Questions tagged [finite-difference-method]
The finite-difference-method tag has no usage guidance.
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Books on Finite Differences by Duffy
There is a well-known book from 2006 by Daniel Duffy, which is Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach. You may find it here on Wiley's:
https://...
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Black Scholes PDE explicit Scheme
I am currently working on the implementation of classic schemes to solve the BS PDE and it seems that I make a mistake in my code because the result looks far from the result of the BS formula.
Here ...
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For derivatives pricing, does FEM actually ever outperform FDM?
Simple question that I was wondering about over during the weekend.
I have done a little FEM during the last years and my university time and did not spend a lot of time with FDM. For a new job I have ...
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Convergence rate of Bermudan to American option
When trying to value an American option we often use grid-based methods (e.g. Monte Carlo in combination with Longstaff Schwartz; or Finite Difference Methods). As such, we are in fact estimating the ...
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Finite Difference Application
We all know that the traditional BS equation is:
$$\frac{\partial \mathrm V}{ \partial \mathrm t } + \frac{1}{2}\sigma^{2} \mathrm S^{2} \frac{\partial^{2} \mathrm V}{\partial \mathrm S^2}
+ \...
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How to solve numerically the IDE of GUILBAUD & PHAM model?
By the Guilbaud & Pham model (Optimal high frequency trading with limit and market orders, 2011), the authors said that integro-differential-equation (IDE)
can be easily solved by numerical method....
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Boundary condition issues for Black-Scholes PDE using finite-differences
I have been implementing an, in my opinion, interesting finite difference method (Runge-Kutta-Legendre of second order) to price American options in the standard Black-Scholes model (see "...
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Numerically stable method for estimating $\partial_t \mathbb{E}[f(X_t)]$ where $X_t$ is an n-dim Ito process and $f:\mathbb{R}^n\rightarrow\mathbb{R}$
Assume $(X_t)_{t\geq 0}$ follows an SDE of the form:
$$dX_t = a(t, X_t) dt + b(t, X_t) dW_t$$
where $W$ is a standard $n$-dimensional Brownian motion, $a$ and $b$ are mappings from $\mathbb{R}_+\times\...
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Finite difference methods with discontinuity in the payoff function
I have implemented a finite difference scheme for pricing options using a Black-Scholes-like model. I tested my implementation on a call option, and found that it gave extremely inaccurate results. I ...
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Approximating second derivatives at boundary of finite difference scheme
The Question
I am implementing a finite difference scheme for the Heston-Hull-White PDE:
\begin{align}
\frac{\partial u}{\partial t} &= \frac{1}{2}s^2v\frac{\partial^2 u}{\partial s^2 } + \frac{1}{...
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Dealing with the ru term in an ADI Finite Difference Scheme
I'm trying to code up the algorithm from this paper. The paper presents an ADI algorithm for pricing options in the Heston-Hull-White model.
The starting point is the Heston-Hull-White PDE, given ...
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Finite Difference Method in Greeks (Options)
I need a way to approximate the analytical formula of Greeks of a generic call option using the Finite Difference Method.
For example, the FD method for Delta/Gamma is the following one:
Now, I am in ...
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Question on boundary conditions when using Finite Difference
I have two questions appearing to me (they are not related directly to each other).
My first question is about boundary conditions when using Finite difference methods. There are two ways to do it: a)...
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Calculate error at all spatial indices for a given time step between BS equation and its numerical solution using explicit method
I am using the explicit finite backward difference scheme to discretize and calculate the price of an European call option in a discretization stencil.
My goal is to find the error at a given time ...
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Negative Density in Local Stochastic Volatility (LSV) Model Calibration
I'm trying to calibrate Local stochastic volatility model using finite difference method, and I'm mainly following this referece: Tian (2015).
I met a problem when calibrating leverage function - the ...
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Boundary Conditions for Call Option in Black Scholes Model
Let $C(t,S)$ be the value function of a call option. I want to price that option using (explicit) finite differences and the Black Scholes PDE. I consider the grid $0=t_0<t_1<...<t_{N-1}<...
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Implicit Scheme for Cox-Ingersoll-Ross Model PDE
I am considering the PDE for the price of a bond $V(r,t)$ with maturity $T$ under the Cox-Ingersoll-Ross model,
$$V_t+\frac12\sigma^2rV_{rr}+\nu(\theta-r)V_r-rV=0\quad r>0, t\in(0,1)$$
with ...
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Maximum norm stability for implicit Black-Scholes equation
I am trying to prove maximum norm stability for the following implicit approximation to the Black-Scholes equation
$$\frac1{\Delta t}\left(U_j^{(n+1)}-U_j^{(n)}\right)+\frac{rS_j}{\Delta S}\left(U_{j+...
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Quick Discretization question for finite difference and finite element methods
Assume we have the discretization in space $x_1, x_2, ... , x_M$ and time $t_1, t_2, ... , t_N$ for a finite difference or finite element method for option pricing and we want to solve for the option ...
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boundary conditions in finite element method
In the appendix A of this paper, https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.227.5073&rep=rep1&type=pdf, a finite element method is demonstrated to price a straddle. The same ...
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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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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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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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659
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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,
$$
Gamma = \frac{CallPrice(S^{up}_{T})...
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Operator splitting method on three assets black scholes equation
Currently I am studying finite difference method on derivatives with three (or more) underlyings and little bit confused on operator splitting method because two papers have different result.
For the ...
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2
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Implicit finite difference method always guarantees positive and stable price of derivative?
For the following black scholes pde
$$
f_t + rSf_S+\frac{1}{2}\sigma^2S^2f_{SS} = rf
$$
By denoting $f_{i}^{n} = $ Price of derivative at price node $i$ and time node $n$ and assume uniform grid, the ...
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Improve Finite Difference Scheme
I understand how to derive and implement standard finite difference schemes. I wonder how to improve such a standard FD scheme? For example, when solving the standard Black-Scholes equation, the ...
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Stability of Finite Difference method for Breeden-Litzenberger
I am trying to derive a risk-neutral density from European call option prices using a second order finite difference scheme. Let $C(K,T)$ be the price of a European call with strike $K$ and expiry $T$ ...
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Canonical text on numerical PDEs in finance
I am looking for a text similar to Glasserman's Monte Carlo Methods in Financial Engineering, but with a focus on numerical methods for PDEs. Glasserman's book seems to cover a lot for what is ...
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Benchmark a Libor Market Model implementation
Assume I have implemented a solution of the Libor Market model PDE in terms of the Finite Difference method. What is a good strategy for validating and benchmarking the results of this implementation?
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Numerical Solution to 3 Dimensional Backward BS PDE
I have a three dimensional backward BS PDE.
$$ \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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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