European and American Option Pricing; Black-Scholes and Binomial Models

Transcription

1 Misha William Brooks Registration number European and American Option Pricing; Black-Scholes and Binomial Models Supervised by Dr Christopher Greenman University of East Anglia Faculty of Science School of Computing Sciences

2 Abstract Options are financial derivatives that allow for an asset to be bought or sold at a fixed price on a specified date. In 1973, Fisher Black and Myron Scholes, developed the Black-Scholes Model. This was the first recognised numerical method for deducing, the fair price for a European style option. Since 1973, option markets have grown dramatically, leading to the creation of more complex exotic option contracts. To calculate the value of these Exotic option, new models had to be developed. Within this project we will develop a detailed understanding of European and American style options. This will include the use of two models, the Black-Scholes and the Binomial. Using historical market data, we will aim to compare and evaluate the effectiveness of both models. European options can be modelled by both the Black-Scholes and Binomial models, whereas American options are slightly more complex. For this we will be using an adjusted binomial model. Acknowledgements Firstly and most importantly, I would like to thank, Dr Christopher Greenman. His guidance and advice throughout this project has been invaluable. guiding me through a challenging and complex project, while allowing me to develop my understanding at my own pace and level. I would also like to thank the UEA Library and the facilities of the Computer Science department. They have allowed me to work at unsociable hours, and have access to the necessary resources. Lastly, I would thank my family and friends for their ongoing support and understanding.

3 Contents Contents 1 Introduction Why Trade Options Basic Exotic Options Option Trading Strategies The Black-Scholes Model Black-Scholes Equation Solving the Black-Scholes Equation The Binomial (Cox-Ross-Rubinstein) Model Derivation of Binomial Model Pricing An European Option Pricing An American Option Analysis Estimating Volatility and Risk-Free Rate Pricing XEO (European style) Options European Call Option European Put Options Pricing OEX (American style) Options Conclusion 45 6 References 47 7 Appendix European Style Options American Style Option Reg: iii

4 1 Introduction 1 Introduction Options were first used in ancient Greece, to speculate on the outcome of the olive harvest. In modern times they are mostly linked to stock/equity. Options give the holder the right, but not the obligation to buy or sell at a given price, known as the strike price, on a predetermined date, this is the expiration date. The first modern use of options was in the 1920 s, Bucket Shops. These allowed traders to speculate on the price increase or decrease of a companies stock, the owner of the bucket shop, would then pick the other side. For example say the owner of the stock thought it would decrease by next week, he would set up a contract with the bucket shop. This would be a simple option of, you can buy my stock next week if the price decreases. This was similar to Boiler Rooms, where brokers create artificial demand for a company. This will cause any stock they own in the company to rise in value, but in reality it is worth next to nothing. As you can see when options were first traded, the market was rife illegal activity and financial deceit. Nowadays options are recognised financial derivatives, usually they are based on stock, but can also be used in other ways. Options can be on traded either in financial markets, such as Chicago Board of Options Exchange (CBOE), or by over-the-counter markets. According to (Hull,2006) there are six factors that affect the price of a stock option, Current stock price, S Strike price, K Time till expiration, T Volatility of the stock price, σ Risk free interest rate, r Reg:

5 1 Introduction Expected dividend payout We will be treating all our options as vanilla options, this means that they don t pay dividends. But we will be looking at the other five factors and how they affect price. There are two types of option, these are put and call options. A call option gives the owner, the right, but not the obligation to buy stock on a given date. This is called the expiration date. They will buy the stock at a predetermined price, known as the strike price. A put option gives the owner the right, to sell his stock on a specific expiration date, for a predetermined strike price. A trader will aim to exercise their call option when the strike price, is lower than that of the stock price. This means they will be buying the stock below the market price and can instantly sell for a profit. On the other hand, a trader will exercise his put option, when the stock price is lower than that of the strike price. By selling at the strike price, he prevents further losses. This can also works in reverse, the trader will make a loss and most likely won t exercise the option. If in the case of a call option, the strike price is higher than that of the stock price. When the put option strike price is less than that of the stock price, then he will chose not to exercise, because the trader can wait out the contract, then sell on the open market. The difference between the strike price and the exercise price, shows the trader how much loss or profit he will make from exercising the option. But in reality you must also take into account, trading costs and taxes involved with trading. Below are the intrinsic values of a European call, C(S,T ) and put, P(S,T ) option at time expiration, time T : C(S,T ) = max(0,s T K) P(S,T ) = max(0,k S T ) At first markets only traded European options, but once these were accepted and the markets started to expand, traders began to think of new and more complex options. These are known as Exotic options; they provide a better fit, to suit the financial position and needs of a traders. They can vary in many ways, but all are originally derived from the European Put and Call option. The creation and acceptance of exotic options gave birth to string of new trading strategies. Variations on timings, strike prices and Reg:

6 1 Introduction payouts, have given investors the opportunity to customise portfolios. The most popular exotic options is an American option, we will explore this in greater detail. But first here is a list of Exotic options and how they differ from the original European option. Over-The-Counter(OTC) Option Markets have also grown rapidly, and it is now viewed as larger than the exchange-traded market. These are usually traded over the phone by financial institutions and firms, the main advantage is these options can be structured to each individual clients needs. These can be variations on strike price, expiration dates and contract size, this has lead to most exotic options to be traded in the OTC market. But it does present the disadvantage of option writer default, which refers to the purchaser experiencing some credit risk. Time Value, is the value of an option, which is not represented by its intrinsic value. This is basically the premium that is required to sell or buy the option, it is found by complex calculations that are based on the stocks volatility. The greater the expected movement in the stocks price, the higher volatility, which gives a higher intrinsic value. As the option nears expiration, the time value decreases, this because the stock has less time to move in a favourable direction. The Moneyness concept describes what position an option holds. This is determined by comparing the strike price to the current market price of the asset. Below are some simple examples, in reality you will need to account for contract cost and premiums. It is also essential to consider the Payoff of an option. This is where the premium paid for the option is deducted from the intrinsic value. If this is not considered an option that is slightly In the Money, could become unprofitable or may even make a loss. 1. In-the-Money The option is profitable at this point, if exercised immediately. A call option is in the money if the strike price is lower than that of the underlying asset price. Say stock A has a market price of $50 and the options strike price is $40. This means it is $10 in the money, because that s the profit the trader will make if he exercises the option. A put option is in the money when the strike price is greater than that of the underlying asset price. Say the stock A is at $40, Reg:

