Geometric Brownian Motion

Geometric Brownian Motion Note that as a model for the rate of return, ds(t)/s(t) geometric Brownian motion is similar to other common statistical models: ds(t) S(t) = µdt + σdw(t) or response = systematic component + random error. Without the stochastic component, the differential equation has the simple solution S(t) = ce µt, from which we get the formula for continuous compounding for a rate µ. 1

An Intuitive Examination of Geometric Brownian Motion in Prices What rate of growth do we expect for S in the geometric Brownian motion model ds(t) = µdt + σdw(t)? S(t) Should it be µ because that is the rate for the systematic component, and the expected value of the random component is 0? Consider a rate of change σ, that is equally likely to be positive or negative. What is the effect on a given quantity if there is an uptick of σ followed by a downtick of equal magnitude (or a downtick followed by an uptick)? The result for the two periods is σ 2. (This comes from the multiplication of the given quantity by (1+σ)(1 σ).) The average over the two periods is σ 2 /2. The stochastic component reduces the expected rate of µ by σ 2 /2. This is the price of risk. 2

Ito s Lemma We can formalize the preceding discussion using Ito s formula. Ito s lemma: Suppose X follows an Ito process, dx(t) = a(x, t)dt + b(x,t)dw(t), where dw is a Wiener process. Let G be an infinitely differentiable function of X and t. Then G follows the process ( G G dg(t) = a(x, t) + X t + 1 2 ) G 2 X 2b2 dt+ G b(x, t)dw(t). (1) X 3

Ito s Lemma Thus, Ito s lemma provides a formula that tells us that G also follows an Ito process. The drift rate is G G a(x, t) + X t + 1 2 G 2 and the volatility is G b(x, t). X X 2b2 This allows us to work out expected values and standard deviations of G over time. 4

Derivation of Ito s Formula First, suppose that G is an infinitely differentiable function of X and an unrelated variable y, and consider a Taylor series expansion for G: G = G G X + X y y + 1 ( ) 2 G + 2 G 2 X 2( X)2 y 2 ( y)2 + 2 2 G X y X y + (2) In the limit as X and y tend to zero, this is the usual total derivative dg = G X G dx + dy, (3) y in which the terms in X and y have dominated and effectively those in ( X) 2 and ( y) 2 and higher powers have disappeared. Now consider an X that follows an Ito process, or dx(t) = a(x, t)dt + b(x, t)dw(t), X(t) = a(x, t) t + b(x, t)z t. Now let G be a function of both X and t, and consider the analogue to equation (2). The factor ( X) 2, which could be ignored in moving to equation (3), now contains a term with the factor t, which cannot be ignored. We have ( X(t)) 2 = b(x, t) 2 Z 2 t + terms of higher degree in t. 5

Consider the Taylor series expansion G = G G X + X t t + 1 2 ( 2 G X 2( X)2 + 2 G t 2 ( t)2 + 2 2 G X t X t ) + (4) Now under the assumptions of Brownian motion, ( X(t)) 2 or, equivalently, Z 2 t is nonstochastic; that is, we can treat Z 2 t as equal to its expected value as t tends to zero. Therefore, when we substitute for X(t), and take limits in equation (4) as X and t tend to zero, we get dg(t) = G G dx + X t dt + 1 2 G 2 X 2b2 dt (5) or, after substituting for dx and rearranging, we have Ito s formula ( G G dg(t) = a(x, t) + X t + 1 2 ) G 2 X 2b2 dt + G b(x, t)dw(t). X Equation (5) is also called Ito s formula. Compare equation (5) with equation (3). 6

Applications of Ito s Formula Ito s formula has applications in many stochastic differential equations used as models in finance. In the differential equation for geometric Brownian motion for S, ds(t) = µs(t)dt + σs(t)dw(t), we can let G = log S, and so substituting in Ito s formula we have ) dg(t) = (µ σ2 dt + σdw(t). 2 Using previous results we have that the difference in G at time zero and time T is normally distributed with mean ) (µ σ2 T 2 (recognize this term from our intuitive discussion) and variance σ 2 T. 7

The Distribution of Stock Prices The geometric Brownian motion model is the simplest model for stock prices that is somewhat realistic. Looking at it somewhat critically, we can see certain problems. First, is the form of the differential equation reasonable? Next, we have the big questions: are µ and σ constant? We will explore these issues later. The basic distributional assumption in the geometric Brownian motion model is that the rates of change of stock prices in very small increments of time are identically and independently normally distributed. 8

