Models of Option Pricing: The Black-Scholes, Binomial and Monte Carlo Methods

Transcription

1 Registration number 65 Models of Option Pricing: The Black-Scholes, Binomial and Monte Carlo Methods Supervised by Dr Christopher Greenman University of East Anglia Faculty of Science School of Computing Sciences

2 Abstract If options are correctly priced in the market, it should not be possible to make profits by creating portfolios of long and short positions in options and their underlying stocks. Using this principle, the Black-Scholes formula for options was derived. However, the Black-Scholes has several limitations. A range of numerical methods such as the Binomial method and Monte Carlo simulation can be used as suitable alternatives, which we will discuss in this course of work. We will compare these methods against market data and analyse the benefits and drawbacks of each method in the pricing of options. Acknowledgements I would like to take this opportunity to express my gratitude to my supervisor Dr Christopher Greenman. Without his guidance and support, the completion of this project would not have been possible. Dr Christopher Greenman s advice and assistance has proved invaluable throughout the course of this work. I would also like to acknowledge the encouragement and motivation my friends and family have given me. In particular, I would like to thank my poor housemates who showed me nothing but patience when I vented my distress and to my wonderful mother who had to proof read this again and again.

3 Contents 1. Introduction 1 2. Options Styles of Options Trading Strategies Spreads Combinations Arbitrage and Put-Call Parity Asset Pricing Wiener Processes Itô s Lemma and the Lognormal Property Black-Scholes Model Derivation of the Black-Scholes Formula Solving the Black-Scholes equation Binomial Pricing Binomial Convergence to the Black-Scholes Monte Carlo Simulation Monte Carlo convergence and the Black-Scholes Data Analysis Estimating the parameters Analysis of data Conclusion 40 References 41 A. Historical Share Price Data for Russell Reg: 65 iii

4 B. MATLAB Code for Black-Scholes Call Option 43 C. MATLAB Code for Black-Scholes Put Option 43 D. MATLAB Code for Binomial Call Option 44 E. MATLAB Code for Binomial Put Option 45 F. MATLAB Code for Monte Carlo Call Option 46 G. MATLAB Code for Monte Carlo Put Option 46 Reg: 65 iv

5 1. Introduction In 1973, Professor Fischer Black and Professor Myron Scholes published The Pricing of Options and Corporate Liabilities. This paper introduced the option pricing formula and the models used have set the standard as a modern valuation tool. The crux of their discoveries was the Black-Scholes Model, which was developed with the help of Robert Merton. The Black-Scholes Model is one of the most important concepts in modern financial theory and regarded as one of the best ways of determining fair prices of options (Investopedia, 2014) also saw the establishment of the first centralized exchange for the trading of options - the Chicago Board Options Exchange [CBOE]. Together, these developments fostered a booming options industry (Jacobs, 1999). The CBOE began with call options on 16 heavily traded common stocks and has subsequently evolved into one of the largest exchanges in the world (Jarrow and Rudd, 1983). Hull (2011) says the CBOE now trades options on over 2500 stocks and many different stock indices. As the key component to option pricing, we will look at the Black-Scholes formula and alternative pricing models in comparison with real world data. 2. Options An option is a security giving the right to buy or sell an asset, subject to certain conditions, within a specified period of time as defined by Black and Scholes (1973). The asset in question is typically shares, a stock index, exchange-traded fund or similar product. We will focus on options where the asset is stock. The key aspect of an option, is that the holder has the right, but not the obligation to buy or sell the asset. This distinguishes options from other financial derivatives, such as forward and future contracts. Where it costs nothing to enter into a forward or future contract, there is a cost to acquiring an option. Options have a predetermined time frame, with the end date known as the expiration or maturity date. Option contracts also specify a predetermined price for which the option holder can either buy or sell the underlying asset. This is known as the strike or exercise price. The purchase price of the option is called the premium. Reg: 65 1

6 There are two basic types of options; calls and puts. Edwards and Ma (1992) define a call as an option giving the buyer the right to purchase the underlying asset at a fixed price. If the current asset price, S, is higher than the option s strike price, K, the option buyer would exercise their right and buy the asset. The value of the call will therefore, be the difference between the current market price and the strike price, S K. If the current value of the asset is less than the strike price, the option buyer would not exercise the option; they would let it expire as it is deemed worthless. In contrast to call options, a put option gives its buyer the right to sell the underlying asset at a fixed price (Edwards and Ma, 1992). If the current asset price is lower than the strike price, the option holder will want to exercise the put, in order to sell the underlying asset at the strike price. The value of the put will therefore be K S. If the strike price is lower than the current market price, the option holder would let the contract expire, as it would be worthless. The value of both a call option, C, and a put option, P, at the expiration date can be mathematically expressed as: C(S,T ) = max(s T K, 0) P(S,T ) = max(k S T, 0) (2.1) where, T, denotes the time to expiry. The value of an option is determined by its moneyness. Moneyness is considered the position of the current price of an underlying asset (Neftcci, 2008). There are three stages of moneyness: 1. In-The-Money: An option is in-the-money when a profit can be made. A call option is in-the-money when the asset price is greater than the strike price, S > K. A put option is in the-the-money when the strike price is greater than the asset price, K > S. 2. At-The-Money: An option is at-the-money when the strike price is equal to the asset price. This is applicable to both calls and puts. 3. Out-Of-The-Money: An option is out-of-the-money when no profits can be made, and the contract is worthless. A call option is out-of-the-money when the strike price is greater than the asset price, K > S. A put option is out-of-the-money when the asset price is greater than the strike price, S > K. Reg: 65 2

7 A further point to make, is that there are two sides to every option contract. Investors can choose to buy or sell options. Sellers of option are also referred to as writers. The buyer of an option is said to have a long option position, whereas the writer of the option is said to have a short position (Redhead, 1997). An investor can take a short or long position in either a call or a put. If an investor who has taken the long position chooses to exercise their right to buy or sell the underlying asset, the writer must deliver the asset. Edwards and Ma (1992) state that the potential loss to an option seller is unlimited. In contrast, if the buyer chooses not to exercise his right, but allows the option to expire, his loss is limited to the premium paid. The potential profits and losses for the option buyer, are hence the reverse of that for the writer. The diagrams in Figure 2.1 demonstrate the payoffs from an option depending on the investors position. (a) long call; (b) short call; (c) long put; (d) short put. Strike price = K; Asset price at expiry = S T Figure 2.1: Payoff diagrams from different European option positions Reg: 65 3

8 2.1. Styles of Options There are several different styles of options. These styles define the class in to which the option falls. The style of the option effects the way in which they are priced and can affect the way in which investors exercise their rights and their potential payoffs. The main option styles are as follows: American Options: options can be exercised at any time prior to and including the predetermined expiry date. Asian Options: the option s payoff depends on the average price of the underlying asset over the predetermined time frame. Barrier options: the option s payoff depends on whether or not the underlying asset has exceeded a predetermined price. Bermuda Options: a combination of American and European options. Options are exercisable only on the date of expiry and on certain specified dates that occur between the purchase and expiration date. Digital Options: the option s payoff is fixed after the underlying asset exceeds the strike price. European Options: options that can only be exercised on the date of expiry. Exotic Options: options that generally trade over the counter. They are more complex than options traded on an exchange (e.g. American and European options). Vanilla Options: a normal put/call option with a predetermined price and predetermined expiry date. The most common types of options are American and European. For simplicity, we are only considering European options. Reg: 65 4

9 3. Trading Strategies There are a wide variety of different trading strategies that an investor can use involving a single option on a stock and the stock itself. A popular strategy is a protective put. Protective puts are created by buying stock and then purchasing a put option on that particular stock. Investors might use this when they feel that investing in the stock alone is too risky. Investors can therefore protect themselves by buying the put, as no matter what happens to the stock price, the investor is guaranteed a payoff equal to the put options s exercise price. This shows that despite the common misconception that derivatives mean risk, derivative securities can be used effectively for risk management. In fact, such risk management is becoming accepted as part of the fiduciary responsibility of financial managers (Bodie et al., 2013). Another popular strategy is a covered call. A covered call position is the purchase of a share of stock with the simultaneous sale of a call option on that stock. The written option is covered because the potential obligation to deliver stock can be satisfied using the stock held in the portfolio. The profit diagrams for both protective puts and covered calls are illustrated in Figure 3.1. Note that throughout this chapter, the dashed lines in the graphs indicate the profits from the individual option or stock positions and the solid line represents the profits from the whole strategy. Many investors also use a variety of different trading (a) Protective Put (b) Covered Call Figure 3.1: Profit diagrams with positions in options and stock strategies by combining puts and calls with various exercise prices. These strategies are Reg: 65 5

10 often split into two classes; spreads and combinations. Below we will briefly discuss some of the more common strategies Spreads Jarrow and Rudd (1983) define a spread position as a portfolio of two or more options of different series, but belonging to the same class in which some options are held long and some short (i.e. two or more calls, or two or more puts). The main types of spreads that we will consider are bull, bear and butterfly spreads. These spreads all involve taking option positions on the same asset with the same expiry date. Bull Spreads A bull spread is created by taking the long position of a European call option on a stock with a certain strike price and taking the short position of a European call option, but with a higher strike price. A bull spread strategy limits the investor s potential profits, but more importantly, it limits the investor s potential risk. Bull spreads can also be created by taking the long position of a European put with a certain strike price and taking the short position of a European put option on the same stock, but with a higher price. An investor who enters into a bull spread is hoping that the stock price will increase (Hull, 2011). These strategies can be seen in Figure 3.2. (a) Bull spread with puts (b) Bull spread with calls Figure 3.2: Profit diagrams of a Bull Spread Strategy Bear Spreads Bear spreads are created by investors taking a long position in a European put with one strike price and then taking a short position in another put with a lower strike price. Reg: 65 6

11 This is the reverse of a bull spread. Investors who enter into bear spreads hope that the market price decreases. Bear spreads can also be created using call options. An investor using European calls would take the long position in a call with a high strike price and then take the short position in a call with a lower strike price. Bear spreads enable the investor to limit the potential risk, but also limits the potential profits. The profit diagram of bear spreads are illustrated in Figure 3.3 (a) Bear spread with calls (b) Bear spread with puts Figure 3.3: Profit diagrams of a Bear Spread Strategy Butterfly Spreads Butterfly spreads are a little more complex than bull and bear spreads. Butterfly spreads involve an investor taking several option positions. A butterfly spread can be created by an investor taking a long position in a European call option with a low strike price, K 1, taking another long position in a call option with a high strike price, K 3 and then taking the short position in two European calls, with a strike price K 2 that is halfway between K 1 and K 3. Butterfly spreads can also be created in the exact same way using puts. Investors using butterfly spreads do not expect large changes in price of stock. The profit diagrams for butterfly spreads can be seen in Figure Combinations Combinations are portfolios containing options of different types (but on the same underlying security), which are either all held long or written (Jarrow and Rudd, 1983). The main types of combinations we will discuss are straddles, strips, straps and strangles. Reg: 65 7

