The Black-Scholes formula

Transcription

1 Introduction History Revolution Aftermath V = SN(d + ) Ke rt N(d ) SCUM Math Night, December 7th, 2004

2 Introduction History Revolution Aftermath Markets and risk Options The Midas formula V = SN(d + ) Ke rt N(d ) It appears to be a simple, harmless formula, but it has been responsible for the making - and the losing - of unimaginable riches. It is a mathematical formula, and the ideas behind it are subtle. How is it that beautiful mathematics managed to get mixed up in the business of making money? And why aren t mathematicians doing better out of it?

3 Introduction History Revolution Aftermath Markets and risk Options Aristotle: B.C. The discussion of [wealth-getting] is not unworthy of philosophy, but to be engaged in [it] practically is illiberal and irksome. Thales call option: he pays a small deposit up front guaranteeing him the first call on a wine press (at an agreed rent). If the harvest is bad, he won t bother to exercise his option. But if the harvest is good, he does, makes a lot of money, and has his story told by Aristotle.

4 Introduction History Revolution Aftermath Markets and risk Options Chance and skill Aristotle thought that in the making of wealth too much was down to chance and not enough to human skill. Play Pause Resume Stop But what about the successes of traders who seem to have enough skill to pick the right stocks and beat the market?

5 Introduction History Revolution Aftermath Markets and risk Options Play Pause Resume Stop Markets are risky!

6 Introduction History Revolution Aftermath Markets and risk Options

7 Introduction History Revolution Aftermath Markets and risk Options Risk protection: selling Suppose you hold a stock and want to sell it in a year s time... A put option allows you to lock-in a minimum price for your stock, but to keep the unlimited upside. For example: the stock is currently at $50 and you protext yourself by purchasing a put option with strike price $50. When the contract expires, if the stock has risen to $60, you can sell it for $60. But if it has fallen to $40, you have the right to sell it for $50.

8 Introduction History Revolution Aftermath Markets and risk Options Risk protection: buying On the other hand, if you think you will want to invest in a particular stock, say in a month s time... A call option allows you to guarantee a maximum price you will have to pay. For example: the stock is currently at $50 and you protext yourself by purchasing a call option with strike price $50. When the contract expires, if the stock has risen to $60, you have the right to buy it for $50. But if it has fallen to $40, the you will buy it for $50.

9 Introduction History Revolution Aftermath Markets and risk Options (The initial stock price and strike price are both $1.)

10 Introduction History Revolution Aftermath Markets and risk Options Risk protection Option contracts come in many varieties puts, calls, butterfly spreads, condors, digital options, up-and-out options, swaptions,... They all involve the exchange of risk. How much is that uncertainty worth? Some people trade in options because want to be protected from risk. Others, because they want to take advantage of the increased leverage options provide. In the previous example, suppose the premium for the option is $2. If the stock goes up, the payoff is $10 and the profit is $8. A 400% profit! If the stock goes down, the payoff is zero and the premium is lost. A 100% loss!

11 Introduction History Revolution Aftermath Options trading Bachelier Gaining understanding A long,disreputable history 1600s - Holland. Tulip dealing is big business, and growers and dealers are trading in options to guarantee prices. Soon speculators are joining in and a thriving options market is born. But the market crashes, many speculators fail to honour their commitments, and the Dutch economy is brought to its knees. 1700s - London. Options are declared illegal! USA. Investment act legitimises options. Annual volume < 300, 000 contracts by April Chigaco. The CBOT starts trading listed call options on 16 stocks, with a first-day volume of 911 contracts Chigaco. The daily volume grows from 20,000 to over 200,000 contracts. So what happened to cause this explosive growth? - Mathematics!

12 Introduction History Revolution Aftermath Options trading Bachelier Gaining understanding The mathematicians are coming The search for a mathematical understanding of the behaviour of the market, and options pricing, has its beginning in the 1900 thesis of Louis Bachelier. Bachelier s thesis was titled Théorie de la Spéculation. He studies the movements of bond prices and associated options on the Parisian Bourse. He derives an analogy between the probability distribution of prices and the flow of heat. His price model is an example of Brownian motion - five years before Einstein s work on the subject. But his thesis was hardly noticed at the time - his career faltered and his work lay waiting until it was rediscovered more than fifty years later.