The Distribution of Stock Prices The compounded rate of return is 1 log(s(t + t)/s(t)). t From t 0 to T we can write this as 1 log(s T ) 1 log(s t0 ). T t 0 T t 0 If this has a normal distribution, then log(s T ) has a normal distribution; that is, S T has a lognormal distribution. *** properties 9

Solution of Stochastic Differential Equations The solution of a differential equation is obtained by integrating both sides and allowing for constant terms. Constant terms are evaluated by satisfying known boundary conditions, or initial values. In a stochastic differential equation (SDE), we must be careful in how the integration is performed, although different interpretations may be equally appropriate. For example, the SDE that defines an Ito process dx(t) = a(x, t)dt + b(x,t)dw(t), when integrated from time t 0 to T yields X(T) X(t 0 ) = T t 0 a(x, t)dt + T t 0 b(x, t)dw(t). The second integral is a stochastic integral. We will interpret it as an Ito integral. 10

The nature of a(x, t) and b(x, t) determine the complexity of the solution to the SDE. In the Ito process ds(t) = µ(t)s(t)dt + σ(t)s(t)dw(t), using Ito s formula for the log as before, we get the solution ( T S(T) = S(t 0 )exp (µ(t) 1 ) ) T 2 σ(t)2 dt + σ(t)dw(t). t 0 t 0 In the simpler version of a geometric Brownian motion model, in which µ and σ are constants, we have S(T) = S(t 0 )exp ((µ 1 ) ) 2 σ2 t + σ W. 11

Expected Values of Solutions of Stochastic Differential Equations Given a solution of a differential equation we may determine the mean, variance and so on by taking expectations of the random component in the solution. Sometimes, however, it is easier just to develop an ordinary (nonstochastic) differential equation for the moments. We do this from an Ito process dx(t) = a(x, t)dt + b(x,t)dw(t), by using Ito s formula on the powers of the variable. So we have dx p (t) = (px(t) p 1 a(x, t) + 12 ) p(p 1)X(t)p 2 b(x, t) 2 dt + ** exercise px(t) p 1 b(x,t)dw(t). Taking expectations of both sides, we have an ordinary differential equation in the expected values. 12

Geometric Brownian Motion in Prices Although the geometric Brownian motion model for rates of returns is quite useful, ds(t) S(t) = µdt + σdw(t), it has limitations. As we have mentioned, one problem is the assumption of constancy of µ and σ. problem of stochastic volatility There are other considerations also. shocks, market and stock 13

Derivatives: Basics A derivative is a financial instrument whose value depends on values of other financial instruments or on some measure of the state of the economy or of nature. The instrument or measure on whose value the derivative depends is called the underlying. A derivative is an agreement with two sides. One of the most important questions in finance is how to price a derivative. 14

Forward Contracts One of the simplest kinds of derivatives is a forward contract, which is an agreement to buy or to sell an asset at a specified time at a specified price. The agreement to buy is a long position and the agreement to sell is a short position. The agreed upon price is the delivery price. The consumation of the agreement is an execution. Forward contracts are relatively simple to price, and their analysis is important for developing pricing methods for other derivatives. 15

Analysis of Forward Contracts Using standard notation, let K be the delivery price or strike price, at time T, and let X t be the value of the underlying. The result at settlement is shown below. payoff 0 K X T long position payoff 0 K X T short position The important question is what is the price of the derivative as a function of X T and of time to settlement. For a forward contract it is easy. The answer is F 0 = X 0 e rt K, where r is the (annual) riskfree rate of growth, and T is the time (in years). 16

Derivatives: Basics There is a variety of modifications to the basic forward contract that involve nature of the underlying asset investment consumable income producing nonasset (e.g., index, price of electricity, weather) negociability of the instrument (market, possibility of short positions, intermediate party, etc.) 17

Derivatives: Basics Other modifications to the basic forward contract involve nature of the agreement (right, that is, contingent claim, or obligation) flexibility of time of execution (at a specified time or up to a specified time) method of settlement (cash or delivery) dependence of the derivative value on the path of the value of the underlying 18

