Questions tagged [parabolic-pde]

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Market models of implied volatility and no arbitrage

Something has been bugging me for a while, and I can't really find an answer to it in papers. Maybe somebody can help me out. In addition to modelling the instantaneous vol, or modelling forward ...
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2 votes
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113 views

Numerical scheme for this HJB equation

Without dwelling on details on how to obtain the HJB equation for this problem, I would like to know if the scheme I wrote for solving it numerically is viable or did I miss something. I need to solve ...
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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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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
471 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 answer
486 views

Linear Or nonlinear Black Scholes Equation

I have been going through the analytical solutions of black scholes equation which transforms it to a heat equation. $$u_{t}=\frac{1}{2}\sigma^{2}u_{xx}$$ Now if the volatility is constant , then its ...
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2 votes
1 answer
92 views

Numerical Solution to 3 Dimensional Backward BS PDE

I have a three dimensional backward BS PDE. $$ \frac{\partial V}{\partial t} + a(t) S \frac{\partial V}{\partial S} + \frac{1}{2} \sigma(t, S)^2 \frac{\partial^2 V}{\partial S^2} + b(t, M) \frac{\...
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Black-Scholes equation to Heat equation .(Boundary conditions)

I have been given a problem to code the heat equation which is transformed from B-S equation (European call option) . Now the boundary conditions are for European call option: $$C(S,T)=\max(S-K,0)$$...
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How do you numerically solve the Dupire Local Volatility PDE in log moneyness-time space?

I am trying to implement a numerical solution to price vanilla calls. I am using the Dupire equation in log moneyness-time (k = ln(F/T)) space as per below PDE I have tried solving it using a fully ...
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4 votes
1 answer
303 views

Bond PDE under an Affine Jump Diffusion model

Under the Jump extended Vasicek model, the dynamics of the short rate are as follow : $$dr_t=\kappa(\theta-r_t)dt+\sigma\sqrt{r_t}\,dW_t+d\left(\sum\limits_{i=1}^{N_t}\,J_i\right)$$ where $N_t$ ...
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2 votes
1 answer
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Unable to obtain correct Finite Difference Results

A rather general question regarding a specific problem I am facing with my Matlab implementation of the implicit FD method for this PDE: \begin{equation} \frac{\sigma_s^2}{2}\frac{\partial^2 V}{\...
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What is the domain of the Black-Scholes operator?

By the Black-Scholes operator I mean the following. $$L_{BS}u(x) = \frac{1}{2}\sigma^2x^2\frac{\partial^2}{\partial x^2}u(x) + rx\frac{\partial}{\partial x}u(x) - ru(x)$$ Obviously, the domain of $...
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Prove that $F(s,x_0)=0$, $F(t,x)=1$ and $\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial x^2}=0$

Using the Dynkin's formula, prove that $F(s,x_0)=0$, $F(t,x)=1$ and $\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial x^2}=0$ where $F(s,t)=2\int_{x-x_0}^{\infty}\frac{1}{\sqrt{2\...
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Which PDE is satisfied by the function of Wiener process $u(t,x)$?

Suppose you have the following function: $u(t,x)=\mathbb{E}[f(xe^{W_t+\frac{1}{2}t})]$, where $W_t$ is a Wiener process. Let us first differentiate: $du=\mathbb{E}[f'(xe^{W_t+\frac{1}{2}t})(e^{W_t-\...
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Black-Scholes PDE - Change of Variables

In the derivation below, I cannot figure out how to solve for "Step 3". Can anyone help me walk through the steps in detail? Derivation:
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9 votes
1 answer
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Advantage of solving the Fokker-Planck equation over Monte-Carlo simulations

For a standard Ito process $$dX_t = \mu(X_t, t) \,dt + \sigma(X_t, t) \,dW_t,$$ the Fokker-Planck or forward Kolmogorov equation gives an equation for the probability density $p ( x , t )$ of the ...
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3 votes
0 answers
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Ill-posed problem: Local volatility calibration. Regularization vs Smoothing

I have asked my question on Mathematics site of Stack Exchange but maybe I will get the answer rather here. I am working on inverse problem - calibration of local volatility (financial application). ...
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1 vote
1 answer
197 views

Gatheral's change of variables for stochastic volatility PDE

This is taken from Gatheral's book "The Volatility Surface", where he tries to go from equation 2.3 to equation 2.4. We have the following PDE, $$ \frac{\partial V}{\partial t}+\frac{1}{2}vS^2\frac{...
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0 votes
1 answer
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pricing a derivative with a running cost

Assume I pricing some commodity derivative that has a running cost with $\$c$ being paid per unit of time. So I define the price under the risk neutral measure to be $$P(t,S_t)=\tilde{\mathbb{E}}\left[...
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2 votes
2 answers
3k views

Feynman Kac and choice of measure

I seem to be confused on this topic. So I write my SDE without a drift to make it simple: $$dX_t=dW_t$$ and before I get to any finance there is a relation that the solution to $$u_t+0.5u_{xx}-ru=0$$ ...
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  • 425
0 votes
1 answer
75 views

parabolic pde with source term

I was wondering if someone is aware of the application when pdes of the form arise $$u_t+u_{xx}+g=0$$ i.e. there is a source term now. Any financial instruments that have this type of pde?
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5 votes
1 answer
409 views

Other numerraire choices when applying Feynman Kac

all of the books and notes I have seen on the Feynman Kac formula mostly applied to Risk neutral measure, i.e. different interest rate models, stochastic volatility, etc. I think risk neutral measure ...
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2 votes
1 answer
151 views

Black Derman Toy short rate and PDE

I am looking at the Black Derman Toy local short rate model as $$d\log r(t)=\alpha(t)(\theta (t)-\log r(t))dt+\sigma dW(t)$$ under RN measure. I would like to derive the bond price PDE. For that I ...
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4 votes
1 answer
659 views

Solving a backwards heat equation using stochastic calculus

Given the PDE $$\frac{\partial F}{\partial t} + \frac{1}{2}\sigma^2 \frac{\partial^2 F}{\partial x^2} = 0$$ with condition $F(T,x) = x^2$, one can use the Feynman-Kac formula to arrive at $$F(t,x) =...
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4 votes
0 answers
93 views

European call option delta and maximum principle

From comments, the maximum principle for parabolic PDE can be used to show that the European call option delta cannot be greater than 1. I am looking forward to such derivations.
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