Questions tagged [black-scholes-pde]

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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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1answer
95 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 ...
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What's the intuition behind the transformation of Black-Scholes into Heat equation?

A sequence of transformations can be used to turn the Black-Scholes PDE into the heat equation. Let $C(S, t)$ be the price of a vanilla European option at time $t$, maturing at time $T$, where the ...
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1answer
65 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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1answer
148 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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26 views

Call Option on the Square of a Log-Normal: Process of Underlying under Stock Measure and Risk Neutral Measure

I'm working on some quant interview questions from the book called Quant Job Interview Questions And Answers (by Mark Joshi and other authors). Here are the questions from the bookd, and the answers ...
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0answers
79 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 ...
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1answer
112 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 ...
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1answer
83 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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2answers
128 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—...
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52 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/...
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1answer
51 views

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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1answer
493 views

Brennan-Schwartz algorithm for pricing American options

I'm reading Pricing American Options using LU decomposition by Ikonen and Toivanen (IT). They reference The valuation of American put options by Brennan and Schwartz, and cast it as method that uses ...
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1answer
81 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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1answer
86 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) ...
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1answer
59 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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0answers
69 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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42 views

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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1answer
129 views

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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165 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 ...
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283 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: $$C(S,T)=\max(S-K,0)$$...
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1answer
137 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}{\...
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51 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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1answer
65 views

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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214 views

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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2answers
790 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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1answer
176 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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1answer
76 views

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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2answers
150 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 \...
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1answer
1k views

Solving Black-Scholes PDE using Laplace transform

I'm trying to obtain the Laplace transform of Call option price with repect to time to maturity under the CEV process. The well known Black scholes PDE is given by $$ \frac{1}{2}\sigma(x)^2x^2\frac{\...
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2answers
803 views

The greeks: where do they come from?

I’m studying the BSM model and having a look at the greeks. I was reading Derivatives, by Paul Wilmott, and he gives the closed form solutions without making the reader see where these solutions come ...
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94 views

AFV Model Implementation for Convertible Bonds

I am reading the original AFV model paper for pricing convertible bonds. https://cs.uwaterloo.ca/~paforsyt/convert.pdf The paper is very technical and I am having trouble finding the actual PDE's to ...
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4answers
247 views

The reason behind the selection of a 1 standard deviation movement for self financing delta hedge

I'm learning this material and I can follow the derivation of the BSM PDE fairly well. The only problem I have is there is an assumption in the derivation (that I am reading) that a stock price ...
3
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2answers
326 views

von Neumann boundary in the transformed PDE

I have transformed the BSM PDE $$\frac{\partial V}{\partial t} + \frac{\sigma^2}{2}S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV = 0 $$ to $u(\tau,x) = V(T-\tau,S_{0}...
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1answer
148 views

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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1answer
59 views

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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2answers
1k views

Black-Scholes Equation - Riskless portfolio derivation

The following is a summary of the derivation of the Black-Scholes equation as given on wikipedia (http://en.wikipedia.org/wiki/Black-Scholes_equation#Derivation) - I have a question regarding the ...
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1answer
236 views

Black Scholes in the case of dividends

Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a ...
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0answers
105 views

Theta from Black-Scholes PDE - is it possible to use implied volatility?

There is a need to derive theta $\theta$ of an option out of standard Black-Scholes PDE. In usual notation ($P$ - price of an option, $S$ - underlying spot): $\theta=r_dP−Sr_d\delta−\frac{1}{2}\...
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493 views

Black-Scholes equation for barrier options

I would like to write down the PDE for the price of an up-and-in call option under the Black-Scholes model as follows. The payoff of the option at expiry $T$ is $$C_T := \max(S_T-K,0)1_{M_T \geq L}$$ ...
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0answers
37 views

Transforming and minimisation of the BS PDE

I'm trying a novel numerical substitution/fitting method to solve the BS PDE, but the issue is that due to the large range of magnitude of prices $V(s,t)\in[10^{-20},10^1]$, when I try to minimise the ...
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0answers
630 views

Black-Scholes PDE - Change of Variables

In the derivation below, I cannot figure out how to solve for "Step 3". Can anyone help me walk through the steps in detail? Derivation:
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1answer
206 views

How to price up-out-call by solving heat equation like down-out-call

We know that by changing the variables we can obtain the Black-Scholes formula of vanilla call through solving the ...
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0answers
37 views

Spectral Analysis for European Put Options

I am trying to implement the spectral analysis on European Put Options. My code is designed to change the number of nodes(basis functions) accordingly, but the boundary condition and thus the range of ...
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1answer
86 views

Assumption in black scholes solution

Under the usual notations, In most textbooks on Quantative Finance, for deriving the Black-Scholes solution I find that authors, while setting up the riskless portfolio, assume that, $$\text{d} (\...
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1answer
378 views

Rigorous derivation of $d\Pi$ for stock with continuous dividend

Suppose we are holding a replicating portfolio $\Pi_t$ of long an option $f(S,t)$ and short some stock, so $$\Pi_t=f(S_t,t)-\Delta_t S_t$$ Suppose the stock follows geometric Brownian motion and pays ...
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2answers
206 views

The PDE of the probability hitting the barrier before T

Suppose: $$d S=\mu S dt+\sigma Sd W$$ $Q(t,S)$ is the probability that $S$ hit the barrier $B(S_t<B)$ before $T,$ then $Q$ satisfies following PDE $$Q_t+\dfrac{1}...
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2answers
204 views

Implications of Black Scholes Plot

I'm pretty new to finances, but I'm heavily into scientific computation. For my scientific computations class, I need to have at least a basic understanding of finances for the presentation I'm going ...
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1answer
298 views

Finite Difference Method for Black-Scholes-Formula

Using finite difference method for the Black-Scholes-Partial Differential Equation one need to impose some boundary conditions on the edge of the grid, i.e for a Grid on $D=[a,b]\times R^+$ one need ...
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104 views

probability of default for Kolomogorov backward equation

suppose $$dA = \mu Adt + \sigma AdX.$$ is a geometric Brownian motion. One says that the Probability $P(A,t)$ of $A$ reashing the critical level $K(t)$ before maturity: $$\dfrac{\partial P}{\partial ...