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Questions tagged [monte-carlo]

Monte Carlo simulation methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results.

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36
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2answers
12k views

How useful is Markov chain Monte Carlo for quantitative finance?

Naively, it seems that Bayesian modeling, structural models particularly, would be quite useful in finance because of their ability to incorporate market idiosyncrasies and produce accurate ...
21
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1answer
2k views

Portfolio optimization with monte carlo sampling from predictive distribution

Let's say we have a predictive distribution of expected returns for N assets. The distribution is not normal. We can interpret the dispersion in the distribution as reflection of our uncertainty (or ...
16
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3answers
15k views

Is there a step-by-step guide for calculating portfolio VaR using monte carlo simulations

I am trying to determine a step-by-step algorithm for calculating a portfolio's VaR using monte carlo simulations. It seems to me that the literature for this is extraordinarily opaque for something ...
15
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5answers
7k views

How to get greeks using Monte-Carlo for arbitrary option?

Let's assume I have an arbitrary option that I can price using Monte-Carlo simulation. What is the general approach (i.e. without relying on specific option type) to calculating the greeks in this ...
14
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5answers
2k views

Monte carlo methods for vanilla european options and Ito's lemma.

I understand that by applying Ito's lemma to the following SDE $$dX=\mu\,X\,dt+\sigma\,X\,dW$$ one obtains a solution to the above SDE which is as follows: $${X}\left( t\right) =\mathrm{X}\left( 0\...
14
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1answer
1k views

Is creating constrained random portfolios a hard problem?

Creating random portfolios with weights $x_i$ can be thought of as sampling from the surface of a simplex given by $$Ex = f$$ and $$Ax \le b$$ Where $E$ and $A$ are constraint matrices for equality ...
14
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1answer
553 views

Consistency of economic scenarios in nested stochastics simulation

I am interested in references on research regarding the consistency of economic scenarios in nested stochastics for risk measurement. Background: Pricing by Monte-Carlo: For pricing complex ...
13
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4answers
3k views

Methods for pricing options

I'm looking at doing some research drawing comparisons between various methods of approaching option pricing. I'm aware of the Monte Carlo simulation for option pricing, Black-Scholes, and that ...
12
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4answers
2k views

Are these steps correct to calculate Value-at-Risk with a Monte Carlo simulation?

I have a problem calculating VaR with the Monte Carlo Simulation. I followed the next steps and would like know if it is a right way to calculate VaR or if I need something more? The steps Generate ...
12
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2answers
986 views

Distribution of Geometric Brownian Motion

Please let me know where I have been mistaken! Let the SDE satisfied by the GBM $S(t)$ be $$ \frac{dS(t)}{S(t)} = \mu dt + \sigma dW(t). $$ Then, the underlying BM $X(t)$ will satisfy $$ dX(t) = \...
11
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3answers
5k views

Black-Scholes under stochastic interest rates

I'm trying to implement the Black-Scholes formula to price a call option under stochastic interest rates. Following the book of McLeish (2005), the formula is given by (assuming interest rates are ...
11
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3answers
1k 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?
11
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2answers
3k views

How to transform process to risk-neutral measure for Monte Carlo option pricing?

I am trying to price an option using the Monte Carlo method, and I have the price process simulations as an inputs. The underlying is a forward contract, so at all times the mean of the simulations is ...
10
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2answers
7k views

When to use Monte Carlo simulation over analytical methods for options pricing?

