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Continuous flows Perpetual maturity cap on Exchange Options PDE Change of variable

Im trying to do a change of variable on the following PDE Using the following change of variable $$ V(P^1,P^2) = P^2 W(C), C=\frac{P^1}{P^2} $$ I get this for the homogeneous part of the equation: ...
f4bby's user avatar
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2 votes
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50 views

Areas of research in calibration of stochastic volatility models

I am working on a thesis in deep calibration of the Heston model, and I wanted to include a section on the historical work, before the use of neural networks in this area. Thus, I was wondering what ...
sxminho's user avatar
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0 answers
61 views

How are opitons greeks computed for models that require numerical PDE solving [closed]

I am often told that options priced under SLV models, the Greeks cannot be exactly replicated by finite differences, but are computed at the level of the grid used to solve the PDE. Can someone please ...
AIEA's user avatar
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3 votes
0 answers
94 views

Option pricing boundary condition

I am currently working on this paper "https://arxiv.org/abs/2305.02523" about travel time options and I am stuck at Theorem 14 page 20. The proof is similar to Theorem 7.5.1, "...
Valentin's user avatar
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2 votes
1 answer
148 views

Pricing PDE of Asian option by Shreve

I am currently working on "Stochastic Calculus for finance II, continuous time model" from Shreve. In chapter 7.5 Theo 7.5.1 he derives a pricing PDE with boundary conditions for an Asian ...
Valentin's user avatar
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1 answer
224 views

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 ...
pierrot's user avatar
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1 vote
0 answers
66 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....
athos's user avatar
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1 answer
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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 ...
David's user avatar
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1 vote
1 answer
292 views

Is stochastic control with the HJB equation used in market making/algo trading at institutions?

In chapter 5 of https://www.maths.ed.ac.uk/~dsiska/LecNotesSCDAA.pdf, they use stochastic control and the Hamiltonian Jacobi Bellman (HJB) equation in attempt to measure bid-ask spreads and optimal ...
THATS MY QUANT MY QUANTITATIVE's user avatar
2 votes
0 answers
45 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 ...
Luca Gi's user avatar
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3 votes
1 answer
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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 ...
probablysid's user avatar
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1 answer
240 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 ...
Andrei's user avatar
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1 vote
0 answers
237 views

Black and Scholes PDE in terms of Future Price [closed]

I was trying to understand why the Black and Scholes PDE for the value of an option, $V (F , t)$, with the forward price, $F$, as underlying is $$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2F^2\...
Eduardo Contreras's user avatar
2 votes
0 answers
68 views

crank nicolson - forward vs backward equations

I have implemented a script in python to solve a differential equation in $(x,t)$ using the crank-nicolson method. To start with, I was testing a case in which I know a solution: $$u_{xx} + u_{x} - u -...
Si Mo's user avatar
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1 answer
142 views

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 ...
userPrimeNumber's user avatar
1 vote
1 answer
615 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} ...
J. Lin's user avatar
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2 votes
3 answers
635 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 ...
userPrimeNumber's user avatar
0 votes
0 answers
90 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 ...
Ratanna's user avatar
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88 views

Can we proof the boundary condition for the Black Scholes derived from a replicating Portfolio?

So for Black Scholes we know that the PDE is the follwing: ${\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}=rV-rS{\frac {\partial V}{\partial S}}...
Nikolai Kl's user avatar
8 votes
1 answer
1k 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 ...
Alex's user avatar
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1 vote
0 answers
81 views

boundary conditions in finite element method

In the appendix A of this paper, https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.227.5073&rep=rep1&type=pdf, a finite element method is demonstrated to price a straddle. The same ...
sigma1988's user avatar
1 vote
0 answers
143 views

How to solve this particular PDE using Feynman-Kac formula?

I have to solve the PDE $$ \begin{align} \frac{\partial F}{\partial t} + \frac{1}{2}\frac{\partial^2 F}{\partial x^2} + \frac{1}{2}\frac{\partial^2 F}{\partial y^2} + \frac{1}{2}\frac{\partial^2 F}{\...
shot22's user avatar
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1 vote
1 answer
138 views

Canonical text on numerical PDEs in finance

I am looking for a text similar to Glasserman's Monte Carlo Methods in Financial Engineering, but with a focus on numerical methods for PDEs. Glasserman's book seems to cover a lot for what is ...
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2 votes
1 answer
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Trying to measure "radius of diffusion" in the stock market

Good evening! I'm quite new to quantitative finance (coming from the math world!), so please excuse me if I'm not familiar with every concept! I am currently studying the Black-Scholes equation, ...
Azur's user avatar
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