12 (a) Butterfly spread with calls (b) Butterfly spread with puts Figure 3.4: Profit diagrams of a Butterfly Spread Strategy Straddles A straddle is the purchase of a European call and a European put that have the same exercise price and expiration date. Straddle strategies are useful when investors think there will be a large change in the stock price, but are unsure of which direction the price will move in. The profit from a straddle strategy is demonstrated in Figure 3.5. Figure 3.5: Profit diagram of a Straddle Strips A strip is defined as a long position in two European puts and one European call with the same strike price and the same expiration date. Strips are often used when investors are unsure which the way the market is going to go, but feel there is a higher chance of a decrease. The profit of a strip can be seen in Figure 3.6. Reg: 65 8

13 Straps A strap is very similar to a strip, except that its used when investors think that an increase in stock prices is more likely than a decrease. Straps are created when investors take a long position in two European calls and take a short position in a European put. The profit of a strap can be see in Figure 3.6 Strip (one call and two puts) and Strap(two calls and one put) Figure 3.6: Profit diagrams of Strips and Straps Strangles A strangle strategy is similar to that of a straddle. An investor purchases a European put and a European call with the same expiration date, but different strike prices. The strike price of the call is higher than the strike of the put. Investors use this strategy when they expect a large movement in the stock price, but are uncertain which direction it will go. The profits made from a strangle is dependant on how close the different strike prices are. The profit from a strangle is illustrated in Figure Arbitrage and Put-Call Parity An arbitrager is a type of investor whose job it is to seek out and exploit irregularities in the market. Arbitrage involves making riskless profits from mispricing; relatively underpriced options are bought and relatively overpriced ones sold (Redhead, 1997). Arbitrage with derivatives takes the form of buying/selling the derivative while simultaneously taking an opposite position in an otherwise identical derivative. Dubofsky Reg: 65 9

14 Figure 3.7: Profit diagram of a Strangle (1992) argues that in well functioning markets, options and other securities cannot be priced to yield arbitrage opportunities. It is often and loosely stated as "there s no such thing as a free lunch." If any arbitrage opportunities do arise, investors quickly exploit them and in the process, prices will adjust until all arbitrage opportunities are eliminated. A very important relationship that prevents arbitrage is the put-call parity equation. Although call and put options are superficially different, they can be combined in such a way that they are perfectly correlated. This is because prices of European put and call options are linked together in an equation known as the put-call parity relationship. To derive the parity relationship we consider a long position in a call option and a short position in a put option; each with the same strike price K and the same expiration date T. At expiration, the payoff on the investment will equal the payoff on the call minus the payoff on the put. The payoff for each option will depend on whether the stock price at expiry, S T, exceeds the strike price. Table 3.1: Payoff from a long call and a short put option S T K S T > K Payoff of long call 0 S T K - Payoff of short put (K S T ) 0 Total S T K S T K Reg: 65 10

15 Table 3.1 illustrates this payoff pattern. If we also consider a second portfolio, buying one share and borrowing K bonds, the payoff of this portfolio is also S T K. Due to the portfolio s having identical payoffs, the cost of establishing them must also be equal. The net cash outlay necessary to establish the option position is C P. The call is purchased for C, and the written put generates an income of P. The cost of establishing the second portfolio is S 0 Ke rt. Therefore we get the equation: C P = S 0 Ke rt (3.1) This creates the put-call parity equation. If the parity relationship is ever violated, an arbitrage opportunity arises (Bodie et al., 2013). This equation also proves to be very useful in the pricing of options. 4. Asset Pricing Before discussing how options are priced, it is vital to understand the pricing of the underlying asset. Any variable whose value changes over time in an uncertain or random way, is said to follow a stochastic process. Stochastic processes can be classified as either discrete time or continuous time. It is often stated that asset prices must move randomly because of the efficient market hypothesis (Wilmott et al., 1993). The efficient market hypothesis makes two important assumptions: The past history is fully reflected in the present price, which does not hold any further information; Markets respond immediately to any new information about an asset. Therefore, the modelling of asset prices is about modelling the arrival of new information that affects the price. With the two assumptions above, unanticipated changes in the asset price are a Markov process. A Markov process is a particular type of stochastic process where only the current value of a variable is relevant for predicting the future. Stock prices are usually assumed to follow a Markov process (Hull, 2011). Consider an asset that at time t has a value of S. With each change in asset price, a return is associated, defined as the change in value divided by the original price. Suppose that during a subsequent time interval t, where represents a small increment, Reg: 65 11

16 the asset price changes from S to S + S. The return, S/S, is composed into two parts. The first part is a predictable and an anticipated return, µ t, where µ is the expected rate of return per unit of time from the asset. This is also known as the drift. The second part, σ z, models the random change in the asset price in response to external effects. It is represented by a random sample drawn from a normal distribution with mean zero. σ denotes the volatility, which measures the standard deviation of the returns and z is the sample from a normal distribution. Together, we obtain the stochastic differential equation S = µ t + σ z (4.1) S which is used for generating asset prices (Wilmott et al., 1993). This is the discrete-time version of the model Wiener Processes The term z in equation (4.1) is known as a Wiener process (or Brownian motion) and is a particular type of Markov stochastic process. One of the properties of the Wiener process is that the change in z during a small period of time t is z = ε t where ε has a standardised normal distribution φ(0,1). It follows from this that z has a normal distribution itself with mean of z = 0 standard deviation of z = t variance of z = t Another important property is that the values of z for any two different short intervals of time, t, are independent. This property further implies that z follows a Markov process. In a generalised Wiener process, the drift rate and variance rate remain a constant. However, this is inappropriate for modelling stock prices as it fails to capture the key aspect of stock prices. The key aspect of stock prices is that the expected percentage Reg: 65 12