I've been using Monte Carlo simulation (MC) for pricing vanilla options with non-lognormal underlyings returns. I'm tempted to start using MC as my primary option-valuating technique as I can get ...
10
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1answer
208 views

Simulating property price index

I am trying to write a Monte Carlo simulation to calculate risk associated with some property based products. What is the most reasonable stochastic process to model property price index? Do people ...
9
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2answers
4k views

Simulation of GBM

I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem: Given a GBM of the form $dS(t) = \mu S(t) dt + \...
9
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1answer
411 views

Simulating the joint dynamics of a stock and an option

I want to know the joint dynamics of a stock and it's option for a finite number of moments between now and $T$ the expiration date of the option for a number of possible paths. Let $r_{\mathrm{s}}$ ...
9
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1answer
977 views

Monte carlo portfolio risk simulation

My objective is to show the distribution of a portfolio's expected utilities via random sampling. The utility function has two random components. The first component is an expected return vector ...
8
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4answers
4k views

Stock Price Behavior and GARCH

In my (limited) understanding, the behavior of a stock price can be modeled using Geometric Brownian Motion (GBM). According to the Hull book I'm currently reading, the discrete-time version of this ...
8
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2answers
9k views

Two correlated brownian motions

Is it true (see here, footnote 2, p.22 / p.14, without proof) that we can obtain two discretized brownian motions $W_t^1, W_t^2$ with correlation $\rho$ by doing $$d W_t^1 \sim \mathcal N(0,\sqrt{dt}...
8
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5answers
2k views

portfolio optimization from empirical return distributions

I'd like to do a portfolio optimization of a set of ETF's but want to avoid traditional problems with normality assumptions in returns etc. Are there techniques that let me sample 'draws' from the ...
8
votes
1answer
784 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
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2answers
701 views

Practical implementation of Least Squares Monte Carlo (tweaks and pittfalls)

The Longstaff-Schwartz LSM approach is nowadays ubiquitous(at least in the academic literature) in pricing path dependant derivatives. Up to now I have mostly worked with lattice methods. My ...
8
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1answer
1k views

Simple model for option premium (for covered call simulation)?

Given a historical distribution of weekly prices and price changes for a stock, how can I estimate the the option premium for a nearly at-the-money (ATM) option, say with an expiration date 3 months ...
8
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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
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2answers
799 views

Is there any research on applying state-space or dynamic linear models to forecasting equity risk premia?

Is there any research on applying state-space or dynamic linear models to forecasting equity risk premia on a security-by-security basis with a medium term horizon (say 3 month to 12 months horizon)? ...
8
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2answers
1k views

Estimation of Empirical Expected Shortfall of a heavy tailed distribution

Assume that you have a portfolio for which you have estimated a parametric model to the underlying instruments, but the distribution of the portfolio as a whole is too complicated to compute ...
8
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3answers
3k views

Simulate correlated Geometric Brownian Motion in the R programming language

In response to this question: How to simulate correlated Geometric brownian motion for n assets? One of the responses provides an implementation in MATLAB: http://www.goddardconsulting.ca/matlab-...
8
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1answer
5k views

Longstaff-Schwartz (Least Squares Monte Carlo) applied to American Options

I'm working on an implementation in R of Longstaff & Schwartz method from the this 2001 article. I've managed to build code that replicates their prices in table 1 (p. 127), but only for the ones ...
8
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1answer
1k views

Simulating conditional expectations

There is a multidimensional process X defined via its SDE (we can assume that its a diffusion type process), and lets define another process by $g_t = E[G(X_T)|X_t]$ for $t\leq T$. I would like to ...
8
votes
1answer
329 views

Advantage of solving the Fokker-Planck equation over Monte-Carlo simulations

For a standard Ito process $$dX_t = \mu(X_t, t) \,dt + \sigma(X_t, t) \,dW_t,$$ the Fokker-Planck or forward Kolmogorov equation gives an equation for the probability density $p ( x , t )$ of the ...
8
votes
1answer
579 views

Do you have a validation set for Libor Market Model implementation?

I'm trying to calibrate a Libor Market Model (LMM) in Matlab with my user-defined function, not their package. I already fitted the market volatilities using SABR but failed to simulate the ...
7
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2answers
662 views

Multithreading Monte-Carlo pricing in QuantLib for a single product

I've been actively using QuantLib for structured product pricing using Monte Carlo. Due to the fact that at a great deal of paths are often needed and one needs to speed up the calculation and all ...
7
votes
2answers
6k views

How do I estimate convergence in monte carlo methods?

