Questions tagged [finite-difference-method]

The tag has no usage guidance.

0
votes
0answers
21 views

Crank–Nicolson for Discrete Type Barrier: Backward propagation

The boundary conditions for Discrete Type Barrier (e.g. Up-and-Out) are: - Dirichelet boundary condition (set to 0 when spot is bigger than Barrier) on Barrier event dates - Otherwise (the other sides ...
1
vote
0answers
32 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{\...
2
votes
2answers
301 views

Local Volatility calculation in Python

I am trying to price Local Volatility in Python using Dupire (Finite Difference Method). I have following set of information Spot: 770.05, Strike: 850, Type: 'C', rfr: 0.0066, time to maturity = ...
1
vote
0answers
85 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
1answer
88 views

Finite difference methods for (continuously) strike-resettable American options

For simplicity, let us consider an American call/put with a continuously resettable strike price. Current time is $t=0$, maturity is at $t=T$, and the initial strike is $K_0$. We consider a "...
2
votes
2answers
96 views

Is it possible to model path-dependent clauses using finite difference methods?

I'm trying to build a convertible bond pricer. In my case a convertible bond is a complex derivative with call, put and conversion price reset clauses, and all of the clauses are triggered in a path-...
1
vote
1answer
58 views

Explicit Euler stability for the Heat Equation (FDM)

Why the Explicit Euler scheme for the Heat Equation is stable only if $k \leq h^2/2$ ? Here is the difference equation: \begin{equation} \frac{U_j^{n+1}-U_{j}^n}{k} = \frac{1}{h^2}(U_{j+1}^n-2U_j^n+...
3
votes
1answer
136 views

Binomial Trees vs FDM

Binomial trees as the number of time steps is increased (or equivalently as the time step tends to 0), converge to the exact value for an option. So why do people use FDM for pricing options (for ...
2
votes
0answers
33 views

American-Bermudan-Asian option fixed strike using finite differences

I'm trying to price the same American-Bermudan-Asian option described in Longstaff Schwartz (2001). Specifically, using finite difference methods with an explicit scheme to solve $\begin{aligned} \...
3
votes
1answer
89 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}{\...
2
votes
0answers
114 views

How do you numerically solve the Dupire Local Volatility PDE in log moneyness-time space?

I am trying to implement a numerical solution to price vanilla calls. I am using the Dupire equation in log moneyness-time (k = ln(F/T)) space as per below PDE I have tried solving it using a fully ...
1
vote
0answers
41 views

Optimal allocation problem by finite differences

I am attempting to apply implicit finite difference to solve Merton's problem of optimal portfolio allocation for constant parameters. The equation to solve is the Hamilton-Jacobi-Bellman equation: $$...
1
vote
1answer
39 views

How many decimals of accuracy can I expect from FDM and MC (both valuation and risk)

I have implemented some Monte Carlo and FDM code. I can then get greeks by bumping. I am comparing to to exact formulas of price + greeks, and am wondering how many decimals of accuracy I can expect ...
1
vote
1answer
63 views

Finite difference: move forwards or backwards?

In finite differences for the black scholes method, you move backwards in time, since of course you know the prices at time $t = T$, and then you iterate until you get to time $t = 0$. However, why ...
2
votes
3answers
104 views

For using finite difference on PDE, what should the grid be?

If I wish to use finite difference methods to approximate the pricing function $F(t, s)$ for an option (say, a call), what size grid should I use? I mean, it seems to make sense to start the grid at ...
1
vote
1answer
38 views

How are FDE's implemented when one wants one particular price?

Say I want to price a particular call option in the Black Scholes model using finite difference methods. The value process of this option $V(s, t)$ satisfies a PDE. I can use finite difference ...
1
vote
0answers
29 views

Oscillating errors in finite difference Black Scholes

I am writing an implementation of the explicit finite difference method to price a standard european call option, and comparing the results to the corresponding analytical value to gauge the error ...
0
votes
1answer
362 views

how to price barrier option under local vol model using QuantLib

I use QuantLib in Python. Now I have implied volatility surface data. How can I get the local vol surface than using finite difference method to price a barrier option in QuantLib?
3
votes
0answers
149 views

Finite Difference with SVI Vol Model

I am attempting to implement a local vol pricing model in finite difference for equity index options. I have followed Gatheral's Lectures and fitted an SVI Model bringing me to the following local ...
2
votes
1answer
191 views

SABR PDE spot/forward upper boundary condition implementation

When running my Finite Difference code, I observe something odd. Although implementing a classical (non-reverting) SABR model, I initialized the variables such that it should be equal to Black-...
1
vote
1answer
126 views

Finite Difference implicit scheme

I'm trying to solve the following PDE numerically using an implicit FD scheme: \begin{equation} \frac{\sigma_s^2}{2}\frac{\partial^2 V}{\partial S^2} + \rho \sigma_S \sigma_\alpha\frac{\partial^2 V}{\...
2
votes
1answer
132 views

Unable to obtain correct Finite Difference Results

A rather general question regarding a specific problem I am facing with my Matlab implementation of the implicit FD method for this PDE: \begin{equation} \frac{\sigma_s^2}{2}\frac{\partial^2 V}{\...
4
votes
1answer
231 views

Finite Difference method in Matlab for SABR volatility model fails to provide correct option values

Currently, I'm trying to implement a Finite Difference (FD) method in Matlab for my thesis (Quantitative Finance). I implemented the FD method for Black-Scholes already and got correct results. ...
1
vote
1answer
311 views

Black Scholes Theta Finite difference

I am trying to obtain the Theta from Closed Formula by using Finite Difference methods and I observe some discrepancies. For instance, here with the following parameters: Spot:50, Strike:50, Rate: 0....
3
votes
1answer
655 views