Types of Derivatives These variations on the basic forward contract are all interesting. Only a few of them are actually available. Some variations are much easier to analyze than others. The simple ones are interesting for classroom analyses and they may provide useful approximations for derivatives that are actually available. For individual investors, there is a relatively small set of derivatives available (realistically). For all of them there is a ready market (always) through a third party, and short sales are possible. Most of the readily traded derivatives are options, or contingent claims. That kind of derivative is a right; not an obligation. Therefore, a long position is a right and a short position (in the derivative) is an obligation. The right expires at the settlement date. 19

Derivatives That Have Markets The common types of derivatives are Stock options Index options Commodity futures Rate futures 20

Uses Stock options are used by individual investors and by investment companies for leverage, hedging, and income. Index options are used by individual investors and by investment companies for hedging and speculative income. Commodity futures are used by individual investors for speculative income, by investment companies for income, and by producers and traders for hedging. Rate futures are used by individual investors for speculative income, by investment companies for income and for hedgins, and by traders for hedging. 21

Types of Common Derivatives The variations depend on the nature of the underlying. Stock (investment asset, possibly income-producing). The buy side is a call and the sell side is a put. Settlement (exercise) is by delivery. Exercise can be any time prior to expiration date ( American style ). Index (investment asset, not income-producing). The buy side is a call and the sell side is a put. Settlement is by cash. Exercise can only be at expiration date. Commodity (consumable asset, not income-producing). Settlement is by delivery. Exercise can only be at expiration date. 22

U.S. Stock Options A market for stock options in the U.S. is one of the national security exchanges: Amex, CBOE, NYSE, Pacific Exchange, and Philadelphia Exchange. All (almost all) options are initiated with and through the Options Clearing Corporation, owned by the exchanges and headquartered on LaSalle St. Options (and also futures) are regulated by the Commodity Futures Trading Commission (the analogue of the SEC). The SEC also regulates options through its regulatory oversight of the exchanges. 23

Analysis of Stock Options Another important difference between stock options and forward contracts is that stock options are rights, not obligations. The payoff therefore cannot be negative. Because the payoff cannot be negative, there must be a cost to obtain a stock option. The profit is the difference between the payoff and the price paid. Another difference in stock options and forward contracts is that (real-world) stock options can be exercised at any time (during trading hours) prior to expiration. We will, however, often consider a modification, the European option, which can only be exercised at a specified time. (There are some European options that are actually traded, but they are generally for large amounts, and they are rarely traded by individuals.) 24

Analysis of Stock Options The two sides of a forward contract result in the two types of stock options. The results at expiration are profit 0 K X profit K 0 X call option put option For short positions, just flip the graphs. 25

Market Models for Derivative Pricing A simple model of the market assumes two assets: a riskless asset with price at time t of β t, and a risky asset with price at time t of X t. The price of a derivative can be determined based on trading strategies involving these two assets. The price of the riskless asset follows the deterministic ordinary differential equation dβ t = rβ t dt, where r is the instantaneous riskfree interest rate. The price of the risky asset follows the stochastic differential equation dx t = µx t dt + σx t dw t. 26

Pricing Derivatives: Basics Start with a European call option. How do we value it? Current time: t 0 Expiration: T Strike price: K Price of underlying: S(t 0 ),..., S(t),..., S(T) Value of the call: C(t 0 ),..., C(t),...,C(T) P for put; V for either We have been (and in this course will continue to be) vague about the pricing unit. In general, we call the price, or the pricing unit, a numeraire. A more careful development of this concept rests on the idea of a pricing kernel. We will usually call them dollars. 27

The Price of a European Call Option A European call option is a contract that gives the owner the right to buy a specified amount of an underlying for a fixed strike price, K on the expiration or maturity date T. The owner of the option does not have any obligations in the contract. The payoff, h, of the option at time T is either 0 or the excess of the price of the underlying S(T) over the strike price K. Once the parameters K and T are set, it is a function of S(T): h(s(t)) = { S(T) K if S(T) > K 0 otherwise 28

The Price of Call Options The price of the option at any time is a function of the time t, and the price of the underlying s. We denote it as P(t, s). What is the price at time t = 0? It seems natural that the price of the European call option should be the expected value of the payoff of the option at expiration, discounted back to t = 0: P(0,s) = e rt E(h(S(T))). Likewise, for an American option, we could maximize the expected value over all stopping times, 0 < τ < T: P(0, s) = sup τ T e rτ E(h(S(τ))). 29