17 return required by investors from a stock is independent of the stock s price states Hull (2011). Therefore, the assumption of a constant drift rate is inappropriate and needs to be replaced by the assumption that the expected return S/S is constant. If the coefficient of z is zero, so that there is no uncertainty, this model implies that S = µs t In the limit, as t 0, ds = µsdt or ds S = µdt Integrating between time 0 and time T, we get S T = S 0 e µt where S 0 and S T are the price of the stock at time 0 and time T. This demonstrates that, when there is no uncertainty, the stock price grows at a continuously compounded rate of µ per unit of time. However, in practice, there is uncertainty. A reasonable assumption is that the variability of the percentage return in a short period of time, t, is the same regardless of the stock price. This suggests that the standard deviation of the change in a short period of time t should be proportional to the stock price and leads to the model ds = µsdt + σsdz (4.2) or ds = µdt + σdz (4.3) S The latter equation, (4.3) represents the stock price in the real world. In a risk-neutral world, µ equals the risk-free rate r, which will be considered later Itô s Lemma and the Lognormal Property A further type of stochastic process, known as an Itô process is now discussed. An important result in this area was discovered by K. Itô in 1951 and is known as Itô s Reg: 65 13

18 lemma. Wilmott et al. (1993) argue Itô s lemma is the most important result about the manipulation of random variables that we require. It relates to the small change in a function of a random variable to the small change in the random variable itself. An Itô process can be defined as a generalized Wiener process in which the parameters a and b are functions of the value of a variable x and time t. Algebraically, this can be written as dx = a(x,t)dt + b(x,t)dz The variable x has a drift rate a and a variance rate of b 2. Itô s lemma shows that a function G of x and t follows the process ( δg dg = δx a + δg δt + 1 δ 2 ) G 2 δx 2 b2 dt + δg δx bdz where dz is the same Wiener process. Thus, G also follows an Itô process, with a drift rate of and a variance rate of δg δx a + δg δt + 1 δ 2 G 2 δx 2 b2 ( ) δg 2 b 2 δx We can apply this to our model for stock price movements in equation (4.2). From Itô s lemma, it follows that the process followed by a function G of S and t is ( δg δg dg = µs + δs δt + 1 δ 2 ) G 2 δs 2 σ 2 S 2 dt + δg σsdz (4.4) δs Both S and G are affected by the uncertainty, dz. This proves to be very important in the derivation of the Black-Scholes formula. We can now use Itô s lemma to derive the process followed by ln(s) when S follows the behaviour of equation (4.2). We define G = ln(s) From this we can derive δg δs = 1 S, δ 2 G δs 2 = 1 S 2, δg δt = 0 Reg: 65 14

19 It follows from equation (4.4) that the process followed by G is dg = (µ σ 2 ) dt + σdz 2 Since µ and σ are both constant, it is evident from this equation that G = ln(s) follows a generalised Wiener process. It has a drift rate µ σ 2 /2 and a variance rate σ 2. This indicates that the change in lns between time 0 and time T is normally distributed with mean (µ σ 2 /2)T and variance σ 2 T. This means that lns T lns 0 φ [(µ σ 2 ) ] T, σ 2 T 2 or lns T φ [ lns 0 + (µ σ 2 2 ) ] T, σ 2 T (4.5) where S T is the stock price at a future time T, S 0 is the stock price at time 0 and φ(m,v) denotes a normal distribution with mean m and variance v. Equation (4.5) shows that lns T is normally distributed. A variable has a lognormal distribution if the natural logarithm of the variable is normally distributed. Equation (4.5) therefore implies that a stock s price at time T, given it s price today, is lognormally distributed. 5. Black-Scholes Model The Black-Scholes option pricing model is probably the most well-known and widely used pricing model in finance (Edwards and Ma, 1992). The idea behind the model was to fairly price options and to find a way to perfectly hedge an option by buying and selling stock to eliminate any risk. Dubofsky (1992) defines a hedge as a trade or set of trades, made in order to lessen or even eliminate, the risks of price fluctuations. The hedge ratio is the number of shares of stock required to hedge the price risk of holding one option (Bodie et al., 2011). The Black-Scholes equation is a partial differential equation governing the price changes of either a European call or European put option under the Black-Scholes Model. As stated by Hull (2011), there are six factors affecting the price of an option: 1. The current stock price, S Reg: 65 15

20 2. The strike price, K 3. The time to expiration, T 4. The volatility of the stock price, σ 5. The risk-free interest rate, r 6. The dividends that are expected to be paid. Before we can introduce the Black-Scholes model we must first make some assumptions. Black and Scholes (1973) define the "ideal conditions" as: The short-term interest rate is known and is constant through time. The stock price follows a random walk in continuous time and is lognormal. The variance rate of the return on the stock is constant. The stock pays no dividends or other distributions. The option is European and therefore can only be exercised at maturity. There are no transaction costs in buying or selling the stock or the option. There are no penalties to short selling. There are no riskless arbitrage opportunities Derivation of the Black-Scholes Formula The stock price process we are assuming is ds = µsdt + σsdz Suppose we have an option whose value f depends only on S and t. Using Itô s lemma, we can rewrite equation (4.4) as ( δ f d f = δs µs + δ f δt + 1 δ 2 ) f 2 δs 2 σ 2 S 2 dt + δ f σsdz (5.1) δs Reg: 65 16

21 This gives the random walk followed by f. The discrete versions these equations are and f = S = µs t + σs z (5.2) ( δ f δs µs + δ f δt + 1 δ 2 ) f 2 δs 2 σ 2 S 2 t + δ f σs z (5.3) δs where f and S are the changes in f and S in a small time interval t. It follows that a portfolio of the stock and the option, with value Π, can be constructed so that the Wiener process is eliminated. This is accomplished with a short position in one option and a long position in δ f /δs shares. The value of the portfolio is therefore Π = f + δ f δs S (5.4) The change Π in the value of the portfolio in time interval t is given by Π = f + δ f S (5.5) δs Substituting equations (5.2) and (5.3) into equation (5.5) we get ( Π = δ f δt 1 δ 2 ) f 2 δs 2 σ 2 S 2 t (5.6) As z has been eliminated, the portfolio must be riskless during t. From our assumptions, the portfolio must earn the same rate of return as other short-term securities. Otherwise, there are opportunities for arbitrage. It follows that Π = rπ t (5.7) where r is the risk-free interest rate. Substituting from equations (5.4) and (5.6) into equation (5.7), we obtain ( δ f δt + 1 δ 2 ) ( f 2 δs 2 σ 2 S 2 t = r f δ f ) δs S t Rearranging this gives δ f δt + rsδ f δs + 1 δ 2 f 2 δs 2 σ 2 S 2 = r f (5.8) This is the Black-Scholes partial differential equation. Reg: 65 17

22 5.2. Solving the Black-Scholes equation To solve the Black-Scholes equation, we must consider the boundary and final conditions to obtain the unique solution. For this, we will look at a European call option, with a value C(S,t), a strike price K and expiry date T. We know from equation (2.1) that the value of a call option at t = T is C(S,T ) = max(s K, 0) where C(S,t) is defined over the domain 0 < S <, 0 t T. This is the final condition of the partial differential equation. The boundary conditions are applied when S = 0 and as S. From our stochastic differential equation (4.1), we can see that if S = 0, then ds = 0 and therefore S can never change. If S = 0 at T, then the payoff will be zero, as the call option will expire worthless. Hence, when S = 0 we have C(0,t) = 0 As the stock price increases without bound, it is more likely to be exercised and the magnitude of the strike price becomes increasingly insignificant. Therefore as S the value of the option becomes that of the stock, and we have C(S,t) S as S We now have the boundary conditions. Finally, if we rearrange the Black-Scholes partial differential equation and substitute C for f we get δc δt + rsδc δs σ 2 S 2 δ 2 C δs 2 rc = 0 We can turn this into a forward equation by a change of the independent variables S = Ke x, t = T τ 1 2 σ 2 C = Kυ(x,τ) Giving the equation δυ δτ = δ 2 υ + (y 1)δυ δx2 δx yυ where y = r/ 1 2 σ 2. The initial condition then becomes υ(x,0) = max(e x 1, 0) Reg: 65 18

23 This demonstrates that the only essential factor controlling the option value is r/ 1 2 σ 2. We can now turn the equation above into a diffusion equation by a simple change of variable. Using υ = e αx+βτ u(x,τ), where α and β are some constants to be found, differentiating gives us βu + δu δτ = α2 u + 2α δu We can eliminate u by choosing δx + δ 2 u δx ( + (y 1) 2 αu + δu δx ) yu β = α 2 + (y 1)α y, and eliminate δu/δx by choosing 2α + (y 1) = 0 These equations for α and β give α = 1 2 (y 1), β = 1 (y + 1)2 4 We then have the equation υ = e 1 2 (y 1)x 1 4 (y+1)2τ u(x,τ), where with δu δτ = δ 2 u for < x <, τ > 0, δx2 u(x,0) = u 0 (x) = max(e 1 2 (y+1)x e 1 2 (y 1)x, 0) (5.9) which represents the payoff. The solution to the diffusion equation is the well-known formula where u 0 is given in equation (5.9) above. u(x,τ) = 1 2 u 0 (s)e (x s)2 /4τ ds (5.10) πτ Reg: 65 19

24 All that remains is to evaluate the integral given in equation (5.10). For convenience, we make change of the variable x = (x s)/ 2τ so that we can write u(x,τ) = 1 2π = 1 2π = I 1 I 2 Looking at I 1, we can see that I 1 = 1 2π = e 1 2 (y+1)x 2π u 0 (x 2τ + x)e 1 2 x 2 dx x/ 2τ 1 2π x/ 2τ x/ 2τ = e 1 2 (y+1)x+ 1 4 (y+1)2 τ 2π e 1 2 (y+1)(x+x 2τ) e 1 2 x 2 dx x/ 2τ e 1 2 (y 1)(x+x 2τ) e 1 2 x 2 dx e 1 2 (y+1)(x+x 2τ) e 1 2 x 2 dx e 1 4 (y+1)2τ e 1 2 (x 1 2 (y+1) 2τ) 2 dx = e 1 2 (y+1)x+ 1 4 (y+1)2τ N(d 1 ) x/ 2τ 1 2 (y+1) 2τ e 1 2 ρ2 dρ where d 1 = x 2τ (y + 1) 2τ, and N(d 1 ) = 1 d1 e 1 2 s2 ds 2π is the cumulative distribution function for the normal distribution. We can calculate the value of I 2 in exactly the same way as I 1, except that (y + 1) is replaced with (y 1) throughout. Finally, retracing through the steps, we write υ(x,τ) = e 2 1 (y 1)x 1 4 (y+1)2τ u(x,τ), and substitute our values x = ln(s/k), τ = 1 2 σ 2 (T t), and C = Kυ(x,τ) to get C(S,t) = SN(d 1 ) Ke r(t t) N(d 2 ) Reg: 65 20

