Questions tagged [black-scholes-pde]

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American Swaption Pricing with PDE discretization

So I am still trying to price an american swaption. (MC approach here: American Swaption Pricing with Monte-Carlo method) I've found in Paul Wilmott, The mathematics of financial derivatives, a PDE ...
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1answer
853 views

Pricing log-contract with Black-Scholes PDE

I was wondering if someone could help me with a problem, regarding the Merton Black Scholes PDE. I have an exam soon and this question on an old exam has been bothering me and a friend for quite a ...
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1answer
95 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
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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1answer
120 views

Why can't we use Finite Differences with non-parabolic PDEs?

The title of the question says it all. Why can we only apply the method to parabolic PDEs like the heat equation, and not to ordinary PDEs?
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1answer
65 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
128 views

Can someone check this boundary condition for me?

At the moment I'm comparing plots between the implicit numerical Black-Scholes PDE and the Monte-Carlo Method for the Black-Scholes equation. However, for the particular boundary condition I'm using I'...
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1answer
227 views

Initial/Boundary Conditions for a Butterfly Option?

What are the initial and boundary conditions for a Butterfly Option? I want to write up a PDE program for it and I have a rough idea of what the payoff should be (is it just a call and a put at the ...
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2answers
460 views

Black-Scholes PDE & Terminal Condition

Just a quick question I was hoping someone could shed light on. So far I am familiar with the Black-Scholes PDE with the terminal condition at time $T$ been $V(t=T,S)=(S-K)^+$. I also understand that ...
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1answer
81 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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1answer
50 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
58 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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2answers
449 views

Pricing of Binary or Digital Options or more generally options with discontinuous payoffs using PDEs

I am trying to find references (books, papers, etc.) for calculating $\mathbb E f(X_T)$, where $X_T$ is a diffusion and $f$ is a real function that is not continuous, by means of solving a PDE or ...
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47 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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40 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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146 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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185 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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83 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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445 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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36 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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119 views

Can someone try this Boundary Condition for the Black-Scholes PDE out for me?

I have a bit of a favor to ask and if anyone could help me out with this I'd really appreciate it. At the moment I'm trying to use the triangle wave formula as the payoff for the Black-Scholes PDE i.e....
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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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2answers
705 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
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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1answer
84 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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23 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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50 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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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 ...
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474 views

Show that the equation solves the Black-Scholes PDE

I have the solution as given Based on this, I have to show that this solves the Black-Scholes formula It means that I should take the partial derivatives of the solution above and then receive the ...