Topic 10: Black-Scholes

1. Topic 10:Black-Scholes

2. Introduction • In this lecture (or series of lectures) we will lay the basic foundation that you need to begin working with the standard continuous-time option pricing model, the Black-Scholes model. • This is really just an introduction to the mathematics of continuous-time finance, obviously in the Math 6202, 6203, and 6204 you will study this material in much greater detail. • This lecture begins with a quick tour through Hull’s Chapter 11, and then moves into the Black-Scholes analysis itself, which comes from Hull Ch. 12.

3. Introduction • Our first concern is to develop a statistical model of how stock prices behave over time. • The term stochastic process simply means that the variable defined by it is random.  • Stochastic processes can be classified in several ways. • By time interval: • continuous time: changes can occur at any point in time: ex. temperature • discrete time: changes can only occur at specific points in time. • By change increment: • Continuous variable: can take on any value in range. • Discrete variable: can take on only specific values. • Stocks are generally assumed to follow a continuous time, continuous variable stochastic process.

4. The Markov Property of Stocks • A Markov property is a particular type of stochastic process where only the present value of a variable is relevant for predicting the future. The past history of the variable and how it got to that point are irrelevant. • There is overwhelming evidence that stock prices are Markov. If they were not, then chartists could make money above and beyond the risk adjusted rate of return. They cannot, and if fact we kind of define the markets as being weak-form efficient: this says the stock prices fully incorporate all public information. This is caused by competition.

5. Wiener Processes • A Wiener process is a particularly useful type of Markov process: it has a mean change of zero and variance • The general format of a Wiener process is very adaptable. In fact it is it’s flexibility that makes it such a good candidate for describing many different phenomena, including stock prices. • Let’s begin by defining a variable, z, which follows a Weiner process. Define Δt as a small length of time (nearly instantaneous) and define Δz as the change in z during time Δt.

6. Wiener Processes • A Wiener process, it will have these properties: Property 1: Δz is related to Δt by the equation: where ε is a random draw from a standard normal distribution (i.e. ε~N(1,0)). Property 2: The values of Δz for any two different short intervals are independent. • It follows from property 1 that Δz itself is distributed normally with mean 0 and variance Δt. It’s standard deviation is (Δt).5

7. Weiner Processes • What are the statistical properties of z during a non-short interval of time, T? • Denote this as z(T)-z(0). It is simply the sum of N small time intervals of length Δt, where : N = T/Δt. • Thus, • Thus, since the mean of a sum of random variables is simply the sum of the means, mean of [z(T) - z(0)] = 0 • Further, since the draws from ε are independent, the variance of the sum of those draws are additive (another way of saying this is that they are uncorrelated).

8. Weiner Processes • Thus, σ2 = N*Δt = T and hence σ = T.5. • So far we have dealt with a discrete time case. We are interested in a continuous time case. Basically nothing changes except we use continuous time notation:

9. Generalized Wiener Process • So far we have dealt only with a simple Wiener process. It has a drift rate of 0 and a variance rate of 1.0. We can add non zero drift rates and non-unitary variance rates. • To get to such a process we build off of the basic Weiner process we already have: dx = a dt + b dz where a and b are constants, and dz is our previously defined Weiner process. • To see this, consider the two components of this equation separately. Note that the first portion (a dt) is not random. It merely says that dx is generally drifting upward at some constant rate.

10. Generalized Wiener Process • Another way of looking at this is to ask the question, what if I divide through by dt?  dx/dt = a. • The second term, however, does contain a random component. What we have is the white noise process (dz) multiplied by some non-random constant. • Recall that the mean of the product of a random variable times a constant is the product of the mean of that random variable times the constant amount. • In this case, i.e. for b dz since dz~N(0,1), the mean of this term is zero. • Thus the drift of the entire process, is simply (a*dt).

11. Generalized Wiener Process • Similarly, the variance of the product of a random variable times a constant is the variance of the random variable times the square of the constant. • So adding this all together we get: dx ~ N(a dt, b2 dt) and hence the st. deviation of dx is given by: b (dt).5. • Similarly over a non-small length of time T: • Mean of change in x = aT • variance of change in x is = b2T • standard deviation of change in x is = b T.5 • If a and b are allowed to be functions of both x and t, then we have the most general Markov, stochastic process, the Ito process: dx - a(x,t) dt + b(x,t) dz

12. The Process for Stock Prices • It is at first tempting to say that stocks follow a generalized Weiner process, that is, they have a constant drift and variance rate. However, this would not take into account that investors demand a rate of return that is independent of a stock’s price. • That is, investors want a return based on the risk of a stock, regardless if it is worth $10 or $100. • Because of this we cannot use the constant expected-drift rate, but rather have to have a constant proportional drift rate. This means that if S is the stock price, investors want a constant proportional return μS where μ is a constant.