25 where and d 1 = ln(s/k) + (r σ 2 )(T t) σ, T t d 2 = ln(s/k) + (r 1 2 σ 2 )(T t) σ T t = d 1 σ T t This is the Black-Scholes formula for the pricing of European call options. The corresponding calculation for a European put is very similar. Its transformed payoff is υ(x,0) = max(e 1 2 (y 1)x e 1 2 (y+1)x, 0) and we proceed through the steps as shown above. However, a far simpler way is to use the put-call parity formula, as shown in equation (3.1) r(t t) C P = S Ke which rearranges to r(t t) P(S,t) = C(S,t) S + Ke and substituting for C(S,t) gives P(S,t) = S[N(d 1 1)] + Ke r(t t) [1 N(d 2 )] Finally, using the identity N(d) + N( d) = 1, we obtain the formula for pricing European put options P(S,t) = Ke r(t t) N( d 2 ) SN( d 1 ) The Black-Scholes formula not only provides a measure of the fair value of the option, but also can be used to calculate the "risk" exposure (Jarrow and Rudd, 1983). This exposure is indicated by the option s hedge ratio H and is given by H = δc δs This shows the sensitivity of the option price to the stock price. An instantaneous unit price change in the stock causes a change of δc/δs in the price of the call. We can Reg: 65 21

26 calculate the value of the hedge ratio by differentiating the equation above which gives us the answer N(d 1 ). The same can be done in relation to the hedge ratio of a put option, giving the answer H = δp δs = N(d 1) 1 These quantities are vital if an option position is to be hedged correctly (Wilmott et al. 1993). Risk-Neutrality Argument As we have just seen, under the Black-Scholes equation, call and put prices depend on the stock price, S, the strike price, K, the time to expiry, T, the volatility σ and the risk-free interest rate, r. Surprisingly, the price of the option does not depend on the expected return from the stock µ. This being the case, the value of an option in a riskneutral economy must be the same as the value of an option in a risk-averse economy. The risk-neutrality argument states that when valuing a derivative, we can make the assumption that investors are risk-neutral (Hull, 2011). This assumption assumes that investors do not increase the expected return that they require for different levels of risk. In a risk-neutral world there are two features that simplify the pricing of derivatives: 1. The expected return on a stock, µ is the risk-free rate 2. The discount rate used for the expected payoff on an option is the risk-free rate. However, in the real world investors are not risk-neutral. The higher the risks investors take, the higher the return that they require. Nevertheless, it turns out that assuming a risk-neutral world gives the right option price for both the real and risk-neutral world. It is fortunate that µ drops out of the equation in the derivation, causing the Black- Scholes model to be independent of risk preferences. If risk preferences do not enter the equation, they do not affect the solution. When moving from the risk-neutral world to the real, risk-averse world, two things happen. The expected growth rate in the stock price changes and the discount rate that must be used for any payoffs from the derivative changes. It is very convenient that these two offset each other. Reg: 65 22

27 6. Binomial Pricing The first major advance in option pricing was made by Black and Scholes (1973). Arguably the next was made by Cox, Ross and Rubinstein (1979) and takes the form of the binomial model argues Redhead (1997). Binomial models approximate the distribution of continuous time random processes of a stock price through discrete time processes. The binomial model is a very useful technique as it can be used to value American options and options where the the volatility and interest rates vary, which the Black- Scholes cannot. The Binomial method follows a simple process in steps and assumes a discrete random walk. Each step represents a possible price of the stock at a point in time. In each time-step, there is a certain probability that the price of the asset will move up and a certain probability that the price will move down. First we will consider a one-step binomial model. One-Step Binomial Model Consider a situation where an investor takes a long position in shares, with a price S 0 and a short position in one option on that particular stock, with a current price f. The option s time to expiry is denoted as T and during T the share price S 0 will either increase in price to S 0 u, where u > 1, or decrease in price to S 0 d, where d < 1. If there is an upward movement in the share price the percentage increase in the share price is u - 1 and if there is a downward movement the percentage decrease is 1 - d. We also denote the payoff from the option in the case of an upward movement as f u and f d in the case of a downward movement. This situation is demonstrated in Figure 6.1 in the form of a binomial tree. Using these values, we can calculate the value of that makes this portfolio riskless. If the share price does increase, the value of the portfolio at expiry will be S 0 u f u. If the share price decreases, the value of the portfolio will be S 0 d f d. It follows that the two are equal when S 0 u f u = S 0 d f d or = f u f d S 0 u S 0 d (6.1) Reg: 65 23

28 Figure 6.1: Option and stock prices in a one-step Binomial model Equation (6.1) therefore shows that is the ratio of the change in option price to the change in share price during T. For the portfolio to be riskless, it must earn the risk-free interest rate. Denoting the risk-free rate by r, the present value of the portfolio, in the case of an upward movement in stock price, is (S 0 u f u )e rt The cost of setting up the portfolio is S 0 f It follows that S 0 f = (S 0 u f u )e rt rearranging this leads to f = S 0 (1 ue rt ) + f u e rt If we substitute from equation 6.1, we get which leads to or f u f d f = S 0 S 0 u S 0 d (1 ue rt ) + f u e rt f = f u(1 de rt ) + f d (ue rt 1) u d f = e rt [p f u + (1 p) f d ] (6.2) Reg: 65 24

29 where p = e rt d (6.3) u d p therefore denotes the probability that the share price increases, while (1 - p) denotes the probability of a decrease in price. Equations (6.2) and (6.3) are used for pricing options when there is one time step. The equations assume no possible arbitrage opportunities. Two-step Binomial Model We can now consider two-step binomial trees. As discussed previously, the value of a stock, with an initial price of S 0, either increases or decreases each time-step. Therefore, after two-steps, the value of the stock will either be S 0 u 2, S 0 ud or S 0 d 2 and the payoffs from the option denoted as f uu, f ud or f dd respectively. This is illustrated in Figure 6.2. Figure 6.2: Option and stock prices in a two-step Binomial model Supposing that the length of the time step is t years, where t = T /n and n is the number of time-steps, equations (6.2) and (6.3) now become: f = e r t [p f u + (1 p) f d ] (6.4) Reg: 65 25

