EC4024 Lectures 14 and 15

Transcription

1 EC4024 Lectures 14 and 15 Options Stephen Kinsella 1. Introduction Terminology, History, & Notation The basic types of financial assets are bond, stock, etc and derivatives - forward, futures and option, etc. They are categorised into underlying assets and derivative assets. Underlying Assets The underlying assets can be stocks, bonds, currency, commodities, and other financial assets, or combinations of these. The traditional stock and bond markets raise necessary capital for corporations and governments, and the foreign exchange market facilitates international trade and investment. The market, essentially, transfers money from one side of the economy---those who have money---to the other side---those who want to use that money to make more. The price of this transfer is the interest rate charged. Stocks Stocks represent the claim of the owners of a firm. Stocks are issued by corporations and can be traded in the stock market. Common stock usually entitles the shareholder to vote in the election of directors and other matters. Preferred stock generally does not confer voting rights but it has a prior claim on assets and earnings: dividends must be paid on preferred stock before any can be paid on common stock. Bank of Ireland, for example, has both preferred and normal stock, as we can see above. Bonds Bonds are issued by anyone who borrows money - firms, governments, etc. They are fixed-income instruments because they promise to pay fixed sums of cash in the future. Bondholders have an IOU (I owe you) from the issuer, but no corporate ownership privileges, as stockholders do. Derivative Assets Derivatives are financial instruments that derive their value from the prices of one or more other assets such as stocks, bonds, foreign currencies, or commodities. Derivatives serve as tools for managing risks associated with these underlying assets. The most common types of derivatives are options and futures. We'll have a lot more to say about derivatives later on in the course. Forwards and Futures A forward is a financial contract in which two parties agree to buy and sell a certain amount of the underlying commodity or financial asset at a prespecified price at a specified time in the future. The specified time is called the time-to-maturity of the forward contract and the price specified in the contract is called the forward price. A futures contract is a standardised forward contract traded in an exchange. To avoid the shortcomings of the forwards that each party cannot change his/her mind to reverse the position specified in the contract, in a futures contract, both the time to maturity and the amount of the underlying asset to be delivered in each contract are standardised so that the futures contract can be traded in a market place. Thus, if one party wants to change position, he/she can buy or sell in the futures market. Options An option is a financial contract, which provides the holder with the right to buy or sell a certain amount of the underlying asset at a prespecified price at or before a specified time in the future. Similar to the forward and futures

2 2 CurrentValue[FileName] underlying asset at a prespecified price at or before a specified time in the future. Similar to the forward and futures contracts, the time specified in an option contract is called the time-to-maturity of the option. The price specified in the contract is called the exercise price or the strike price of the option. Unlike a forward or futures contract, an option contract gives its holder the right, but not the obligation to buy or sell the underlying asset. The two most common types of options are calls and puts. A call option is an option to purchase the underlying asset, while a put option is an option to sell. Everyone would like to hold options because they provide a positive likelihood for the holder to make a profit. But he/she has to pay the price. The cost for this likelihood is the money paid by the buyer of option to the seller to compensate the latter's possible losses. This money is called the premium of the option, or simply the option price. As an example, let's say you decide to buy a new car. You select a certain type of car, and the dealer tells you that if you place the order today and place a deposit, then you can take delivery of the car in 3 months time. If in 3 months time the price of the model has decreased or increased, it doesn't matter. When the agreement between you and the dealer is reached, you have entered into a forward contract: you have the right and also the obligation to buy the car in 3 months. Instead, suppose the car you selected is on offer at 30,000 but you must buy it today. You don't have that amount of cash today and it will take a week to organize a loan. You could offer the dealer a deposit, for example 200, if he will just keep the car for a week and hold the price. During the week, you might discover a second dealer offering an identical model for a lower price, then you don't take up your option with the first dealer. At the end of the week the 200 is the dealer's whether you buy the car, or not. In this case, you have entered an option contract, a call option here. It means that you have the right to buy the car in a week, but not the obligation. The expiration time is one week from now, the strike price is 30,000. In this example, for both the forwards and option contracts described, delivery of the car is for a future date and the prices of the deposit and option are based on the underlying asset - the car. Another example comes from Aristotle ( Aristotle tells us how the philosopher Thalus profited handsomely from an option - type agreement around the 6 th century b.c. According to the story, one - year ahead, Thalus forecast the next olive harvest would be an exceptionally good one. As a poor philosopher, he did not have many financial resources at hand. But he used what he had to place a deposit on the local olive presses. As nobody knew for certain whether the harvest would be good or bad, Thalus secured the rights to the presses at a relatively low rate. When the harvest proved to be bountiful, and so demand for the presses was high, Thalus charged a high price for their use and reaped a considerable profit. Forwards, Options, and Futures Definition 1. An option is a financial product which gives the holder the right but not the obligation to buy or sell the underlying ass set at a predetermined price. Another definition: Defintion 2. Derivatives are financial instruments that derive their value from the prices of one or more other assets such as stocks, bonds, foreign currencies, or commodities. Derivatives serve as tools for managing risks associated with these underlying assets. The most common types of derivatives are options and futures. We'll have a lot more to say about derivatives later on in the course. Yet another definition from Neftci (2000), is Definition 3. A financial contract is a derivative security on a contingent claim if it's value at expiration time T is determined exactly by the market price of the underlying cash asset at time T. Let's define some notation right away. You'll find this is consistent across much of the literature. T: Expiration date of derivative contract; F HTL: Price of the contract at time T; S T :Value of the underlying asset at time T; d : payout from derivative. Because options are defined on an underlying asset or assets, they belong to a class of financial products known as derivatives. The types of derivatives which exist are forwards and futures, options, and swaps. Options were first traded in 1972 in Chicago, but have quite a long philosophical pedigree, as you can see from above. The underlying assets can be stocks/currencies/interest rates/indexes and commodities.

3 CurrentValue[FileName] 3 Some derivative instruments are written on products of cash-and-carry markets. Gold, silver, and T-bonds are some examples of cash-and-carry markets (Neftci, 2004). One can borrow at a risk free rate by collateralizing the underlying asset, buy and store the product, and insure it until the expiration date of the contract. The cash and carry market is an excellent introduction to forms of arbitrage, where securities are bought in the cash market and a short forward contract entered into. Forwards and Futures Definition. A futures contract is an agreement between two counterparties to exchange a specified amount of a financial security (bond, bill, stock, or currency) at a fixed future date at a predetermined price (Pilbeam, 2004, pg. 273). A forward is a future whose final amount of the underlying asset to be delivered is negotiable the future's value is fixed. Margins are required for futures but not forwards, and most importantly, the size of the forward is much larger to begin wih: 5 million minimum versus 50, ,000. A great example of the futures market is the Eurodollar market. Essentially this market is a way for the market to bet on what will happen to interest rates over a 1 year horizon. Definition. Eurodollars are time deposits denominated in U.S. dollars that are deposited in commercial banks outside the U.S., and they have long served as a benchmark interest rate for corporate funding. (ref) We price euro futures using a version of the arbitrage pricing formula: Z % = + r d * t d ê + r n * t n ê 365D - 1> * 365 t d - t n * 100 Z is the implicit interest rate for the future, r d is the interest rate on the forward period. r n is the interest rate on the near period, and t d,n are the number of days remaining until the near or distant periods, respectively. Let r d = r n = 0.03, t d = 10, t n = 90. Example (1) rd = 0.05; rn = 0.03; td = 10; tn = 90; z = H1 + rn * td ê 365L H1 + rn * tn ê 365L - 1 * 365 td - tn * 100 Exercise Let r d = 0.04, r n = 0.025, t d = 20, t n = 50. Calculate Z. 2. Valuing options When the valuation is a monotonic function of volatility (strictly increasing or decreasing), there is no problem in describing it here. A simple vanilla European Call option, developed by Black-Scholes (1973) and others, looks like the figure below. Let's manipulate it and have a look at it's properties. The American put was defined correctly as a moving boundary problem by McKean in We have a function, which satisfies Black Scholes PHS, TL = Max@0, K - SD Implemented from Black and Scholes (1973) paper, which took several key ideas from papers by Thorp and other gamblers, as well as physicists (See Taleb 2007), the Black Scholes model allows pricing of options over time. There are five crucial factors that determine the likelihood of a call option being exercised (Pilbeam, pg. 326). 1. The current price of the share. The higher the price of the stock, the more likely the share is to be exercised for ay given strike price. 2. The strike price. The higher the strike price of a call option, the less likely it is that it will be exercised, and hence the lower its price. (2)

4 4 CurrentValue[FileName] 3. The time left to expiration. The longer the time left to expiration, the higher the chance of the option being exercised, and hence the higher its price. 4. The volatility. The more volatile an option is the more likely that its price at the time of expiration will exceed the strike price, and hence the higher its price. 5. The risk free rate of interest. The purchaser of a call option is paying an issuer cash for an option which can be exercised to buy an undrlying security at a later date. The option holder then benefits from the difference betwee the option premium and the underlying security can be invested in at a risk free rate of interest until the option expires. A rise in the risk free rate of interest makes it more attractive to buy the option rather than the underlying security. So, a call option needs to be priced more highly when interest rates are low than when they are high. The higher the risk free rate of interest, the higher a call option price. These are summarised in table X below, taken from Pilbeam, (2007, pg. 326). 2.1 Intrinsic Value & Time Value Factor a rise in Eu Call Eu PUT Current Price + - Strike Prie - + Time to Expiration + - Volatility + + Risk Free Rate + - Table X. Factors affecting the price of options. The option premium is the sum of the intrinsic value of an option and the time value of that option. The intrinsic value of an asset is the gain one would realise if an option is exercised immediately, or Intrinsic Value (Call) = Cash Price - Strike Price Intrinsic Value (Put) = Strike Price - Cash Price. When the intrinsic value of an option exists, then the option is said to be 'in the money'. A call option whose strike price is below the cash price is said to be in the money. If the call option's strike price is above the cash price, the option is out of the money. When the strike price is equal to the cash price, the option is at the money. The Time value of the option is the option premium less the intrinsic value, or Time Value = option premium - intrinsic value. (1) Example Consider a call option valued at 20 cent in the Apple stock, with a strike price of 92 cent and a cash price for the underlying asset at 100 cent. The option is in the money for 8 cent and it's intrinsic value is 12 cent plus the other 8 cent which is the time value of the asset. Exercise Consider the call option valued at 14 cent in Microsoft, with a strike price of 90 cent and a cash price for the underlying asset at 70 cent. Is this option in the money or out of the money? Why? What is the intrinsic value of the option? Lognormal Distributions The Lognormal distribution is assumed to hold for returns in the BS model. The idea of the lognormal is that the log of the price is normally distributed, rather than the price itelf. The lognormal looks like the figure below, and here we see the distribution generating so-called 'fat tails', of which more later. HcodeL

5 CurrentValue[FileName] 5 Out[6]= percentile from top 21 mean standard deviation random seed size BestFit Ø x RSquared Ø ran Lognormal distributions are important in the distribution of the option premium between time and intrinsic values. Say an option is way out of the money. The lognormal distribution supplies a description of the movement of the actual price paid around the spot price, depending on what happens in reality. There are many prices that might end up being paid---what we are interested in are the most likely prices. The distribution of the logs of these prices is approximately normal. Consider a 'deeply in the money' option shown in figure X below. The spot price is quite far above the exercise price, so it's very likely the holder of the option will make money from it. As one moves from a deeply in-the money position to a deeply out of the money position, the likelihood of one's option being exercised decreases markedly. Pilbeam (2006, pages ) provides a good summary of these effects. 3. Developing Black Scholes BS assumes quite a few things to make it work. Among these are 1. The underlying asset pays no dividends or interest over it's lifetime; 2. The option is European; 3. The risk free rate is fixed for the lifetime of the security;

6 6 CurrentValue[FileName] 4. The financial markets are perfectly efficicent; 5. The price of the underlying asset is log normally distributed; 6. One can sell short the underlying at all times to keep the market liquid with no transactions costs; 7. The price of the underlying asset is continuous. The basic idea of BS is that a long position in an underlying stock is neutralised by a short position in options. This combination allows the trader with a combined position to make a return equal to the risk free rate. 3.1 Derivation Begin from the assumption that the intrinsic value of the call option on expiration is the spot price S minus the exercise price X, if the option is in the money. If it isn't, the value is zero. If we knew the intrinsic value on expiration with certainty, and we knew the expiration value was above the stock price, then the call premium would be valued by C = S - X > 0. C is the call premium, S is the cash or spot price, and X is the exercise price. The holder of the call can use the money he has to borrow while the call matures. He should do this at the risk free rate, r, and so doesn't need the full amount of X, only the amount he'll make when the call matures. He can use the risk free rate to get him there, so he needs to have X =rt in euros, or whatever the option is denominated in. Here is the natural number , r is the riskless interest rate, and T is the time left to maturity. Substituting, we have (2) (3) C = S - X -rt. (4) Which says the value of the call will be equal to the price of the share on expiration, less the present value of the exercise price. Now, we need to modify this equation slightly, to take account of the fact that we don't know the future with any certainty. If we include a term to describe the expected value of the underlying security at expiration SN(d1), and allow the value of X -rt to be normally distributed, we have C = SNHd1L - X -rt NHd2L. Let the value of d1 be given by (5) d1 = lnhs ê XL + Ir + s2 ë 2 ë T s T (6) and d2 = d1 - s T. (7) The BS model says, in effect, that the current value of a call is the present value of the expected cash price, minus the expected value of the strike price. Example Consider a call pricing problem, where the spot price is 100 cent, the investor buys the call option to purchase at 90 cent a share. Let the risk free rate be 5%, and let the value of recent historical volatility be given by s=30%. If the option has 83 days to expiry, calculate the price of the option. S = 100 X = 90

7 CurrentValue[FileName] 7 r = 0.05 T = 83 ê 365 = 0.23 s 2 = 0.49, so s = 0.7. d1 = lnh100ê90l+i ë2m = Look up the value of the normal distribution using the normal tables at N(d1)=N(0.52)=-.6986, and, because the option is positive, we don't need to subtract the number from one, we can calculate d2 as d2 = = Looking this up in the normal tables gives us N Hd2L = N H0.17L = Plugging these values into the BS formula, we have C = 100 H0.6985L - 90 H2.718L 0.05 H0.23L H0.5675L= Of the price, 10 cent is intrinsic, and 8.18 is time value. Exercise Consider a call pricing problem, where the spot price is 110 cent, the investor buys the call option to purchase at 93 cent a share. Let the risk free rate be 7%, and let the value of recent historical volatility be given by s=30%. If the option has 71 days to expiry, calculate the price of the option. Sensitivities BS won Scholes a Nobel prize for showing that the price of options is determined by the time left to maturity, the volatility of the underlying security, and the spot price of the underlying security. These underlying effects mean the BS model is sensitive to changes in them. These sensitivities can be measured, by what are called the greeks. q: The theta of an option measures it's sensitivity to changes over time. d: The delta of an option measures it's sensitivity to changes in the price of the underlying asset. g: The gamma of an option measures it's sensitivity to changes in q. Ó or k: The vega or kappa of an option measures it's sensitivity to changes in the volatility of the underlying share. Further reading Salih N. Neftci, An introduction to the mathematics of financial derivatives, 2nd ed, Elsevier Press, Nassim N. Taleb, Dynamic Hedging, John Wiley & Sons, 1st edition, Haug, Espen Gaarder and Taleb, Nassim Nicholas, "Why We Have Never Used the Black-Scholes-Merton Option Pricing Formula (fourth version)" (January 2008). Available at SSRN: Bouchaud J.-P. and M. Potters (2003): Theory of Financial Risks and Derivatives Pricing, From Statistical Physics to Risk Management, 2nd Ed., Cambridge University Press. Mandelbrot, B. and N. N. Taleb (2007): Mild vs. Wild Randomness: Focusing on Risks that Matter. Black, F and Scholes, M., 1973, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 81, p Thorp, E. O., and S. T. Kassouf (1967): Beat the Market. New York: Random House Demonstrations from "Power Law Tails in Log Normal Data" from The Wolfram Demonstrations Project and

8 8 CurrentValue[FileName] Implementation of a Simple European Call Option Let us plot the option value as a function of the volatility, for an option where the strike price is 10, the underlying asset is priced at 11, for a vanilla European call with a one year expiry, 5% risk free rate continuously compounded and zero dividends. (link) d1@s_, k_, r_, d_, s_, T_D = ILog@S ê kd + Ir - d + s 2 ë 2M TM í Js T N; d2@s_, k_, r_, d_, s_, T_D = ILog@S ê kd + Ir - d - s 2 ë 2M TM í Js T = J1 + ErfBz í 2 FN í 2 ; BSCall@S_, k_, r_, d_, s_, T_D := S -d k, r, d, s, TDD - k -r k, r, d, s, TDD; BSPut@S_, k_, r_, d_, s_, T_D := k -r k, r, d, s, TDD - S -d k, r, d, s, TDD; BSCallDelta@S_, k_, r_, d_, s_, T_D = D@BSCall@S, k, r, d, s, TD, SD; BSPutDelta@S_, k_, r_, d_, s_, T_D = D@BSPut@S, k, r, d, s, TD, SD; BSCallGamma@S_, k_, r_, d_, s_, T_D = D@BSCall@S, k, r, d, s, TD, 8S, 2<D; BSPutGamma@S_, k_, r_, d_, s_, T_D = D@BSPut@S, k, r, d, s, TD, 8S, 2<D; BSCallTheta@S_, k_, r_, d_, s_, T_D = -D@BSCall@S, k, r, d, s, TD, TD; BSPutTheta@S_, k_, r_, d_, s_, T_D = -D@BSPut@S, k, r, d, s, TD, TD; BSCallRho@S_, k_, r_, d_, s_, T_D = D@BSCall@S, k, r, d, s, TD, rd; BSPutRho@S_, k_, r_, d_, s_, T_D = D@BSPut@S, k, r, d, s, TD, rd; BSCallVega@S_, k_, r_, d_, s_, T_D = D@BSCall@S, k, r, d, s, TD, sd; BSPutVega@S_, k_, r_, d_, s_, T_D = D@BSPut@S, k, r, d, s, TD, sd;

9 CurrentValue[FileName] 9 Manipulate@ Plot@BSHedge@S, k, r, d, s, td, 8S, 50, 150<, AxesLabel Ø 8"stock price", "option sensitivity"<, ImageSize Ø 400, ImagePadding Ø 8825, 30<, 825, 25<<, AxesOrigin Ø 8Automatic, 0<D, 88BSHedge, BSCallDelta, "call sensitivities"<, 8BSCallDelta Ø "call delta", BSCallGamma Ø "call gamma", BSCallRho Ø "call rho", BSCallVega Ø "call vega", BSCallTheta Ø "call theta"<<, 88BSHedge, BSPutDelta, "put sensitivities"<, 8BSPutDelta Ø "put delta", BSPutGamma Ø "put gamma", BSPutRho Ø "put rho", BSPutVega Ø "put vega", BSPutTheta Ø "put theta"<<, 88k, 100., "strike price"<, 50., 150.<, 88r,.05, "interest rate"<,.02,.1<, 88d,.01, "dividend yield"<, 0.0,.1<, 88s,.3, "volatility"<,.1,.5<, 88t,.1, "time to maturity"<,.001, 1<, SaveDefinitions Ø True, AutorunSequencing Ø 84, 6, 7<D call sensitivities call delta call gamma call rho call vega call theta put sensitivities put delta put gamma put rho put vega put theta strike price interest rate dividend yield volatility time to maturity option sensitivity stoc

10 10 CurrentValue[FileName] := 1 ê 2 * H1 + Erf@x ê Sqrt@@2DDDL; done@s_, s_, k_, t_, r_, q_d := HHr - ql * t + Log@s ê kdl ê Hs * Sqrt@tDL + Hs * Sqrt@tD ê 2L; dtwo@s_, s_, k_, t_, r_, q_d := done@s, s, k, t, r, qd - s * Sqrt@tD; SetDelayed::write : Tag Norm in Norm@x_D is Protected. à BlackScholesCall@s_, k_, v_, r_, q_, t_d := s * Exp@-q * td * Norm@done@s, v, k, t, r, qdd - k * Exp@-r * td * Norm@dtwo@s, v, k, t, r, qdd; dox = Plot@BlackScholesCall@11, 10, vol, 0.05, 0, 1D, 8vol, 0.05, 0.4<, PlotRange Ø AllD Let' s change these values in our lecture, using the demonstration below. Implementation of the Black Scholes model (Code)

11 CurrentValue[FileName] 11 option call put strike price 100. interest rate 0.05 dividend yield volatility 0.3 time to maturity 0.1 ption price stock p