# Methods of financial security

### Introduction

A financial option is a security that grants its owner the right, but not the obligation, to trade another financial security at specified times in the future for an agreed amount. The financial security that can be traded in the future is called the underlying asset. An option is part of the derivatives family, as its value is derived from the underlying asset. It is the most versatile trading instrument ever invented. Since it costs less than its underlying asset, it provides a high leverage approach to trading that can significantly limit the risk of trading and provide additional income. It also allows for hedging against one-sided risk, providing an efficient management of risk. Another advantage that benefits options users is the capability to handle information asymmetries by attaching put options exercisable at the IPO price to the new stocks being sold.

Malaysia's first exchange traded option contract was introduced on 1 December 2000 by Bursa Malaysia Derivatives Berhad's (BMDB), the Kuala Lumpur Options and Financial Futures Exchange (KLOFFE). The first options contract was introduced almost five years after the launch of KLOFFE's first product, the KLCI Futures Contract. This delayed launch was due to the currency crisis and the economic turmoil of the late 1990s. The options are called KLCI options with the Kuala Lumpur Composite Index as its underlying asset. The introduction of index options represent an extension of KLOFFE's only other product, the index futures which has the same underlying asset. As an index linked derivative, the KLCI options are heavily dependent on market performance and other macro factors. Having introduced index options, the exchange is next scheduled to introduce stock options, whereby the underlying asset would be its own individual stock.

Options have been considered one of the most difficult financial products to price. The underlying asset may not be traded, making it difficult to estimate value and variance for the underlying asset. Variance may not be known and may change over the life of the option, which can make the option valuation more complex. Also exercise may not be instantaneous, which will again affect the value of the option. Over the years, various options pricing models have been created to price options. The classic and pioneer model developed by Nobel laureates Fischer Black, Robert Merton and Myron Scholes, aptly named the Black-Scholes Model has been widely used to price specific types of options, especially European options. However, this model does not produce exact prices for all types of options, which resulted in other pricing methods. There are three pricing option pricing methods widely used - binomial methods, finite difference models and Monte Carlo models.

The binomial option pricing model, published by John Cox, Stephen Ross and Mark Rubenstein in 1979 is an iterative solution that models the price evolution over the whole option validity period. More precisely, it represents the price evolution of the option's underlying asset as the binomial tree of all possible prices at equally-spaced time steps from today under the assumption that at each step, the price can only move up or down at fixed rates and with respective pseudo-probabilities. It is considered more accurate than Black-Scholes because they are more flexible as discrete future dividend payments can be modeled correctly at the proper forward time steps, and American options can be modeled as well as European ones.

The finite difference methods were first applied to option pricing by Eduardo Schwartz in 1977. It can solve option pricing problem that have the same level of complexity as those problems solved by tree approaches, and are therefore usually employed only when other approaches do not prove to provide feasible solutions. The finite difference method is created due to the fact that the option value can be modeled using a partial differential equations using Taylor series expansions of functions near the point or points of interest. It should be noted that binomial model has the same complexity as this method. However, this approach is limited in terms of underlying variables.

For many types of options, traditional valuation techniques are intractable due to the complexity of the instrument. In these cases, a Monte Carlo approach may often be useful. It was first introduced by Boyle in 1977. It is a very flexible valuation methodology and it can be shown to converge to the true option price. It can solve problems of several state variables, making it a multi-dimensional model. It can also easily accommodate different stochastic processes, multiple underlying assets, path dependence, and particular features of exotic options.

This paper attempts to apply Monte Carlo simulation model to price the KLCI options and compare the results with the Black-Scholes Model and Binomial Option Pricing Model. It will also review the best variance reduction technique to be used for KLCI option pricing.

### Background of Study

### SGX MSCI Taiwan Index (SGX TiMSCI) Options

The Morgan Stanley Capital International (MSCI) Taiwan Index is an index containing 60% of Taiwan market value of Taiwan Stock Exchange. It is a free float-adjusted, market capitalization-weighted index representing a sampling of large, medium and small enterprise shares. The index is calculated in Taiwan Dollars on a real time basis and disseminated every 15 seconds during market trading hours. The index has a base date of January 1, 1988. On January 9, 1997, futures contracts on the MSCI Taiwan Index began trading on the Singapore Exchange.