13 Introduction History Revolution Aftermath Options trading Bachelier Gaining understanding Brownian motion In 1827 Robert Brown observed pollen particles floating in water under the microscope and noted their jittery behaviour. In order to make sure that the motion was not due to the pollen being alive he did the same thing with dust particles. Bachelier models bond price movements in the same way Einstein later models the motion of particles under bombardment.(in fact he derives his results in three different ways.) Between any two points in time (t and t + t), the change in the bond price is a normally-distributed random variable following a bell-curve law. Any non-overlapping changes are independent. For times that are close together, the curve is peaked, and for longer times it is smeared out. This produces random, infinitely-long, but continuous curves.

14 Introduction History Revolution Aftermath Options trading Bachelier Gaining understanding Brownian motion in pictures

15 Introduction History Revolution Aftermath Options trading Bachelier Gaining understanding Rediscovery and enhancement In the 1930s and 40s, A. N. Kolmogorov, Kiyoshi Itô, Paul Lévy and Norbert Wiener put the mathematical description of Brownian motion on a much firmer basis, and Itô figures out how to do calculus on these random functions. Brownian motions are now called Wiener processes by the mathematicians. In 1955 Paul Samuelson turns his attention to option pricing. He and his students discover Bachelier s thesis. They also redefine Bachelier s model so that it refers to the logarithm of the stock price - this prevents the model from generating negative stock prices.

16 Introduction History Revolution Aftermath Options trading Bachelier Gaining understanding Geometric Brownian motion

17 Introduction History Revolution Aftermath Options trading Bachelier Gaining understanding The hunt is on In the period , people were working very hard indeed to try to solve the option pricing problem. Perhaps they had an inkling of how important such a discovery might be. Paul Samuelson - almost gets there. Guynemer Giguere - figures out boundary conditions Case Sprenkle - his model requires estimates of growth rates and investors risk-aversion. James Boness - translated Bachelier s thesis, creates an option model based on a discounted expected payoff. Henry McKean - wrote a book with Itô, a paper with Samuelson. Ed Thorp - he s even closer, building on Boness.

18 Introduction History Revolution Aftermath Options trading Bachelier Gaining understanding Almost there Play Pause Resume Stop

19 Introduction History Revolution Aftermath Convergence Balance The formula Publication Convergence In the late 1965 Fischer Black make the journey from physics to finance, joining the consulting firm Arthur D. Little. A couple of years later, Myron Scholes (hailing from our beloved northern goldmining regions) joins the faculty at MIT, and meets Black. Black and Scholes work on the option pricing problem. They realise that risk is the key - it is what is at the root of all the problems others are having, and it is what options are all about. They work on the idea of creating a small portfolio, consisting of just three items: S the stock. B a risk-free bond (a costless bank account). V the option.

20 Introduction History Revolution Aftermath Convergence Balance The formula Publication Make it go away... Their idea is to try to balance this porfolio (S, B and V ) so that the risk goes away. If the worth of the option is independent of individual preferences, then it just might be possible.... Here s what you can do. Start out by borrowing some money and investing it in S and V in some ratio. (Zero net investment.) Tomorrow, or when you next come to trade, the values of B, S and V have all changed. Your portfolio could be worth anything. But... if you choose your initial balance to minimise the uncertainty (the risk), could you get rid of it altogether? If you could, then you would know for sure the value of your portfolio tomorrow. Given that you invested nothing in it today, if its value is going to be anything but zero, you have found a money-making machine.

21 Introduction History Revolution Aftermath Convergence Balance The formula Publication Easy street? Your money-making machine is what is known as an arbitrage opportunity. The problem is that once word gets around, everybody wants a piece, and the effect of this is to puch prices the other way. The gap closes, and your machine does not work any more. Black and Scholes adopted the standard assumptions: that the grapevine works perfectly and instantaneously that there are no barriers to anyone entering into a trade, no matter how small or how often. The result of this is that these arbitrage opportunities do not exist. But this means that your perfectly-balanced portflio must still be worth nothing tomorrow. This is going to give you a handle on how the value of your option is changing with time.

