Questions tagged [feynman-kac]

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Hitting time of Brownian motion with drift using Feynman-Kac

I was studying this question from "A Practical Guide to Quantitative Finance Interviews" and was having some trouble understanding one solution. Please advise if misunderstood anything or if ...
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1 vote
1 answer
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Black Scholes PDE in forward log space

In BS world, we have the stock process in log space $dS_t=(r-\frac{1}{2}\sigma^2)dt+\sigma dW$. Let's say we want to price $f(t,x)=\mathbb{E}_{t,x}[h(S(T)]$. Using Feynman-kac, we get \begin{equation} ...
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4 votes
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Deriving the Heston-Hull-White PDE

I'm trying to derive the Heston-Hull-White PDE. The correct backwards PDE is equation (1.3) of this paper on page (2). I will begin deriving the forward PDE, but switching between the two is trivial. ...
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2 votes
1 answer
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Feynman-Kac representation of Black-Cox model

Consider the standard setup from Black and Cox (1976, Journal of Finance). A firm issues a defaultable coupon bond to finance a productive asset that follows a geometric brownian motion: $$dx_t = \mu ...
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Derivation defaultable bond price in Leland 1994 (Merton)

Consider the model in Leland (Journal of Finance, 1994). The partial differential equation that describes the price of the (perpetual coupon defaultable) bond is: $$\frac12 \sigma^2 V^2 F_{vv}(V,t) + \...
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3 votes
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Explicit form for forwards Feynman-Kac formula

This might be a simple question, but I'm having trouble with it. Consider the Cauchy problem with final condition. \begin{equation} \begin{cases} \frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) ...
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7 votes
1 answer
399 views

Hyperbolic and Elliptic PDEs in Quant Finance

Parabolic PDEs (e.g. heat equation) are closely linked to finance via the Feynman Kac Theorem. Do other types of PDEs appear in quant finance? Elliptic PDEs don't contain a time dimension (so perhaps ...
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1 vote
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Feynman Kac: Perpetual Bond

I would like to derive a PDE for a perpetual bond. Suppose we have a bond that will pay a coupon $C$ until there is a default event that occurs. Take the time of default as $\tau$ and consider the ...
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2 votes
2 answers
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Proving $\mathbb{P}(S_t<0|S_0=s_0)=0$ for Geometric BM

I am trying to prove that for the geometric Brownian motion of a stock $\textrm{d}S_t=\mu S_t\textrm{d}t+\sigma S_t\textrm{d}B_t$ with strictly positive constants $\mu$ and $\sigma$ and and $S_0=s_0&...
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1 vote
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An example of Feynman-Kac

I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems. For example, assume that $S_t$ ...
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How to solve this particular PDE using Feynman-Kac formula?

I have to solve the PDE $$ \begin{align} \frac{\partial F}{\partial t} + \frac{1}{2}\frac{\partial^2 F}{\partial x^2} + \frac{1}{2}\frac{\partial^2 F}{\partial y^2} + \frac{1}{2}\frac{\partial^2 F}{\...
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1 vote
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Feynman-Kac formula for $\mu(t,x)=-\frac{1}{1-t}, \sigma(t,x)=1$ and $g(t,x)=x^2$

Consider the following PDE on $[0,T]\times \mathbb{R}$: $$ \begin{cases} \dfrac{\partial F}{\partial t}+\mu(t,x) \dfrac{\partial F}{\partial x}+ \frac12 \sigma^2(t,x)\dfrac{\partial^2 F}{\partial x^2}...
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