Reference no: EM132364400 , Length: word count : 5500

The topic needs  Data collection , analysing and discussion

You need to compare between  MONTECARLO method and BLACK SCHOLES method on AMERICAN OPTIONS.

Please, pricing American options using montecarlo simulation) should be done first. By the way, data can be accessed either by QUENDL  or by THOMPSON REUTERS.

RESEARCH METHODOLOGY

This chapter will look into the Monte Carlo method and the formulation on how it can be used in pricing of options. It will then explore the variance reduction techniques that aid in reducing the errors in Monte Carlo estimation.

The data that will be used will be gotten from the Quandl website on a single stock listed in the S&P 500 and then do the analysis over the pricing over a one year 2018 period to compare the price realizations and the estimated prices from the simulation to identify how good the LSM Monte carlo method is in pricing American options.

1. Monte Carlo Methods

The theoretical understanding of Monte Carlo methods draws on various branches of mathematics. In this version, the increase in the complexity of derivative securities in recent years has led to a need to evaluate high-dimensional integrals. Monte Carlo becomes increasingly attractive compared to other methods of numerical integration as the dimension of the problem increases. A great number of Monte Carlo simulation models are known and used in practice.

Monte Carlo technique for valuation of derivatives securities is a method, which is based on the probability distribution of complete histories of the underlying security process. The Monte Carlo method lends itself naturally to the evaluation of security prices represented as expectations. Generically, the approach consists of the following steps:

Simulate sample paths of the underlying state variables (e.g., underlying asset prices and interest rates) over the relevant time horizon. Simulate these according to the risk-neutral measure.

Evaluate the discounted cash flows of a security on each sample path, as determined by the structure of the security in question.

Average the discounted cash flows over sample paths.

However, a difficulty occurs for Monte Carlo valuation of American options, Monte Carlo methods are required for options that depend on multiple underlying securities or that involve path dependent features. Since determination of the optimal exercise time depends on an average over future events, Monte Carlo simulation for an American option has a "Monte Carlo on Monte Carlo" feature that makes it computationally complex.

1.1 Mathematical Background

Monte Carlo (MC) simulation is an alternative to the numerical PDE method. Boyle (1977) is the first researcher to introduce Monte Carlo simulation into finance. The method itself is simple and easy to implement. Monte Carlo (MC) simulation is the primary method for pricing complex financial derivatives, such as contracts whose payoff depends on several correlated assets or on the entire sample path of an asset price. We can simulate as many sample paths as desired according to the underlying stochastic differential equation that describes the stock process.

For each sample path, the option value is determined and the average from all paths is the estimated option price. The option price μ is written as an integral that represents the mathematical expectation of the discounted payoff under a so-called risk-neutral probability measure. This expectation is usually with respect to a non-uniform density over the real space, but with a change of variables, it can be rewritten as an integral over the s-dimensional unit hypercube

[0,1)^t={u=(u₀,...,u_(t-1) ):0<u_j<1 for all j}:

μ=μ(ƒ)=₀∫¹... ₀∫¹ ƒ(u₀,...,u_(t-1) ) du₀...du_(t-1)=_[0,1)^t∫^u ƒ(u)du=E[f(U) ], (3,1.

For some function f:[0,1)^t→R, where u represents a point in [0,1)^t, and U~U[0,1)^t is a random point with the uniform distribution over the unit hypercube. In this paper, we assume that the integral is already written in the form (3.1) for a fixed positive integer t, and we want to estimate μ. This t represents the number of calls to the underlying random number generator used in our simulation. In situations where this number of calls is random and unbounded, t can be taken as infinite, with the usual assumption that with probability one, only a finite number of coordinates of U need to be explicitly generated.

For the European option, the MC method works well. In fact, we even have an analytical solution, e.g., using the Black-Scholes formula. More importantly, the value is determined only by the terminal stock price if one assumes a given starting point, time, constant interest rate and volatility. It is easy to see that Monte Carlo simulation must work in a forward fashion.

However, for the American option, because of early exercise, in contrast to a partial differential equation, we would also need to know the option value at the intermediate times between the simulation start time and the option expiry time. In Monte-Carlo this information is harder to obtain, therefore, even though it is simple and capable of handling multi-factor problems, once we have to solve a problem backwards, Monte Carlo simulation becomes awkward to implement.

2 Least Square Monte Carlo method (LSM)

There is basically one way to value American-style options, instead of determining the exercise boundary before simulation, this approach focuses on the conditional expectation function; see e.g., Carriere (1996), Tsitsiklis and Roy (1999). Longstaff and Schwartz (2001) proposed the Least-Squares Monte Carlo (LSM) method, an easy way to implement this approach. Clement et, al. (2001) studied related convergence issues. Tian and Burrage (2002) discussed the accuracy of the

LSM method. Moreno and Navas (2003) further discussed the robustness of LSM with regard to the choice of the basis functions.

Longstaff and Schwartz (2001) introduce the use of Monte Carlo simulation and least squares algorithm of Carriere to value American options since nothing more than simple least square is required. At each exercise time point, option holders compare the payoff for immediate exercise with the expected payoff for continuation. If the payoff for immediate exercise is higher, then they exercise the options.

