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12 votes
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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}...
Kevin's user avatar
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5 votes
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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 ...
Jan Stuller's user avatar
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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 $$f(t,s)=\mathbb{E}(S_T^4|S_t=s),$$...
Nap D. Lover's user avatar
5 votes
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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 ...
Frido's user avatar
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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 ...
Hans's user avatar
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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 ...
ir7's user avatar
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1 vote
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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 ...
ir7's user avatar
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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 ...
jherek's user avatar
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