Local Volatility implementation

The Dupire equation is well-known and mentioned in thousands of articles. Although I could not find a lot of documentation about a consistent and proper way of implementing the formula (The difficulty ...
1
vote
0answers
354 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 ...
1
vote
1answer
122 views

MATLAB exercise on an European call option with time-varying volatility

I have to solve the following exercise: compute and plot the value $V = V(S, t),\ t<T$, ($T=$ maturity) of an European CALL option (with arbitrary $t$, $T$, $K$ (strike price), $r$ (risk-free ...
1
vote
0answers
205 views

Trinomial Tree and finite difference methods

I want to understand the connection between the trinomial tree and the finite difference methods. As far as I understood so far is, if we transform the Black-Scholes-PDE to heat equation, the explicit ...
0
votes
0answers
110 views

Finite Difference Method for Black-Scholes differential equation

I am using the backward finite difference Method to simulate the price. The algorithm is the follows: Suppose we know $V_T(S_T)=payoff$, we can use a backward recurrence:$$V_{T-\delta t}=V_T-\frac{\...
0
votes
0answers
52 views

what is the meaning of $U^{n+1/3}$ ADI method

For the ADI in numerical method $$\frac{U^{n+1/3}-U^n}{k/3} = \Delta^2_x U^{n+1/3} + \Delta^2_y U^n + \Delta^2_z U^{n+2/3}$$ $$....$$ $$....$$ don't like $U^{n+1/2} = \dfrac 1 2 (U^{n+1}+U^n),$ I ...
0
votes
2answers
1k views

Which options do not have a closed form pricing formula like BS?

Q1 Are there options that doesn't have a closed form pricing formula like BS? Is this the situation when we have to use Finite Difference Method? Can someone give an example? (I hope this option ...
4
votes
2answers
276 views

Smoothing of the payoff function as a terminal condition for numerical option pricing

I am interested in using a 4th order finite difference method in (underlying asset) space to price a European call basket option. I have developed the solver and everything works as expected, except ...
4
votes
0answers
329 views

binomial trees and finite differences

I was reading Tavella Randall book and their explanation why binomial trees are a particular example of finite differences. I started having additional questions. So, they way they do that is saying ...
0
votes
0answers
119 views

zero curvature boundary condition

Assume I am solving numerically Black Scholes PDE $$u_t+0.5\sigma^2s^2u_{ss}+rsu_s-ru=0$$ and I decided to have boundary condition on the right boundary as $u_{ss}=0$. One way is to write the discrete ...
1
vote
1answer
201 views

Pricing an American derivative with finite differences

I have a basic fundamental question on pricing an American option in the Black-Scholes (BS) framework: I seem to confuse two different approaches to price any early exercise, Write down a linear ...
5
votes
1answer
245 views

Other numerraire choices when applying Feynman Kac

all of the books and notes I have seen on the Feynman Kac formula mostly applied to Risk neutral measure, i.e. different interest rate models, stochastic volatility, etc. I think risk neutral measure ...
1
vote
2answers
414 views

Full value function of an American option with QuantLib FD

I am looking at the Equity Option example of QuantLib: http://quantlib.org/reference/_equity_option_8cpp-example.html and more particularly the FDAmericanEngine. However, I am not interested in the ...
2
votes
1answer
205 views

pricing american put option with fdm

Assume I use some finite difference solver to solve for American type of exercise in BS framework where stock pays dividend discretely. Then at every time iteration, for call option, I firstly adjust ...
1
vote
0answers
105 views

methodology confirmation for computing implied risk-neutral CDF from option prices

In this question, the risk-neutral probability distribution $q(S_T=s)$ for the underlying at time $t = T$ is given by the Breeden-Litzenberger identity as: $$ \frac{1}{P(0,T)} \frac{ \partial^2 C }{\...
0
votes
1answer
1k views

Calculation of option Greek (sensitiviety) theta via finite difference

I am able to get good approximations for delta, gamma, and rho via finite difference method, but not theta. I believe my issue is the value of h. Theta is basically the difference between the price ...
3
votes
2answers
293 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}...
2
votes
0answers
47 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) +...
1
vote
0answers
35 views

Methods Available for Derivative Pricing in Mathematica? [closed]

I am using Mathematica to price options (built in functions, no need to reinvent the wheel, right?). In the documentation, the Binomial method is used as an example of specifying a non-standard method....
-1
votes
1answer
498 views

Is the code of my binary call option pricer (using explicit finite difference, backward scheme) correct? [closed]

I am using explicit finite difference (backward scheme) to price a binary call option. Here is my MATLAB code: ...
1
vote
1answer
110 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
95 views

Solving a Non-Linear PDE using a Finite Difference Scheme

I have the following non-linear PDE and I have no idea how to go about solving it using a finite difference scheme in Python. Can someone get me started and/or point me to an algorithm for doing this? ...
2
votes
1answer
60 views

What different techniques exist for modeling exotics near payoff discontinuities in Finite Difference method?

If you are modeling an exotic, like a binary or a barrier, and hedging it with vanillas that have strikes quite close to the exotic's strike, then a large asset step size, for example, $\delta S = \...
1
vote
2answers
4k views

calculate gamma value using finite difference method

I try to use the finite difference method to get the approximately gamma value, but there is an issue I can't solve. First, I set $h$ to 1 basis point of underlying asset value, but the result is not ...
2
votes
0answers
85 views

Multivariate interpolation for estimating FDM in-between grid points

After implementing some FDM to price some option, there are gaps between our grid points that may be of interest. From reading around, it appears common to use bilinear interpolation to estimate ...