Otherwise, they will leave the options alive. The expected payoff for continuation is conditional on the information available at that time point. The key insight underlying this approach is that this conditional expectation can be estimated from the cross-sectional information in the simulation by using least squares.

This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference techniques cannot be used. To find out the conditional expectation function, we regress the realized payoffs from continuation on a set of basis functions in the underlying asset prices.

The fitted values are chosen as the expected continuation values. We simply compare these continuation values with the immediate exercise values and make the optimal exercise decisions, then we obtain a complete specification of the optimal exercise strategy along each path. We recursively use this algorithm and discount the optimal payoffs to time zero. That is the option price.

The method starts with Nrandom paths (_nS^k,t_n )for 1≤k≤N and t_n=ndt. Valuation is performed by rolling back on these paths. Suppose that _(n+!)F^k=F(_(n+!)S^k,t_(n+1)) is known. For points (_(n+!)S^k,t_(n+1) ) set X=_(n+!)S^k the current equity value and Y=e^(-rdt) F(_(n+!)S^k,t_(n+1) ) the value of deferred exercise. Then perform regression of Y as a function of the polynomials X,X²,...,X^m for some small value of m which is called basic function; i.e. approximate Y^k by a least squares fit of these polynomials in X. Hence we use this regressed value in deciding whether to exercise early.

For the most part there is nothing that can be done to overcome the rather slow rate of convergence characteristic of Monte Carlo. However the precision of the estimator may be improved by reducing the standard deviation. The techniques used for this task are known as variance reduction techniques and a pair of them is discussed next.

3 Variance Reduction Techniques

In this section we first discuss the role of variance reduction in meeting the broader objective of improving the computational efficiency of Monte Carlo simulations. We then discuss two specific examples of variance reduction techniques: control variate and antithetic variates and illustrate their applications in pricing problems

3.1 Control Variates

The method of control variate is among the most widely applicable, easiest to use and effective of the variance reduction techniques. It generally exploits information about the errors in estimates of known quantities to reduce the error in an estimate of an unknown quantity.

Suppose again we want to estimate the derivative price α. The Monte Carlo estimator from n independent and identically distributed replications α₁,...,α_n is α ^=(α₁+?+α_n)/n. Imagine now that for each replication it is possible to calculate another output X_i along with α_i. The pairs (X_i,α_i )i=1,...,n are i.i.d and suppose that the E[X] is known. Thus for any fixed b it is possible to calculate
α_i (b)=α_i-b(X_i-E[X])

For each replication i. The control variate Monte Carlo estimator would then be

α ^(b)=α ^-b(X_i-E[X] )=1/n _(i=1)∑ⁿ (α ^-b(X_i-E[X] )

and the observed error X ^-E[X] is used to control the estimate E[α]. The control variate estimator (2.3) is unbiased and consistent.

The variance of each replication α_i(b) is

Var[α_i (b) ]=Var[α_i-b(X_i-E[X] ) ]= σ_α²-2bσ_α σ_x ρ+b² σ_x²≡σ² (b)

Replacing the population parameters in the above equation with their sample counterparts yields the estimate

(b_n) ^= (_(i=1∑)ⁿ (X_i -X ^)(α_i-α ^))/ ((_(i=1∑)ⁿ (X_i-X ^)² )

3.2 Antithetic Variates

The method of antithetic variates is one of the simplest and widely used techniques in financial pricing problems. It attempts to reduce the estimator variance by introducing negative dependence between pairs of replications. The core observation is that if a random variable U is uniformly distributed over [0, 1] then 1-U is too. Thus if we generate paths using as inputs U_i,...,U_n it is possible togenerate additional paths using 1-U_i,...,1- U_n without changing the law of the simulated process. The combinations of variables (U_i,1-U_i )constitute antithetic pairsin the sense that generally a large value of one is followed by a small value of the other.

These observations extend to other distributions through the inverse transform method. Specifically, in a simulation of stochastic processes based on independent standard normal random variables, antithetic pairs may be constructed by pairing a sequence Z_i,...,Z_n of i.i.d. N(0,1) variables with the sequence -Z_i,...,-Z_n of i.i.d. N(0,1) variables. If the Z_i s are used to simulate the incrementsof a Brownian path the simulate the increments of the reflection of the path about the origin.

To analyze the variance reduction produced by the method imagine we have to estimate an expectation E[Y] and that using antithetic sampling we produce a series of pairs of observations (Y₁,Y ~₁ ),...,(Y_n,Y ~_n ). The procedure presents the following characteristics:
The pairs (Y₁,Y ~₁ ),...,(Y_n,Y ~_n ) are i.i.d.;

For each i,Y_i,and (Y_i ) ~ has the same distribution, though they are not independent.

The antithetic variates estimator is defined as

(Y_AV ) ^= 1/2n (_(i=1)∑ⁿ Y_i +_(i=1)∑ⁿ (Y_i ) ~ )=1/n _(i=1)∑ⁿ ((Y_i+(Y_i ) ~)/2)

The method of antithetic variates increases efficiency in pricing options that depend monotonically on inputs (e.g. European, American or Asian options).