Questions tagged [numerical-methods]

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54 views

Accuracy of Explicit Euler method (finite difference) decreases as Δx decreases, shouldn't it increase?

The price of a commodity can be described by the Schwartz mean reverting SDE $$dS = \alpha(\mu-\log S)Sdt + \sigma S dW, \qquad \begin{array}.W = \text{ Standard Brownian motion} \\ \alpha = \text{ ...
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0answers
128 views
+50

Solve the Schwartz mean reverting PDE for option pricing using Euler explicit method (matlab)

Objective: Implement the Euler Explict Method for solving the PDE for option prices under the Schwartz mean reverting model. The price evolution of a commodity can be described by the Schwartz SDE $$...
-1
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1answer
53 views

Good ways to select best decision among N decisions, each with a profit/loss distribution? [closed]

I'm working on a problem where an asset owner (e.g., owner of a factory, power plant, etc.) can take a number of possible decisions (say 10). Each of those 10 decisions entails certain actions, but ...
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0answers
110 views

CIR model. Is there a closed-form solution or even a good proxy of analytical solution?

Is there a closed-form (analytical) solution for the Cox-Ingersoll-Ross SDE \begin{equation} dr_t=k_r(\theta_r-r_t)dt+\sigma_r\sqrt{r_t}dW_t\tag{1} \end{equation} ? Notice that $\{r_t\}$ is our ...
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1answer
55 views

How to compute returns from cumulative returns in Python? [closed]

If X is a $T\times N$ pandas DataFrame of multivariate asset returns, the cumulative returns can be computed in python as (1 + X).cumprod() - 1 How can I reverse this operation so that I go ...
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1answer
73 views

How to simulate asset prices/returns that display market regimes?

Are there any techniques that can make a multivariate random number generating process for stock prices/returns, like geometric Brownian motion via Cholesky, also include the simulation of a finite ...
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0answers
13 views

Is it easier to estimate the mean of volume/dollar bar returns than time bars?

Financial data, by default, is sampled using constant intervals of time, i.e. we collect the prices of assets every day, and then take the log-differences of prices to compute daily returns. These are ...
8
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2answers
133 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 ...
1
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1answer
137 views

How to find characteristic function in Fourier Cosine method (COS method) by Fang and Oosterlee

Fang and Oosterlee (2009) introduced Fourier-Cosine method (COS method) in their paper. The formula to price an option is approximately $$e^{-r\Delta t} \sum_{k=0}^{N-1}' Re\left\{ \phi\left( \frac{k\...
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1answer
76 views

Overview of frequentist, likelihood and Bayesian approaches to finance problems

In quantitative finance tasks (asset pricing, portfolio optimization, option pricing, volatility forecasting, etc), there are frequentist, likelihoodist and Bayesian approaches or interpretations to ...
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2answers
107 views

Are asset return means difficult to predict because they have no lower bound?

In finance, it is widely known that the volatility of asset returns ($\sigma$) are easier to predict than the expected value of asset returns ($\mu$) , otherwise known as the average return or mean. ...
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0answers
51 views

Risk-Neutral covariance matrix of arbitrage-free Nelson Siegel

For my thesis on a Bayesian sampling routine for a modification on arbitrage-free Nelson-Siegel I came across an equation that involves a matrix exponential within an integral, i.e. $\int_{0}^{\Delta ...
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0answers
46 views

Longstaff Schwartz method (LSM): how to increase accuracy?

In the LSM method, I am currently (as they do in the paper) using weighted Laguerre polynomials as basis functions, about 3-5 of them. If I wish to increase the accuracy of my model, what should I do?...
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0answers
48 views

How is it possible that the measurement uncertainty in Kalman Filter is less than 0?

In Euan Sinclair's Option Trading, Pricing and Volatility Strategies and Techniques, it mentions that the true value of the price can be estimated via Kalman Filter: $$S_\mathrm{new} = S + k (S_b − S)...
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1answer
100 views

L2 Assumptions of the Longstaff Schwartz method

In page 121 of the original LS Paper they use the fact that the space of functions they are dealing with (payoffs of American options), belong to the $\mathcal L^2$ space. They use this assumption ...
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0answers
46 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). ...
1
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1answer
183 views

QuantLib returns slightly different bondYield when backtested

I am just starting to get familiar with QuantLib (in particular, fixed rate bond pricing functions). I read a number of examples, from which I am able to calculate bond price and bond yield. The ...
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0answers
29 views

How to approximate expectation and variance of an integral from a discrete Time series financial dataset?