7 1 Introduction and the strike price is $50. The put option is therefore $10 in the money, if the trader exercises the option now, a $10 profit will be made. 2. Deep-In-the-Money, This is an In-the-Money option, but with a larger difference between the strike price and stock price. Because of this, these options are traded closer to their actual intrinsic value. This means the option changes at an equivalent rate, simultaneously to that of the stock price. Thus, investing in the option is similar to investing in the underlying asset, except the option holder will have the benefits of lower initial investment, providing limited risk. 3. Out-the-Money, This is an option that will make a loss if exercised immediately. A call option is out-the-money when the underlying asset price is lower than the strike price. When this happens, the option won t be exercised, but in some rare cases the position is ignored. A put option is viewed as out-the-money when the strike is less than that of the underlying asset price. This is because the trader can get more for selling it on the open market. 4. At-the-Money The Strike price is equal to that of the underlying asset price. This is the same for put and call options. It s very rare that this happens in an open market. For example if stock A is valued at $50 in the market, and the strike price is $50 for the put and call option, then the option is at-the-money. But due to trading charges and commission it will probably make a loss if exercised. When an option is at-the-money trading activity increases as the option price is very low, hence easy to speculate and low cost. The decision whether to exercise an option, is based on the strike price compared to the stock price at expiration. Below are some simple graphs taken from Investopedia (Investopedia, 2014), that show the option position, provided the contract is held to expiration, then exercised. These are based on European style contracts, later on I will go on to show more complex strategies. Let S be the final price of the underlying asset, X is the strike price and P is the option price at time T, which is the expiration date. Reg:

8 1 Introduction Diagrams Option Profit Option Profit Diagrams, at Expiration; Strike price = X; Underlying asset price, at expiry = S T 1. Long Call - Is where an trader buys a call option. This will have a strike price that s equal to the current stock price, but with expiration in a months time. If the stock price rises, he will exercise the option, then resell the stock at the current market price. If it falls, he will wait out the contract and he loses what he paid, to buy the original call option. 2. Short Call - The trader expects the price of the option to go down. By selling a call option with a strike equal to the current stock price. Provided that the stock price goes down, the option will become worthless and he will make a profit, but if it goes up he is making a loss. 3. Long Put - Similar to Long Call, but the trader buys a put option instead. This has a strike price equal to the current stock price and expiration in a months time. If the stock price falls, as he expects, then the put option now provides a method Reg:

9 1 Introduction of mitigating losses. Therefore it s now in-the-money and can be sold on. If the strike price goes up, then the put option becomes worthless, this gives the trader a loss of the original price paid to acquire the put. 4. Short Put - The trader expects the stock price to increase, so he sells a put option with a strike price equal to the current stock price. If it does increase the option becomes worthless and he makes a profit, but if it decreases he will make a loss. 1.1 Why Trade Options Option contracts can be used in many ways and by many different companies. In the real world firms that deal in commodities will set up physical options. For example a Mining company has found an area of land rich in gold, but the cost of extracting the gold is greater than the gold will sell for in the commodities market. But this doesn t stop the company from buying the land, therefore if the market price rises above the cost of extraction, they can then start mining. Hence they have the option to extract the gold, if it is financially profitable. In the financial markets options have been used for over thirty years, but in many ways it s still an emerging market. Traders in the past have been put off, due to options being seen as complicated and risky, when in reality it offers some distinct advantages and if used properly can be much safer than the normal stock market. Below are some the advantages described in (Hull, 2006). 1. Cost Efficiency - A trader can buy an option that will mirror the underlying stocks position, but for a fraction of the price of buying the stock. For example an investor who believes the price of the stock will go up over the next six months could be a call option, with an expiration date for six months time, but with the current stock price as the strike price. Hence if it does go up, he s made a significant profit. This is without having to spend the considerably larger amount it would have cost to buy the stock at the start. The amount the trader could lose is therefore a lot less, as his original investment was less. 2. Less Risk - This is very much a situation of being used correctly. There are some situations where buying an option is riskier than owning the underlying asset. Reg:

10 1 Introduction Options can provide reduced risk in three ways. These are firstly they require less financial commitment, which means less can be lost in the investment. Secondly they can also provide protection from gap openings. This is where a stocks price changes quickly and a big demand or losses occurs. Quiet often the change in price isn t reflective of the actual value of the stock. Finally, options can be used for hedging. Hedging is the process of betting against a stock you already own. Hence if a trader owns Stock A, and wishes to hedge against it making a loss, they will buy a put option to sell at the current stock price at a future date. This way he won t make a loss if the stock price falls, and will make a profit if it rises. In theory the only way he can make a loss is if the option cost is greater than the rise in stock price over the period of the contract. The major advantage of this above a Stop-Loss Order is that a Stop-Loss can only be done during trading hours, a Stop-Loss order is where a trader will sell stock when it drops below a certain price. Options do not shut down, when the markets close for the day. Hence if a stock crashes overnight the option position is unaffected. 3. Higher Potential Returns - Very simple, to purchase a option cost a lot less then to buy the stock. But the profits almost the same. Therefore the return on investment is much higher. For example if a option is bought for $10, with a strike price of $50, and the stock is sold for $50. Suppose the stock price rises to $100 over 6 months. The investor who purchased the option makes a $40 profit, and the investor who bought the stock makes a $50 profit. That would be only $10, for $40 less original investment. 4. More Strategic Alternatives - Options provide investors with a means of position synthetics. This is the ability to recreate other positions. Options are very flexible tools, with many helpful investment alternatives. In reality most stocks don t make big moves over short periods of time, hence if an investor can take advantage of stagnation in a market, but predicting correctly the next move, then they can make large profits. Options make this possible, but with less cost. Reg:

11 1 Introduction 1.2 Basic Exotic Options These are further explained along with more complex Exotic options in Exotic Option Pricing and Advanced L/hatevy Models(Kyprianou, 2006). American - The option can be exercised up to and on the expiration date, allowing for Flexible exercise times and more complex strategies, making it easier to mitigate losses early. Asian - The payoff is determined by the average underlying stock price over a pre-set period of time. They can be either Fixed or Floating. On average they are cheaper as the volatility of the average price is less than that of the spot price. The trader can also cover many transactions with one option, simplifying the portfolio. Binary (all-or-nothing) - These can be European or American, the payoff is either a fixed amount or nothing. They can be Cash based or Asset based. Once the underlying asset value crosses a set point, they become active or inactive, dependent on the contract. Normally Cheaper than plain American or European, the trader is usually confident of a certain movement in price. This Can be risky due to Knock out/in, leaving the option worthless. Lookback - The trader can "look back" for the optimal values, they can also be based on fixed or floating system. fixed means the payout is the difference between the strike and the maximum or minimum value of the underlying asset. Floating means the payout is the difference between the underlying asset at the end of the contract and the maximum or minimum value during the contract. It eliminates timing problems and maximises profits, but is expensive. Bermudan - Can be exercised on specific dates before expiration, a less flexible version of the American. Usually Cheaper than American and still allows for Reg:

12 1 Introduction flexibility. Reg:

13 1 Introduction 1.3 Option Trading Strategies There are many different trading strategies an investor can adopt, I will briefly describe some below. All images are taken from (Investopedia, 2014), with edit descriptions using (Ross, 2012) and (Brealey, 2011). Covered Call, is a simple strategy. once an assets is bought outright, a call option is written to the equivalent amount of the asset. It s a simple one for one principle, every unit of stock bought a option is written to sell at a set price. This is usually associated with a short-term position Married put, is also a one for one principle. But this time for every unit of the asset owned, a put option is purchased to sell at the price it was bought at. This is used to prevent short term losses, especially when a trader is uncertain about the assets position Reg:

14 1 Introduction or price. Bull Call Spread, is where an investor will buy a call option, while simultaneously selling a call option. These will have the same expiration date and be for the same quantity, but the one he sells will have a higher strike price. If the assets prices rises as he expects and somebody buys his issued call option, the profit is the difference in strike prices. This is a type of vertical spread strategy. Bear Put Spread is also a type of vertical spread. An investors will buy a put option, and simultaneously sell a new put for the same amount and with the same expiration date, but with a lower strike price. If the assets price decreases and somebody buys the put option, the profit is the difference between the two strike prices. Reg:

15 1 Introduction Protective Collar, is a strategy used to lock in profit from a stock that s recently experienced substantial gains. This is achieved from simultaneously buying out-the-money put options and writing an out-the-money call option for the same stock. The put protects the investor from large downward moves, hence locking in the profit from the original gain. While the sale of the call the cost of purchasing the put is covered. Long Straddle is when an investor will simultaneously buy a put and call option with the same strike price, expiration date for a stock. This is used when he is uncertain which direction the stock price will go, but is sure that it will move. By doing this he covers himself each way, he can make unlimited profit with a maximum loss of just the price of the options. Reg:

16 1 Introduction Long Strangle is when an investor will buy out-the-money put and call options on stock, with the same expiration date. Usually the strike price of the put will be lower than that of the call. This is a good strategy if the investor is unsure on what way the stock will move. Also due to both options being out-the-money, it is usually cheaper than a straddle. Butterfly Spread is a combination of Bull and Bear Spreads. It uses three strike prices, K1,K2 and K3. K1 is the a low strike, with K3 as the high strike. K2 will then be halfway between K1 and K3. For example the investor purchases a call option with a strike price of K1, and writing three call options, two at the strike price of K2, and one at K3. This strategy is based on the assumption the stock price will rise, but protects the investor if it falls again before reaching K3. Reg:

17 1 Introduction Iron Condor involves using four different options, these are based on both, a short and a long strangle. A strangle is when an investor buys one out-the-money call and put option, which have the same expiration date and strike price. The four options have different strike prices, but all have the same expiration date. The investor will sell two options, these are a middle strike priced, out-the-money call and put. While buying two options that are high strike priced, out-the-money call and a put. This strategy protects him from losses, but does limit potential profit as more profit is made when there is little movement in stock price. Reg:

18 2 The Black-Scholes Model 2 The Black-Scholes Model Fisher Black and Myron Scholes first proposed their equation for pricing of a European style option, in their paper "The Pricing of Options and Corporate Liabilities" published in The Black-Scholes equation was derived from a stochastic partial differential equation, it estimates the price of an option over time. Before 1973, options where seen as risky and unreliable, with markets that couldn t be trusted. The Black-Scholes equation made it possible for every trader to estimate what the price of the option should be, hence preventing miss-pricing. This in turn legitimised the activities of markets such as the Chicago Board of Options Exchange. Since then there has been a dramatic increase in the use of options within derivative trading. Robert Merton, also provided key contributions, but his approach was very different. He worked backwards basing his theory on producing a risk-less portfolio of stock and Options on those stocks, and therefore the return on the portfolio is risk free. The Black-Scholes equations is based on some simple assumptions, these are important as they gave a basis for the equation to be derived. They are as follows: No Arbitrage Investors and Firms can borrow and lend cash at a constant risk free rate Any amount of a stock can be bought or sold There are no transaction fees, costs or taxes incurred Stock prices follow a Geometric Brownian Motion, with a constant drift and volatility The Stock doesn t pay dividends Security trading is continuous Arbitrage refers to a situation where risk-less profit can be made. If a trader makes two simultaneous transactions, which yield an instant profit, while avoiding all market Reg:

19 2 The Black-Scholes Model risk, this would be a form of arbitrage. Arbitrage can happen if any of the following situations are ignored; First is the law of same price, which states that an identical asset must be valued the same in every market. If two assets have the same cash flow, they therefore must trade at the same price. Lastly an assets that is, trade at its known future price, must be discounted back to the present time, at the current risk free rate. Any of these could be a reason for arbitrage advantage to occur, hence it is theoretical and very difficult to regulate. Adjustment of other traders usually means, if one person finds a way to have arbitrage advantage, others will copy, hence the market average is restored. 2.1 Black-Scholes Equation I will now explain the notation used when deriving the Black-Scholes equation, this references (Cox, 1979), (Black and Scholes, 1973) and (Hull, 2006); S - Stock Price V (S,T ) - Payoff function of an option at maturity K - Strike Price r - Risk Free Rate µ - This represents the drift rate of S (the rate at which the mean changes) σ - The volatility of the Stock return t - The time in years till the contract reaches expiration date Π - Value of the portfolio The Black-Scholes equation is based on the theory of stock prices following a geometric Brownian motion, this can be represented as; ds S = µdt + σdz (2.1) Reg:

20 2 The Black-Scholes Model This shows that the log of the underlying asset follows a random walk, with a volatility of σ, and with the return on the stock being represented by µdt. Random walk is where a variable jumps between values or sites on a lattice. Most models for option pricing use this as the basis for valuing the underlying stock price. V (S,T )can be used to represent the pay-off function of the option at maturity. But to find it s value at an earlier time, we need to understand how V changes as a function of the Stock Price and Time. Itô calculus can be used here, it allows us to find the differential of time dependent options. As the variable dt decreases, the closer d 2 approaches dt. By multiplying equation 1 through by the Stock Price, S, you get ds = µsdt + σsdz (2.2) Itô Lemma shows that V evolve as a function of S and T, this gives dv = ( V S µ + V t V 2 S 2 σ 2 S 2 )dt + V σsdz (2.3) S Robert Merton s theory of creating a risk-less portfolio now comes into play; Consider a portfolio containing two positions, A short option and a long amount V S of shares. Now as stated, let Π be the value of the portfolio, hence Π = V + V S S (2.4) Now consider, as the time interval t changes, the value of the portfolio will change, giving Π. To make this clearer I will represent the small changes by. Π = Π t = t V = V This means equations 1.1, 1.2 and 1.3 become; S = µs t + σs z (2.5) V = ( V V µs + S t V 2 S 2 σ 2 S 2 ) t + V σs z (2.6) S Reg:

21 2 The Black-Scholes Model Π = V + V S (2.7) S To find the value of the portfolio we must substitute S and V into equation 2.6, this gives us, Π = (µ V V S t + t t V This simplifies down to, σ 2 S 2 ) t + σs V S 2 S V z) + µs S t + σs V S Π = ( V t 1 2 σ 2 S 2 2 V ) t (2.8) S2 Hence you can see that the variable dz is no longer in the equation. This means that the portfolio must be risk-less during t,π earns the same risk free rate of return that other investments may offer. Arbitrage would be earned if the portfolio earns more than the risk free rate. Now considering r as the risk free rate of return over the time period t t + t, this leads to the equation, Π = rπ t (2.9) showing that with no opportunities of arbitrage, the best rate the portfolio can reach is r. Hence when investors are using this strategy, portfolios must be continuously updated. This is to ensure that the ratio in 2.8 is maintained at all times, allowing the portfolio to grow at the rate r. If an investor finds a strategy or combination that gives a higher return, than the risk free rate, then markets will adjust. This is to say the market adjusts and adopts new strategies, hence the improved rate of return now becomes the risk free rate. Now to form the Black-Scholes differential equation, just substitute for Π, giving ( rv + rs V V S ) t = ( t 1 2 σ 2 S 2 2 V ) t S 2 Reg:

22 2 The Black-Scholes Model This can be rearranged to equal 0, 2.2 Solving the Black-Scholes Equation V t σ 2 S 2 2 V S 2 + rs V rv = 0 (2.10) S Black-Scholes equation, is generally only used to find the price of European put and call options. In this section, we will consider how to find explicit solutions to the Black- Scholes equation, when considering European options. Firstly we must revisit the derived equation and the boundary conditions, Consider a call option, V t + 2 1σ 2 S 2 2 V + rs V S 2 S rv = 0 V (S,T ) = C(S,t), with the value of this call option being represented as, C t σ 2 S 2 2 C S 2 + rs C rc = 0 (2.11) S From the arbitrage condition, you can establish that the option will be exercised when S > K, hence we can let C(S,T ) = max(s E,0) Next we consider the case of S = 0 at expiry, this would yield a payoff of 0. Hence the call option is worthless hence we have C(0,t) = 0. This also works in reverse, as S, the value of the option will tend to that of the asset. C(S,t) S, as S Reg:

23 2 The Black-Scholes Model Now we can establish the boundary conditions, C(0,t) = 0,C(S,t) S as S and C(S,T ) = max(s E,0) To derive solutions to the Black-Scholes PDE, we first must convert our equation into the form of a heat equation.the first step is to remove the S and S 2 terms in the equation 2.1. But before we do this we have to transform some of the variables; U(x,τ) = V (S,t) K x = ln S K S = Ke x τ = σ 2 2 (T t) t = T 2τ σ 2 = 1 k V (Kex,T 2τ σ 2 ) Now we apply the chain rule to the partial derivatives in the Black-Scholes PDE. This gives us equations for V t, V S and 2 V S 2 V t = K U τ = Kσ 2 2 τ t U τ (2.12) V S = K U x x S = K U S x = e x U x (2.13) Reg:

24 2 The Black-Scholes Model 2 V S 2 = K U S 2 x + K S = K S 2 U x + K S = K S 2 U x + K S = e 2x K ( 2 U x 2 U x ) S ( U x ) x ( U x ) x S 2 U x 2 (2.14) Next we substitute our values for V t, V S and 2 V S 2 into the Black-Scholes PDE, Kσ 2 2 This can be simplified to, U τ + rkex e x U x σ 2 K 2 e2x e 2x K ( 2 U U x 2 x ) ru = 0 U τ + (k 1) U x + 2 U ku = 0 (2.15) x2 k = 2r σ 2 The coefficients x and τ have been removed from the PDE. Hence there is now a change in boundary conditions for V is V (S T,T ) = (S T K) +. When t = T and S t = S T we have, x = ln S T K, which we can write as x T, and τ = 0. Giving boundary conditions for U as U 0 (x T,0) = 1 K (S T K) + = (e xτ 1) + We also need to make the additional transformation for W(x,τ), Where, W(x,τ) = e αx+β 2τ U(x,τ) (2.16) α = (k 1) 2 β = (k+1) 2 Reg:

25 2 The Black-Scholes Model From this we can form the heat equation. But first we need to work out the partial derivatives of U in terms of W 2 U x 2 Substituting these, back into U τ U τ = eαx+β 2τ ( W τ W(x,τ)β 2 ) U x = eαx+β 2τ ( W τ αw(x,τ)) = e αx+β 2τ (α 2 W(x,τ) 2α W x + 2 W ) x 2 + (k 1) U x + 2 U x 2 ku, to give β 2 W(x,τ) W τ W W +(k 1)[ αw(x,τ)+ x ]+αw(x,τ) 2α x + 2 W kw(x,τ) = 0 x 2 Which can finally be simplified to the heat equation, W τ = 2 W x 2 (2.17) Looking back at equation 2.2, the boundary conditions for W(x,τ) is since β = α + 1. W 0 (x T ) = W(x T,0) = e αx T U(x T,0) Therefore we get the transformation of V to W, = (e (α+1)x T e αx T ) + = (e βx T e αx T ) + (2.18) V (S,t) = 1 K e αx β 2τ W(x,τ) (2.19) Obtaining the Black-Scholes Call Price;Since W(x,τ) follows the heat equation, it solution will be give by u(x,τ), with the boundary conditions of W 0 (x T ), this will give the solution, W(x,τ) = 1 4πτ (x ξ )2 e 4τ W 0 (ξ )dξ = 1 4πτ (x ξ )2 e 4τ (e βξ e αξ ) + dξ (2.20) Reg:

26 2 The Black-Scholes Model Make the change of variables, z = ξ x 2π ξ = 2τz + x dξ = 2τdz Now we have, W(x,τ) = 1 2π e z2 2 e β( 2τz+x) α( 2τz+x) (2.21) The integral is non-zero, when the second exponent is greater than zero, when β[ 2τz+ x] > α[ 2τz + x], this is can be rearranged to z > x 2π. We can now break this up into two pieces. W(x,τ) = 1 2π e z2 x 2τ 2 e β( 2τz+x) dz 1 2π e z2 x 2τ Complete the square in the first integral I 1, with the exponent 2 e α( 2τz+x) dz = I 1 I 2 (2.22) z2 2 + β 2τz + βx = 1 2 (z β 2τ) 2 + βx + β 2 τ this makes the I 1 I 1 = e βx+β 2 τ 1 2π x 2τ e 1 2 (z β 2τ) 2 dz With the transformation of y = z β 2τ, the integral becomes I 1 = e βx+β 2 τ 1 2π e 1 2 z2 dy x 2τ β 2τ = e βx+β 2τ (1 Φ( x 2τ β 2τ)) = e βx+β 2τ x Φ( + β 2τ) 2τ (2.23) Second integral is the same except β = α, this gives us Reg:

27 2 The Black-Scholes Model I 2 = eαx + α 2 τφ( x 2τ + α 2τ) Replacing now the variables, to get back to our original notation, This leaves us with, α = 1 2 x = ln S K, k = 2r σ 2, sigma2 r 2 (k 1) =, sigma 2 β = 1 sigma2 r+ 2 2 (k + 1) = sigma 2 τ = σ 2 2 (T t) and, x 2τ + β 2τ = ln K S sigma2 +(r+ 2 )(T t) σ = d T t 1 Therefore we get the integrals, x 2τ + α 2τ = d 1 σ T t = d 2 Now we have the solution, I 1 = e βx+β 2τ Φ(d 1 ) I 2 = e αx+α2τ Φ(d 2 ) W(x,τ) = I 1 I 2 = e(βx + β 2 τ)φ(d 1 ) e αx+α2τ Φ(d 2 ) But we want the solution to the call price, V (S,t), hence why we must put it back into the equation, V (S,t) = Ke αx β 2 τw(x,τ) = Ke αx β 2 τ(i 1 I 2 ) Reg:

28 2 The Black-Scholes Model Our final equation is given by V (S,t) = SΦ(d 1 ) Ke r(t t)φd 2 (2.24) Example:Imagine a stock, with a current price of $42, and a European option with a strike price, of $40 in 6 months time. The risk free rate is at 10% and a stock volatility of 20% per annum. This gives us, S = 42 K = 40 r = 0.1 T = 0.5 σ = 0.2 Put these into the equations for Φd 1 and Φd 2, Φd 1 = Φ( Φd 2 = Φ( we also need to calculate, Ke rt, ln( 40 )+( ) ln( 40 )+( ) ) = ) = Ke rt = 40e 0.05 = Using the equation for finding the call option we get, C = 42N(0.7693) N(0.6278) = 4.76 We can also work out the put option price, P = 42N( ) N( ) = 0.81 Reg:

29 3 The Binomial (Cox-Ross-Rubinstein) Model 3 The Binomial (Cox-Ross-Rubinstein) Model The binomial model is a numerical method for valuing options; it s based on a discrete time lattice model. It starts by setting up two binomial trees; one models the value of the underlying asset (stock) over the lifetime of the option contract, while the second tree models the options value based on the movements of the underlying assets price. The binomial method is slower than the Black-Scholes, but it offers some distinct advantages. Firstly when deriving solutions, it doesn t require the complex stochastic processes required with Black-Scholes and similar models. The binomial method also allows us to derive solutions for more complex option contracts, a perfect example is the adjustment made for American style options contracts. Due to American style options offering continuous opportunities for early exercise, they can t be accurately modelled by the Black-Scholes. The Black-Scholes only provides the price at given time. But because of the many small steps that make up the binomial tree, it can be compute the value at each time step. By increasing the number time steps a more accurate solution is achieved, the below figure is a perfect example of this. Reg:

30 3 The Binomial (Cox-Ross-Rubinstein) Model Binomial trees are based on the assumption that the underlying stock price movements consist of a large number of small binomial movements. Assume at t = 0, the stock price is S, over the period of time dt the stock price could either move up or down. These are represented by S u for an upward movement and S d for the downward, the probability of u is p, while d has a probability of (1 p). Hence an asset starting at time T, can take the value of either S d or S u at (T + dt). This is known as the one-step valuation. Binomial trees work on a risk-neutral assumption, this basically means that the expected return from traded assets is equal to the risk-free interest rate. Hence when calculating previous nodes in the binomial, you discount back using the risk-free interest rate. For the call value we denote this movement as, C u representing the upwards movement and C d the downwards movement C u = max(us K,0) C d = max(ds K,0) As you can see, it gets more complicated when there is more than one step. To start, consider the stocks position at (T + 2dt), there are three different options; 1. Su 2 - The stock price has increased in value, from S u Su, 2 over the time period T + dt T + 2dt. Reg:

31 3 The Binomial (Cox-Ross-Rubinstein) Model 2. S - The stock price has returned to its original value at the start of the contract (T = 0). This can happen either from S d S du = S or S u S ud = S, taking place over the time periodt + dt T + 2dt. 3. Sd 2 - The stock price has decreased in value, from S d Sd 2, over the time period T + dt T + 2dt. This process can be used for each step till the contract reaches expiration, forming a binomial tree of values. As in the black scholes, we replicate a risk-less portfolio using a combination of stock and options, which have the same cash flows. With the principles of arbitrage we can say that, the value of the option must be equal to that of the portfolio. First we look at risk-neutral positions, using the no arbitrage assumption. us c u = ds c d = c u c d (u d)s Hence the initial investment has to be the same as the present value of the portfolio, giving us this equation let, S c = Ae rdt c = S Ae rdt c = ( erdt d (u d) c u + u erdt (u d) c d)e rdt p = erdt d (u d) Therefore the risk neutral call option price at the present time is (3.1) c = (pc u + (1 p)c d )e rdt Reg:

32 3 The Binomial (Cox-Ross-Rubinstein) Model p is the risk neutral probability of the share price going up, and respectively (1 p) is the probability of it going down. This leads on to assume the expected value of the stock price can be shown as, E[S] = pus + (1 p)ds = Se rdt 3.1 Derivation of Binomial Model All mathematics involved is further explained in (Buchanan, 2006), (Hull, 2006), (Redhead, 1997) and (Wilmott, 1995) To do this we determine u and d. To start with we will look at equation 3, This can also be written as, p = erdt d (u d) e rdt is also the expected value for E[ S dt S ], when t = dt. pu + (1 p)d = e rdt (3.2) E[ S dt S ] = pu + (1 p)d = erdt Now the diffusion of this process is determined by the variance, var[ S dt S ] = E2 [ S dt S ] E[ S2 dt S ] var[ S dt S ] = pu2 + (1 p)d 2 (pu + (1 p)d) 2 = pu 2 + (1 p)d 2 e 2rdt (3.3) Taking the variance for Black-Scholes model, var[s t ] = σ 2 dt and σ is volatility of stock price. var[ S dt S ] = e2rdt (e σ 2 dt 1) (3.4) Reg:

33 3 The Binomial (Cox-Ross-Rubinstein) Model Where σ is the volatility. We are left with two possible options, to find u and d. But first we re going to attach and simplify 3.2 and 3.3 when simplified this becomes, e 2rdt (e σ 2 dt 1) = pu 2 + (1 p)d 2 e 2rdt e 2rdt+σ 2 dt = pu 2 + (1 p)d 2 (3.5) Now substitute this into equations 3.1 into the right hand side of 3.4. erdt d u d u2 u erdt u d d2 = (u2 d 2 )e rdt (u d) = (u + d)e rdt 1 (3.6) u d This finally leaves us with, Now assume that u = d 1, giving us e 2rdt+σ 2 dt = (u + d)e rdt 1 u 2 e rdt u(1 + e 2rdt+σ 2 dt ) + e rdt = 0 but we need to find the value for u, hence we set it using the quadratic formula as it s in the form, AX 2 BX +C = 0 This gives us the root equation, u = (1 + e2rdt+σ 2dt ) + (1 + e 2rdt+σ 2dt ) 2 4e 2rdt 2e rdt (3.7) this is very complicated to solve in one go, so I m going to try and simplify into sections. First, is the equation that is square rooted Reg:

34 3 The Binomial (Cox-Ross-Rubinstein) Model (1 + e 2rdt+σ 2 dt ) 2 4e 2rdt (2 + (2r + σ 2 )dt) 2 (4 + 8rdt) 4σ 2 dt now equation 3.5 with the substituted variables for the numerator This can be expanded and simplified to 2+(2r+σ 2 )dt+ 4σ 2 dt 2e rdt u (1 + rdt + σ 2 dt 2 + σ dt)(1 rdt) 1 + rdt + σ 2 dt 2 + σ dt rdt = 1 + σ dt + σ 2 dt 2 As this is a second order, the expansion is e σ dt, this gives us the parameters p = erdt d u d u = eσ dt d = e σ dt Now we have solutions for u and d we just multiply though the steps in our binomial tree, this is know as The Binomial (CRR) Model. 3.2 Pricing An European Option As shown in (Ross, 2012) to find the explicit solution, we first need to consider the binomial tree described in the previous section. Let f i, j be the value of the option at position (i, j). j represents the time instance, it s the jth period between 0 N. i is a reference to the nodes position within the jth period. N represents the amount of steps, hence there are N + 1 layers in the lattice, with Ndt = T when the contract reaches maturity. Hence for a European option N = j. For maturity of the contract we get, f i,n = max(0,su i d N i K) Reg:

35 3 The Binomial (Cox-Ross-Rubinstein) Model But if we now consider working backwards to the present, we get the value for the option at the specific time f i, j = e rdt (p f i+1, j+1 + (1 p) f i, j=1 ) 3.3 Pricing An American Option Pricing an option using the binomial model is fairly simple, but so far we have only looked at exercising an European style option. We will now consider an American style option, which as explained earlier differs from a European option, because it allows for early exercise. This means it can be exercised at any time up to and on, the expiration date. During this section I will only look at non-dividend paying put options. I won t be looking at the call, because it is never optimal to exercise when being held to maturity or if the purchased stock will be hold to maturity. It is more beneficial to sell the option, rather than exercising. This is further explained in (Cox, 1979) and (Kyprianou, 2006). Now consider the put option, start by looking at the lowest value node in the binomial tree, f i,n = max(k S i,n,0) the price at this node is valued at S i,n = Su i d N i. Next consider a point in the layer before, if S i,n 1 > K then the option is out-the-money. But if S i,n 1 < K then there is a choice to make, do you exercise early or wait to see if a greater profit can be made in the future. This problem presents the question of when is the optimal stopping point, when is the best time to exercise an in-the-money option? To start with consider our N subintervals of length dt, now take the jth node and lets say this at the i dt. We refer to this as the (i, j) where 0 i N and 0 j i, hence the options value is referred to as f ( i, j) with the price of the underlying asset being Su j d i j. Looking at a call option, its value at expiration is max(s T K,0), this gives the equation for its value as, f ( N, j) = max(su j d N j K,0). We can also value the put as f ( N, j) = max(k Su j d N j,0). Now we ve worked out the value at a specific node, we can look at what happens when it moves from that point. Once again the probability remains p of (i, j) becoming (i + 1, j + 1) over the period dt, hence the time becomes (i + 1)dt. Also just like before the probability of moving down is (1 p), if this happens then it becomes (i 1, j). Without early exercise we get f ( i, j) = e rdt (p f i+1, j+1 + (1 p) f i+1, j ). Now we will look at what happens when Reg:

36 4 Analysis early exercise is available, early exercise should be executed when an options intrinsic value is greater than its continuation value. So we compare the previous equations and get, f i, j = max(su j d i j K,e rdt (p f i+1, j+1 + (1 p) f i+1, j )) for the American call. f i, j = max(k Su j d i j,e rdt (p f i+1, j+1 + (1 p) f i+1, j )) for the American put. An exact value for the American put can be obtained as dt 0. Hence the more steps, higher value for N, then the better estimate of the option price. 4 Analysis The focus of this section is to look at the effectiveness of the Black-Scholes and Binomial option pricing models. This will involve analysing how accurately both models estimate the option price, using the historical data. I obtained all my option data from a free sample available at The data represents a range of option contracts, that all have a start date 14/10/2013, but with varying expiration dates. This data contained values for the Greeks; This is a reference to the letters used to represent a range of volatilities. This meant I didn t have to estimate σ, which is representative of the stocks volatility. I will quickly explain how you would go about calculating σ if it isn t available. 4.1 Estimating Volatility and Risk-Free Rate Volatility is essential for traders to determine the fair price of options. Options and strategies involved make more profit if the stock price makes significant jumps in the right direction. Due to the constant movements of stock prices, volatility becomes very difficult to estimate accurately. For example if you take the assumption used in Black- Scholes, that all stock prices follow a Brownian Motion, how do you find a constant rate? Reg:

