 1,370
Only top scored, non community-wiki answers of a minimum length are eligible