1. What is an Option? An option gives one party the right, but NOT THE OBLIGATION to perform some specific investment action at a future date and for a defined price. For example, the right to purchase 100 shares of IBM on May 12, 2000 for $200 a share would be an option. On the other hand, a contract to purchase 100 shares of IBM on May 12, 2000 for $200 is a forward contract, not an option. Options & Contingent Claims Stephen Chadwick May 12, 1999

2. Option Terminology Underlying Asset:Whatever the option gives you the right to buy/sell. Call:The right to buy an asset at a certain price. Put:The right to sell an asset at a certain price. Vanilla Option:A basic call or put with a fixed strike price and maturity (and no funky special conditions). Strike Price:The price at which the asset is to be traded (this may be variable or otherwise complicated). Also known as “exercise price” Maturity:The time when the option expires. In The Money: The option would make money if exercised now. Out of the Money:The option would lose money if exercised now. Options & Contingent Claims Stephen Chadwick May 12, 1999

3. More Option Terminology Volatility:A measure of the degree to which the underlying asset’s value fluctuates. Usually defined as the standard deviation of asset returns (denoted by σ) Delta:Also known as the “hedge ratio,” this tells you the amount of the underlying asset you must sell (or buy, if delta is negative) in order to hedge the option. The absolute value of delta is also used as a rule-of-thumb of the probability that a “vanilla” option will end up in the money. Mathematically delta is the first derivative of the option price with respect to the asset price. Gamma:The change in delta with respect to the change in the asset’s value [i.e., the 2nd derivative of the option value with respect to asset value.] This is analogous to convexity in bonds. You probably don’t need to know this for the exam! Options & Contingent Claims Stephen Chadwick May 12, 1999

4. Yet More Option Terminology American Option:An option that can be exercised at any time. European Option:An option that can only be exercised at maturity. Options & Contingent Claims Stephen Chadwick May 12, 1999

5. How Do We Value Options? Suppose we have a 1 year American vanilla call option on Acme Tortilla, Co. (symbol ATX) ordinary stock. The current stock price is $100, but we obviously do not know what the price will be in a year. Since the call is struck at $110, the option will be in the money if the price is above that. The figure below shows a possible distribution of ATX share prices in one year: Options & Contingent Claims Stephen Chadwick May 12, 1999 $100 (Spot Price) $110$90$120$80 In the money! Out of the money :(

6. Volatility: From the diagram on the previous page we can see that increasing volatility (i.e. flattening and widening the distribution) will increase the chance that the option ends up in the money. This leads to a first pricing axiom: Vanilla Options Always Increase in Value With Increasing Volatility (σ) Options & Contingent Claims Stephen Chadwick May 12, 1999

7. Time to Maturity: The volatility axiom is important but its only a start. Now suppose the Acme option was extended by a year. Since we can exercise at any time (it’s an American option), this must obviously increase the value of the option since the longer we wait the more likely it is that ATX will end up in the money. This leads to another axiom: Vanilla Options Always Increase in Value With Increasing Time to Maturity Options & Contingent Claims Stephen Chadwick May 12, 1999

8. A Binomial Example: Now lets back up and use some hard numbers. For simplicity, suppose that ATX had only two possible outcomes; it could either rise to $130 dollars or it could fall to $90. We think each outcome is equally likely. Suppose the risk-free yield curve is flat at 5%. It is tempting to simply take the expected payoff and call that the option’s value: E(Option) = (0.5 x $20 + 0.5 x $0) / 1.05 = $9.52 Note, however, that 1.05 is the risk-free discount rate and with a payout of $20 or $0 we are hardly talking about a risk-free investment here! Since we can’t determine an appropriate cost of capital for the option (as opposed to the stock, which we might get from CAPM), it is not possible to use this method. Recommendation: Read B&M pg. 573: “Why discounted cash flow... Options & Contingent Claims Stephen Chadwick May 12, 1999

9. Binomial Option Valuation Example: We’ll now value the option using an “arbitrage portfolio” method. Suppose we put together a portfolio of A shares of the stock, financed by selling B worth of bonds. Our goal is to make this portfolio have the same value as the call option C: C = 100A - B Now, since the stock can only take on one of two values after one year, we must constrain A & B to make sure that the portfolio is worth the same as the possible payouts of the call option ($20 or $0). Remember that B will have appreciated to 1.05 because Rf=5%: 130A - 1.05B = 20 90A - 1.05B = 0 Options & Contingent Claims Stephen Chadwick May 12, 1999

10. Binomial Option Valuation Example: Solving for A and B we get: A = 0.5 Shares B = $42.86 borrowed at 5% And:C = 100A - B = $7.14 Note that A is actually the delta (hedge ratio) of this option. Since it is 0.5, we know that the one option can be hedged by shorting 0.5 shares of ATX stock. Options & Contingent Claims Stephen Chadwick May 12, 1999

11. Black-Scholes The binomial method is an elegant but limited solution to the discount rate problem mentioned earlier. The famous Black-Scholes (Merton) formula extends a similar method to price continuously fluctuating assets. Skipping to the chase (take 15.450: Analytics of FE if you want to see do the derivation), the formula is: R f = continuously compounded risk free rate expressed in terms of period t (e.g. years) P = stock price K = strike price t = number of periods to exercise date  = volatility (standard deviation of continuously compounded rate of return on P) d1 = log [e t*Rf * P / K] /  t + (  t) / 2 d2 = d1 –  t Call price = N(d1) * P + N(d2) * K / e t*Rf [N(d) is the cumulative normal probability density function.] Options & Contingent Claims Stephen Chadwick May 12, 1999

12. Option Pricing: Important Points If you look at the Black-Scholes formula you will notice that the expected return (μ) on the stock does not appear. The only parameters are spot, strike, Rf, T, and σ. This is analogous to the way that the up/down probabilities did not affect the price in the binomial example. Note that the call price varies (approximately) as the square root of T, and is (approximately) directly proportional to σ. This is in line with the two pricing axioms mentioned earlier. The effect of Rf is harder to interpret, but in general the higher Rf is, the higher the option price will be. This is similar to forwards/futures. Higher strike prices lower the value of calls, while higher spot prices raise them. The reverse is true for puts. Options & Contingent Claims Stephen Chadwick May 12, 1999

13. Option Position Math: Options & Contingent Claims Stephen Chadwick May 12, 1999 + =

14. Put - Call Parity: A slightly different way to express the previous slide is called put-call parity. This relation always holds true for European puts and calls of the same maturity with the same strike price. It DOES NOT HOLD TRUE OTHERWISE! Put - Call = PV(Strike) - Spot Options & Contingent Claims Stephen Chadwick May 12, 1999

15. Uses of Options: Options are useful for a variety of reasons. Some people use them to speculate since they offer large payoffs for a relatively low investment (large downside risks too, of course). Others use them to hedge current positions. One example of this is “portfolio insurance,” which is essentially a put option on a portfolio. This provides limited downside risk but retains the upside. Of course, options aren’t free and you do get what you pay for. For example, a long volatility position (such as a long straddle/strangle) will gradually lose money if stock prices prove not to be volatile after all. This is called “theta decay,” where theta represents the amount of value that an option loses if nothing happens to stock prices. Options & Contingent Claims Stephen Chadwick May 12, 1999

16. Combined Option Positions: Options & Contingent Claims Stephen Chadwick May 12, 1999