Questions tagged [finite-difference-method]

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Determining the early exercise curve of an American option

When I have found the price of an American option using, say, a finite difference scheme - how do I find the early exercise curve from this solution? Here is my idea: What I have is the price of the ...
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1answer
118 views

Negative theta for a short put

I am getting a negative theta for a short put deal Is it possible and if yes then under what conditions. Kindly explain I am just learning these concepts so my question may sound vague to some of you ...
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14 views

Backward difference approximation (BDF-2) for Options

I am working on a project for compound options and the assignment is as following: ...
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57 views

What are the boundary conditions for the Forward contract PDE?

European call When solving the PDE for the value $V$ of a European call option under the Black-Scholes model using a finite difference scheme, we have that Initial/terminal condition. $V(S_T,T) = \...
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Issue in Understanding the Boundary Conditions for European Call Option in Implicit Finite Difference Method

I have a working Python code which prices European call option in Implicit Finite Difference setting. However, I am unable to understand the Boundary Conditions implemented on the coefficient matrix ...
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15 views

Discrete Geometric Average Methodology for Pricing Asian Option using Finite Difference in Python

I am able to price Asian Options using Discrete Arithmetic Average in Finite Difference scheme and implementing the same in Python. However I am struggling to write the code for the same in Discrete ...
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43 views

Error in Call Option Valuation using Implicit Finite Difference implemented in Python

I am trying to valuate call option using implicit Finite difference method (Forward Marching) implemented in Python. However I am getting the error in the code. Following is the code I have developed: ...
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1answer
122 views

Greeks: Estimate gamma by Monte Carlo finite difference

When I was using Monte Carlo to calculate the gamma of a vanilla call option by finite difference method, I stuck in this weird situation as below. Consider this, $$ Gamma = \frac{CallPrice(S^{up}_{T})...
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73 views

Operator splitting method on three assets black scholes equation

Currently I am studying finite difference method on derivatives with three (or more) underlyings and little bit confused on operator splitting method because two papers have different result. For the ...
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2answers
182 views

Implicit finite difference method always guarantees positive and stable price of derivative?

For the following black scholes pde $$ f_t + rSf_S+\frac{1}{2}\sigma^2S^2f_{SS} = rf $$ By denoting $f_{i}^{n} = $ Price of derivative at price node $i$ and time node $n$ and assume uniform grid, the ...
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143 views

Improve Finite Difference Scheme

I understand how to derive and implement standard finite difference schemes. I wonder how to improve such a standard FD scheme? For example, when solving the standard Black-Scholes equation, the ...
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44 views

Stability of Finite Difference method for Breeden-Litzenberger

I am trying to derive a risk-neutral density from European call option prices using a second order finite difference scheme. Let $C(K,T)$ be the price of a European call with strike $K$ and expiry $T$ ...
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1answer
48 views

Canonical text on numerical PDEs in finance

I am looking for a text similar to Glasserman's Monte Carlo Methods in Financial Engineering, but with a focus on numerical methods for PDEs. Glasserman's book seems to cover a lot for what is ...
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31 views

Black Scholes PDE with explicit finite difference implementation

I am implementing the explicit finite difference method to solve Black Scholes PDE, but I have difficulty to find out why it is not working (my python code below) Is it an issue of instability in my ...
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heat equation : why my code does not generate enough decay in time?

I am new to solving PDEs with finite difference methods. I implemented the code below to solve the heat equation following the explicit scheme, but when I plot the result I am suprised that the decay ...
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115 views

Benchmark a Libor Market Model implementation

Assume I have implemented a solution of the Libor Market model PDE in terms of the Finite Difference method. What is a good strategy for validating and benchmarking the results of this implementation? ...
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1answer
69 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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2answers
2k 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 = ...
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260 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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1answer
102 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 "...
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2answers
188 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-...
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1answer
66 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+...
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1answer
247 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 ...
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83 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} \...
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1answer
206 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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233 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 ...
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63 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: $$...
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1answer
44 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 ...
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1answer
85 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 ...
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3answers
124 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 ...
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1answer
42 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 ...
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43 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 ...
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1answer
1k 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?
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200 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 ...
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1answer
230 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-...
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1answer
144 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}{\...
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1answer
142 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}{\...
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1answer
293 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. ...
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1answer
512 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....
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1answer
1k 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 ...
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1answer
603 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
156 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 ...
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305 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 ...
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160 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{\...
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58 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 ...
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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 ...
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487 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 ...
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380 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 ...
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143 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 ...
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1answer
221 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 ...