# Questions tagged [pde]

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### Kou model — solving PIDE for European and American options in Python

Toivanen proposed$^\color{magenta}{\star}$ a method to solve the partial integro-differential equation (PIDE) with a numerical scheme based on Crank-Nicolson. In particular, he proposed an algorithm ...
30 views

### Singular Perturbation in Hagan's 2002 SABR paper "Managing Smile Risk"

I'm reading Hagan's 2002 paper Managing Smile Risk originally published on the WILMOTT magazine, and got something confusing. The set up: $P(τ,f,α,K)$ is the solution of the problem as in Equation (A....
62 views

### When to use total derivative and when not to?

as I was trying to teach myself financial mathematics, I came across this topic on transforming black scholes pde to a heat equation. I had the exat same question as this post Black Scholes to Heat ...
1 vote
116 views

### Is stochastic control used in market making/algo trading at institutions?

I have recently completed a class that mirrors these lecture notes expect chapter 5: https://www.maths.ed.ac.uk/~dsiska/LecNotesSCDAA.pdf In chapter 5, they use stochastic control and the Hamiltonian ...
42 views

### Pricing equation with two correlated states

Consider the following asset pricing setting for a perpetual defaultable coupon bond with price $P(V,c)$, where $V$ is the value of the underlying asset and $c$ is a poisson payment that occurs with ...
30 views

### On the operational process of fractional and delay Brownian motions (FGBM/GDBM) governing respective market scenarios

I have some knowledge about the fabrication of a stochastic differential equation (SDE) governing asset price ($S(t)$) dynamics (This answer helped me up to some extend). For instance, I am little bit ...
165 views

### What are the parallels between the Black-Scholes equation and the heat equation?

I'm trying to understand the analogy between the Black-Scholes equation (1) and the heat partial differential equation (2). I understand that (1) can be written in the form of (2) mathematically, but ...
69 views

### Theta discretization PDE

I am trying to understand the validity of why we can theta discretize the solution to a PDE. For a PDE following: $$0 = \partial_tf + A f$$ I understand that for one discrete time step the solution to ...
229 views

### What is the PDE for this interest rate derivative?

We have the following model for the short rate $r_t$under $\mathbb{Q}$: $$dr_t=(2\%-r_t)dt+\sqrt{r_t+\sigma_t}dW^1_t\\d\sigma_t=(5\%-\sigma_t)dt+\sqrt{\sigma_t}dW^2_t$$ What is the PDE of which the ...
1 vote
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### Power option's PDE

I am looking to understand the PDE of Power Options in Paull Willmot on Quantitative Finance (2nd Ed), Ch. 8.9 - Formulae for Power Options (p. 149). Suppose the payoff depends on the asset price at ...
1 vote
498 views

### 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} ...
1 vote
411 views

### Any book which is intro to PDEs but prioritises techniques useful for solving Black-Scholes?

Summary: Can you recommend any book which is: Intro/first course in PDEs Covers solution methods useful for Black-Scholes model? Background I have just started learning about PDEs (after studying ...
86 views

### What is the difference between "stochastic" heat equation and just heat equation?

I am trying to understand the difference between the "stochastic" heat equation and the heat equation. Will i be wrong to say the stochastic heat equation is just the heat equationg with the ...
71 views

1 vote