13. The Process for Stock Prices • If a stock had a zero variance, this would imply: ds = μS dt, or, in another, more convenient, format: dS/S = μ dt. Thus, ST = S0 e μt. • Of course, stocks do exhibit volatility. A reasonable assumption (based on empirical observation) is that the variance is roughly constant over any relatively short time period. Define σ2 as the variance rate of the proportional change in the stock price. • This means that σ2 Δt is the variance of the proportional change in the stock price over time Δt while the variance of the actual change is σ2 S2 Δt. Thus the instantaneous variance rate of S is therefore σ2 S2 (i.e. as Δt -> 0).

14. The Process for Stock Prices • This leads us to represent S by an Ito process of the form: dS = μS dt + σ S dz or, in the more generally seen form: dS/S = μ dt + σ dz • Generally μ is referred to as the stock’s expected return, and σ as it’s volatility. • It is really easy to build a simulation of this process. • Let’s assume that the at stock priced at 50 had an expected return ( μ) of 10% and that the volatility of that stock (i.e. standard deviation of the return, i.e. σ) was 20%. Let us divided the year into increments of 1 week (i.e. dt=1/52) and chart the movements over 3 years:

15. The Process for Stocks

16. The Parameters • Now notice that we have defined a process for stock prices that only involve two parameters: μ and σ2. Those parameter’s are usually expressed in time units of one year. • What are the properties of those parameters? • μ: Generally, investors demand a higher value of μ for riskier stocks. Historically, the “average” stock’s μ has been 8 percentage points higher than the risk-free rate of return. Fortunately, we don’t have to worry too much about this parameter, since the expected rate of return does not matter for pricing derivatives. (Recall that we can price based on arbitrage arguments.) • σ: This is an important parameter, and we will discuss how to estimate it from historical data later. It is important to note that σ(T).5 will be approximately (but not exactly) the standard deviation of stock returns during the period T.

17. Ito’s Lemma • In the early 1950's Ito developed a method for describing how functions based on random variables evolve. • Assume that G is a function of time, t, and a random variable x, i.e. G(x,t). Let x be a general Ito process: dx = a(x,t)dt + b(x,t) dz then Ito’s lemma states that where dz is the typical Wiener process.

18. Ito’s Lemma • Thus G itself follows an Ito process with drift of and variance rate of • Now, applying this to stocks, if a stock follows a proportional Ito process: dS = μS dt + σ S dz 

19. Ito’s Lemma • with constant μ and σ, a function G of S (such as a derivative) will follow: • This is the key to pricing all derivatives. Notice that both S and G have one common (and sole) source of uncertainty: dz. This is what allows us to price G in terms of S!

20. Application to a Forward Contract • To see how this works, let’s look at a forward contract. From earlier in the book we defined a forward price F to be: F = Ser(T-t). • So we can define F as a function of S and t: F(S,t), and then taking some standard derivatives:

21. Application to a Forward Contract • Appling Ito’s lemma, F should evolve as dF, • which in this case will be • Since by definition F = Ser(T-t), dF becomes: • Note that this is basis for the notion that the growth rate in a stock index futures price is the excess return over the risk free rate.

22. Application to the Logarithm of a Stock Price • Now it turns out that in many cases we will want to work with the log of stock price, not with the stock price itself. • What process does ln S follow? • Again, define a function G(S,t) = ln S. • Taking normal derivatives:  • Substituting into equation Ito’s lemma:

23. Application to the Logarithm of a Stock Price • So it has mean (over interval [t,T]): (μ - σ2/2)(T-t)  and variance: σ2(T-t) • Thus the change in the stock price is given by ln ST - ln S0 and this is distributed: ln ST - ln S0 ~ N[(μ - σ2/2)(T-t), σ(T-t).5] or written slightly differently:   ln ST ~ N[ln S0(μ - σ2/2)(T-t), σ(T-t).5] • The net effect is to show that ln ST is normally distributed. This means that a stock’s price at time T, given its price today, is lognormally distributed, with standard deviation of σ(T).5

24. The Lognormal Property of Stock Prices • What is neat (and convenient) is that our best evidence is that the lognormally distribution does a pretty decent job modeling real stock prices. • Indeed, Hull has a nice quote: “our uncertainty about the logarithm of the stock price, as measured by its standard deviation, is proportional to the square root of how far ahead we are looking.”