Principles Basic principle of Black-Scholes: We seek a portfolio with zero expected value, that consists of short and/or long positions in the option, the underlying, and a risk-free bond. There are two key ideas in developing pricing formulas for derivatives: 1. no-arbitrage principle 2. replicating, or hedging, portfolio 30

An arbitrage is a trading strategy with a guaranteed rate of return that exceeds the riskless rate of return. In financial analysis, we assume that arbitrages do not exist. 31

Example of the No-Arbitrage Principle Consider a forward contract that obligates one to pay K at T for the underlying. At time t, with t < T, the price of the underlying is S(t). What should the price of the contract be or, equivalently, What should K be so that the price of the contract is 0? Its value at expiry is S(T) K, and of course we do not know S(T). If we have a riskless rate of return r, we can use the no-arbitrage principle to determine the correct price of the contract. To apply the no-arbitrage principle, consider the following strategy: take a long position in the forward contract and sell the underlying short. With this strategy, the investor immediately receives S(t). At time T this amount can be guaranteed to be S(t)e r(t t). 32

The No-Arbitrage Principle If K < S(t)e r(t t), a long position in the forward contract and a short position in the underlying is an arbitrage. Conversely, if K > S(t)e r(t t), a short position in the forward contract and a long position in the underlying is an arbitrage. Therefore, under the no-arbitrage assumption, the correct value of K is S(t)e r(t t). 33

The replication approach is to determine a portfolio and an associated trading strategy that will provide a payout that is identical to that of the underlying. This portfolio and trading strategy replicates the derivative. A replicating strategy involves both long and short positions. 34

Pricing Derivatives If every derivative can be replicated by positions in the underlying (and cash), the economy or market is said to be complete. We will generally assume complete markets. The Black-Scholes approach leads to the idea of a self-financing replicating hedging strategy. The approach yields the interesting fact that the price of the call does not depend on the expected value of the underlying. It does depend on its volatility, however. 35

Self-Financing Replicating Hedging Strategy Neil Chriss s example of a casino. Game is to flip a fair coin 3 times. Casino pays $1 if heads occurs 3 times in a row, HHH. How much should it cost to play? (Casino will then add operating and profit margin.) Relation to options; who s short and who s long Big player: $100,000. Bet broker: casino bets $12,500 on H on first toss; casino is even If H occurs, casino has $25,000 and bets on H on second toss 36

Analysis of Casino Hedging Example Hedging in general Special properties: Self-financing Replicating. Roles of three participants; who s short and who s long bid-ask spread (will broker charge the expected value of the bet?) other transaction costs... Assumptions: market impact, complete market 37

Expected Rate of Return on Stock XYZ selling at S(t 0 ); no dividends. What is its expected value at time T > t 0? It is merely the forward price for what it could be bought now. Forward price: e r(t t 0) S(t 0 ), where r is the risk-free rate of return, S(t 0 ) is the spot price, and T t 0 is the time interval. This is the no-arbitrage principle. The expected value of the stock does not depend on the rate of return of the stock (that s µ in some of the models we ve used). This is true because of the cost of a forward contract. 38

Call Option Holder of the forward contract (long position) on XYZ must buy stock at time T for e r(t t 0) S(t 0 ). Holder of a call option buys stock only if S(T) > K. Role of volatility on forward contract holder (volatility not good) on call option holder (volatility good) enhances the value of the option Assumptions... Conclusion under assumptions: expected volatility of the underlying affects the value of an option, but expected rate of return of the underlying does not. 39

Hedging Hedging risk: defray the risk of one investment by making an offsetting investment. Cost of hedge. Return of hedge (reduced risk). Perfect hedge: returns exactly the amount needed to cover any loss, and no more. Example: write call (i.e., go short). what is a hedge? owning enough stock to cover is a hedge, but not perfect (it costs too much, and writer is not protected against drop in price of underlying). 40

Hedging Black-Scholes approach constructs a portfolio consisting of some underlying and some risk-free (zero-coupon) bonds to offset the short call. (discuss dividends, coupons, etc.) 41

Hedging In a perfect hedge, the hedging instrument behaves exactly the way the hedged instrument does. Call option vs. hedging instrument: value at T. they re equal... called payoff replication. 42

Dynamic Hedging process of managing the risk of options hedging portfolio s value at any time is equal to the value of the option at that point in time. 1. replicates the payoff 2. has fixed and known total cost weighted portfolio, balancing hedging strategy produces a synthetic version of the option. 43