37 4 Analysis To do this we use Standard Deviation. This is a vital statistical measurement, it shows how much variation there is from the mean value. To put this into financial terms, it is the level of fluctuation experienced on a stocks price in relation, to the expected value of the stock, over a given time period. Hence the value for standard deviation is obtained by comparing previous stock prices. But just because a stocks previously had a low volatility, it doesn t mean that the same is guaranteed in the future. Estimating volatility from historical data, involves looking at the position of stock over a fixed time interval. To find this we must revert back to the equation, from this we can take, ds S = µdt + σdw mean( ds S ) = µdt ds var( S ) = σ 2 dt We ve been working with the assumption that option trading can take place any day and any time, hence dt = Now with some simple rearrangement we can get equations for µ and σ. µ = ds mean( S ) dt σ = var( ds S ) dt Now we have our equation for volatility, it s simply a matter of putting values from historical data in. Risk-Free Rate is known as the minimum rate of return earned by a risk-less investment, in a no arbitrage market. In reality this doesn t exist because all investments carry a certain level of risk, but it s small enough to be considered risk-less. Risk-free rate, is most commonly calculated using Capital Asset Pricing Model (CAPM), which is further explained in (Buchanan, 2006) and (Brealey, 2012). But you can also considered a treasury bill from a country with AAA credit rating, such as the USA or England. Capital Asset Pricing Model states that the expected risk equals the beta times the market risk, it can represented as, E[r] = r f + β(e[r m ] r f ) Reg:

38 4 Analysis E[r] = Expected rate of return on the asset E[r m ] = Expected market rate of return r f = Risk-free rate of return β = Sensitivity of the asset, in relation to the market Hence with some simple rearrangement we can then find our value for the risk-free rate(r f ). But it s much simpler and quicker to take it from readily available sources such as The Bank of England and other financial institutions. I got a rate of 0.05 for the start date of my data, 14/10/2013. But this is an estimation taken at one date, it could have changed over the lifetime of the option, but this wouldn t be by sizeable amounts. Now we return back to our market data, this was a free sample, that contained over 600,000 options. All the option had the same start date, but varying expiration dates, along with being for many different companies and markets. Next steps was narrowing down the data into European and American Options, from this I found two of the most actively traded options; These where XEO (European style) and OEX (American style). Both are S&P 100 index linked options. S&P 100 is an index linked stock, combining 100 of the largest and most established companies, within the S&P 500. Below are two graphs modelling the stock price of OEX and XEO from the start date of the date. Both graphs have been taken from Chicago Board Options Exchange; OEX Stock Price XEO Stock Price As you can see the stock price for XEO and OEX follow the same trend and carry identical values. Both are index stocks on the same 100 companies, hence the price is Reg:

39 4 Analysis determined by the same underlying assets. This general increase in stock price, will be reflected in the option price. But this is where the style of contract becomes relevant. American option offer early exercise, hence they are more flexible than European, this advantage means they are generally more expensive. With S&P 100 index linked stocks, the general trend in the long run is expected to always be positive. When traders become confident in a trend, early exercise becomes less relevant. 4.2 Pricing XEO (European style) Options When modelling the European option price, we can look at the Black-Scholes and Binomial (European) Model. We will analyse both models numerical methods, by comparing our calculated values to the market values. As my stock data only had the Ask Price, I used this as the closest to the actual present value. An options Ask Price, is the price at which the owner is willing to sell at. By doing these comparisons we can evaluate the strengths and weakness of both models, while also looking at how different parameters may effect each models accuracy. With just over 800 European options, with varying expiration dates and strike prices, we will be able to see accurate trends in the pricing using each model. All models were computed in MATLAB, this allowed me to calculate the numerical values quickly, while being accurate. There are functions available in the MATLAB financial package, but for this project they where unavailable. These are known as blsprice for Black-Scholes and binprice for the Binomial method. But from using (Brandimarte, 2013) My own versions are included in the appendix. Firstly we will look at the call option, with over 400 call option, it was easier to represent my results using graphs. I compared my results against the ask price, using the variables T and K. When looking at the Ask Price, we must assume that the owner of the call option is looking to make as much profit as possible, hence I expected the price to be slightly higher than the actual value. Reg:

40 4 Analysis European Call Option Black-Scholes for European Call From the graph we can see that when the contract is close to expiration (low value of T ), then the expected price is similar to the Ask Price. But as T increase, so does the difference in the two prices. Looking at the Strike price in comparison to the present value of the option, we get the expected trend, as strike price increases option value decreases. Reg:

41 4 Analysis Binomial for European Call This graph shows similar results for the Binomial, to the Black-Scholes model, with prices being accurate in the short run position, But as T increase, the difference in price also increases. Again we see the clear trend of present value decreasing as the strike price increases and vice versa. Reg:

42 4 Analysis Black-Scholes and Binomial for European Call From this final graph you can clearly that the results from both models are significantly close. While the Black-Scholes and Binomial have predicted nearly identical values, they are still off that of the current market Ask Price, when the options are far from expiry. As before mentioned though, the trend is as expected and both models seem accurate for the short run position of the option. Overall when we look at these graphs we can see significantly similar trends in results for each model, with the value increasing in price as the option moves deeper in-the-money. We also see that time till expiration has a big effect on the value of a option, with both models being most accurate using low values of T. Personally I feel this is down to a mix of speculation and hedging, when traders predicting big changes in price, they will hedge or buy accordingly. Reg:

43 4 Analysis European Put Options Black-Scholes for European Put For the Put option you expect the opposite trend to that of the call, as strike price increases the present value of the option also increases. This is due to the put option being in-the-money when the owner can exercise above the stocks market value. Again we see the trend, as T increases the difference between the predicted value and current ask price also increases. With the model shows significant accuracy for low values of T. Reg: