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Questions tagged [black-scholes-pde]

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421 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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75 views

Time discretisations, FDM vs FEM

I am interested in adaptive mesh methods for numerical solution of PDEs with applications to finance. As part of a school project, I have been pricing vanilla European call and put options using 2D ...
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261 views

PDE and Black Scholes problem

Consider Black Scholes problem $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$ with boundary condition $V(S,T)=f(S)$, ...
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39 views

BSM PDE and diagonal dominance

I'm using the Thomas Algorithm to solve the Crank-Nicolson FD scheme. However, I've found that the matrix can occasionally fail to be diagonally dominant particularly when flat/local vol is very small....
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91 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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83 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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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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How does the diameter of the spatial grid affects the solution of a Crank Nicholson algorithm?

this is my first question so I hope I express myself clearly. I'm trying to implement an Implicit and a Crank Nicolson algorithm for the generic PDE $\partial_\tau u(\tau,x)+a \partial_x^2 u(\tau,x) +...
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411 views

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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49 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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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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56 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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290 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
29 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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304 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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0answers
100 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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60 views

American option - Black Scholes equation

I know that the Black-Scholes equation for an American option should satisfy the following inequality: $$ V_t + \frac{1}{2}\sigma^2 S^2 V_{SS} + r S V_S - rV \leq 0. $$ Actually we have 3 ...
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84 views

Black-Schole PDE for european put option under Cox-Ingersoll-Ross model

For a european put option, starting from the classical Black-Scholes PDE (assuming constant rate), how do we come up with the Black-Scholes PDE under the Cox-Ingersoll-Ross model (CIR) such as 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 ...