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Suppose we have a standard Ito process $dX_{t}=\mu\left(X_{t},t\right)dt+\sigma\left(X_{t},t\right)dW_{t}$.

As far as I know, there are two approaches to solve this numerically: to frame it as a PDE and solve it, or to simulate random paths using Monte Carlo methods, and from there calculate expectations value which will give us prices of financial instruments. The accuracy will then scale like $C/\sqrt{N}$, where $N$ is the number of samples.

I would like to understand the limitations of Monte Carlo methods for this type of simulations, and more specifically to know whether there are problems in finance where this type of simulations are unfeasible. I know that, when we have many dimensions, the PDE method is unfeasible, and the only option is to use Monte Carlo methods. I'm also aware that the convergence rate is not too good, since there is a $\sqrt{N}$, so it might be expensive to get solutions with high accuracy.

However, I would like to understand what is the behaviour of $C$. I would assume that it depends a lot on the specific problem, but in general, I would expect that $C$ could depend on the simulation time $t$, or on the dimensions $d$. Can we say anything about the scaling though? For example, could there be any cases where $C$ increases exponentially with $t$? If that were the case for a specific problem, it would be unfeasible to simulate it with Monte-Carlo techniques for large times, since the number of samples needed to maintain a certain accuracy would also increase exponentially with time. Are there any cases in finance where $C$ scales badly, therefore making it hard to simulate the process using Monte Carlo techniques? Also, do you know of any reference where I could read about this?

Thank you very much.

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  • $\begingroup$ Monte Carlo Methods in Financial Engineering is an often recommended book. $\endgroup$ – Oscar Nov 26 '20 at 14:40
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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 is indeed the standard deviation of the Monte-Carlo estimator for the expectation of $f(X)$. For an iid sample of paths of the process $(x_i)$ this estimator is the average of $f$ over the paths and $$ \text{Var}\left[ \frac{1}{n}\sum_{i=1}^n f(x_i)\right] = \frac{1}{n}\text{Var}\left[ f(x_1)\right].$$

So $C = \sqrt{\text{Var}\left[ f(x_1)\right]}$ and properties of $X_t$, such as the interval over which $X_t$ is defined or the dimension of its state space, do matter only to the extent they influence this variance.

One typical - yet important - case where Monte-Carlo is inefficient, is rare events. Take $f$ to be an instrument which pays $1$ with probability $\delta$ and 0 otherwise. Then the relative error (as standard deviation divided by expectation) is $$ \text{rel. err} = \frac{\sqrt{\text{Var}\left[f\right]}}{\text{E}\left[f\right]}=\frac{\sqrt{\delta(1 - \delta)}}{\delta}=\sqrt{\frac{1-\delta}{\delta}}.$$ As $\delta$ becomes small you obviously get in trouble.

The standard textbook for all topics Monte-Carlo in finance is Glasserman.

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  • $\begingroup$ Such a dependence is easily seen when using the standard method of antithetic sampling - it will make your variance worse if the function is even, and better if it's odd. $\endgroup$ – rubikscube09 Nov 26 '20 at 21:59
  • $\begingroup$ Strong agreement on Glasserman - a great and highly readable textbook! $\endgroup$ – StackG Nov 27 '20 at 6:15
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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 the Monte Carlo Method was getting xerocopied like "samizdat". Now this book is in its 3rd edition (2016) already, and is good. It is not finance-focused, which is also good. I would not have understood what I was doing without this book.

The classic paper (which I haven't read myself) is "The Monte Carlo Method," Nicholas Metropolis and Stan Ulam, Journal of the American Statistical Association, Vol. 44, No. 247, September 1949, pp. 335 - 341. An interesting read on its history is Stan Ulam, John Von Neumann, and the Monte Carlo Method by Roger Eckhard.

Specific to finance, you may like:

Paul Glasserman. Monte Carlo methods in financial engineering. Springer (2004) - the best on this list

Peter Jäckel. Monte Carlo Methods in Finance. Wiley (2002)

Ralf Korn, Elke Korn, Gerald Kroisandt. Monte Carlo methods and models in finance and insurance. CRC (2010)

Paolo Brandimarte. Handbook in Monte Carlo Simulation: Applications in Financial Engineering, Risk Management, and Economics. Wiley (2014)

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    $\begingroup$ Upvote for pointing at the Metropolis gang. $\endgroup$ – Kermittfrog Nov 26 '20 at 15:51
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    $\begingroup$ I would second reading Glasserman. It is indeed a tough read (at least I took some time to chew through), but it has all the nitty gritty details. $\endgroup$ – Kermittfrog Nov 26 '20 at 15:52
  • $\begingroup$ The Eckhard paper mentions "the Chip" and the Connection Machine at the end, and how they will revolutionalize MC. I used to code FFT in starlisp for the Connection Machine. It was very cool hardware for its times. $\endgroup$ – Dimitri Vulis Nov 26 '20 at 15:54
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I just can't believe nobody recommended, so far, one of the best books:

Numerical Solution of Stochastic Differential Equations by Peter E. Kloeden and Eckhard Platen.

Hope it helps! Thanks!

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