Chapter 12: Basic option theory

1. Chapter 12: Basic option theory Investment Science D.G. Luenberger

2. Before we talk about options • This course so far has dealt with deterministic cash flows and single-period random cash flows. • Now we’d like to deal with random flows at each of several time points, i.e., multiple random cash flows. • Multiple random cash flow theories are generally very difficult. Here, we’d like to focus on a special case: derivatives, i.e., those assets whose cash flows are functionally related to other assets whose price characteristics are assumed to be known. • Options are an important category of derivatives.

3. Option, I • An option is the right, but not the obligation, to buy (or sell) an underlying asset under specified terms. Usually there are a specified price, called strike price or exercise price (K), and a specified period of time, called maturity (T) or expiration date, over which the option is valid. • An option is a derivative because whose cash flows are related to the cash flows of the underlying asset. • If the option holder actually does buy and sell the underlying asset, the option holder is said to exercise the option. • The market price of an option is called premium.

4. Option, II • An option that gives the right to purchase (sell) the underlying asset is called a call (put) option. • An American option allows exercise at any time before and including the expiration date. • An European option allows exercise only on the expiration date. In this course, we will focus on European options because their pricing is easier. • The underlying assets of options can be financial securities, such as IBM shares or S&P 100 Index, or physical assets, such as wheat or corn.

5. Option, III • Many options are traded on open markets. Thus, their premiums are established in the market and observable. • Options are wonderful instruments for managing business and investment risk, i.e., hedging. • Options can be particularly speculative because of their built-in leverages. For example, if you are very sure that IBM’s share price will go up, betting $1 on IBM call options may generate a much higher return than betting $1 on IBM shares.

6. Notation • S: the price of the underlying asset. • C: the value of the call option. • P: the value of the put option.

7. Value of call option at expiration

8. Value of put option at expiration

9. Value of option at expiration • The value of the call at expiration: C = max (0, S – K). • The call option is in the money, at the money, or out of money, depending on whether S > K, S =K, or S <K, respectively. • The value of the put at expiration: P = max (0, K – S). • The put option is in the money, at the money, or out of money, depending on whether S <K, S =K, or S >K, respectively.

10. 11.1 Binomial model, PP. 297-299 • A binomial model can be a single-period model or a multiple-period model. • A basic period length can be a week, a month, a year, etc. • If the price of an asset is known at the beginning of a period, say S, the price of the asst at the end of the period is one of only two possible values, S*u and S*d, where u > 1 > d > 0. S*u (S*d) is expected to happen with probability p (1 – p). • That is, we have uncertainty, but in the form of two possible values.

11. Binomial model

12. Calibration, I • Binomial models provide an uncertain structure for us to model the underlying asset’s price dynamics. • This modeling is necessary because option value is a function of the underlying asset’s price dynamics. • Thus, to obtain accurate valuation for an option, we need to do a good job on modeling the underlying asset’s price dynamics. • That is, we need to choose binomial parameters, i.e., p, u, and d, carefully such that the binomial-based price dynamics is consistent with the observable (historical) price characteristics, e.g., average return and standard deviation, of the underlying asset.

13. Calibration, II • Let v be the expected (average) annual return of the underlying asset, v = E [ln(ST /S0 )], say 12%. • Let  be the yearly standard deviation of the underlying asset, 2 = var [ln(ST /S0 )], say 15%. • Note that 12% return and 15% standard deviation are about the kind of numbers that you would expect from a typical S&P 500 stock. • Now we need to define the period length relative to a year. If we define a period as a week, a period length of t is 1/52.

14. Calibration, III • Then, the binomial parameters can be selected as: • p = ½ + ½ *(v / )* (t)1/2. • u = e * (t)1/2. • d = 1/u. • e is exponential and has a value of 2.7183. • With these choices, the binomial model will closely match the values of v and  (see pp. 313-315 for the proof).

15. Calibration, IV

16. 1-period binomial option theory, I • We assume that it is possible to borrow or lend at the risk-free rate, r. • Let R = 1 + r, and u > R > d. • Suppose that there is a call option on the underlying asset with strike price K and expiration at the end of the single period. • Let Cu (Cd) be the value of the call at expiration.

17. 3 related lattices

18. No-arbitrage • The key to price the call option at time 0 is to form a portfolio at time 0: (1) the portfolio consists of the underlying asset and the risk-free asset, (2) the portfolio’s value at time 1 is equal to the value of the call at time 1, regardless whether it is up or down. • This portfolio is called a replicating portfolio: x dollar worth of the underlying asset and b dollar worth of the risk-free asset. • No-arbitrage: because the replicating portfolio and the call yield the value at time 1 regardless what might happen, the value of the replicating portfolio and the call at time 0 must be the same. • That is, x + b = C.

