Questions tagged [numerical-methods]

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25 votes
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How to quickly estimate a lower bound on correlation for a large number of stocks?

I would like to find stock pairs that exhibit low correlation. If the correlation between A and B is 0.9 and the correlation between A and C is 0.9 is there a minimum possible correlation for B and C? ...
Joshua Chance's user avatar
23 votes
3 answers
4k views

When do Finite Element method provide considerable advantage over Finite Differences for option pricing?

I'm looking for concrete examples where a Finite Element method (FEM) provides a considerable advantages (e.g. in convergence rate, accuracy, stability, etc.) over the Finite Difference method (FDM) ...
Alexey Kalmykov's user avatar
14 votes
3 answers
2k views

Effective Euro-USD (EURUSD) Exchange Rate Prior to Euro's Existence

Motivation: I am running a quantitative analysis that requires long-term, exchange rate data. Problem: Does anyone have methods for dealing with the EURUSD exchange rate prior to the Euro's existence?...
agathocles's user avatar
13 votes
2 answers
21k views

How to numerically obtain delta?

The delta in option pricing, also called the hedge ratio, is expressed as the sensitivity of the option price to the underlying price change. The analytical solution for the most common option ...
JohnAndrews's user avatar
12 votes
3 answers
2k views

What tools are used to numerically solve differential equations in Quantitative Finance?

There are a lot of Quantitative Finance models (e.g. Black-Scholes) which are formulated in terms of partial differential equations. What is a standard approach in Quantitative Finance to solve these ...
Roman's user avatar
  • 529
12 votes
1 answer
685 views

What weights should be used when adjusting a correlation matrix to be positive definite?

I have a correlation matrix $A$ for an equity market that is not positive definite. Higham (2002) proposes the Alternating Projections Method, minimising the weighted Frobenius norm $||A-X||_W$ where $...
Nemis's user avatar
  • 699
11 votes
3 answers
2k views

Reference on Markov chain Monte Carlo method for option pricing?

I have to implement option pricing in c++ using Markov chain Monte Carlo. Is there some paper which describes this in detail so that I can learn from there and implement?
Shane Wingard's user avatar
9 votes
5 answers
4k views

Is there a good closed-form approximation for Black-Scholes implied volatility?

While the solution for IV can certainly be reached using numerical search methods, I wonder if a high precision closed-form approximation exists. For example, there is a very robust (precise within ...
sashkello's user avatar
  • 989
9 votes
1 answer
820 views

QuantLib and exact numerical simulation

I've just downloaded quantlib and started playing around with it, and it looks like it's designed primarily to use Euler discretizations for everything -- so far as I can tell, there's not even a ...
user3296's user avatar
  • 278
9 votes
3 answers
6k views

Usage of Brownian Bridge?

I was recommended to read something about Brownian Bridge. Could someone familiar with BB give some recommendation? It was mentioned that BB benefits in 2 places BB could reduce the simulation paths,...
athos's user avatar
  • 2,231
9 votes
1 answer
1k views

Speeding up computations: when to use Quasi and standard Monte-Carlo in pricing

I am familiar with the theory of Monte-Carlo techniques in the numerical integration, and recently I have started my experiments with these methods applied to derivatives pricing. I am using ...
Ulysses's user avatar
  • 1,484
8 votes
5 answers
914 views

Heston Model Integration Oscillations

Is there a way to reduce oscillations for the numerical integration when evaluating the Heston model. I am pricing a series of 5000 options scattered over the Heston model parameter space and I find ...
Sam Palmer's user avatar
8 votes
2 answers
1k views

What is Quantization?

I have asked myself many times about Quantization Numerical Methods, is anyone here familiar with the subject and could give a reasonable insight of what Quantization concepts are about, and what are ...
TheBridge's user avatar
  • 4,563
8 votes
1 answer
1k views

Hyperbolic and Elliptic PDEs in Quant Finance

Parabolic PDEs (e.g. heat equation) are closely linked to finance via the Feynman Kac Theorem. Do other types of PDEs appear in quant finance? Elliptic PDEs don't contain a time dimension (so perhaps ...
Alex's user avatar
  • 83
8 votes
4 answers
20k views

What does "convergence" in Monte Carlo simulation mean?

I have read about convergence in terms of MC simulation for derivative pricing, but I am not clear on what it exactly means. Let us suppose I price an option 100,000 paths twice and both result in the ...
Karthik Balasubramaniam's user avatar
8 votes
2 answers
582 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 ...
Alex's user avatar
  • 688
8 votes
1 answer
446 views

When pricing options, what precision should I work with?

