19 votes
Accepted

Black-Scholes under stochastic interest rates

We assume that the short interest rate $r_t$ follows the Hull-White model, that is, the short rate $r$ and the stock price $S$ satisfies a system of SDEs of the form \begin{align*} dr_t &= (\...
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  • 20.4k
15 votes
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Is there a step-by-step guide for calculating portfolio VaR using monte carlo simulations

You have the correct approach. (1) The simulation generates sampled portfolio values, $P_1,P_2, \dots, P_n$ at time $t=T$. VaR is specified as a left-tail percentile. Order the sample as $$P_{(1)} ...
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  • 3,240
12 votes
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Least Squares Monte Carlo

To compute the price of an American option or a callable instrument in general, at each potential exercise date, one is required to compare its continuation value (discounted risk-neutral expectation ...
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  • 13.9k
10 votes
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Practical implementation of Least Squares Monte Carlo (tweaks and pittfalls)

LSM is very fiddly. The most important things in my view are 1) don't believe anyone who says that the choice of basis functions doesn't matter. 2) implement an upper bounder, eg Andersen--...
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  • 6,743
10 votes

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

the problem is that the pay-off has discontinuous first derivative. Try a contract with pay-off that is twice differentiable and it will probably work. The problem is that all the value comes from ...
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  • 6,743
10 votes

Two correlated brownian motions

Here is the general approach you can follow to generate two correlated random variables. Let's suppose, X and Y are two random variable, such that: $$X \sim N(\mu_1, \sigma_1^2)$$ $$Y \sim N(\mu_2, \...
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  • 2,108
10 votes
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How to simulate Levy processes

You have many different options. Firstly, you know the characteristic function for the log stock price and, using inversion, you can recover the (inverse) distribution and density function and ...
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  • 13.7k
9 votes
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Two correlated brownian motions

First you need to correct the formula to: $$ W_t^2 = \rho W_t^1 + \sqrt{1-\rho^2} Z_t, $$ where $Z_t$ is a BM independent of $W_t^1$ If you calculate the variance and the covariance, then you see ...
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  • 13.2k
8 votes
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When to use the real world drift and when the risk neutral one for a Monte-Carlo simulation?

In general these are the two basic approaches to QuantFinance: Sell side (market maker, risk neutral): You use risk-neutral probabilities ("$\mathbb{Q}$") e.g. in option pricing (to e.g. calculate ...
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  • 26.7k
8 votes
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Speeding up computations: when to use Quasi and standard Monte-Carlo in pricing

By definition the fair value of an option is given by an expectation value of the payoff, $\mathbf{E}\left[\textrm{payoff}(\textit{paths})\right]$. The probability distribution of the ...
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  • 1,809
8 votes

Stopping Monte Carlo simulation once certain convergence level is reached

You have the right idea, but it seems you don't know $\mu$, so using it in your error check doesn't seem correct. Also, checking the result every 10,000 iterations may not be optimal for deciding ...
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  • 2,748
8 votes
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rate of convergence for Monte Carlo

The estimation error is a random variable and not a simple scalar. As such, when performing one-shot assessments, you could always end up observing that using $6400$ paths provides a "better" price ...
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  • 13.9k
8 votes

Least Square Monte Carlo - american Call Option

If implemented properly, least-squares Monte Carlo as originally suggested by Longstaff-Schwartz should allow you to identify sub-optimal exercise dates and a lower bound of the true option price. ...
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  • 13.9k
8 votes
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How are Brownian Bridges used in derivatives pricing in practice?

Yes, the term Brownian Bridge seems to be used loosely. I assume you are talking about continuously monitored barriers by the way, since you mention the probability of the barrier being crossed in ...
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  • 511
8 votes
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Multithreading Monte-Carlo pricing in QuantLib for a single product

Yes, it can work. However, keep in mind that: you'll be safer if you don't share any objects between threads; see my answer here, in particular the last point; even if you use different seeds, there'...
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8 votes

Limitations of Monte Carlo simulations in finance

The properties of standard Monte-Carlo are not determined solely by the underlying process. You need to include the instrument $f$ you want to price in your analysis as well. One measure for accuracy ...
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  • 1,933
7 votes
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What are the merits of pseudo random numbers over quasi random numbers in monte-carlo simulation?

Quasi Random Numbers are more tricky than it might seem, using them as a black box like with PRNGs is risky. E.g. an unscrambled Sobol' sequence is uniform only asymptotically, while for realistic ...
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  • 1,543
7 votes
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What are some examples of non-solvable SDE where Monte Carlo discretization is necessary

the LIBOR market model the Heston model -- Euler and Milstein are actually bad for this and much more sophisticated methods are necessary local volatility models
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  • 6,743
7 votes
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Calibration by monte carlo, should I fix my seed?

It is not cheating. It allows you to make your results (e.g. prices, calibrated parameters) 'reproducible' which is good. However, fixing the seed can hide convergence issues. When the variance of ...
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  • 13.9k
7 votes
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Why are Interest Rate Swaps not valued using Monte Carlo Simulations?

Forward rates are determined from current spot rates bootstrapped from traded instruments. The reason is that if the forwards were different from the ones inferred from the spot rates, there would be ...
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  • 5,285
7 votes

Limitations of Monte Carlo simulations in finance

When I was first tasked with implementing VaR using MC in the 1990s, I knew little about MC, and there were no good books. The draft manuscript of Reuven Y. Rubinstein, Dirk P. Kroese. Simulation and ...
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6 votes
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What does "convergence" in Monte Carlo simulation mean?

To keep things simple let's assume you have a perfect random number generator (i.e. I will discuss only the statistics not the numerics of the problem). I will also focus on the practical matter and ...
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  • 1,933
6 votes

Importance Sampling - where to center the sampling distribution?

importance sampling is well known to be tricky. See the extensive discussion in Glasserman's book. I presume that you are simply meanshifting and multiply by the ratio of normal densities. For this ...
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  • 6,743
6 votes
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How many monte carlo runs do I need for pricing a Call?

It really, really, really depends on your parameters, i.e. $r$, $\sigma$, $K$, $T$, $S_0$. For example, here are some results from implementing the stopping criteria I explain in my answer here. ...
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  • 2,748
6 votes

Pricing a log-contract using Monte Carlo

By definition, the payoff of a log-contract of maturity $T$ writes $$ \phi(S_T) = \ln\left(\frac{S_T}{S_0}\right) $$ Let $\Pi_t$ denote the $t$-value of such a contingent claim. We are interested in ...
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  • 13.9k
6 votes
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Interpret simulation results ($P$ and $Q$ measures)

I believe that the confusion arises because of the wrong treatment of NIG. The answer to the question you link is misleading, as it simulates under P which is not appropriate for option pricing. None ...
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  • 4,217
6 votes
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Deep ITM Call Implied Vol via Monte Carlo

[Short answer] IMHO there is a fundamental problem with wanting to extract a sound implied volatility figure out of a deep ITM option's price. You should use out-of-the-money forward options (OTMF) ...
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  • 13.9k
6 votes
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How to perform Monte-Carlo simulations to price Asian options?

Instead of simulating the spot price, simulate its logarithm since the latter can be simulated exactly for any time step. \begin{equation} \ln S_{t + \Delta t} = \ln S_t + \left( r - \frac{1}{2} \...
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