25. The Distribution of the Rate of Return • If the previous section tells us about the properties of the stock price, we are also concerned with the statistical properties of the rate of return on the stock. • That is, if previously we were concerned with the way the price of the stock changed over time, here we are concerned with the way the rate of return of the stock changes over time. • Define η as the annual continuously compounded rate of return realized between t and T. • By definition ST = Seη(T-t) And η= 1/(T-t) ln (ST/S) • and since ln(ST/S) ~ φ((μ- σ2/2)(T-t),σ(T-t).5) • this implies η~φ((μ- σ2/2),σ/(T-t).5).

26. The Distribution of the Rate of Return • Indeed, the last slide shows that our model assumes that stock returns are normally distributed. • It might be nice to have some validation that this is indeed what we see in the real world. • Short answer: For most options pricing, it appears that stocks returns are “normal-enough”, although some researchers argue that real returns violate normality in two related ways: • The tales of the distributions are too fat to be a true normal, they are closer to a logistic distribution. • There seems to be a higher probability of large downturns in price than upturns.

27. The Distribution of the Rate of Return • Let us examine the returns to two stocks to at least visually examine how they are distributed. • General Electric (GE) and the parent company of American Airlines (AMR). • Data collected from March 1994 through October 2002. • First, let’s look at prices over that time:

28. The Distribution of the Rate of Return

29. The Distribution of the Rate of Return • Now, let’s look at a histogram of returns for each of these firms.

30. The Distribution of the Rate of Return

31. Estimation of Volatility from Historical Data • Sometimes we want to estimate volatility from historical data. When we do this, we typically examine a series of closing prices. Let’s adopt the following notation: • n+1: the number of observations of closing prices. • Si: the stock price at the end of the ith interval. • τ: length of time interval in years. • Define: μi = ln (Si/Si-1)for i=1,2,...,n. • By definition we have constructed a series of continuously compounded return values. The usual estimator that we get in stats class look like

32. Estimation of Volatility from Historical Data • Or • Of course we know from equation 11.1 that the standard deviation of μi is σ(τ.5), and so we must adjust our estimate of the standard deviation, we do this by dividing s by (τ.5), s* = s/(τ.5)

33. Assumptions Behind the Black-Scholes Differential Equation • Although the mathematics appear a bit more complex and a bit more intimidating, the basic idea behind the Black Scholes analysis is the same as in the binomial. • That is, if you can set up a portfolio that consists of the option and the stock, and that portfolio is perfectly risk-free, you can back-out the price of the option. • To get to this point we make a number of assumptions. Most of these assumption are not very binding. • The stock price follows the process developed in Chapter 10 with μ and σ constant. • The short selling of securities is allowed. • There are not transactions costs or taxes, all securities are perfectly divisible. • There are no dividends during the life of the option. • There are not arbitrage opportunities. • Securities trade continuously. • The risk-free rate is constant.

34. Derivation of the Black-Scholes Differential Equation • Recall that we assume a stock price S which follows the process: dS = μS dt + σS dz • working with a discrete time version of this: ΔS = μS Δt + σS Δz. • Let f represent the value of any derivative written on S. From Ito’s lemma, it must be the case that:

35. Derivation of the Black-Scholes Differential Equation • Now clearly the only “risk” from this derivative security comes from the Δz term. • If I construct a portfolio consisting of -1 shares of the derivative and (∂f/∂s) shares of the stock, will have an instantaneously riskless portfolio. To see this note that the value of the portfolio (Π) is: • and so the change in this portfolio is: • which by substitution gives us:

36. Derivation of the Black-Scholes Differential Equation • Notice that we have managed to remove all of the Δz’s from the equation. As a result we know that it must grow at the risk-free rate. It follows, therefore, that ΔΠ = r Π Δt • where r is the risk-free rate. Substituting back into previous equations gives: • which leads us to the general equation:

37. Derivation of the Black-Scholes Differential Equation • Notice that f is ANY derivative based on S, not just a call or a put. In fact, to price any particular derivative, you must first impose upon this equation appropriate boundary conditions. • These literally specify the boundary values of S and t, and how the derivative behaves at those boundaries. • Fortunately, these boundary conditions are basically our payoff conditions: • call option: f = max(S-X,0) when t=T • put option: f = max(X-S,0) when t=T

38. Risk-Neutral Valuation • Notice in the PDE that the drift rate of the stock does not appear. It has fallen out in all of the algebra. • Thus, how investors feel about μ are irrelevant when it comes to pricing the option f. We can choose to work in a world with any set of investor attitudes about μ that we want to, and values obtained in those worlds are valid in any other world. • This allows us to make a hugely simplifying assumption. Just assume that investors are risk-neutral. In such a world, all securities would grow at the risk-free rate because investors do not demand a premium to induce them to take risk.

