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3
votes
1answer
44 views

Solve Black scholes PDE without using any transformation

I know that one of the methods of solving the black scholes PDE given by : $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV ...
3
votes
0answers
58 views

Solution for american perpetual put

I have been attempting an exercise in which I have to determine the value of an american perpetual put, $P$ in terms of the asset value $S$. The solution to the exercise says: When $S>S_f$ (the ...
4
votes
1answer
74 views

Deriving Black Scholes PDE under stock as a numeraire

There are many ways to derive the Black Scholes PDE. The Martingale way would be to demand the option price is driftless according to particular measures. Below I derive the correct PDE using the bank ...
0
votes
0answers
30 views

Coupled Black-Scholes equations

Could someone provide me some information about the modelling of several options at the same time by using Black-Scholes (probably coupled) equations? Specifically I am wondering if in finance one has ...
-1
votes
1answer
85 views

Black_scholes formula for a butterfly option

Im wondering if I can apply Black-Scholes formula to valorate a butterfly option, i.e: $$B(T)=Vcall(S(T)-K,0)+Vcall(S(T)-K',0)-2Vcall(S(T)-K'',0)$$ with $K<K''<K'$, just evaluating each call ...
2
votes
0answers
30 views

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) ...
2
votes
2answers
81 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 ...
0
votes
0answers
28 views

Solving Black Scholes PDE using Laplace transform with barrier up and in, up and out call option

I tried to finish the option pricing in european barrier up and in, up and out call option using Laplace transform. The barrier option there is a boundary condition. Can you explain step by step ...
3
votes
1answer
114 views

Boundary conditions: Dirichlet vs Neumann

I'm thinking about the interplay of Dirichlet and Neumann BCs in a FDM scheme. Let's assume a simple Black-Scholes call option problem, with BS PDE with constant coefficients, i.e. instead of $S$, in ...
1
vote
1answer
45 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?
2
votes
1answer
182 views

Feynman Kac Formula for path-dependent options

Consier geometric Brownian motion: $dS_t/S_t=\mu dt+\sigma dW_t$ Feynman Kac theorem tells us that the conditional expectation $v(t,x)=E[ e^{-rT}\Psi(S_T) | S_t=x]$ can be computed by solving the ...
2
votes
3answers
186 views

Analytical soluton to the Black-Scholes equation with a modified European Call Option

Please consider the following modified European Call Option where $ 0 < a \leq 1$. When $a = 1$ the modified European call option is reduced to the standard European call option. ...
1
vote
3answers
198 views

Numerical Solution to BS PDE - Digital Option

Here is a relatively simple question about PDE's pricing. Assume that we are within the BS framework and moreover that interest rate is zero. The price $V(t,S_t)$ of the digital is known to be ...
2
votes
0answers
55 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 ...
4
votes
1answer
229 views

Closed form solution of PDE of Option Price

Let $V=V(S_t,t)$ be the option price and \begin{align} V_t+\mu\,S\,V_S+\frac{1}{2}\sigma^2\,S^2\,V_{SS}=0\\ V(S_T,T)=\ln (S_T)^{2}. \end{align} My question: How can I obtain a closed form solution of ...
8
votes
2answers
649 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 ...
7
votes
4answers
299 views

Why is $C(t,S_t)/B_t$ a martingale?

In the derivation of the Black-Scholes formula given by Joshi (extract below), he says $C(t,S_t)/B_t$ is a martingale. Why? I understand this can be deduced from the Black-Scholes PDE since the drift ...
-2
votes
1answer
89 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 ...
1
vote
2answers
145 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 ...
4
votes
1answer
263 views

generalized black scholes

I understand how to derive the black scholes solution if $dS_t$ = $\mu S_tdt$ + $\sigma S_tdW_t$ and r is constant. The solution is c(t, x) = $xN(d_{+}(T - t), x))$ - K$e^{-r(T - t)}N(d\_(T - t), x))$ ...
2
votes
0answers
157 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 $V(S,T)=f(S)$, ...
6
votes
1answer
665 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 $$ ...
1
vote
2answers
458 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 ...
5
votes
1answer
209 views

Heat/Diffusion Equation

I am working on a problem where I have successfully reduced a version of Black Scholes to the Heat Equation and then shown the solution to be: ...
0
votes
1answer
277 views

Does Implied Volatility always exist?

I am considering a simple Heston Model Market with one risky and one riskless asset. The dynamics of the riskless asset is simply $dB_t=r*B_t*dt$ The dynamics of the risky asset is as follows, $ ...
1
vote
0answers
137 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 ...
2
votes
1answer
449 views

American Swaption Pricing with Monte-Carlo method

I want to price an American swaption but I am not sure about what I am doing. Tree methods and PDE discretization seem difficult to adapt to a swaption. I am trying a Monte-Carlo approach. (in ...
1
vote
0answers
87 views

Approximating the PDE price of an option with a binomial model

I'm trying to replicate, with a binomial model, the price of an option obtained with a PDE. It doesn't really work, so I was wondering, if there are some caveats when doing that. The PDE model use a ...
7
votes
3answers
683 views

Black--Scholes hedging argument

I'm trying to understand the standard hedging argument to derive the Black--Scholes PDE. There's one aspect of the derivation which I can't get passed and I'd be very grateful for some clarification ...
2
votes
3answers
186 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 ...