19. Outcome matching • The value of the replicating portfolio equals to the value of the call at time 1 when it is up: u * x + R * b = Cu. • The value of the replicating portfolio equals to the value of the call at time 1 when it is down: d * x + R * b = Cd. • Solve for x and b from the two equations.

20. 1-period binomial Solution • x = (Cu – Cd) / (u – d). • b = (u * Cd – d * Cu) / (R * (u – d)). • Based on no-arbitrage, we know that C = x + b. • C = (Cu – Cd) / (u – d) + (u * Cd – d * Cu) / (R * (u – d)). • After some algebras, we have C = (1/R) * [q * Cu + (1 – q) * Cd], where q = (R – d) / (u – d). • Note that p is not in the pricing equation because no trade-off among probabilistic events is made.

21. 1-month IBM call option, I • Consider IBM with a volatility of its logarithm of  = 20%. The current price of IBM is $62. A call option on IBM has an expiration date 1 month from now and a strike price of $60. The current interest rate is 10%, compounded monthly. Suppose that IBM will not pay dividends.

22. 1-month IBM call option, II

23. When no information aboutv and  • If we have a primitive binomial problem in which we have no information on expected return and standard deviation, we then must know the two possible outcomes. • Example: suppose the market price of a stock is $50. The two possible outcomes for the stock price is either $60 or $40 in a year. • A call option with a one-year expiration and a $50 exercise price. • Interest rate is 5%.

24. Duplicating portfolio • We need to duplicate the call with the strategy of buying stocks and borrowing monies. • The duplicating strategy is to buy ½ share of the stock and borrow $19.05. Why this particular combination? We will talk about this later.

25. When the stock price is up • When the stock price is up, the payoff of buying a call is $10 ($60 - $50). • When the stock price is up, the payoff the duplicating strategy is also $10. The sum of the following two positions is $10 ($30 - $20): (1) buying ½ share: ½ * $60 = $30, and (2) borrowing $19.05 at 5%: -$19.05 * 1.05 = -$20.

26. When the stock price is down • When the stock price is down, the payoff of buying a call is $0. • When the stock price is down, the payoff the duplicating strategy is also $0. The sum of the following two positions is $0 ($20 - $20): (1) buying ½ share: ½ * $40 = $20, and (2) borrowing $19.05 at 5%: -$19.05 * 1.05 = -$20.

27. No arbitrage • The call and the duplicating strategy generate identical payoffs at the end of the year. • No arbitrage principle implies that the current market price of the call equals to the current market price of the duplicating position. • The market price of the duplicating position is $5.95 ($25 - $19.05). Buying ½ share costs $25 (1/2 * $50). • The call price is $5.95.

28. Why ½ share? • The duplicating portfolio was given to be buying ½ share and borrowing $19.05 earlier. • Why ½ share? This amount is called the delta of the call. • Delta = swing of the call / swing of the stock = ($10 - $0) / ($60 - $40) = ½.

29. Why borrowing $19.05? • Buying ½ share gives us either $30 or $20 at expiration, which is exactly $20 higher than the payoffs of the call, $10 and $0, respectively. • To duplicate the position, we thus need to borrow a dollar amount such that we will need to pay back exactly $20. • Given the future value is $20, the interest rate is 5%, and the number of the time period is 1, we have the present value to be $19.05 (use your financial calculator).

30. Multiple-period pricing • The usefulness of single-period binomial pricing is that it can be applied to multiple-period problems in a straightforward manner. • That is, the single period pricing, C = (1/R) * [q * Cu + (1 – q) * Cd], is repeated at every node of the lattice, starting from the final time period and working backward toward the initial time.

31. 5-month IBM call option, I

32. 5-month IBM call option, II

33. 1-month vs. 5-month • The call price for 1-month IBM call is $3.14. The call price for 5-month IBM call is $5.85. • Holding other factors constant, the longer the maturity, the higher the call premium. • The reason for this is that additional time allows for a greater chance for the stock to rise in value, increasing the final payoff of the call option. • See Figure 12.3, p. 324. • Along the same line of seasoning, the higher the standard deviation, the higher the call premium. You should verify this numerically.

34. How about put option pricing? • So far, we have focused on call pricing. • The reason for this is that for European options one can calculate the value of a put option, P, based on value the call option, C, when they have the same strike price and maturity. • P = C – S + df * K, where df is risk-free discount factor. • This relationship is called the put-call parity.

35. Put option pricing • Consider a GM call option and a GM put option, both have 3 months to expiration and the same strike price, $35. The current price of GM shares is $37.78. The call premium is $4.25. The interest rate is 5.5%, so over 3 months, the discount factor is 0.986 (= 1 / (1 + 0.055/4). • P = C – S + df * K = 4.25 – 37.78 + 0.986 * 35 = $1.00.

36. Options are interesting and important • A combination of options can lead to a unique payoff structure that otherwise would not be possible. • Options make it happen! • Example: a butterfly spread, p. 325. • Question: who would hold a butterfly spread?