I am experimenting with Monte Carlo methods. I'd like to measure/estimate convergence with a graph/chart. How do I do that? Can anyone please direct me to relevant documentation/links or even give me ...
7
votes
2answers
605 views

Interpret simulation results ($P$ and $Q$ measures)

I am struggling in interpreting results of my simulations. I use Monte Carlo algorithm to simulate stock paths and calculate option price. The notation: $r$ is a risk free interest rate, $T$ is time ...
7
votes
3answers
1k views

What are the merits of pseudo random numbers over quasi random numbers in monte-carlo simulation?

I understand that quasi-random numbers have much better convergence, but are there any reasons for me to use pseudo-random numbers instead?
7
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2answers
2k views

Principle Component Analysis vs. Cholesky Decomposition for MonteCarlo

Let's assume we have a portfolio containing large number (~500) of risk factors. We want to simulate the portfolio dynamics. PCA based simulation would be faster as we can reduce the dimensionality. ...
7
votes
2answers
849 views

Vanilla European options: Monte carlo vs BS formula

I have implemented a monte carlo simulation for a plain vanilla European Option and I am trying to compare it to the analytical result obtained from the BS formula. Assuming my monte carlo pricer is ...
7
votes
2answers
726 views

Least Square Monte Carlo - american Call Option

An American Call Option on an non dividend paying stock has the same value as a european one. I tired to compare the results given by the LSM with the results given by the B&S formular. It seems ...
7
votes
1answer
1k views

Sanity check - How to price callables

This question is meant as a sanity check whether i got the workflow right for pricing callable bonds. If anyone finds a mistake, or has a suggestion, please answer. The workflow is: For every call ...
7
votes
2answers
725 views

How to minimize the difference between a parametric VaR and a MC-VaR with lognormal assumption?

Given that we want to find the Value at Risk for a portfolio of stocks only, there are two main methods to proceed. In the problem, we also assume that stocks follow a geometric Brownian motion. A ...
7
votes
2answers
538 views

How to reduce variance in a Cox-Ingersoll-Ross Monte Carlo simulation?

I am working out a numerical integral for option pricing in which I'm simulating an interest rate process using a Cox-Ingersoll-Ross process. Each step in my Monte Carlo generated path is a ...
7
votes
1answer
840 views

Libor Market Model Calibration

Currently I am doing a research on the plain vanilla multi-curve framework Libor Market Model meaning that no stochastic volatility is involved. I had the idea to calibrate to the swaption market. In ...
7
votes
2answers
1k views

How to estimate the greeks with a Monte Carlo simulation?

I am simulating the path of three indices to price a 1 year basket option. All the indices are domestic, so there is no currency component. At each time step I am using the local volatility ...
7
votes
1answer
6k views

How to simulate a jump-diffusion process?

I would like to price Asian and Digital options under Merton's jump-diffusion model. To that end, I will have to simulate from a jump diffusion process. In general, the stock price process is given ...
7
votes
2answers
619 views

Whites Reality Check for Pair Trading

I want to use the Monte Carlo Method described in Aronsons book Evidence based Technical Analysis to test if a given pairs trading strategy is useless. First step there is to randomize the returns of ...
7
votes
1answer
147 views

Should we apply practical constraints on the distribution of monte carlo paths?

to limit interest rate paths to a 'reasonable' range (if we could define reasonable). Now we calibrate log-normal skew and mean reversion monthly to robust basket of atm swaptions and in and out caps....
6
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4answers
9k 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
3answers
3k views

How to perform Monte-Carlo simulations to price Asian options?

If I wish to price a fixed-strike Asian Call option via Monte-Carlo (This has no early-exercise), are my following steps correct?: 1) Simulate random asset prices. (Milstein) $\ d S(t) = \ rS(t)dt + ...
6
votes
3answers
3k views

Greeks: Why does my Monte Carlo give correct delta but incorrect gamma?

For a vanilla European call, my Monte Carlo method gives the right option price and delta but the wrong gamma. In particular, the value of gamma varies wildly each time I run the method. I estimate ...