The Application Behind Options Pricing Models

Derivatives are financial products whose values are derived from one or more underlying asset or financial instrument. In today’s world derivative products are being used extensively as a means of managing financial risks. These products are used to transfer risk from one party who cannot take that risk to counterparty who has the appetite to assume that risk .Some types of derivatives called Exchange Traded Derivatives are traded publicly on exchanges while others called Over the Counter Derivatives are traded between private parties in private transactions. Options are a kind of derivative product which gives the holder the right but not the obligation to buy or sell an asset at a price called Exercise price on or before a certain date called the Exercise Date. They are traded publicly over exchanges around the world. The underlying asset can range from foreign exchange, futures, commodities, etc.

Apart for its use as hedging instruments it is also used by speculators and arbitrageurs in the markets. This report aims to provide a basic understanding into the various options pricing models which are used to calculate the theoretical fair value of an option. The report also aims to give technical insights on using those pricing models with reference to the option pricing model developed by the author in Microsoft Excel using Visual Basic Programming (VBA). The Excel application developed by the author can be used by traders, speculators or anyone participating in the options markets to determine the theoretical fair price of an option and then compare it with the actual option price to determine whether the option is overpriced or underpriced. This will enable the user take appropriate positions in the market. The author has developed the application so that one can price an option according to three popularly used models i.e. Binomial Model, Black Scholes Model and Monte Carlo Simulation Model. The Author has also provided for a functionality which automatically downloads latest stock price data of stocks in the Dow Jones Index and gives user option to select the stock to price, automatically loading the stock price in the model.

Introduction

Objectives

In finance, the wide number of potential outcomes adds uncertainty and risk in estimating the future value of financial instruments. Monte Carlo simulation (MCS) is one technique that helps to reduce the uncertainty involved in estimating future outcomes. This simulation method can be used to evaluate the accuracy and performance of other models. Apart from MCS , Black Scholes and Binomial Model are the other commonly used options pricing models in the financial markets.

Aim of the Project:

1.A A A A A  To research the efficiency of the Monte Carlo Simulation models when applied to pricing derivative options. 2.A A A A A  To compare the Monte Carlo Simulation technique with other options pricing derivative techniques like Binomial Pricing Model and Black Scholes Model. 3.A A A A A  To Develop an Excel Based Application which prices options using Monte Carlo Simulations, Binomial Model, Black Scholes Models also providing additional information like volatility, options Greeks which are useful for options traders and structurers.

Scope

There are various options pricing models that have been developed over time some of which focus on certain specific types of derivatives. This project is not intended to be a comprehensive guide to all the options pricing models. It is rather a means of understanding the options pricing models most commonly used in the market. Equipped with this knowledge one should be able to understand the other pricing models available which are not within the scope of this report. The basic scope of this report is to get a basic overview of the options terminology and basic pricing models which may serves as a base for further academic research in detail.

Literature Review

Introduction to Derivatives

In the financial markets there are various types of risk faced by market players, corporations etc. The ability of a firm to get low cost financing can be hindered because of the highly volatile interest rates. Exporters have to hedge their currency risks as their profit strongly depends on how undervalued their currency is. Companies may be needed a commodity which is a vital input in the future but uncertainty in the future price might affect its profitability. These are some of the risks that can be mitigated by use of derivatives. They have been around for a long time but its importance and use has been growing exponentially in the past few decades because of the rapid industrialisation and maturing of the world financial markets. As the name suggests Derivatives are products that derive their value from an underlying asset or commodity. The underlying assets can be real assets like commodities, metals, etc. As well as financial assets such as bonds, stocks, stock index, futures, etc.

Forwards and Futures

An asset can be sold today as per the decided price or later in the future according to a price which is decided today. Forward contracts are used for the latter contracts. They are contracts between two parties who decide to trade an asset based on a price decided today, the transfer of which will take place sometime in the future. One of the parties is which agrees to take the delivery of the underlying asset in the future for a certain specified price is called the long. The other party which agrees to make the delivery of the asset in the future at the agreed price is called the short. Neither party pays any amount to get into the contract but the contract is binding and to cancel the contract the party has to enter into an offsetting contract with the same counterparty or it may also assume an offsetting contract with another counterparty in which case it assumes and additional counterparty risk. These forward contracts are mostly done in private transactions and hence its terms are also customised to the needs of both the counterparties. In Most of the cases the asset to be traded is actually delivered by the Short Party to the Long Party.