I'm wondering if there's any point at all in double-precision calculations, or whether it's ok to just do everything in single-precision, seeing how the difference on non-Tesla GPUs for single and ...
Dmitri Nesteruk's user avatar
7 votes
3 answers
8k views

What is an efficient method to find implied volatility?

I have a code that finds the implied volatility using the Newton-Raphson method. I set the number of trial to 1000 but sometimes it fails to converge and doesn't find the result. Is there a better ...
opt's user avatar
  • 559
7 votes
2 answers
1k views

Black-Scholes fastest computation method

What is the fastest way to numerically compute Black-Scholes-Merton option prices? I'm trying to find fastest and still precise method. Currently I'm using numerical approximation of Normal cdf with ...
Ilya's user avatar
  • 328
7 votes
0 answers
357 views

Solving option market making problem

I am currently working on a paper for quoting option as a market maker from Bastien Baldacci , Philippe Bergault & Olivier Guéant Without dwelling on details on how to obtain the HJB equation for ...
Kupoc's user avatar
  • 98
7 votes
2 answers
298 views

Likelihood ratio and pathwise sensitivity method for coupled SDEs

I have two coupled SDEs \begin{align*} dS_t=rS_tdt+V_tdW_t^{(1)},\\ dV_t=aV_tdt+b(V_t)dW_t^{(2)},\\ \end{align*} where $W_t^{(1)}$ and $W_t^{(2)}$ are independent Brownian motions, initial input data ...
user107224's user avatar
6 votes
2 answers
552 views

What is a cubature scheme?

Ideally an intuitive explanation with an example, please.
user40's user avatar
  • 2,687
6 votes
1 answer
1k views

How to apply quasi-Monte Carlo to path-dependent options?

Following up on my recent question on variance reduction in a Cox-Ingersoll-Ross Monte Carlo simulation, I would like to learn more about using a quasi-random sequence, such as Sobol or Niederreiter, ...
Tal Fishman's user avatar
  • 13.4k
6 votes
2 answers
755 views

Architecture of a global pricing library with immutable payoffs

By global pricing library I mean a library handling equity, rate etc, hybrid products having several models (BS, LV, SV, LSV) having several numerical methods (analytic formula, MC, PDE FD/FE) I ...
Olórin's user avatar
  • 1,223
6 votes
2 answers
12k views

Value of American Call vs Value of European Call when using implicit finite differences

I calculated values for put options (european and american) using the implicit finite difference method and compared them to black/scholes values. The values for american put options are higher than ...
FreshF's user avatar
  • 301
6 votes
0 answers
206 views

SABR-LMM: best way to perform a MC simulation

I am working on a SABR-LMM model with the following system of SDEs under a numeraire $N$: $$ \begin{align} &\mathrm{d} F_i(t) = \sigma_i (t) (F_i(t) + s)^{\beta} \Big( \mu^f_i (t) \mathrm{d}t ...
BEQuant's user avatar
  • 428
6 votes
1 answer
410 views

Importance sampling for Monte Carlo with local volatility in practice

I am given a diffusion with a local volatility to price barrier options: $$dX(t)=X(t)\mu dt+X(t)\sigma(t,X)dW_t$$ I want to use Importance Sampling to price barrier options "far" out of the ...
user56787's user avatar
  • 125
6 votes
0 answers
255 views

Use of Local Times in Option Pricing

I know two applications of local time in option pricing theory. First, it allows a derivation of Dupire's formula on local volatility in a neat way (i.e. without resorting to differential operator ...
TheBridge's user avatar
  • 4,563
5 votes
1 answer
4k views

Least Squares Monte Carlo

Could you explain to me in words (no formulas) the concept of the Least Squares Monte Carlo method to price an American style option?
Nasser Bin's user avatar
5 votes
2 answers
4k views

How to remove outliers in financial times series?

I have a bunch of time series; i need to clean them before modelling. So far I just know the “filtering/smoothing” method : -Ex: moving average methodology (filter the data with a moving average (...
Malick's user avatar
  • 2,572
5 votes
1 answer
592 views

In Carr-Madans option pricing method, why do they use FFT?