On July 3, 2006, SGX Taiwan Index Options was introduced to the exchange. The contract size is one SGX MSCI Taiwan Index Futures contract, with the contract months being the 2 nearest serial months and 12 quarterly months in March, June, September and December cycle. The price of an option contract is quoted in index points. The minimum fluctuation of the contract is one hundredth of an index point, equivalent to 1 U.S. Dollar per contract. The strike price is stated as an integer divisible by 5 without remainder. No new options are listed if there are less than three calendar days to the last trading day of the options, which is usually on the second last business day of the contract month. Trading will come to a halt when the underlying futures contract is bid or offered at its initial daily price limit, or at its expanded daily price limit. A person is only allowed to own or control any combination of MSCI Taiwan Index Futures and Options Contracts of less than 5000 contracts on the same side of the Market and in all the contract months combined. The option is exercised European style, whereby it may be exercised only upon expiration. In the absence of contrary instructions delivered to the clearing house, an option that is in-the-money at expiration shall be automatically exercised.

### SGX MSCI Singapore Index (SGX SiMSCI) Options

The MSCI Singapore Free Index is a market capitalization-weighted index of large and medium capitalization stocks traded on the SGX. The index is compiled by Morgan Stanley Capital International (MSCI). To ensure investability of the index, MSCI conducts regular reviews on the component stocks. As of September 27, 2007, the index comprises 36 stocks. The MSCI Singapore Index Futures was launched in September 1998, whereas the Options Contracts was introduced much later on in April 2009. The introduction of the Options Contracts will further enhance the Futures' market depth.

The options contract size is equivalent to one MSCI Singapore Index Futures Contract, with the contract months being the 2 nearest serial months and March, June, September and December months on a 1-year cycle. The minimum price fluctuation is one tenth of an index point, equivalent to S$20. Strike price is stated as an integer divisible by 5 without remainder and in terms of the futures contract which is deliverable upon exercise of the option. No new options are listed if there are less than three calendar days to the last trading day of the options, which is usually on the second last business day of the contract month. Trading will come to a halt when the underlying futures contract is bid or offered at its initial daily price limit, or at its expanded daily price limit. However, there will be no trading halt on an option's last trading day. A person is only allowed to own or control any combination of SGX MSCI Singapore Index Futures and Options Contracts of less than 10000 contracts on the same side of the Market and in all the contract months combined. The option is exercised European style, whereby it may be exercised only upon expiration. In the absence of contrary instructions delivered to the clearing house, an option that is in-the-money at expiration shall be automatically exercised.

### Binomial Option Pricing Model (BOPM)

The Binomial Option Pricing Model or BOPM in short, is a simple discrete-time model for valuing options made popular by John Cox, Stephen Ross and Mark Rubinstein in 1979. The reason behind the invention of this model was that the Black-Scholes Option Pricing Model requires advanced mathematical tools and it limits to pricing European options only. In their paper "Option Pricing: A Simplified Approach", they stated that "Our formulation, by its very construction, leads to an alternative numerical procedure that is both simpler, and for many purposes, computationally more efficient".

The BOPM framework is illustrated by initially assuming that a stick price follows a multiplicative binomial process over discrete periods. Assume that ?t = T/M denoting the spacing between successive time points, where T is the expiry date. So the stock prices will be considered at times ti = i?t, for 0 ? i ? M. The rate of return on the stock over each period can have two possibilities: u with probability q, or d with probability 1 - q. It is also important to take note that the price can either move up or down between successive levels time levels. Therefore, if the current stock price is deemed as S, the prevailing stock price at the end of the time period will be

For a European call option, the payoff at expiry has the form of = Max[0, - S). The objective is to compute the option value at time zero, assuming we have the option values corresponding to time t = ti+1 and all possible asset prices. As the assumption that the stock price movement goes up or down, the stock price comes from either with probability p, or from with probability 1 - p. Thus, the option value scaled with the appropriate factor that allows for interest rate, r is given by the equation

Once the parameters u,d, p and M have been chosen, the formulas completely specify the binomial model.

