# Pricing Options using Binomial and Trinomial Methods

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Published in the 1970s, the Black-Scholes-Merton model provided an entirely new definition for the financial option market, half a century later the Binomial tree option pricing model was published, and that is the true key that allows the option market to be generalized to the world. Based upon the Binomial model, the Trinomial option pricing model was built to reduce possible errors and persons thus expected it to be a better approach. Still how much better is the Trinomial model, and is it worth spending the time on calculations? These will be the key comparisons provided in this dissertation. The comparisons are based upon computer calculating time used, and approximation error. An illustrative example is used to build the data base for further comparison of the convergence speed of these two models. All the values are calculated using the Matlab program and Casio calculators in order to provide examples of the assumption that the Trinomial option pricing model is a better model in reducing the approximation error, but takes much longer than the Binomial tree model to get the results.

## Chapter 1 introduction

The emergence of financial derivatives in the 1970s marked a highly significant and exciting event in the history of finance. Options trading began in the United States and European markets in the late eighteenth century, and over the last 20 years, options played a key role in all financial derivatives.

The option price was an old question for the financial world. Back in the 1900s Louis Bachelier published his academic dissertation 'ThÃ©orie de la speculation' (Theory of Speculation), which became known by the public as the milestone of modern finance. The 'random walk theory', which built a random model of the stock price's changing pattern and how it follows in the stock market, was first applied in his paper. In 1964, Paul Samuelson (Nobel Prize in Economic Science winner) revised L.Bachelier's model, and instead of the stock price he used stock returns to eliminate the negative figures which might occur in L.Bachelier's model. Based upon this new model P.Samuelson also studied the Call Option pricing problem, and built a pricing equation for it. Although the equation was quite a beauty to watch, it could not be used in real world dealings since two of the main factors depended upon the investor's personal predilection.

Futures and options are traded actively on many exchanges throughout the world. Before any certain systematization models of the option had been created it was impossible for people to evaluate any kind of option price in a common way. Any approximations of the price based trader's personal experience would well likely result in mistakes. The only method to maximize the good of the option price would be to build a standard and systematization model and find the quantification of the option trading.

This was an important event in the financial world at that time. Since the emergent of option trading, and especially of securities options trade, researchers have been busy in the studies of options pricing.

In 1973, Fischer Black and Myron Scholes published 'The Pricing of Options and Corporate Liabilities' at the University of Chicago, where they presented the famous Black-Scholes model for options pricing (B-S model for short). They derived a partial differential equation, which governs the price of the option over time. Once it has been published, the B-S model received strong responses and gained a breakthrough in this field. While some researchers conducted thorough tests on the model's accuracy, many others presented various opinions on the problems in the model and expanded on them for the purposes of improvement and extension. Because of this glary partial differential equation and all of the contribution that it had created, M.Scholes and R.Merton (F.Black was deceased) both won the Nobel Prize for Economics.

In 1979, Cox, Ross and Rubinstein published a paper called Option Pricing: 'A Simplified Approach', and in a simple manner obtained the pricing formula using the Binomial model, which was applied widely. This is the event that really changed the option trading market because it made option trading more transparent to most traders, and advanced the improvement of the market. During the option, the trading market developed more and more different sorts of option models, with the most famous and widely used models being the European option and American option. As these two options were named, they were mainly applied in Europe and America and the main difference between the two options is when the option will be fulfilled (I will fully explain this at a later stage).

The Binomial option pricing model is essentially a Binomial Tree which shows possible values that an underlying asset or stock initial stock price can take, and the resulting value of the option price at each individual stage of the asset. The main idea of the tree is constructed by assuming that the stock can only go up or down by a factor related to the length of time period, and volatility of the stock. Trinomial model was developed by Prelim Boyle in 1986; it is an adjusted and improved version of the Binomial Tree. Instead of assuming the stock can only go up or down, the Trinomial Tree allows a third choiceâ€”the stock remains constant.

Compared to the Binomial and Trinomial tree model, the Black-Scholes model is a more mathematical and theoretical model:

V = SN (d1) - N (d2) (Will be explained at later stage)

Although the binomial option pricing model and trinomial tree values converge on the Black-Scholes formula value as the number of time steps increases. With these two simplified methods the option pricing theory and option market became more generalized and easier for the public. With the time flows, the option market began to prevail all over the world, and therefore more and more specific different types of options were created to adapt to the disparate country.

In this dissertation I will mainly study and present the relation and difference between the Black-Scholes model, the Binomial Option Pricing model and the Trinomial Tree model, in both a mathematical and financial way.

Chapter 1: This chapter is mainly about the Black-Scholes model's differential equation, including every valuable deduction I provide a few interesting examples to give a straight forward view of this method.

Chapter 2: In this chapter I will explore the Binomial pricing model with European and American options. By presenting the formulas and equations I will study how to calculate the option price and explain some basic financial terms. At the main time I will also compare the results of the Binomial Tree model to the Black-Scholes model.