30 p = e r t d (6.5) u d However, to find the value of f we need to find the values of both f u and f d. We can do this through the repeated application of equation (6.4), giving f u = e r t [p f uu + (1 p) f ud ] (6.6) f d = e r t [p f ud + (1 p) f dd ] (6.7) Finally, substituting from equations (6.6) and (6.7) into equation (6.4), we get f = e 2r t [p 2 f uu + 2p(1 p) f ud + (1 p) 2 f dd ] where p 2, 2p(1 p) and (1 p) 2 are the probabilities that the value of the stock price will reach the upper, middle and lower, final nodes. Solving u, d and p The binomial method assumes a risk-neutral world in which the random walk for S is lognormally distributed. We can approximate this continuous random walk with a discrete random walk having the same mean and variance (Wilmott et al., 1993). Thus S n e r t = E[S n+1 ] = S n (pu + (1 p)d) and var[s n+1 ] = E[Sn+1] 2 (E[S n+1 ]) 2 = (S n ) 2 (pu 2 + (1 p)d 2 + (pu + (1 p)d) 2 ) where S n is the stock price at time n. This gives us two equations for the parameters, u, d and p. Assuming that the time-steps and probabilities are constant for the entire random walk, we find that pu + (1 p)d = e r t (6.8) pu 2 + (1 p)d 2 = σ 2 t + e 2r t (6.9) Reg: 65 26

31 As expressed by Wilmott et al. (1993), another equation that is frequently used, because it leads to a lattice with particular useful properties, is the condition that u = 1/d (6.10) This condition is useful as it leads to a balanced lattice model, where after two timesteps the stock price can return to its starting value e.g. S 0 ud would simply equal S 0. We can now solve u, d and p with equations (6.8), (6.9) and (6.10), and find that u = e σ t, d = e σ t, (6.11) p = er t e σ t e σ t e = e r t d σ t u d as proposed by Cox, Ross and Rubinstein (1979) to match volatility. (6.12) 6.1. Binomial Convergence to the Black-Scholes As the number of time-steps tends to infinity, we see that the Binomial option pricing model converges to Black-Scholes Model. Suppose we are valuing a European call option, with n time steps. Each step is of length T /n. If there have been j increases in stock price and n j decreases in stock price on the binomial tree, the final stock price is S 0 u j d n j, where u and d are the proportional upward and downward movements and S 0 represents the initial stock price. The payoff from the call option is therefore max(s 0 u j d n j K, 0) From the properties of the Binomial distribution, the probability of exactly j upward movements and n j downward movements can be defined as n! (n j)! j! p j (1 p) n j It then follows that the expected payoff from the option is n j=0 n! (n j)! j! p j (1 p) n j max(s 0 u j d n j K, 0) Reg: 65 27

32 As the binomial model demonstrates movements in a risk-neutral world, we can discount this at the risk-free rate r to obtain the option price: C = e rt n j=0 n! (n j)! j! p j (1 p) n j max(s 0 u j d n j K, 0) (6.13) The terms in equation (6.13) are nonzero when the final stock price is greater than the strike, when S 0 u j d n j > K or ln(s 0 /K) > j ln(u) (n j)ln(d) Given that u = e σ T /n and d = e σ T /n, as seen in equation (6.11), this condition then becomes or Equation (6.13) can therefore be written as where ln(s 0 /K) > nσ T /n 2 jσ T /n j > n 2 ln(s 0/K) 2σ T /n C = e rt n! j>α (n j)! j! p j (1 p) n j (S 0 u j d n j K) For convenience, it is appropriate to define α = n 2 ln(s 0/K) 2σ T /n n! U 1 = j>α (n j)! j! p j (1 p) n j u j d n j (6.14) so that n! U 2 = j>α (n j)! j! p j (1 p) n j (6.15) C = e rt (S 0 U 1 KU 2 ) (6.16) It is well known that the binomial distribution approaches a normal distribution as the number of trials tends to infinity. More specifically, when there are n trials with a Reg: 65 28

33 probability of success p, the probability distribution of the number of success if approximately normal with mean np and standard deviation np(1 p). When considering U 2 in equation (6.15), the variable is the probability of successes being greater than α. From the properties of the normal distribution, it follows that, for a large n ( ) np α U 2 = N np(1 p) (6.17) where N is the cumulative normal distribution function. If we now substitute for α, we obtain U 2 = N From equation (6.12) we have ( ) ln(s 0 /K) n(p 1 2σ T p(1 p) + 2 ) p(1 p) p = ert /n e σ T /n e σ T /n e σ T /n (6.18) By expanding the exponential functions in a series, we see that when n tends to infinity, p(1 p) tends to 4 1 and n(p 2 1 ) tends to (r σ 2 /2) T 2σ so that in the limit, as n tends to infinity, equation (6.18) now becomes ( ln(s0 /K) + (r σ 2 ) /2)T U 2 = N σ T (6.19) We now consider variable U 1. From equation (6.14), we have n! U 1 = j>α (n j)! j! (pu) j [(1 p)d] n j (6.20) Define It then follows that p = 1 p = pu pu + (1 p)d (1 p)d pu + (1 p)d (6.21) Reg: 65 29