Self-Financing Dynamic Hedging Cost of hedge: 1. infusion of funds cost 2. transaction costs (bid-ask, friction, inability to execute trades at exactly the price specified by the strategy, etc.) A hedging strategy is self-financing if its total to-date cost at any time (excluding transaction costs) is equal to the setup cost. Setup cost: initial outlay; e.q. strategy requires going long $1,000 and short $700; setup is $300 To do this, we need a formula for the relative rates of change of the price of the call and that of the underlying, = dc ds. 44

Delta of an Option The delta of an option is the rate of change of the option s value with respect to the change in the underlying s price. Consider times t 0 and t 1. If the value of an option at time t is V (t), and the price of the underlying is S(t), the delta at t 0 is approximated by t0 = V (t 0) e r(t 1 t 0 ) V (t 1 ) S(t 0 ) e r(t 1 t 0 ) S(t 1 ). We essentially neutralize the change in time by the risk-free rate. comments: zero in denominator; time value 45

Market Models for Derivative Pricing A simple model of the market assumes two assets: a riskless asset with price at time t of β t, and a risky asset with price at time t of S(t). The price of a derivative can be determined based on trading strategies involving these two assets. The price of the riskless asset follows the deterministic ordinary differential equation dβ t = rβ t dt, where r is the instantaneous riskfree interest rate. The price of the risky asset follows the stochastic differential equation ds(t) = µs(t)dt + σs(t)dw t. 46

Preliminary Formula C(t) = t S(t) e r(t t) B(t), where B(t) is the current value of a riskless bond. 47

We speak of a portfolio as a vector p whose elements sum to 1. The length of the vector is the number of assets in our universe. Sometimes we limit this to assets whose values are independent of each other; that is, we may exclude derivatives. The no-arbitrage principle can be stated as: There does not exist a p such that for some t > 0, either p T s 0 < 0 and p T S(t)(ω) 0 for all ω, or p T s 0 0 and p T S(t)(ω) 0 for all ω, and p T S(t)(ω) > 0 for some ω. 48

A derivative D is said to be attainable (over a universe of assets S = (S (1), S (2),..., S (k) )) if there exists a portfolio p such that for all ω and t, D t (ω) = p T S(t)(ω). Not all derivatives are attainable. The replicating portfolio approach to pricing derivatives applies only to those that are attainable. 49

Dynamic and Self-Financing Portfolios The value of a derivative changes in time and as a function of the value of the underlying; therefore, a replicating portfolio must be changing in time or dynamic. In analyses with replicating portfolios, transaction costs are ignored. Also, the replicating portfolio must be self-financing; that is, once the portfolio is initiated, no further capital is required. Every purchase is financed by a sale. 50

A Replicating Strategy Using our simple market model, with a riskless asset with price at time t of β t, and a risky asset with price at time t of S(t), (with the usual assumptions on the prices of these assets), we can construct a portfolio whose value will almost surely the payoff of a European call option on the risky asset at time T. At time t, the portfolio consists of a t units of the risky asset, and of b t units of the riskless asset. Therefore, the value of the portfolio is a t S(t)+b t β t. If we scale β t so that β 0 = 1 and adjust b t accordingly, the expression simplifies, so that β t = e rt. The portfolio replicates the value of the option at time T if it has value K S(T) if this is positive and zero otherwise. If the portfolio is self-financing d(a t S(t) + b t e rt ) = a t ds(t) + rb t e rt dt. 51

The Black-Scholes Differential Equation Consider the fair value V of a European call option at time t < T. At any time this is a function of both t and the price of the underlying S t. We would like to construct a dynamic, self-financing portfolio (a t, b t ) that will replicate the derivative at maturity. If we can, then the no-arbitrage principle requires that for t < T. a t S t + b t e rt = V (t, S t ), Further, if V (t, S t ) is continuously twice-differentiable, we can use Itô s formula to develop an expression for V (t, S t ). We assume no-arbitrage and we assume that a risk-free return is available. These are big if s, but nevertheless, let s proceed. 52

First, differentiate both sides of the equation that represents a replicating portfolio with no arbitrage: ( a t ds t + rb t e rt V dt = µs t + V S t t + 1 2 ) V 2 St 2 σ 2 St 2 dt + V (σs t )db t S t By the market model for ds t the left-hand side is (a t µs t + rb t e rt )dt + a t σs t db t. Equating the coefficients of db t, we have a t = V S t. From our equation for the replicating portfolio we have b t = (V (t, S t ) a t S t ) e rt. 53