Chapter 3: In this chapter I will demonstrate the Trinomial model with examples and large amount of figures by using the Matlab software. The European and American options will be compared with the Trinomial model.

Chapter 4: In the last chapter in my dissertation I will look at how effectively the Trinomial tree model is improved based on the Binomial model. The Matlab code I wrote will help me process this comparison up to a million steps. This will be my thesis of this dissertation and this project.

## 1.1 Risk

Many of the valuation and risk management principles apply across all financial options.

In this section, I will first briefly introduce some basic concepts and features of risk management and financial derivatives, especially the option pricing problems.

Risk---Uncertainty of the result

The risks obtained and a person's unexpected profit is the same as bringing loss or even damage to a person. In the financial market, risk is ubiquitous with: asset risk (stock), currency risk (exchange rate): credit risk, and so on.

There are two ways of facing the risks.

Risk Avoidance

Risk-taking

The process of selecting investments with higher risk in order to profit from an anticipated price movement, is called speculation.

Financial derivatives are types of risk management instruments whose payoff depends upon the behaviour of the underlying assets. The most common derivatives are forward contracts, futures and options.

Forward contract: A cash market transaction in which delivery of the commodity is deferred until after the contract has been made. Although the delivery is made in the future, the price is determinedÂ on the initial trade date.

The party agreeing to buy the underlying asset in the future is called a long position, and the party agreeing to sell the asset in the future is called a short position.

The value of a forward position at maturity depends upon the relationship between the delivery price (K) and the underlying price (ST) at that time.

For a long position this payoff is: fT = ST âˆ' K

For a short position, it is: fT = K âˆ' ST

Forward contract is normally traded over-the counter, OTC.

Futures contracts are very similar to forward contracts, except they are not exchange-traded or the contract is standardized, and thus does not have the interim partial payments due to marking to market.

Before studying the Binomial Tree method, I will look at what options are.

## 1.2Options

An option is a derivative financial instrument that gives the buyer or holder the right, but not the obligation, to buy or sell an underlying financial asset or commodity. The buyer of the option gains the right, but not the obligation, to engage in some specific transaction on the asset.

An option which conveys the right to buy something is called a call option, and an option which has the right to sell is called a put option. The reference price at which the underlying may be traded is called the exercise price or strike price.

Most options have an expiration date. The process of activating an option is called exercise. If the option is not exercised by the expiration date, it becomes void and worthless.

The options and related concepts can be classified into the following types:

1. Exchange-traded options

Exchange-traded options (also called "listed options") are a class of exchange-traded derivatives. Exchange traded options have standardized contracts, and are settled through a clearing house with fulfillment guaranteed by the credit of the exchange. Since the contracts are standardized, accurate pricing models are often available. Exchange-traded options include:[4][5]

stock options,

commodity options,

bond options and other interest rate options

stock market index options or, simply, index options and

options on futures contracts

callable bull/bear contract

2. Over-the-counter

Over-the-counter options (OTC options, also called "dealer options") are traded between two private parties, and are not listed on an exchange. The terms of an OTC option are unrestricted and may be individually tailored to meet any business need. In general, at least one of the counterparties to an OTC option is a well-capitalized institution. Option types commonly traded over the counter include:

Interest rate options

Currency cross rate options, and

Option on swaps or swaptions.

3. Option styles

Some options with complex financial structures are called exotic options, and these include:

Barrier option - any option with the general characteristic that the underlying security's price must pass a certain level or "barrier" before it can be exercised.

Double barrier option-A double barrier option involves a mechanism where if either of two 'limit prices' is crossed by the underlying, the option either can be exercised or can no longer be exercised.

Cumulative Parisian barrier option -A cumulative Parisian barrier option involves a mechanism where if the total amount of time the underlying asset value has spent above or below a 'limit price', the option can be exercised or can no longer be exercised.

Standard Parisian barrier option-A standard Parisian barrier option involves a mechanism where if the maximum amount of time the underlying asset value has spent consecutively above or below a 'limit price', the option can be exercised or can no longer be exercised.

Binary option-A binary option pays a fixed amount or nothing at all, depending on the price of the underlying instrument at maturity.

An Asian option is an option where the payoff is not determined by the underlying price at maturity but by the average underlying price over some pre-set period of time.

Bermudan option - an option that may be exercised only on specified dates on or before expiration.

For a cleaner view, I summarized various types of options in to a table below:

standard of classification

Types of options

Option buyer's right

Call option and put option

Excises time of option buyers.

European option and American option

intrinsic value

In the money options, out of the money options and at the Money options

Trading place

Exchange-traded options and OTC options(Over-the-counter)

Structures of options

exotic options and vanilla options

Margin of option.

Unsecured and secured options

There are two main reasons why investors would use options: to reduce risk and to

gain more profit such as to speculate and to hedge. These will be discussed later.

There are two main types of options, one is the European option the other is American option. The European option can only be exercised on the expiry date, while the American options may be exercised at any time before or on the expiry date.