### Black-Scholes Option Pricing Model (BSOPM)

The celebrated Black-Scholes Option Pricing Model, BSOPM was articulated by Fischer Black and Myron Scholes in their 1973 paper, "The Pricing of Options and Corporate Liabilities". In their research paper, they stated that "Most of the previous work on the valuation of options has been expressed in terms of warrants. For example, Sprenkle (1961), Ayres (1963), Boness (1964), Samuelson ( 1965), Baumol, RIalkiel, and Quandt ( 1966), and Chen (1970) all produced valuation formulas of the same general form. Their formulas, however, were not complete, since they all involved one or more arbitrary parameters". The fundamental insight to the solution is that the option is implicitly priced if the stock is traded.

The BSOPM has a few underlying assumptions of the market in deriving the value of an option in terms of the price of the stock (Black and Scholes, 1973):

- The short-term interest rate is known and is constant through time.
- The stock price follows a random walk in continuous time with a variance rate proportional to the square of the stock price. Thus the distribution of possible stock prices at the end of any finite interval is lognormal. The variance rate of the return on the stock is constant.
- The stock pays no dividends or other distributions.
- The option is "European," that is, it can only be exercised at maturity.
- There are no transaction costs in buying or selling the stock or the option.
- It is possible to borrow any fraction of the price of a security to buy it or to hold it, at the short-term interest rate.
- There are no penalties to short selling. A seller who does not own a security will simply accept the price of the security from a buyer, and will agree to settle with the buyer on some future date by paying him an amount equal to the price of the security on that date.

With the assumptions made, the value of the option will only depend on the price of the stick and time and the variables involved. Therefore, a hedged position is possible, consisting of a long position in the stick and a short position in the option, whereby the value will not depend on the price of the stock but instead on time and the variables known as constants.

The Black-Scholes formula can be used to calculate both the values of European call and put options. The derivation of the formula is obtained by solving the Black-Scholes partial differential equation:

It is a relationship between the price of the stock (S), the price of a derivative as a function of time and stock price (V), the annualized risk-free interest rate continuously compounded (r), and certain partial derivatives of V.

From the Black-Scholes PDE, we obtain a solution of

where

As for the value of European put option, the formula below is used,

for all 0 ≤ t ≤ T.

For extremely large S the payoff is almost certain to be zero.

### Monte Carlo Framework

The application of Monte Carlo methods to option pricing was pioneered by Phelim Boyle in 1977. It is categorized as a sampling method because the inputs are randomly generated from probability distributions to simulate the process of sampling from an actual population. The reason it is more sought after than the aforementioned option pricing models is that it solves a problem by simulating directly the physical process, and is not necessary to write down the differential equations that describe the behavior of the system. It permits simulations of several sources of uncertainties that affect the value of options. An example of a problem would be the curse of dimensionality, where there are more than three or four state variables. Monte Carlo simulations are able to address that problem.

The Monte Carlo framework will be explained in a general setting. Suppose we want to estimate some ?, and we have

θ = E (g(X ))

where g(X) is an arbitrary function such that E(|g(X)|) < ?, where X is a uniform variable on the interval (0,1). A series of n independent random observations {Xi} can be generated from the probability function f (X). The expected value is estimated by a sample mean that is a random variable.

The estimator of θ is given by

Since E(|g(X)|) < ∞, we can say that

θ as n

As the Monte Carlo simulation never provides exact results, one always has to take the sample variance into account. It can be expressed as

Central limit theorem tells us that

We could also say that is approximately a standard normal variable scaled by . For large n we have

Hence we get a way to approximate and its variation.

### Generating Pseudorandom Variates

The generation of pseudorandom numbers are variates from the uniform distribution on the interval (0,1). Suitable transformations are then applied to obtain samples from the desired distribution. Among the popular transformation methods used are linear congruential generators (LCGs), inverse transform method, and the acceptance-rejection approach.

