Questions tagged [feynman-kac]

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1answer
89 views

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. ...
2
votes
1answer
72 views

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 ...
1
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0answers
29 views

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) + \...
3
votes
0answers
87 views

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) ...
7
votes
1answer
247 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 ...
1
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0answers
38 views

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 ...
2
votes
2answers
333 views

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&...
0
votes
0answers
77 views

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 $S_t$ ...
1
vote
0answers
88 views

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}{\...
1
vote
0answers
44 views

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}...