39. Risk-Neutral Valuation • This provides us with a rather simple way of valuing a derivative in this world: simply assume that the stock will grow at the risk free rate r until time T. • At that point see what would be the payoffs to the derivatives, and then discount those payoffs at the risk-free rate back to today. This is the value of the derivative in both the risk-neutral and risk-averse world.

40. The Black-Scholes Pricing Formulas • Let’s start with the call option. As you know, the value of a call option at maturity in a risk-neutral world is: E’[max(ST-X,0)] • where E’ represents the risk-neutral world. This means that the value of this option at time (T-t) is simply that payoff discounted back to time (T-t) at the risk free rate (just like any other discounted cash flow): c = e-r(T-t)E’[max(ST-X,0)] • So now, all we have to deal with is determining this expected value.

41. The Black-Scholes Pricing Formulas • As we showed earlier, the rate of return on S in a risk-neutral world is r, not μ, therefore the distribution of ST is given by: • we will avoid what Hull describes as “tedious” algebra and cut to the equation:  c = SN(d1) - Xe-r(T-t)N(d2) • where

42. The Black-Scholes Pricing Formulas • And • N(x) is the cumulative probability distribution function for a variable that is normally distributed with mean zero and a standard deviation of 1. • Note that we can re-write this equation as: c = e-r(T-t)[Ser(T-t) N(d1) - XN(d2)]. • Expressed this way you can see what this is really getting at, let’s break it down:

43. The Black-Scholes Pricing Formulas • First, notice that Ser(T-t) is the expected value of ST in a risk-neutral world. Thus you can think of the term Ser(T-t)N(d1) as the expected payoff to a security which is worth ST is ST>X and zero otherwise. • So this is the first half of your option: how much the asset on which you hold the option is worth at the end, and XN(d2) is the expected amount you have to pay for it times the probability of your making that payout. • So essentially this has two parts: the value of the asset after the payment, and the cost of exercising the option. the e-r(T-t) is simply discounting it back to today’s price. • Note I will provide you with tables for N(x) on the exam. • You can get a similar result for puts, but it is somewhat easier to use put-call parity: p = Xe-r(T-t)N(-d2) - SN(-d1).

44. The Black-Scholes Pricing Formulas • Let’s work an example. Let’s say that you hold a European option with these parameters: S = $42, K = $40, r = .10, σ=.2, and T-t=.5. What is the value of that option today? • First, let’s calculate d1 and d2:

45. The Black-Scholes Pricing Formulas • Obviously the next step is to calculate N(d1) and N(d2), and there are several ways you could do this: • In Excel, use the function normsdist(), • Use a polynomial approximation, or • Use a table of normal values. Let’s look at each of these: • In Excel simply enter: =normsdist(0.7693) = 0.7791=normsdist(0.6278) = 0.7349 • Hull presents a polynomial approximation on page 248 of his book. This approximation is as follows:

46. The Black-Scholes Pricing Formulas • Hull’s approximation: • For d1= 0.7693, the approximate value for N(0.7693) works out to be 0.779142, and for d2=0.6278 the approximate value for N(0.6278) works out to be 0.734933.

47. The Black-Scholes Pricing Formulas • If you look in the back of Hull’s book you can see an example of a standard normal table similar to the one I have created here: • To use it find the row/column combination that most closely equals your value of (x), and then interpolate between that value and the next closest one.

48. The Black-Scholes Pricing Formulas • All three method give you approximately the same answer for N(d1) and N(d2) • The next step is to calculate Ke-r(T-t). • Ke-r(T-t) = 40 e-.1(.5) = 38.049. • If this is a call option that you hold, its value is given by: • And if it is a put, its value is given by:

49. The Black-Scholes Pricing Formulas • Notice that if you wanted to solve for the put value, you could have used put-call parity as well: p = c + Ke-r(T-t) – S0 or p= 4.76 + 40e-.10(.5)- 42 p = 4.76 + 38.049 – 42 = 0.81

50. The Black-Scholes Pricing Formulas • McDonald presents the basic Black-Scholes equations in a slightly different format: • In this case the parameter δ is a continuous dividend yield. This is just a minor modification to Hull’s Black-Scholes model. Note that this is in keeping with the idea of setting the “total” return to the risk-free rate as we have done in previous settings.