Assume k is the strike price; T is the expiry date, and the payoffs Vt:

Vt = (St-K) + (call option)

Vt= (K-St) + (put option)

In this case, S is the spot price of the underlying asset. (t=T)

Next, I will discuss the option pricing problems.

Options are a type of bond derivative; its price depends upon the movement of underlying assets.

The change of price of underlying assets is random because it is a kind of risk asset. Once the price of underlying assets is confirmed, then the option price can be confirmed too. This is saying that at the time the price of the underlying asset is St, the option price will be Vt and there exists function V(S, t) so that Vt= (St, t).

At the expiry date, the value of option VT is the payoffs.

VT = (ST-K) + (call option)

VT= (K-ST) + (put option)

The option pricing problem is to calculate V=V(S, t), (0,

V(S, T) =

Especially when t=0, and let the stock price is S0, what is the premium?

p=V (S0, 0) =?

Therefore, the option pricing problem is a working backward problem.

## 1.3 Types of investors.

Now, I will look at three types of people in the stock market

Hedger: An individual who enters into hedging trades.

Hedging is a way of reducing risk. Hedgers want to avoid exposure to adverse movements in the price of an asset.

For example:

A Chinese company needs to pay a British supplier one million pounds after 90days.The company is facing the risk of fluctuation of exchange rate. If there is a big exchange-rate rise, this will affect its anticipated profit because of the extra cost. If the exchange rate is 12.5 Yuan / pound.

The company considers two Hedging plans in view of the probability that the exchange rate may rise.

Plan 1. Buy a forward contract stated to use 12625000 Yuan to purchase one million pounds after 90days.

Plan 2. Buy a call option contract stated to use 12500000 Yuan to purchase one million pounds after 90days and pay a 250000 Yuan premium (as 2%).

I now list the two hedging strategies in the table below:

Spot exchange rate

(Yuan/pound)

90dayslater exchange rate(Yuan/pound)

Without hedging

Forward contract

Purchase call option contract

12.5

Increase to13

13million Yuan

12,625,000 Yuan

12,750,000 Yuan

Decrease to12

12million Yuan

12,625,000 Yuan

12,250,000yuan

According to the statistics provided, it can be seen that there will be extra costs when the exchange rate rises if the company does not use any hedging strategies. The costs are fixed after90days if they choose the forward contract but they may miss the chance that if the exchange rate goes down, they will gain from unforeseen profit .Meanwhile the company will prevent extra costs (rise in exchange rate) and gain profits (decrease in exchange) if they choose to purchase the call options contract, but they have to pay the premium.

Speculator: An individual who is taking a position in the market.

Usually the individual is betting that the price of an asset will go up or that the price of an asset will go down.

Options like futures provide a form of leverage. For a given investment, the use of options magnifies the financial consequences. Good outcomes become very good, while bad outcomes may cause the whole initial investment being lost.

For example, assume the stock price of X at 30th of April is ï¿¡666. The stock price may go up in August, and there are two investment strategies that investors may take.

Investors spend ï¿¡666000 cash on 1000 shares of stocksï¼›

Investors purchase a call option contract which ends on 22nd of August: strike price is ï¿¡680, 1000shares, assume investors paid ï¿¡39000 premium for that.

We now analyze the investor's investment return in two different situations. (Ignore the interest rate)

Case 1.If the stock price rises up to ï¿¡730 on 22nd August.

For strategy A: The investor sells stocks on 22nd August to get ï¿¡730000 in cash.

Return = (730000-666000)/666000=9.6%

For strategy B: The investor exercises his option and gets profit:

Profit=730000-680000=ï¿¡50000

Return = (50000-39000)/39000

Case 2.If the stock price drop to ï¿¡660 instead of rise on 22nd August.

Strategy A:

Loss =666000-660000=ï¿¡6000

Return= (660000-666000)/666000

Strategy B: The investor's profit is=0

He will lose ï¿¡39000, and the percentage loss is 100%.

Arbitrageur - An individual engaging in arbitrage.

Arbitrage - A trading strategy that takes advantage of two or more securities being mispriced relative to each other.

Arbitrage opportunities cannot last for long. As arbitrageurs interfere in the market, the forces of supply and demand will bring the market back to equilibrium. Therefore, in my project most of the arguments concern financial derivatives such as option prices, and, forward contracts will be based on the assumption that no arbitrage opportunities exist.

## 1.4 The Black Scholes Merton model

There are seven important assumptions we use to derive the Black Scholes Model:

It assumes that percentage changes in the stock price in a short period of time are normally distributed. It is defined as expected return on stock per year and as volatility of the stock price per year. This assumption suggests returns on the underlying stock are normally distributed, which is reasonable for most assets that offer options.

It is possible to buy and sell any amount of stock, this includes short selling.

There are no transactions costs , taxes or other fees.

The stock pays no dividends during the option's life.

There are no arbitrage opportunities.

Markets are efficient and Security trading is continuous.

The risk free interest rate is constant and known.(

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