22 Introduction History Revolution Aftermath Convergence Balance The formula Publication Perfect balance Consider a simplified model with these ingredients: a stock, which is currently at $50 and can move up to $60 or down to $30. a call option with strike price $45. a zero interest rate. Create a portfolio consisting of buying one stock, selling two options, and borrowing $30. If the stock goes up, the net value is $60-2 $15 - $30 = 0. If the stock goes down, the net value is $30 - $30 = 0. The value at the start must be zero - so V = $10.

23 Introduction History Revolution Aftermath Convergence Balance The formula Publication In continuous time In the previous example, we created a portfolio that was perfectly balanced - in all eventualities its value stayed at zero. Can we do this with a more realistic model? Well, the answer is no. Here s the best you can do if you rebalance once a day...

24 Introduction History Revolution Aftermath Convergence Balance The formula Publication

25 Introduction History Revolution Aftermath Convergence Balance The formula Publication Trading more often If turns out that you can do better if you rebalance once an hour...

26 Introduction History Revolution Aftermath Convergence Balance The formula Publication

27 Introduction History Revolution Aftermath Convergence Balance The formula Publication Robert Merton Merton arrived on the scene in 1968 and brought with him expertise in Itô calculus, and an understanding of continuous-time stochastic processes. He met Scholes in 1969 and it was he who figured out that their dream of perfect balance could be achieved by continuously adjusting their portfolio. Here s the result of our previous experiment, rebalancing every minute of the year...

28 Introduction History Revolution Aftermath Convergence Balance The formula Publication

29 Introduction History Revolution Aftermath Convergence Balance The formula Publication The balance equation Achieving perfect balance tells you that the value of your portfolio is stable over time, and the no arbitrage principle then forces the option value to depend on S and B and the time t in a particular way. If r is the continuously-compounded rate of interest earned by B, and σ is a measure of the volatility of S, then V t + rs V S + σ2 2 S 2 2 V S 2 = rv. This is what has become known as the Black-Scholes equation. It had already been solved by McKean, and the solution is the formula everyone had been looking for.

30 The formula Introduction History Revolution Aftermath Convergence Balance The formula Publication The value of a call option on an asset S, expiring at time T, with strike price K is V = SN(d + ) Ke rt N(d ), where N(x) is the cumulative normal distribution function, d ± = S/Ke rt ± σ2 T 2 σ T and r is the risk-free interest rate, continuously-compounded, and σ is the volatility of the asset. The balance is struck by selling N(d + ) units of the asset for every unit option bought. This is known as the delta, or hedge ratio.,

31 Introduction History Revolution Aftermath Convergence Balance The formula Publication The option value surface

32 Introduction History Revolution Aftermath Convergence Balance The formula Publication Getting it out Black and Scholes had a little trouble getting their paper published. They had to try three times the first two times the paper was rejected without even being reviewed! The suspicion is that Black s non-academic position may have had something to do with it. Merton had written his own version, more general than Black and Scholes, but he graciously delayed the publication of his until their paper appeared.

33 Introduction History Revolution Aftermath Nobel prizes LTCM Nobel s for almost all Myron Scholes Robert C. Merton

34 Introduction History Revolution Aftermath Nobel prizes LTCM Death of a dream Merton and Scholes wanted to see their ideas in practice. They teamed up with some of the top investors from Wall Street to form a new company - Long Term Capital Management. They raised $3 billion from investors, including many of the major banks, on the promise of using dynamic hedging (a.k.a. continuous rebalancing) on a huge scale to form a gigantic vacuum cleaner sucking up nickels from around the world. They were enormously successful - returning 20%, 43% and 41% to their investors in the first three years.

35 Introduction History Revolution Aftermath Nobel prizes LTCM Death of a dream But at the tail end of the century things started to go wrong. The trouble started in asia - markets were collapsing and deviating significantly from their historical norms. LTCM carried on as normal, convinced that things would stabilize. When Russia defaulted, the game was up. In order to prevent the global economic collapse that would have resulted from the failure of LTCM (!), the Federal Reserve had no choice but to bail out LTCM - to the tune of $3 billion.

36 Introduction History Revolution Aftermath Nobel prizes LTCM Summing up Mathematics plays an unexpectedly significant role in the operation of financial markets. Mathematics provides powerful tools for understanding and even controlling the nature and effects of uncertainty and risk. But some humility is called for!