Questions tagged [feynman-kac]
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18
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Why is Feynman-Kac formula applicable in Burgard-Kjaers PDE paper?
In the paper Partial Differential Equation Representation of Derivatives with Bilateral Counterparty Risk and Funding Costs by Burgard and Kjaer, they say we may formally apply the Feynman-Kac theorem ...
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Is there a Black Scholes PDE for a GBM with path-dependent volatility?
Question: Is there a known path-dependent Black-Scholes PDE?
To be a little more precise, let $S$ be a stock price under a risk-neutral measure such that $S$ satisfies the SDE with path-dependent ...
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The derivation of vega/gamma relationship
In Lorenzo Bergomi, Stochastic Volatility Modeling, Chapter 5 Appendix A.1, Equation (5.64), as shown below, seems to assume $\hat\sigma$ to be constant. If that is the case, why do we bother to ...
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Feymann Kac pde with correlated process
I have to solve the following PDE:
\begin{equation}
\begin{cases}
\dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{1}{...
- 163
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Feymann Kac for multidimensional pde
I Have to solve the following PDE:
\begin{equation}
\begin{cases}
\dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{\...
- 163
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Can the Feynman-Kac formula be used for asset classes that don’t have options?
So rather than a call option C(S_t,t) we have some type of asset with asset price is given by S(x,t) where x is any type of variable that the asset price depends on. I.e Price of wooden desks, W(x,t) ...
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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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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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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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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) + \...
- 327
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120
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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) ...
- 53
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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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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 ...
- 21
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357
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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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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
$$
\begin{align}
\frac{\partial F}{\partial t} + \frac{1}{2}\frac{\partial^2 F}{\partial x^2} + \frac{1}{2}\frac{\partial^2 F}{\partial y^2} + \frac{1}{2}\frac{\partial^2 F}{\...
- 111
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Feynman-Kac formula for $\mu(t,x)=-\frac{1}{1-t}, \sigma(t,x)=1$ and $g(t,x)=x^2$
Consider the following PDE on $[0,T]\times \mathbb{R}$:
$$
\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}...
- 121