### Linear Congruential Generators(LCGs)

D.H. Lehmer in 1948 proposed a simple linear congruential generator as a source of random numbers. In this generator, each single number determines the next number by a simple linear function followed by a modular reduction. It generates a sequence of non-negative integer numbers Zi, given an integer number Zi-1

where a is the multiplier, c is the shift and m is the modulus. To generate a uniform variate on the unit interval, ui the number Zi is divided by m.

Often, the c is taken to be 0, and in this case the generator is called a 'multiplicative congruential generator'. For c ≠ 0, the generator is called a 'mixed congruential generator'.

If a and m are properly chosen, ui will look like they are randomly and uniformly distributed between 0 and 1.

### Inverse Transform Method

Inverse transform method is also known as the inverse probability integral transform or Smirnov transform. It is used to generate sample numbers at random from any probability distribution givens its cumulative distribution function (cdf).

The probability integral transform states that if X is a continuous random variable with a cumulative function F(x), and if Y = Fx(X), then Y has a uniform distribution on [0,1]. If F is invertible, the following inverse transform method can be used:

- Generate a random number U~U(0,1)
- Return X = F-1(U)

P{X ≤ x} = P{F -1 (U) ≤ x} = P{U ≤ F(x)} = F(x)

where F is monotonic and U is uniformly distributed.

### Acceptance-Rejection Method

It is not always possible to find an explicit formula for F-1(x) for the cumulative distribution function of X we wish to generate F(x) = P(X ? x). Thus, the acceptance-rejection approach is more efficient in generating random numbers. Firstly, assume that the F that is to be simulated from has a probability density function of f(x), which is continuous. The basic idea is to find an alternative probability distribution G, with density function g(x) which is close or similar to f(x). It is assumed that the ratio f(x)/g(x) is bounded by a constant c > 0, with c preferably as close to 1 as possible.

The algorithm is as such:

- Generate a random variable Y distributed as G.
- Generate U (independent from Y).
- If then set X=Y (accept); otherwise go back to step 1 (reject).

### Variance Reduction Method

The Monte Carlo method undoubtedly provides an easier technique to value options. However, it can be expensive. It uses the sample mean to approximate expected value of the random variable X, where the Xi are i.i.d with E(Xi) = E(X). The width of the corresponding confidence interval is inversely proportional to. This makes it an expensive business to improve approximation by taking more samples. To shrink a confidence interval by a factor of 10, it requires 100 times as many samples. However, the confidence interval width also scales with . The idea behind variance reduction is to replace the Xi with another sequence of i.i.d. random variables that have the same mean as Xi but with smaller variance. If the variance in Xi can be reduced by a factor R < 1, the variance reduction method gives confidence intervals of the same width for R times less work.

### Antithetic Variates

The method of antithetic variates for pricing options is based on the fact that if Xi has a standard normal distribution, then so does -Xi. In the case of

I = E(f(U)), where U ~ N(0,1)

the standard Monte Carlo estimate is with i.i.d. Ui ~ N(0,1) and the antithetic alternative is with i.i.d. Ui ~ N(0,1) the variance is and this shows that for monotonic f, the variance in the antithetic samples is always less than or equal to half that in the standard sample.

### Control Variates

Another method that can be used is called the control variate method. A control variate for this context is a somewhat similar option whose true value is known. We then obtain a simulated value of that option. The difference between the true value of the control value and its simulated value is then added to the simulated value of the option we are trying to price. This will add the error in the control variate into the simulated value of the option of interest.

Let

cs - simulated price of the option we are trying to price,

vt - true value of another similar option

vs - simulated value of the other similar option

The control variate estimate is then found to be

cs + (vt - vs)

A simulation will be done on cs - vs and adding vt. The variance will be Var(cs - vs) = Var(cs) + Var(vs) - 2Cov(cs,vs).

This will be less than Var(cs) is Var(vs) < 2Cov(cs,vs), meaning that the control variate method relied on the assumption of a large covariance between cs and vs. The control variate chosen needs to have a very high correlation with the option to be priced. A European call option may be used as a control variate to price a European put option.