Now, equating coefficients of dt and substituting for a t and b t, we have the Black-Scholes differential equation, r ( V S t V S t ) = V t + 1 2 σ2 S 2 t 2 V S 2 t Notice that µ is not in the equation. 54

Delta Hedging The Black-Scholes differential equation can also be derived by constructing a riskless, self-financing portfolio consisting of a long position in the underlying and a short position in the call. Letting A t be the amount long in the underlying, and N t be the amount short in the derivative, we arrive at the equation A t N t = V S t. The ratio A t N t is called the Delta, and this approach to riskless portfolio construction is called delta hedging. 55

The Black-Scholes Differential Equation Instead of European calls, we can consider European puts, and proceed in the same ways (replicating portfolio or Delta hedging). We arrive at the same the Black-Scholes differential equation (rewritten), V t + rs V t + 1 S t 2 σ2 St 2 2 V S 2 t = rv. 56

The Black-Scholes Formula The solution depends on the boundary conditions. In the case of European options, these are simple. For calls, they are V c (T, S t ) = (S t K) + For puts, they are V p (T, S t ) = (K S t ) + 57

The Black-Scholes Formula With these boundary conditions, there are closed form solutions to the Black-Scholes differential equation. For the call, it is C BS (t, S t ) = S t Φ(d 1 ) Ke r(t t) Φ(d 2 ), where d 1 = log(s t/k) + (r + 1 2 σ2 )(T t) σ, T t d 2 = d 1 σ T t, and Φ(s) = 1 2π s e y2 /2 dy. 58

The Black-Scholes Pricing Formula It is interesting to look at the Black-Scholes prices as a function of the price of the underlying. The Black Scholes Call Pricing Function Call Option Price 0 10 20 30 40 60 80 100 120 140 Current Stock Price 59

The Black Scholes Put Pricing Function Put Option Price 0 10 20 30 40 60 80 100 120 140 Current Stock Price 60

Assumptions The Black-Scholes model depends on several assumptions: differentiability of stock prices with respect to time a dynamic replicating portfolio can be maintained without transaction costs returns are independent normal mean stationary variance stationary 61

Stochastic Volatility A condition in which the variance of the returns is random (rather than stationary) is called stochastic volatility. Data on returns provide empirical evidence that the volatility is stochastic. 62

Stochastic Volatility and Implied Volatility Stochastic volatility can also account for other empirically observed failures of the Black-Scholes model. For a given option (underlying, type, strike price, and expiry) and given the price of the underlying and the riskfree rate, the Black-Scholes formula relates the option price to the volatility of the underlying; that is, the volatility determines the model option price. The actual price at which the option trades can be observed, however. If this price is used in the Black-Scholes formula, the volatility can be computed. This is called the implied volatility. 63

Implied Volatility Recall the Black-Scholes formula: C BS (t, S t ) = S t Φ(d 1 ) Ke r(t t) Φ(d 2 ), where d 1 = log(s t/k) + (r + 1 2 σ2 )(T t) σ, T t d 2 = d 1 σ T t, as before. Let c be the observed price of the call. Now, set C BS (t, S t ) = c, and We have f(σ) = S t Φ(d 1 ) Ke r(t t) Φ(d 2 ), c = f(σ). 64

Computing the Implied Volatility Given a value for c, that is, the observed price of the option, we can solve for σ. There is no closed-form solution, so we solve iteratively. Beginning with σ (0), we can use the Newton updates, σ (k+1) = σ (k) (f(σ (k) ) c)/f (σ (k) ). 65

Computing the Implied Volatility We have where f (σ) = S t dφ(d 1 ) dσ Ke r(t t)dφ(d 2) dσ = S t φ(d 1 ) dd 1 dσ Ke r(t t) φ(d 2 ) dd 2 dσ, φ(y) = 1 2π e y2 /2, and dd 1 dσ = σ2 (T t) log(s t /K) (r + 1 2 σ2 )(T t) σ 2, T t dd 2 dσ = dd 1 dσ T t. 66

Discrepancies in the Observed Implied Volatilities The implied volatility from the Black-Scholes model should be the same at all points. It is not. The implied volatility, for given T and S t, depends on the strike price, K. In general, the implied volatility is greater than the empirical volatility, but the implied volatility is even greater for far out-ofthe money calls. It also increases for deep in-the-money calls. This is called the smile curve, or the volatility smile. 67

