# Pricing Options Using Binomial and Trinomial Methods: Online Essay Help

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.

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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.

Contents

• 1 Chapter 1 introduction
• 2 1.1 Risk
• 3 1.2Options
• 4 1.3 Types of investors.
• 5 1.4 The Black Scholes Merton model

## 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 personâ€™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 â€“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 buyerâ€™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 investorâ€™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 investorâ€™s profit is=0 He will lose ï¿¡39000, and the percentage loss is 100%. Arbitrageur â€“ An individual engaging in arbitrage. Arbitrage â€“ 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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