Lecture 2 – Monte Carlo method in finance Option price as a path integral The Monte Carlo method The Monte Carlo method applied to option pricing Random numbers generation Monte Carlo Variance reduction techniques Low-discrepancy sequence 12/3/2018

The option pricing as path integral (I) The price estimate of an exotic option can be reformulated as a path integral (where the paths consist of all possible future evolution of underlying equity prices). Within risk-neutral probability measure, the option fair value, f, can be expressed as: where fp is the option pay-off for a given path, Ê is the expectation value in a risk neutral world, T is the time to maturity and r is the risk neutral interest rate. 12/3/2018

The option pricing as path integral II The path integral formulation of option pricing problem shows up its intrinsic probabilistic nature. In Physics there are many powerful computational methodologies to compute path integrals (e.g. among the others the Monte Carlo method). The application of Monte Carlo method to finance/option pricing, consists to generate a sufficiently high number of estimate of fp in order to compute its mean value (and also standard deviation). 12/3/2018

The Monte Carlo method The Monte Carlo method consists to formulate/solve the problem as a series of sampling, generated according to some random numbers generator, from which ones can extract an estimate of the expected value. The standard deviation can be used to evaluate the error. In other words Monte Carlo is a stochastic methodology based on random numbers generation. 12/3/2018

Problems that can be solved by resorting to Monte Carlo 1. Problems with an intrinsic probabilistic nature, i.e. problems which involve stochastic phenomenon. E.g.: the option pricing the Value at Risk (VaR) estimation of a financial portfolio. 2. Problems with an intrinsic deterministic nature, i.e. problems which do not involve any stochastic variables but where the solution can be equivalently rewritten in term of an expected value of a function of some non deterministic variables. : Integral calculation in D dimension 12/3/2018

Example: Monte Carlo and the calculation of p (I) An extraction from a sample of stochastic random numbers can be used to estimate an integral This integral can be viewed as the expectation value of a function (f) of two random variables, uniformly distributed in the interval [-1, 1]x [-1, 1] Uniform probability density function. 12/3/2018

Example: Monte Carlo and the calculation of p (II) The integral value con be estimated as the arithmetic average of N values of f(xi yi), where each pair (xi yi) is randomly sampled according to a uniform distribution in [-1, 1]x [-1, 1]. I.e. : is an estimator of I=p/4. An estimation of error can be derived by resorting to the central limit theorem, which states that the sum S_n of n independent and identically distributed (i.i.d.) random variables (having mean m and finite variance s^2) is well approximated by a gaussian distribution with mean m and variance s^2/n as n (the sample size) increases. 12/3/2018

Error scaling in Monte Carlo simulations In Monte Carlo simulations the error scales as: This behavior is independent from problem dimension. The error involved by applying lattice method on integral calculation in D dimension, scales as: The Monte Carlo error is independent from dimension, making Monte Carlo one of the most powerful numerical methods for high dimensional problems (typically more than four). 12/3/2018

Monte Carlo method for pricing an option STEP 1 – Define a stochastic process for the underlying STEP 2 – Calculate multiple scenarios by repeatedly sampling the possible path evolution for the underlying equity. First of all divide the option life into m time interval of length t. Compute the future (stochastic) values Si at time ti = i t until the option maturity T is reached. The simulation process of a single path, requires the generation of m independent random numbers sampled from a gaussian distribution. 12/3/2018

Monte Carlo method for pricing an option (II) STEP 3 – Evaluate the pay-off (i.e. the premium paid at maturity) under each scenario (equity path). STEP 4 – Compute the (discounted) mean value (i.e. the option price!) and its error, basing on the above distribution. Stochastic process for the underlying Probabilistic distribution of discounted pay-offs. Compute mean (option price) and standard deviation (=> error). Scenarios 12/3/2018

Monte Carlo method for pricing an option (III) Indeed, for a log-normal process, instead to use Eulero discretization procedure: one can resort to a closed form solution to simulate the stock evolution within an arbitrary long finite time interval Dt: 12/3/2018

Problems to face in random number generation The Monte Carlo method is based on sampling random numbers. Is it possible for a computer (i.e. a deterministic machine) to generate true random numbers (that is real stochastic objects)? The answer is negative! Indeed there are only three possible solutions: True random numbers they are numbers that lack any pattern and are generated by resorting to physical phenomenon that is expected to be intrinsically unpredictable (e.g. radioactive decay, thermal noise etc.). In such a case a specialized hardware or some database are required, being the use of truly random numbers not practical. Pseudo-random numbers they are numbers produced by a computer using computational (deterministic) algorithms. Being such algorithms deterministic, the sequences they produced are only apparently random, which in fact are completely determined by an initial value (known as the seed). Quasi-random numbers they are numbers generated according to a deterministic algorithm. In such a case, however, the sequences produced have a definite pattern that fills in gaps uniformly. As a result, the statistical error is lower than that obtained with pseudo random numbers. 12/3/2018

Algorithms to generate pseudo random numbers. The pseudo random number generators are usually based on linear congruential generator: This generator produces integer numbers in the interval [0, m]. In order to obtain random floating point numbers uniformly distributed between 0 and 1, we set: The parameters: a, b and m characterize the generator quality. a multiplier, b increment, m module. Pros: It is very fast; Cons: The random number sequence will repeat the pattern after some trials (the so called “period”). The period is always lower than m. 12/3/2018