The Smile Curve The Black Scholes Implied Volatility Implied Volatility 0.20 0.21 0.22 0.23 0.24 0.25 96 98 100 102 104 Strike Price The available strike prices are not continuous, of course. This curve is a smoothed (and idealized) fit of the observed points. 68

The Smile Curve The smile curve is not well-understood, although we have a lot of empirical observations on it. Interestingly, prior to the 1987 crash, the minimum of the smile curve was at or near the market price S t. Since then it is generally at a point larger than the market price. 69

Computations of Black-Scholes Implied Volatility The Black-Scholes price depends on being able to evaluate the standard normal CDF, and the implied volatility depends on being able to evaluate the standard normal PDF. The PDF is easy to compute from the definition: f(x) = e x2 /2 / 2π. The CDF is more difficult. In R, the CDF is computed by pnorm and the PDF by dnorm. In Statistics Toolbox of Matlab, the CDF is computed by normcdf and the PDF by normpdf. Without the Statistics Toolbox, in Matlab, the CDF can be computed at the point x by (1 + (erf(x/sqrt(2))))./2. 70

Variation in Volatility over Time The volatility also varies in time. There are periods of high volatility and other periods of low volatility. This is called volatility clustering. The volatility of an index is somewhat similar to that of an individual stock. The volatility of an index is a reflection of market sentiment. (There are various ways of interpreting this!) In general, a declining market is viewed as more risky than a rising market, and hence, it is generally true that the volatility in a declining market is higher. Contrarians believe high volatility is bullish because it lags market trends. 71

Measuring the Volatility of the Market A standard measure of the overall volatility of the market is the CBOE Volatility Index, VIX, which CBOE introduced in 1993 as a weighted average of the Black-Scholes-implied volatilities of the S&P 100 Index from at-the-money near-term call and put options. ( At-the-money is defined as the strike price with the smallest difference between the call price and the put price.) In 2004, futures on the VIX began trading on the CBOE Futures Exchange (CFE), and in 2006, CBOE listed European-style calls and puts on the VIX. Another measure is the CBOE Nasdaq Volatility Index, VXN, which CBOE computes from the Nasdaq-100 Index, NDX, similarly to the VIX. (Note that the more widely-watched Nasdaq Index is the Composite, IXIC.) 72

The VIX In 2006, CBOE changed the way the VIX is computed. It is now based on the volatilities of the S&P 500 Index implied by several call and put options, not just those at the money, and it uses near-term and next-term options (where near-term is the earliest expiry mover than 8 days away). It is no longer computed from the Black-Scholes formula. It uses the prices of calls with strikes above the current price of the underlying, starting with the first out-of-the money call and sequentially including all with higher strikes until two consecutive such calls have no bids. It uses the prices of puts with strikes below the current price of the underlying in a similar manner. The price of an option is the mid-quote price, i.e. the average of the bid and ask prices. 73

Technical Details: Computing the VIX Let K 1 = K 2 < K 3 < < K n 1 < K n = K n+1 be the strike prices of the options that are to be used. The VIX is defined as 100 σ, where σ 2 = 2erT T ( n i=2;i j K i Ki 2 Q(K i ) + K j Kj 2 1 T ( ( Q(Kj put) + Q(K j call) ) /2 F K j 1) 2, T is the time to expiry (in our usual notation, we would use T t, but we can let t = 0), F, called the forward index level, is the at-the-money strike plus e rt times the difference in the call and put prices for that strike, K i is the strike price of the i th out-of-the-money strike price (that is, of a put if K i < F and of a call if F < K i ), K i = (K i+1 K i 1 )/2, Q(K i ) is the mid-quote price of the option, r is the risk-free interest rate, and K j is the largest strike price less than F. 74 )

Technical Details: Computing the VIX Time is measured in minutes, and converted to years. Months are considered to have 30 days and years are considered to have 365 days. There are N 1 = 1,440 minutes in a day. There are N 30 = 43,200 minutes in a month. There are N 365 = 525,600 minutes in a year. A value σ1 2 is computed for the near-term options with expiry T 1, and a value σ2 2 is computed for the next-term options with expiry T 2, and then σ is computed as σ = ( T 1 σ 2 1 N T2 N 30 N T2 N T1 + T 2 σ 2 1 N 30 N T1 N T2 N T1 ) N365 N 30. 75