Questions tagged [parabolic-pde]
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For derivatives pricing, does FEM actually ever outperform FDM?
Simple question that I was wondering about over during the weekend.
I have done a little FEM during the last years and my university time and did not spend a lot of time with FDM. For a new job I have ...
- 113
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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 ...
user34971
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126
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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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2
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1
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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:
$$dx_t = \mu ...
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Explicit form for forwards Feynman-Kac formula
This might be a simple question, but I'm having trouble with it.
Consider the Cauchy problem with final condition.
\begin{equation}
\begin{cases}
\frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) ...
- 53
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Hyperbolic and Elliptic PDEs in Quant Finance
Parabolic PDEs (e.g. heat equation) are closely linked to finance via the Feynman Kac Theorem.
Do other types of PDEs appear in quant finance? Elliptic PDEs don't contain a time dimension (so perhaps ...
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1
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Linear Or nonlinear Black Scholes Equation
I have been going through the analytical solutions of black scholes equation which transforms it to a heat equation.
$$u_{t}=\frac{1}{2}\sigma^{2}u_{xx}$$
Now if the volatility is constant , then its ...
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2
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1
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Numerical Solution to 3 Dimensional Backward BS PDE
I have a three dimensional backward BS PDE.
$$ \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:
$$C(S,T)=\max(S-K,0)$$...
- 53
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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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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 $N_t$ ...
- 63
2
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1
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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}{\...
- 117
2
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1
answer
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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\...
- 1,012
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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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