Thus forward contracts are usually settled by delivery. image1.gif Figure : Payoffs from Forward Contracts As it can be seen in the payoff diagram about the Long will profit only if the price of underlying increases whereas if the price of the underlying decreases the long will be forced to buy at a higher rate than what is available in the market. As far as the Short is concerned it will profit from the transaction only if the price of the underlying decreases in the future as if the price increases it will be forced to sell at a lower price than it could have sold the underlying at current market price. Thus this is a zero sum game; one party’s gain is another party’s loss. Forward contracts are mostly used for mitigation of risk i.e. hedging rather than for speculation purposes. A Common example of forward contracts is a corporation which is due to receive payments in Pounds 2 months later. It is exposed to fluctuations in the currency market and hence sells pounds 2 months in the future thereby assuming a short position and mitigating its risk by locking an exchange risk at which it can sell pounds in the future. As opposed a corporation which has to make payment in pounds in 2 months will assume a long position and enter into a forward contract to buy pounds 2 months later at a price agreed upon today there by eliminating the uncertainty factor and hedging its currency risk.

Some of the problems faced in the forwards markets are that though the contracts being private with the terms customised according to the counterparties offer a lot of flexibility, but it makes them non standardised and difficult to trade. Also a large part of the transaction is risky because of counterparty risk. If one party does not honour the terms of the contract the other party goes in losses. Futures are similar to forward contracts but remove some drawbacks of forward contracts like illiquidity, counterparty risk. The only difference in futures contracts is that the terms of the contract such as the price at which the underlying asset will be delivered, how much of the asset has to be delivered, date of delivery, etc are the same for all contracts i.e. standardised. They are usually traded on public exchanges in which case the risk of the other party defaulting is highly reduced as the exchange is always the counterparty for every trade. For example if one party wants to assume a long position the short position will always be assumed by the exchange and vice versa which highly reduces the problem that the other party will not honour the terms of the contract. Futures Forwards Trade on an organised exchange OTC in Nature Standardised and hence liquid Customised hence less liquid Margin Payment Required No Initial Margin Payment Required Daily Settlement Settlement at End of Day

Options

Options are very much different from the other derivative products like forwards and futures. In forwards and futures once the parties enter in the agreement they are obliged to honour the terms of the contract or complete the transaction as per recorded in the contract irrespective of whether they are in profit in loss. Whereas in options contracts both the parties decide to trade a particular asset sometime in the future at a price which they decide during the initiation of the contract. The difference lies in the fact that the buyer of option i.e. Long party has to pay an initial sum of money called the premium in order to gain the privilege of having a right but no the obligation to exercise the contract. While the party selling the option i.e. Short receives this premium for the risk it takes. Thus the Long party will only exercise the option if the transaction is profitable to him according to the market rates prevalent at the time of expiration of the contract Call Option: A Call Option Gives the buyer a right but not the obligation to buy an asset at a certain price at or before sometime in the future. The seller will receive a form of payment called premium and will still have the obligation to exercise the contract if the buyer of the call option chooses to exercise it. Consider the following at the Exercise Date of the option

Option Pricing Models

Introduction

In the financial world among all the products available derivatives are considered the most complex ones and hence the most difficult to price as well. Of course, some derivatives are far more difficult to price than others. Over the past decades researchers have come up with many numbers of options pricing models and they can be broadly categorised into three categories: Analytical Models Numeric Models Simulation Models The Simulation models refer to the mathematical method called Monte Carlo Simulation which has been named after a famous casino in Monaco on the French Rivera. The Motel Carlo Simulation was initially used to find a reasonable estimate of the probability of winning a game of pure chance. The method works by simulating the outcomes of a process by generating random numbers. In this process the accuracy of the approximation of the results is directly proportional to the number of replications and gives results that are more closer to analytically correct solution. Since the Monte Carlo simulation method requires the process to be run over and over again for sometimes thousands of times this process is quite heavy on the CPU and would require a faster computer for it to run effectively. Monte Carlo simulation can be used in all sorts of business applications whenever there is a source of uncertainty (such as future stock prices, interest rates, exchange rates, commodity prices, etc.). To illustrate the basic concepts, I will be focusing on pricing options, which are generally the most difficult types of Derivatives to value.

Analytical Models:

A Classic example of these kinds of models is the Black Scholes model which is one of the most widely and commonly used option pricing models used in the financial industry. They are the most elegant of the pricing methodologies. In the analytical approach one begins with assumptions about how the input variables of the model will behave and these assumptions are then converted into direct mathematical questions which give the desired result. These mathematical relationships are used as a means to establish a connection between the input variables (eg, Current Stock price, Exercise price etc.) and the output variable ( in this case the theoretical fair value of the option). The model that is developed as the end product using the analytical method takes the form of a formula or equation which connects the input to the output .This formula is often referred to as the “solution.”? The main advantage of using analytical models is the ease with which one can use them to produce a precise valuation. It has some disadvantages as well. First, it is difficult to derive the analytical models if one does not have the knowledge of advance calculus. Second, in certain cases it may not be feasible to use the analytical method of problem solving. Third, the models or equations derives using the analytical models and very much in flexible. That is they can be only applied to calculate the desired assumptions under the set of assumptions that were used initially to design the model. Figure :Categories of Option Pricing Models

The second type of model is Numeric Models which are considered to be much more flexible than the earlier discussed analytical models. In these kind of models we again start with a set of assumptions on how the input variable behave but we employ an algorithmic way using a finite series of steps to arrive at the desired value rather than using an equation which relates the input to the output. Thus an approximate value of the desired output is derived rather than a precise value as in the case of analytical models. The advantages of numeric models are they are not so complex to build as compared to the analytical models and also do not require advanced knowledge of the stochastic calculus to build, understand and critically appreciate them. The second advantage lies in the dat they are much more flexible than the previously discussed analytical models. The assumptions made to build the model can be easily changed later to adapt to the new problem situations. The disadvantage though however lies in the fact that the degree of accuracy of the desired output value is directly proportional to the number of times one runs the models. This means to get a very hig degree of accuracy we would have to run the models and may have to make thousands of calculations. But this disadvantage is no longer valid in current world as rapid technological advances has increased the speed of the microprocessors rapidly and also reduced the cost which has made them more accessible.

Simulation Models:

The third category of options pricing models is the Simulation models. These models are less elegant than the analytical models and also not as fast as the earlier tow models we discussed. But their main advantage lies in the incredible flexibility which can be used to price very complex derivatives that depend on facts which cannot be modelled using either the analytical or the numerical valuation technique. Indeed, the future price of any financial asset can be simulated when it is expressed in the form of an expected value. In the simulation approach, we program a computer to “simulate”? observations on the random variable of interest. But also Care must be taken to ensure that the values that we simulate must be according to a type of distribution (Binomial, Exponential) and the other statistical parameters as each individual problem may demand.

 Binomial Options Pricing Model

In this section we will have a deeper understanding of the binomial option pricing model and also its application using VBA in the option pricing application that the author has developed.

One Period Binomial Model:

In a Binomial Model the stock price is allowed to either go up or down possibly at different rates. A Binomial Probability distribution is one in which there are two outcomes and states. This probability distribution governs the probability of the up and down movement of stock prices. Consider a stock with current stock price S on which options are available. The beginning of the period is today and referred to as time 0 and end of the period is 1. At time 1 the stock can take one of the two values. It can go up by a factor of u or go down by a factor of d. If the stock price goes up the stock price at time 1 will be uS and if stock price goes down stock price at time 1 will be dS. Therefore the intrinsic value of call option at expiration is :

Cu= Max [0, uS-X]

Cu= Max [0, dS-X]

binomial model diagram.jpg Figure :One Period Binomial Model Let r be the risk free rate and we assume that investors can borrow and lend at the risk free rate. The aim of this model is to determine the fare theoretical value of an option which is them compared with its actual price in the market and which tells us whether the option is underpriced or overpriced. We develop the formula for option price C by constructing a risk free portfolio consisting of a mix of stocks and options. Consider a portfolio of h shares of stocks and a single written call. Let V denote the current value of the portfolio .V is given by V=hS-C. We can think of V as the amount of money we require to construct this portfolio. At Expiration the value of the portfolio can be given by Vu if the stock price goes up or Vd if the stock price goes down. Thus

Vu = huS - Cu

Vd = hdS - Cd

We can think of Vu and Vd as the money that we will get back at expiration when we sell the portfolio. According to our earlier assumption we defined this portfolio as risk free which means regardless of the stock price going up or down the portfolio should give the same riskless return i.e. Vu= Vd huS - c = hdS - Cd Solving for h gives us h If the portfolios initial value is growing at the risk free rate its vale after one period at expiration will be

V (1+r) = (hS – C) (1+ r) = Vu = huS - Cu

 

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