34 and we can write equation (6.20) as U 1 = [pu + (1 p)d] n n! j>α (n j)! j! (p ) j (1 p ) n j Since the expected return in the risk-neutral world is the risk-free rate, r, it follows that pu + (1 p)d = e rt /n and U 1 = e rt n! j>α (n j)! j! (p ) j (1 p ) n j This shows that U 1 involves a binomial distribution where the probability of an up movement is p rather than p. Approximating the binomial distribution with a normal distribution, we obtain ( ) U 1 = e rt np α N np (1 p ) substituting for α gives ( U 1 = e rt ln(s 0 /K) n(p N 2σ T p (1 p ) ) ) p (1 p ) Substituting for u and d into equation (6.21) gives ( p e rt /n e σ )( T /n e σ ) T /n = e σ T /n e σ T /n ert /n Once again by expanding the exponential functions in a series, we see that as n tends to infinity, p (1 p ) tends to 4 1 and n(p 2 1 ) tends to (r + σ 2 /2) T 2σ giving the result ( U 1 = e rt ln(s0 /K) + (r + σ 2 ) /2)T N σ T Finally, from equations (6.16), (6.19) and (6.22) we have (6.22) C = S 0 N(d 1 ) Ke rt N(d 2 ) where d 1 = ln(s 0/K) + (r + σ 2 /2)T σ T Reg: 65 30

35 and d 2 = ln(s 0/K) + (r σ 2 /2)T σ = d 1 σ T T Hence, as the number of time-steps n tends to infinity, the Binomial model converges to the Black-Scholes formula for option pricing. This result is illustrated in Figure 6.3. Figure 6.3: Binomial model convergence to the Black-Scholes model 7. Monte Carlo Simulation Another numerical method we shall review for the option pricing problem is the Monte Carlo simulation. The basic idea of the Monte Carlo simulation is to randomly sample the stock process by a large number of times to obtain observations of the stock price at maturity, S T. This data is then used to obtain a large number of possible realizations of the call price at maturity. The price of a European option is just the expectation of the option s payoff under the risk-neutral log-normal distribution (Joshi, 2008). The output is the distribution of call values at expiry which can be used to obtain a value of the current call price by application of the risk-neutrality argument; i.e. E[max(S T K, 0)] is computed as the sample mean of the option price distribution at expiry and then discounted to the present at the risk free rate. The Monte Carlo simulation follows a relatively straightforward process: Reg: 65 31

36 1. Sample a random path for S in a risk-neutral world. 2. Calculate the payoff from the derivative. 3. Repeat steps 1 and 2 to get as many sample values of the payoff from the option as required. 4. Calculate the mean of the sample payoffs to obtain an estimate for the expected payoff. 5. Discount the expected payoff at the risk-free rate, giving an estimate of the option value. Starting with the process derived in equation (4.2), in a risk-neutral world ds = µsdt + σdz where dz is the Wiener process, µ is the expected return and σ is the volatility. To simulate the path followed by S the option life can be divided into N short intervals of length t and approximate the equation above as S(t + t) S(t) = µs(t) t + σs(t)ε t where S(t) denotes the value of S at time t and ε is a random sample from a normal distribution with mean zero and standard deviation of 1.0. This enables us to calculate the value of S at time t from the initial value of S, and the value at time 2 t to be calculated from the value at time t, and so on. One simulation trial involves constructing a complete path for S using N random samples from a normal distribution. It is usually more accurate to simulate lns rather than S. From Itô s lemma the processed followed by lns is d lns = (µ σ 2 ) dt + σdz 2 so that lns(t + t) lns(t) = (µ σ 2 ) t + σε t 2 Reg: 65 32

37 which is equivalent to S(t + t) = S(t)exp [(µ σ 2 ) t + σε ] t 2 If µ and σ are constant it follows that S(T ) = S(0)exp [(µ σ 2 ) T + σε ] T 2 This is the equation used to construct the path of S Monte Carlo convergence and the Black-Scholes The accuracy of the Monte Carlo simulation is dependent on the number of trials. As we can value the option by repeatedly drawing a share price from the risk-neutral lognormal distribution and averaging the resulting payoffs, we know that this will eventually converge to the expected price, by the law of large numbers. However, the Monte Carlo simulation is slow to converge states Joshi (2008) and can often need millions of simulations. Figure 7.1 illustrates the convergence of the Monte Carlo against the Black- Scholes with an increasing number of simulations. As demonstrated, it appears that the higher the number of simulations, the closer the Monte Carlo gets to the Black-Scholes model and converges to its true value. 8. Data Analysis In this section, we will analyse the accuracy of the Black-Scholes equation, the Binomial method and the Monte Carlo simulation against real market data. The market data, provided by Yahoo! Finance (2014), looks at a range of options from the Russell 2000 Index which can be found under the symbol ˆRUT. The data looks at the prices for call and put options on the 11 th April We will be examining options with varying strikes and varying expiry dates. The Russell 2000 Index measures the performance of the small-cap segment of the U.S. equity universe. It includes approximately 2000 of the smallest securities based on a combination of their market cap and current index membership. The Russell 2000 Reg: 65 33

38 Figure 7.1: Convergence of the Monte Carlo simulation compared to the Black-Scholes equation is constructed to provide a comprehensive and unbiased small-cap barometer and is completely reconstituted annually to ensure larger stocks do not distort the performance and characteristics of the true small-cap opportunity set (Russell Investments, 2014) Estimating the parameters In order to analyse the option pricing models we have discussed against our market data, we must gather the details for the determinants of the that influence the option price. From our data gathered from Yahoo! Finance (2014), we are given the current stock price, S 0 = , the strike prices K and time to expiry T. The time to expiry is the number of trading days until the option matures. The number of trading days in a year is assumed to be 252 for stocks (Hull, 2011). An option s life is measured by T years and, assumes 21 trading days in a month. The only parameters we need to estimate are the volatility, σ, and the risk-free rate, r. Estimating the volatility We can estimate the volatility by looking at the past changes in share price, ds/s. Each change in share price is calculated as ds S = S 2 S 1 S 1, S 3 S 2 S 2, S 4 S 3 S 3,..., S n S n 1 S n 1 Reg: 65 34