# The Application Behind Options Pricing Models

Derivatives are financial products whose values are derived from one or more underlying asset or financial instrument. In todayâ€™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.

## Chapter 1-Introduction

## 1.1-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.Â Â Â Â Â To research the efficiency of the Monte Carlo Simulation models when applied to pricing derivative options.

2.Â Â Â Â Â To compare the Monte Carlo Simulation technique with other options pricing derivative techniques like Binomial Pricing Model and Black Scholes Model.

3.Â Â Â Â Â 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.

## 1.2-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.

## 1.3-Report Organisation

## Chapter 2-Literature Review

## 2.1-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.

## 2.2-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 partyâ€™s gain is another partyâ€™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

## Table 1-Distinction between Futures and Forwards

## 2.3-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:

## S= Current Stock Price

## X=Exercise Price

If S>X: Call Option Holder will exercise the option as he is entitled to buy the asset at a lower price than the current market price. The contract is said to be In the Money in this case.

If S<X: Call Option Holder will not exercise the option as he is able to buy the same asset at a lower price in the global markets. The contract is said to be Out of Money in this case.

If S=X: He may or may not opt to exercise the option.

Also an Option price is given by:

## Option Price Value= Time Value + Intrinsic Value

## Where Intrinsic Value= Max [0, (S-X)]

525px-Payoff_call_option_svg.png

Figure -Call Option Payoff Diagram

## Put Option:

A Put option gives its buyer the right but not the obligation to sell a certain asset at some date in the future known as the exercise date at a pre specified price called Exercise Price. Just like in the Call Option the buyer of the contract has the right whereas the seller of the contract has the obligation to buy the asset at the exercise price should the holder of the option decides to exercise it.

Consider the following at the Exercise Date of the option:

## S= Current Stock Price

## X=Exercise Price

If S<X : The Put option holder will exercise his option as he is able to sell the underlying assets at a price higher than what he would have got if he would have sold at current market prices. Such an option at expiration is said to be In the Money.

If S>X: Put option holder will not exercise his option as he will be able to sell the same asset at a higher party if he sells them in the market. In this case put option holderâ€™s loss is only limited to the amount initially he paid to seller of put option as premium. Such an option at expiration is said to be Out Of Money.

525px-Payoff_put_option_svg.png

Figure -Put Option Payoff Diagram

## 2.4: Option Greeks

The Price of an option can be influenced by many factors. These factors help the options traders determine the positions they ought to take and hence their profitability. It is therefore of utmost importance to know about the â€œGreeksâ€? which are a set of risk measures which determine to what extent is an option price dependent on changes in the underlying price, implied volatility and time value. Most commonly used by traders are four Greeks namely Delta, Theta, Vega and Gamma.

## CallÂ

## Options

## Increase in Volatility

## Decrease in Volatility

## Increase in Time to Expiration

## Decrease in Time to Expiration

## Increase in the Underlying

## Decrease in the Underlying

## Long

## +

## -

## +

## -

## +

## -

## Short

## -

## +

## -

## +

## -

## +

Figure : Factors Affecting Long and Short Call Options Price

## PutÂ

## Options

## Increase in Volatility

## Decrease in Volatility

## Increase in Time to Expiration

## Decrease in Time to Expiration

## Increase in the Underlying

## Decrease in the Underlying

## Long

## +

## -

## +

## -

## -

## +

## Short

## -

## +

## -

## +

## +

## -

Figure : Factors Affecting Long and Short Put Options Price

As it can be seen from figure 4 an increase in volatility, increase in time to expiration and increase in price of the underlying will increase the value of the long call option. Whereas a decreased in volatility, decrease in time to expiration and decrease in price of the underlying will increase the value of short call option.

## Vega

## Theta

## Delta

## Gamma

Measures Impact of a Change in Volatility

Measures Impact of a Change in Time Remaining

Measures Impact of a Change in the Price of Underlying

Measures the Rate of Change of Delta

Figure : The Major â€œGreeksâ€?

## Delta:

It can be considered as a value which measures the change in the price of an option i.e. premium paid buy option buyer, which results from the change in the price of the underlying asset (commodity, futures, etc.). For put options the value of delta ranges from -100 to 0 while for call options the value of the delta ranges from 0 -100.

Since put premiums fall with an increase in the price of the underlying i.e. puts have a negative relationship to the underlying, puts have negative delta. The calls have a direct relationship with the price of the underlying and hence have a positive delta.

Consider an example where the delta of an sugar call option is 0.5 and if the price of sugar increases by 10 cents then the premium for the all option will also increase by 5 cents (0.5 x 10).

## Gamma:

The Greek called Gamma is dependent on the value of the earlier Greek discussed i.e. Delta. It is the rate of change of delta and also called the â€œfirst derivative â€œof delta. The value of gamma for a long call as well as a long put is always positive. This means that the value of delta will increase with the increase in the price of the underlying and value of delta will decrease with the decrease in the price of the underlying. The Value of Gamma for at the money options is highest and its values decrease as it goes out of money or in the money which can be seen by the figure below.

Greeks_OTM-ATM-ITM.png

Figure -Effect of Moneyness on the value of Greeks

## Theta:

The third commonly used Greek theta is not much used but is an important theoretical concept. Theta measures the rate of decline of time-premium resulting from the passage of time. In simple terms, an option premium that is not intrinsic value will decline at an increasing rate as expiration nears. The Value of Theta like the other Greeks for at the money options is highest and its values decrease as it goes out of money or in the money which can be seen by the figure below.

## Vega:

Vega, our fourth and final risk measure, quantifies risk exposure to implied volatility changes. Vega tells us approximately how much an option price will increase or decrease given an increase or decrease in the level of implied volatility. Option sellerâ€™s benefit from a fall in implied volatility, and it's just the reverse for option buyers. The further an option goes in the money or out of money the smaller will be the value of Vega.

## Chapter 3- Option Pricing Models

## 3.1-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 â€œsolution.â€?

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

## Option Pricing Models

## Simulation Models

## E.g. Monte Carlo Simulation Models

## Numeric Models

## E.g. Binomial Pricing Models

## Analytical Models

## E.g. Black Scholes Model

## Numeric 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 â€œsimulateâ€? 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.

## 3.2- 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 â€“ C) (1+ r) = Vu = huS - Cu

On substituting the value of h and solving for the option value C we get:

## C =

Where p is given by p=

This is a very basic formula derivation for options price on which the other multi period models are based on which we will go through in the next section.

## Two Period Binomial Models:

In the 2 period binomial models there are three time points as compared to two in the one period model. The option prices at expiration are given by:

## Cu2 = Max [0, u2S â€“ X]

## Cud = Max [0, udS - X]

## Cd2=Max [0, d2S â€“ X]

binomial model diagram.jpg

binomial model diagram.jpg

Figure : Two Period Binomial Model

Using the same method as in the one period model we can obtain the following formulas:

## Cu =

## Cd = [ pCud + (1-p) Cd2 ] / [(1+r)]

Therefore the call price at time zero can be found by calculating the weighted average of the two possible call prices in the next period i.e.

## C =

The probability value of the stock price moving either way is determined using the volatility value of the stock where:

U = e ^. v=volatility

D= e ^.

The value of p can also be derived by the assumption that over a period of time dT the underlying asset will yield the same profit as an riskless investment , so that if it is worth at S at time t, it will be S.e^(r.dT) at the time t+dT .

Thus S.e^(r.dT) = (p.u.S + (1-p).d.S) and solving for p we get

## P=

Thus we can say that the binomial pricing model is an iterative model that models the evolution of options price over a number of periods of time.

## Extending the Binomial Model to n Periods:

We can extend the binomial option pricing model to any number of periods. As the number of time periods increase the option value also gets more closer to the actual precise value. We do assume that there are n periods remaining until expiry of the options and also that the option pays no dividends.

clip_image012.jpg

Figure - Multi period Binomial Model

## Implementation in VBA:

The author has implemented the multi period binomial model using Visual Basic for applications (VBA) in the excel pricing model he has developed. The functionality provides for pricing both American and Europeans types of calls and put. The user can also specify the number of steps or periods and the model will give the value of the Call and put along with the commonly used option Greeks like Delta, Gamma and Theta. Below is a snapshot of the application developed by the author:

Figure : Binomial Application Snapshot

Public Function CRRBinomial(OutPutFlag As String, AmeEurFlag As.jpg

As you can see above in the code a Function named CRRBinomial has been defined to price option using the Binomial Model.

OutPutFlag : used to determine what kind of information the function returns when it is called. Eg if OutPutFlag =â€?pâ€? function will return the option value and so on.

AmeEurFlag: determines whether the option to be priced is an American or a European option.

CallPutFlag: determines whether the option to be priced is Call or Put option.

OptionValue: defined as an array which runs till n+1 where n is the number of periods? This variable is used to calculate the Call option price at various points in time which are then discounted back at the risk free rate of return to give the option price at current time period.

U,d : probability of stock price going up or down

dT: time in each period calculated as (time to expiration)/(number of period)

Df : Discounting factor using riskless rate of return calculated as Exp(-r * dt)

For j = n - 1 To.jpg

The Code above calculates the option price value using the formula for the multi period binomial model. The If condition is to check whether the type of option to be priced is American or European type and accordingly applies the formula to calculate the option value. The variables ReturnValue(1) , ReturnValue(2) , ReturnValue(3) calculate the option Greeks Delta, Gamma and Theta respectively.

## 3.3- Black Scholes Options Pricing Model.

## Assumptions of Black Scholes Model:

As Mentioned earlier all models are based on a certain specific set of assumptions. The following assumptions are made in the Black Scholes model.

Stock Prices evolve according to Lognormal Distribution and behave randomly: We here assume that the behaviour of the stock prices is random which is also a common assumption in pricing most derivatives. Although some fund managers claim that they can predict stock prices uses past trends and other methods. Studies have shown that there may be only small room for predictability and that stock returns are largely unpredictable.

We also mentioned that stock returns are log normally distributed. The logarithm of a number is the power to which a base must be raised in order to equal that number. The base of natural logarithms is e which is equal to 2.718. In finance mostly natural logarithms are used. A lognormal distribution is one in which the log of a variable is normally distributed. This distribution of stock returns has positive skewness i.e. most of the variables lie to the right side of the mean in the distribution. This assumption of lognormal distribution is quite a good approximation of the real prices and also prevents the stock price from being negative.