I have discrete time series financial data, with time($u$), price($S$) and someVariable($q$) which looks something like this. ...
4
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1answer
174 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 ...
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0answers
31 views

How important is it that numerical methods can price for various strikes simultaneously?

I am reading a paper which presents a numerical method to price call options. Call this Method 1. The method can also price several call-options for a range of strikes simultaneously if you want it to,...
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1answer
814 views

Simulation of Geometric Brownian Motion in R

Using R, I would like to simulate a sample path of a geometric Brownian motion using \begin{equation*} S(t) = S(0) \exp\left(\left(\mu - \frac{\sigma^{2}}{2}\right)t + \sigma B_{t}\right), \end{...
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2answers
121 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, ...
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1answer
59 views

Calibrate a model parameter with an error function

Suppose I want to find the implied volatility using an option model from market prices. Surely I can find the implied volatility for each strike price ($k$ different strike prices) for a given ...
3
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1answer
82 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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2answers
96 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 ...
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0answers
79 views

Numerical Solutions to PDEs with Financial Applications

I am reading a paper by Richard White, Opengamma named Numerical Solutions to PDEs with Financial Applications. There is an implementation codes as stated in paper hosted at https://opengamma.com/...
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1answer
59 views

Sensitivity Approximation - Crank Nicolson

I am looking into a new method of calculating sensitivities starting off with a proof of concept with Black Scholes PDE. Suppose I want to calculate Rho and take the derivative of the PDE (heresy!!) ...
1
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1answer
87 views

What are the industry standards and rules of thumb when it comes to numerical methods?

So, as far as I know, we have 3 main numerical methods. Monte Carlo, PDE-methods (FDM), and numerical integration methods (Fourier transforms and so on). How do these methods generally compare to ...
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0answers
73 views

How good is a “good accuracy” in pricing?

Say you want to test various numerical algorithms for purposes of pricing. How close do you need to be to some benchmark value (the "actual" price) for your accuracy to be good? Say I am trying to ...
2
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1answer
140 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....
3
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1answer
400 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?
3
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1answer
257 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
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1answer
598 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 ...
2
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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 "...
5
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1answer
226 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 ...
1
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0answers
159 views

What is the best source to get 10 milliseconds time-series data for numerical computation?

I am working with 4th order Runge-Kutta method to compute a second order differential equation. For the best accuracy, I need a 10 milliseconds ohlcv time-series data. I know that I can build it ...
3
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0answers
221 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. ...
1
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1answer
69 views

In search of nice (approx) function forms of the volatility of cumulative simple returns

Let's consider a period $t\in[0,T]$, and let the simple return over year $t$ ($1\le t\le T$) be $r_t$. Assume $r_t$ are iid normal. The cumualative simple return over the whole period $[0,T]$ is $$R_T=...
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2answers
123 views

Paper recommendation with examples

Could you please advise me some papers/working documents with applications mainly focused on Fixed Income/Financial Engineering (Numerical Methods) as the one in the link below? Preferably on Matlab, ...
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0answers
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: $$...
1
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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 ...
2
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1answer
228 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-...
1
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1answer
143 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}{\...
2
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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}{\...
4
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1answer
289 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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0answers
50 views

Computation of an integral containing d ln x (Scale of Market Shocks)

I am trying to implement a Scale of Market Shocks method (SMS) which was presented in a 1999 working document by Olsen & Associates named Introducing a Scale of Market Shocks and later refined in ...
0
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1answer
58 views

How to simulate a path through its solution and conditional expectation / variance

Hi I want to simulate in Matlab the following stochastic integral: $ x(t) = x(s) e^{-a(t-s)} + \sigma \int_s^t e^{-a(t-u)} dW_1(u)$ with $E[x(t) \vert F_s] = x(s) e^{-a(t-s)}$ $Var[x(t) \vert ...
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0answers
30 views

Transform the payoff to be non-zero

Is there any way to transform the basic call option payoff $V(s,0) = \max(s-K,0)$ such that $g(V(s,0))\neq 0$ $\forall s $, where $g()$ is the transform function of the payoff. This is to use in a ...
2
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0answers
38 views

Transforming and minimisation of the BS PDE

I'm trying a novel numerical substitution/fitting method to solve the BS PDE, but the issue is that due to the large range of magnitude of prices $V(s,t)\in[10^{-20},10^1]$, when I try to minimise the ...
1
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1answer
79 views

4-point Trapezium rule for numerical integration

Background: This is in reference to Mark Joshi's concepts of mathematical finance ch.7 problem 11. Question: We have in the Black-Scholes model: $S_0 = 1, T = 1, \sigma = 0.1, r = 0$. A ...