### Quasi Monte Carlo

### Literature Review

### The Evolution of Option Pricing Theory

The most popular and celebrated option pricing theory has to be the one by Black, Scholes and Merton in 1973, aptly titled the Black-Scholes Option Pricing Model. It provides an analytical solution for the pricing of European options. It is a closed-form solution that regards volatility as a constant term throughout the life of the option. However, this was proved to be untrue by the empirical observation of data provided by the markets, although the model worked pretty well for at-the-money options for relatively small expirations like less than 2 years. Ironically it was Black (1975) who discovered that there were volatility biases displayed by the option market prices with respect to the Black-Scholes Model formula. Out-of-the money put options tend to be overpriced, giving rise to a high volatility implied by the B-S Model while in-the-money put options tend to underpriced, implying low volatility (Latane & Rendlemen, 1976; Mayhew,1995; Corrado & Miller, 1996). This situation is called the volatility "smile" or "smirk" and is proved to be common in equity derivatives market, with foreign exchange derivatives exhibiting volatility smiles in the sense that both in- and out-of-the-money options having higher implied volatilities than at-the-money options.

In order to address the problem of the volatility not being constant, the option pricing theory evolved by producing models that regarded volatility as a stochastic process. The first few stochastic volatility models were by Hull and White (1987), Stein and Stein (1991) and Heston (1993). These models allow for nonzero correlation between the level of the stock return and its variance. The most famous model is however the model derived by Heston, as it assumes a degree of correlation between the stock returns and the volatility itself. Heston also provided a closed-form analytical solution for the pricing of European options.

Stochastic volatility models can also address term structure effects by modeling mean reversion in the variance dynamic. Ho and Lee (1986) were the first to model the whole term structure of interest rates as a stochastic object where the initial term structure coincides with the empirically observed one.

The problem if volatility was addressed by Brenner & Subrahmanyam (1988) and Corrado & Miller (1996). They used numerical methods to obtain implied volatilities. However, the Brenner & Subrahmanyam (1988) formula only applies for at-the-money forward options whilst Corrado & Miller provides a better approximation formula. Using the Corrado & Miller (1996) formula as a benchmark, Hallerbach (2004) derived a formula that exhibits higher approximation accuracy and extends over a wider region of moneyness.

As the B-S Model has a defect, whereby assuming volatility is constant, this has instigated the development of other models. The first being the lattice method was first published by Parkinson (1997) who introduced a trinomial method to price options. Perhaps the best-known and most widely cited original paper on the lattice method is Cox, Ross, Rubenstein (CRR) (1979) The framework is like of a binomial tree where the price can have two options of either going up or down with different probability assigned to each option, It is easy to implement and gives pretty accurate results but is used mainly for pricing 'easy' options with constant volatility and no more than 3 underlying assets, a setback named 'curse of dimensionality'. Some argue that it is wasteful to study the binomial model as it would only served a pedagogical purpose and the B-S Model would be preferred for actual applications. Chance (2007) argued that it would be difficult to consider the binomial method as a method to derive path-dependent option values without first knowing how well it would work for the one scenario in which the true continuous limit is unknown.

Also, CRR do not give enough information to price options in the real world. Cox and Rubenstein (1985) however, do give adequate information to deduce real-world option pricing but the information was instead used to help evaluate option performance in a portfolio theory context. Arnold & Crack (2000) extended the model to allow for option pricing using real-world rather than risk-neutral world probabilities.

Rendleman-Bartter and Jarrow-Rudd-Turnbull also proposed a binomial model which is fitted to the physical process. As their models were similar they later append with the Jarrow-Turnbull approach. However, their model does not recover the variance for finite N using the risk neutral probabilities. It does not prevent arbitrage nor recover the correct volatility. Neil Chriss's model (1996) created a binomial model which did the opposite. It specifies the raw mean and log variance of the physical process. The transformation to risj neutrality returns the no-arbitrage condition and it recovers the volatility for any number of time steps. Paul Wilmott (1998) also developed a model where the raw mean and variance of the process is specified. Thus, it prohibits arbitrage and recovers volatility.

Lattices can be used for relatively complex derivatives as shown by Heston & Zoo (2000) and Alford & Webber (2001). One drawback is that these algorithms require a lot of computational time. This was proved by Broadie and Detemple (1996).