Questions tagged [parabolic-pde]
The parabolic-pde tag has no usage guidance.
24
questions
2
votes
0answers
102 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 ...
2
votes
1answer
85 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:
$$dx_t = \mu ...
3
votes
0answers
101 views
Explicit form for forwards Feynman-Kac formula
This might be a simple question, but I'm having trouble with it.
Consider the Cauchy problem with final condition.
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) ...
7
votes
1answer
345 views
Hyperbolic and Elliptic PDEs in Quant Finance
Parabolic PDEs (e.g. heat equation) are closely linked to finance via the Feynman Kac Theorem.
Do other types of PDEs appear in quant finance? Elliptic PDEs don't contain a time dimension (so perhaps ...
1
vote
1answer
419 views
Linear Or nonlinear Black Scholes Equation
I have been going through the analytical solutions of black scholes equation which transforms it to a heat equation.
$$u_{t}=\frac{1}{2}\sigma^{2}u_{xx}$$
Now if the volatility is constant , then its ...
2
votes
1answer
86 views
Numerical Solution to 3 Dimensional Backward BS PDE
I have a three dimensional backward BS PDE.
$$ \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{\...
2
votes
0answers
574 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
votes
0answers
332 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 ...
4
votes
1answer
290 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
162 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}{\...
2
votes
1answer
202 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
51 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
74 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-\...
4
votes
0answers
1k 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:
9
votes
1answer
734 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
votes
0answers
507 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). ...
1
vote
1answer
188 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,
$$
\frac{\partial V}{\partial t}+\frac{1}{2}vS^2\frac{...
0
votes
1answer
77 views
pricing a derivative with a running cost
Assume I pricing some commodity derivative that has a running cost with $\$c$ being paid per unit of time. So I define the price under the risk neutral measure to be $$P(t,S_t)=\tilde{\mathbb{E}}\left[...
2
votes
2answers
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 $$u_t+0.5u_{xx}-ru=0$$ ...
0
votes
1answer
74 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?
5
votes
1answer
375 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
149 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 ...
4
votes
1answer
624 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) =...
4
votes
0answers
86 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.