In the famous fourier option pricing method by Carr-Madan, (http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.348.4044&rep=rep1&type=pdf), the crucial formula is They evaluate this by ...
ewofeo's user avatar
  • 53
5 votes
4 answers
305 views

Asymptotics of Call Option as $S\to0$

Let $C(S)$ denote the (initial) value of a call option with underlying spot price $S$. I assume that the underlying has continuous sample paths (not necessarily a geometric Brownian motion though). As ...
Alex's user avatar
  • 688
5 votes
1 answer
1k views

Quadratic exponential method (by Andersen) in Heston model

I am having trouble understanding the reasons that led Andersen to define his QE scheme to efficiently simulate Heston Stochastic volatility model (you may check the celebrated scheme here). The ...
Adam's user avatar
  • 473
5 votes
1 answer
266 views

Parameter estimation using martingale measures - include real world data?

Please note: I posted this in nuclearphynance first, but didn't get any replies. For desks which sell exotics it is common practice (as far as I know it) to calibrate the model (Stochastic Volatility,...
user13655's user avatar
  • 215
5 votes
0 answers
145 views

Integrated Delta does not seem to be smooth (ATM, Heston)

I am interested in an integrated call option that removes the dependence on time, $$I(S)=\int_0^\infty C(S,t)\text{d}t.$$ Because the value of a call option is a smooth function, I expect this ...
Kevin's user avatar
  • 15.7k
5 votes
0 answers
173 views

Optimized search for yield-to-worst of a callable bond

Suppose that I need to find the yield-to-worst of a callable bond, and that the option is American (call any time). The bond may have step-up coupons and/or non-constant call price (oprion strike). ...
Dimitri Vulis's user avatar
4 votes
3 answers
515 views

Fastest way to calculate YTM from bond price

I would like to calculate YTM for every top of the book update on the 10-year note traded on Brokertec. There is no closed form solution so have to use a root finding method like Newton-Rhapson. It ...
bloodynri's user avatar
  • 219
4 votes
4 answers
444 views

Is it possible to defend a Computer Science master thesis by writing a project about quantitative finance?

What are features and examples of computational finance (financial computing) problems (for thesis project in Master in Computer Science)? Is it possible to defend Computer Science master thesis by ...
TomR's user avatar
  • 193
4 votes
1 answer
131 views

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+...
user107224's user avatar
4 votes
2 answers
945 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 ...
millovanovic's user avatar
4 votes
1 answer
775 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 ...
Vikash Balasubramanian's user avatar
4 votes
1 answer
611 views

How to determine the order of convergence of the Euler-Maruyama method?

To make this simple let us consider the Geometric Brownian Motions. My questions: 1. How can I show that the Euler-Maruyama Method is convergent using GBM? 2. How can I determine the order of ...
Sanjay's user avatar
  • 1,637
4 votes
1 answer
447 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. ...
Pim's user avatar
  • 117
4 votes
2 answers
571 views

Simulation scheme for SABR beside the standard Euler discretization

QUESTION: Beside Euler Scheme, is there another more robust (and preferably easy to implement) way to simulate asset path with SABR dynamics? Simulation that will withstand even for high volatilities....
Sanjay's user avatar
  • 1,637
3 votes
1 answer
1k views

Anyone has detailed explanation on how to use epstein-zin preferences in asset pricing models

I'd be interested to know how Epstein-Zin preferences are used in, say, consumption-based asset pricing models. I'm looking for specific derivations (how you get the SDF) and possible numerical ...
Stéphane's user avatar
  • 2,456
3 votes
2 answers
310 views

Produce the random variable for an asset from a uniformly distributed random varible

I'm working on a quant interview question from the book called Quant Job Interview Questions And Answers (by Mark Joshi and other authors). I cannot understand the following question(not the answer, ...
M00000001's user avatar
  • 627
3 votes
1 answer
3k views

Covariance matrix and Cholesky decomposition

I am simulating a spread option with stochastic volatility using Monte Carlo simulation. I have the positive-definite covariance matrix $$ \rho = \left( \begin{array}{cccc} 1 & \rho_{1,2} & \...
Alfie's user avatar
  • 223
3 votes
2 answers
87 views

Does discretizing a diffusion model make it look like a jump diffusion model?

Can we distinguish a sample generated from a diffusion model with large time steps from a sample generated from a jump diffusion model. Not mathematically but numerically (if we ask a computer to tell ...
bigInner's user avatar
  • 191
3 votes
1 answer
2k views

Implied volatility in Monte Carlo models

Suppose I want to get the implied volatility for a given option, whose process does not generate a closed-form formula. In that framework, how is the IV calculated, given the fact that bisection ...
alexbougias's user avatar
  • 1,416
3 votes
1 answer
95 views

When is a numerical solution the only way to obtain a solution to BS?

I am only now reading into Mathematical Finance, I understand the derivation of the BS equation with vanilla European options. On the next page of my book it starts to delve into obtaining exact ...
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