Questions tagged [numerical-methods]

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25
votes
2answers
2k views

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? ...
22
votes
3answers
3k 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) ...
14
votes
3answers
1k 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?...
12
votes
1answer
654 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 $...
11
votes
3answers
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 ...
11
votes
3answers
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?
10
votes
2answers
18k 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 ...
9
votes
1answer
682 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 ...
8
votes
5answers
3k 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 ...
8
votes
2answers
893 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 ...
8
votes
1answer
804 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 ...
8
votes
3answers
4k 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,...
8
votes
1answer
402 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 ...
7
votes
2answers
667 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 ...
6
votes
5answers
371 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 ...
6
votes
4answers
10k 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 ...
6
votes
2answers
463 views

What is a cubature scheme?

Ideally an intuitive explanation with an example, please.
6
votes
2answers
9k 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 ...
5
votes
2answers
3k 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 (...
5
votes
3answers
5k 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 ...
5
votes
1answer
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, ...
5
votes
1answer
225 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,...
5
votes
0answers
215 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 ...
4
votes
4answers
228 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 ...
4
votes
1answer
803 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 ...
4
votes
2answers
339 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
1answer
361 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 ...
4
votes
1answer
267 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. ...
3
votes
1answer
146 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 ...
3
votes
1answer
2k 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} & \...
3
votes
1answer
77 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 ...
3
votes
1answer
355 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 ...
3
votes
1answer
209 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 ...
3
votes
1answer
342 views

Finite difference methods

I am simulating the price of a basket option with the help of equations from the report http://www.it.uu.se/edu/course/homepage/projektTDB/vt07/Presentationer/Projekt3/...
3
votes
0answers
165 views

Is there a more efficient data structure to implement binomial trees than 2d array?

I'm just curious what is the "industry standard" for implementing a binomial tree (if "standards" exist in this case). For simplicity, let's just talk about the simplest trees with recombining nodes. ...
3
votes
0answers
75 views

Time discretisations, FDM vs FEM

I am interested in adaptive mesh methods for numerical solution of PDEs with applications to finance. As part of a school project, I have been pricing vanilla European call and put options using 2D ...
3
votes
0answers
422 views

What R-packages for SOCP problems are there?

Currently, I am looking deeper into the topic of second-order cone programming. Could you suggest packages that solve SOCP-problems in R? With your answer, please provide a short description of ...
2
votes
2answers
104 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, ...
2
votes
2answers
144 views

Approximating a function with trignometric polynomials

Let’s say I have a function, which is a time series of data points, I am trying to find a polynomial of fixed sine's and cosines that bests approximate the data points. I know Chebyshev Approximation ...
2
votes
1answer
509 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 ...
2
votes
1answer
219 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-...
2
votes
1answer
135 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}{\...
2
votes
1answer
242 views

Numerical Optimizer Matlab Calibration LMM

I am trying to mimimize the following function in order to calibrate the Libor Market Model $$\sum_{i=1}^{n} \left(\sigma_i^{market}-\sigma_i^{Reb}\left(a,b,c,d,\beta\right)/\sqrt{T_i}\right)^2,$$ ...
2
votes
1answer
55 views

Integration in the context of modelling with the Meixner Process

I failed to evaluate the integral of $\frac{e^{ax}}{x\sinh(bx)}$ with respect to $x$ from negative infinite to positive infinite. What techniques can I use to evaluate the integrals of such kind for ...
2
votes
2answers
175 views

How can I estimate the parameters of an option value model of retirement?

I am modelling an option value model of retirement, see for instance Stock and Wise (1990). I am however not sure to which class of problems this model falls into and hence which optimization method I ...
2
votes
1answer
93 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
1answer
268 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 ...
2
votes
1answer
298 views

Numerical Methods for Merton Model

The stochastic differential equation for an underlying with jumps in Merton model is: $$d{{S}_{t}}=\mu \,{{S}_{t}}dt+\sigma \,{{S}_{t}}\,d{{W}_{t}}^{P}+(J-1){{S}_{t}}d{{q}_{t}}$$ where $t \quad\,\,\, ...
2
votes
3answers
598 views

Numerical Solution to BS PDE - Digital Option

Here is a relatively simple question about PDE's pricing. Assume that we are within the BS framework and moreover that interest rate is zero. The price $V(t,S_t)$ of the digital is known to be $\Phi(...
2
votes
2answers
791 views

Problem when calculating the daily return on a forex trade, what is the best way to do such a calculation?

I intend to calculate the daily return on my investment in forex. Assume a trader invests $\$$40 at a leverage of 100:1, so in total he is trading $\$$4000 worth of currency, and assume the position ...