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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1 vote
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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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Locally riskless

Most derivations of the Black-Scholes formula end up with the following dynamics of some (hedged) portfolio: $$ \int_{t=0}^{T} \left(\frac{\partial f}{\partial \tau}(S(t),t)+\frac{1}{2}\cdot\frac{\...
2 votes
2 answers
164 views

Black-Scholes PDE derivation gap

Most derivations of the Black-Scholes formula end up with the following dynamics of some (hedged) portfolio: $$ \int_{t=0}^{T} \left(\frac{\partial f}{\partial \tau}(S(t),t)+\frac{1}{2}\cdot\frac{\...
4 votes
1 answer
138 views

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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36 views

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?
0 votes
1 answer
49 views

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 ...
0 votes
1 answer
123 views

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 ...
0 votes
1 answer
80 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 ...
1 vote
1 answer
257 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 ...
0 votes
0 answers
64 views

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
2 answers
257 views

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 ...
3 votes
3 answers
423 views

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 ...
1 vote
0 answers
61 views

Finding the PDE and replicating strategy of a european contigent claim [duplicate]

Suppose that we have the Black and Scholes model where the interest rate and the volatility are time varying: $dB(t)=r(t)B(t)dt$ and $dS(t)=S(t)b(t)dt+S(t)\sigma(t)dW(t), S(0)=s>0$ where $r,b,\...
2 votes
2 answers
974 views

Boundary Conditions for Call Option in Black Scholes Model

Let $C(t,S)$ be the value function of a call option. I want to price that option using (explicit) finite differences and the Black Scholes PDE. I consider the grid $0=t_0<t_1<...<t_{N-1}<...
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1 vote
0 answers
68 views

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 ...
0 votes
3 answers
158 views

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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0 answers
58 views

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 ...
3 votes
0 answers
82 views

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 ...
2 votes
2 answers
234 views

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 ...
1 vote
2 answers
947 views

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
2 answers
374 views

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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1 vote
1 answer
137 views

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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1 vote
1 answer
312 views

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 ...
1 vote
1 answer
192 views

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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3 votes
1 answer
273 views

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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3 votes
1 answer
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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 ...
3 votes
0 answers
152 views

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
1 answer
502 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 ...
1 vote
1 answer
253 views

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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3 votes
2 answers
541 views

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—...
1 vote
0 answers
105 views

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/...
2 votes
1 answer
159 views

Zero-coupon bond pricing equation derivation

I'm trying to understand how in Chawla's paper that I've linked below, how he obtains equation (2.5) for the zero coupon bond pricing equation? The equation is: $\frac{\partial B}{\partial t} + \...
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1 vote
1 answer
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Sensitivity Approximation - Crank Nicolson

I am looking into a new method of calculating sensitivities starting off with a proof of concept with Black Scholes PDE. Suppose I want to calculate Rho and take the derivative of the PDE (heresy!!) ...
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1 vote
1 answer
344 views

Modifying Basic Black Scholes Equation For Time Dependent Variables - Per Wilmott?

I am reading Wilmott's book and don't understand why he makes the following step to re-write the PDE. I get equation 8.4, that's just the typical PDE for a dividend yielding stock where r(t), D(t) ...
2 votes
0 answers
111 views

What is the recipe for deriving a PDE for the price of an option?

In the Black Scholes setting, here is how my understanding is of how we derive the PDE for the value of an option. We assume that the price of the option is Markovian in our state variable $S_t$. ...
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1 vote
1 answer
109 views

Is it possible to transform arithmetic-average strike continuous sampling Asian Black-Scholes equation to a heat equation?

By Transformation from the Black-Scholes differential equation to the diffusion equation - and back, we are able to transform vanilla European option into a heat equation. And we know that the ...
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2 votes
1 answer
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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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3 votes
1 answer
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Why does Black Scholes formula give inconsistent dimensional analysis result?

For example, distance = speed * time, m = m/s * s. But this technique gives wrong answer on the Black Scholes formula. The square root in the denominator gives wrong unit inside of the culumulative ...
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1 vote
0 answers
580 views

Pricing Knock Out Barrier Options by solving Black Scholes PDE (MATLAB)

This question is based on MATLAB functions. Suppose there is a stock S following the process $dS_t=(r-q)S_tdt+\sigma(S_t,t)dW_t$ r - risk-free rate, q - dividend yield, W - Weiner process The ...
2 votes
0 answers
648 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: $$C(S,T)=\max(S-K,0)$$...
3 votes
1 answer
368 views

Errors on Finite Differences + Implicit Scheme + Black & Scholes

I'm solving the classical Black & Scholes (BS) PDE for a European option using finite difference and the implicit scheme. In other words, I'm trying to solve $\displaystyle\frac{\partial V}{\...
0 votes
0 answers
131 views

If the value of a call option is not dependent on the drift of the stock, why does a higher stock price mean a higher call option price [duplicate]

I have read that the price of an option is not affected by the drift of the stock since the drift term doesn't appear in the Black Scholes PDE. I become confused because to me, this implies that the ...
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0 votes
1 answer
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Why does a higher stock value imply a higher call option value [closed]

This may seem like a very dumb question, but if the underlying stock price is greater, then why should a call option be worth more. My reasoning is that, if the option price is not affected by the ...
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1 vote
0 answers
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Black Scholes Replicating Portfolio Riskfree Asset

Im having a question about this standard derivation of the Black-Scholes formula: http://www.soarcorp.com/research/BS_hedging_portfolio.pdf The paper states $$C=\Delta S+B$$ and finally $\Delta = ...
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3 votes
1 answer
275 views

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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0 votes
2 answers
3k views

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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1 vote
1 answer
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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 ...
4 votes
2 answers
169 views

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 \...