Questions tagged [finite-difference]
Finite difference is a numerical procedure used to approximate derivatives computation by a linear combination of the value of the function at some specific points. This is particularly useful when solving PDEs and SDEs which involve discretization in both time and state dimensions.
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Gamma for a basket option in Python - Finite Differences vs. AAD Autograd library using Heaviside Approximation
I have been trying to use the Heaviside Approximation for a simple basket option so that I can solve for Gammas with AAD (Adjoint Automatic Differentiation). This routine smooths the payoff function ...
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Theta discretization PDE
I am trying to understand the validity of why we can theta discretize the solution to a PDE. For a PDE following:
$$0 = \partial_tf + A f$$
I understand that for one discrete time step the solution to ...
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Combine standard error in finite difference with Monte Carlo
I'm using Montecarlo to estimate the value of an option,
$$\overline V(S_T, r, \sigma, T;N)=\mathbb{E} \left[V(S_T, r, \sigma, T)\right]$$
which comes with a standard error $SE$.
I'm using "bump-...
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Finite Differences Vega calculation - confirmation on proper approach
I have a MC simulation that uses finite differences to calculate the Greeks. It's for baskets and calendar spreads mostly.
Now the logical (to me anyway) approach to calculate Vega is to increase the ...
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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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How to approximate a delta using monte carlo methods and finite differences via Higham's book?
I'm currently taking a Mathematical Finance module at University and one of the recommended texts is “An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation” by D.J. ...
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Confusion about terminology : Finite difference for option pricing
Consider the following initial-boundary value problem for $u = u(x,t),$
$$u_t - a u _{xx} = f(x,t) \text { for } 0 < x < L \text { and } 0 < t< T$$ along with bunch of initial and boundary ...
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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
$$\frac1{\Delta t}\left(U_j^{(n+1)}-U_j^{(n)}\right)+\frac{rS_j}{\Delta S}\left(U_{j+...
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Local Volatility Model Error
I am implementing my local volatility pricer using the finite difference method in MATLAB. I parametrise the implied volatility surface using the SSVI parametrisation (Gatheral & Jacquier), which ...
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Operator splitting method on three assets black scholes equation
Currently I am studying finite difference method on derivatives with three (or more) underlyings and little bit confused on operator splitting method because two papers have different result.
For the ...
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Canonical text on numerical PDEs in finance
I am looking for a text similar to Glasserman's Monte Carlo Methods in Financial Engineering, but with a focus on numerical methods for PDEs. Glasserman's book seems to cover a lot for what is ...
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Numerical Solution to 3 Dimensional Backward BS PDE
I have a three dimensional backward BS PDE.
$$ \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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Kirk Spread Approximation, Greeks by Finite Difference
I am using finite difference on Kirk's Approximation for Spread Options to estimate greeks of the Spread Option. Now this is creating an problem in the estimation of gamma. For at the money options (...
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Dupire Formula question
I want to calculate the local volatility from Dupire's formula:
$\sigma _{VL}^{2} (K,T,S_{0}) = \frac{\frac{\partial C}{\partial T}}{\frac{1}{2} K^{2} \frac{\partial^2 C}{\partial K^2}}$
So I use ...
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Comparison of various improvements to Hagan's SABR formula?
There has been several papers improving the original Hagan's approximation formula (see this answer) to SABR model. At least, I know three below:
Obloj
Paulot (Also see this thread)
Balland (Download)...
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