# Questions tagged [black-scholes-pde]

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### Feymann Kac for multidimensional pde

I Have to solve the following PDE: \begin{equation} \begin{cases} \dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{\...
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### Technical difficulties with degenerate PDEs

Crossposted at Mathematics SE I have seen lot of discussions in Math.stackexchange platform about 'degenerate partial differential equations'. But I still unclear about the 'technical difficulty' ...
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### Boundary condition issues for Black-Scholes PDE using finite-differences

I have been implementing an, in my opinion, interesting finite difference method (Runge-Kutta-Legendre of second order) to price American options in the standard Black-Scholes model (see "...
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### I am struggling to prove how when volatility tends to infinity, call option is equal to St and pt option = Ke-r(T-t) [duplicate]

I know how to prove when volatility tends to infinity but i am struggling to prove this. Can anone help?
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### Black-Scholes PDE transformation

From "Mathematics of Financial Derivatives" by Wilmott, Howison and Dewynne, section 5.4, p76. How do I start making the transformations to get to the dimensionless equation? I.e. we start ...
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### Black-Scholes differential equation rewritten [closed]

I have seen that the Black-Scholes equation $$\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+ rS\frac{\partial V}{\partial S}-rV=0$$ can also be written in the ...
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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
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### 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 ...
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### Option pricing with risk-neutral approach

Problem Given $Y_t$ price of a stock (no-dividents), and a derivative paying $Y_T^2$ at maturity $T$, evaluate the price of the instrument now using risk-neutral approach and check that it satisfies ...
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1 vote
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### Black-Scholes Portfolio

In the black-scholes model, the hedging portfolio is given (in some textbooks) by $$\Pi_t = V_t - \Delta S_t,$$ i.e., the portfolio consits of a long position in the option $V$ and $\Delta$ units of ...
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### How to derive a pricing PDE for an asset that follows a mean-reverting process?

I want to derive a Black-Scholes type partial differential equation to price options on an asset that follows a mean-reverting process (Schwartz model). My attempt follows the methodology of deriving ...
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### Generalized Black Scholes PDE in a Two Factor model

I'm reading the book of Clewlow and Strickland on Energy derivatives. In the section about the two-factor model, an equation, similar to B&S PDE is presented, but the proof is not presented. Spot ...
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### Intepreting European call option when expiration approaches to infinity

Assume that dividend = 0, then the price of call option is $$C = S\cdot P_{s}[S(T) > K] - e^{-rT}K\cdot P_F[S(T) > K] = SN(d_1)-e^{-rT}KN(d_2)$$ where $P_s[S(T) > K]$ = Probability of ITM ...
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### Derivative Pricing of an Asset

The Stochastic Differential Equation that models the change in an asset price is $$dS = (12S-sin(S))dt+\frac{\sigma S}{S^2+1}dX$$ where dX's are random variables drawn from standard normal ...
1 vote
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### Is there a relation between the so-called volatility drag and the sigma term in Black-Scholes' model? [duplicate]

The closed-form solution of Black Scholes Dynamics $dS_t=S_t(\mu dt +\sigma dW_t$) is $$S_t=S_0 e^{(\mu -\sigma ^2/2) t+\sigma dW_t}.$$ The $-\sigma^2/2$ term is quite similar to the volatility drag ...
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### Operator splitting method on three assets black scholes equation

Currently I am studying finite difference method on derivatives with three (or more) underlyings and little bit confused on operator splitting method because two papers have different result. For the ...
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### Implicit finite difference method always guarantees positive and stable price of derivative?

For the following black scholes pde $$f_t + rSf_S+\frac{1}{2}\sigma^2S^2f_{SS} = rf$$ By denoting $f_{i}^{n} =$ Price of derivative at price node $i$ and time node $n$ and assume uniform grid, the ...
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### May someone please explain the intuition behind the Black-Scholes Equation?

Consider the Black-Scholes equation for a European Call Option, \begin{equation} \begin{cases}\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + r\frac{\...
1 vote
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### Black Scholes PDE boundary conditions

So I'm trying to solve the black scholes equation using a finite difference model, but I'm getting a answer that's off and I'm having trouble understanding why. This is the result for a option with K ...
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### Question About Converting Black Scholes Differential Equation to Heat Equation

I'm reading a book about converting Black Scholes equation to heat equation and I highlighted in bold for those I have doubts, and really appreciate your advice on it. Let $S$,$T$,$V$ denote ...
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### Nonlinear Black-Scholes model Vs linear Black-Scholes

I am working on a project related to Nonlinear BS partial differential equation, with terms for transaction costs and/or discrete hedging. I have two questions: Is there any exact solution to the ...
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### Black Scholes PDE

I seen two variations of the Black-Scholes PDE with either $+{\frac {\partial V}{\partial t}}$ or $-{\frac {\partial V}{\partial t}}$, and wanted to ask why that is? a) https://en.wikipedia.org/wiki/...
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### Forward price vs. futures price - Wilmott

I am reading Paul Wilmott's book PWOQF2, and there is something I don't get in his derivation of the convexity adjustment between forward and futures prices (chap. 30). He models $S$ and $r$ ...
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### Black-Scholes equation Variational / Weak form

I am having difficulty deriving the weak formulation of the Black-Scholes Equation. I have multiplied it with a test function phi and integrated over Omega. But results on the internet suggest ...
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### Alternative derivation of Black Scholes by Merton

I am currently reading the Theory of Rational Option Pricing (1973) by Robert Merton. In the paper, I encountered a section under the title "An Alternative Derivation of the Black- Scholes Model". I ...
1 vote
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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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### Delta hedging: theoretical value vs actual price

One way to derive the Black-Scholes PDE is via the Delta-hedging argument: Suppose that $V_t = V(t, S_t)$, for some function $V: [0,T] \times \mathbb{R} \to \mathbb{R}$. We construct a portfolio by ...
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### What class of derivatives satisfy the Black-Scholes PDE?

The title pretty much sums up the question, but I will provide some context. There is a large class of derivatives—such as those the payoffs from which depend only on the share price at maturity—...
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### Numerical Solutions to PDEs with Financial Applications

I am reading a paper by Richard White, Opengamma named Numerical Solutions to PDEs with Financial Applications. There is an implementation codes as stated in paper hosted at https://opengamma.com/...
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### Alternative derivation of the Black Scholes formula

I encountered the following derivation of the Black Scholes formula for call price. It may very well be an established method but I had never seen it before so I called it an alternative derivation. ...
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### Proof Black Scholes Theta

I saw the following proof of theta in a paper I read, and I thought it looked pretty neat. Unfortunately I don't understand the step that they do. This is what they do: Now, I don't get how they go ...
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### Cash deposit in replicating portfolio for BS equation unnecessary?

The book on Option Valuation Methods that I currently study (Higham 2013) constructs a replicating portfolio $\Pi = A(S,t)S + D(S,t)$ for deriving the BS PDE, where $D$ is a cash deposit. $D$ does not ...
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### Why does it make sense that $S$ and $e^{rt}$ are solutions to the Black-Scholes PDE?
It's readily verified mathematically that $V=S$ and $V=e^{rt}$ are solutions to the Black-Scholes PDE \$\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2} \frac{\partial^2 V}{\partial S^2} + r S \...