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Questions tagged [parabolic-pde]

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2
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0answers
90 views

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: $$C(S,T)=\max(S-K,0)$$...
2
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0answers
78 views

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 ...
3
votes
1answer
141 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 $N_t$ ...
2
votes
1answer
127 views

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}{\...
3
votes
1answer
114 views

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 $...
1
vote
0answers
46 views

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\...
0
votes
1answer
54 views

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-\...
3
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0answers
421 views

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:
7
votes
1answer
256 views

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 ...
3
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0answers
260 views

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). ...
0
votes
1answer
105 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{...
0
votes
1answer
56 views

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[...
2
votes
2answers
2k 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 $$u_t+0.5u_{xx}-ru=0$$ ...
0
votes
1answer
49 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?
6
votes
1answer
222 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 ...
2
votes
1answer
110 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 ...
3
votes
1answer
344 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) =...
3
votes
0answers
65 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.