1 Lecture 17/18/19 Options II Alexander K. Koch Department of Economics, Royal Holloway, University of London February 25, February 29, and March 10 2008 In addition to learning the material covered in the reading and the lecture, students should be able to work out a number of price bounds for options (depending on their type, the time left to expiry, and their strike price); understand the basic notion of binomial option pricing and the idea of arbitrage with a tracking portfolio; be able to price a plain-vanilla option in a two-period binomial framework; understand what the hedge ratio ( Delta ) means. Required reading: Bodie, Kane, and Marcus (2008) (Chapter 21) Supplementary reading: Elton and Gruber (1995) (Chapter 8) M.S. Scholes (1998), Derivatives in a Dynamic Environment, American Economic Review, 88, 350-370. (*) E-mail: Alexander.Koch@rhul.ac.uk.
2 1 Price bounds for options 1.1 Assumptions and notation We will make use of the following notation: strike (exercise) price of an option: X; maturity date: T ; price of the underlying asset at date t: S t ; dividend payments: D; (constant) risk-free rate of return (p.a.): r f ; price of a European call/put at date T: c t /p t ; price of an American call/put option at date T: C t /P t. 1.2 Bounds on the value of a European call The most obvious lower bound on the value of an option is that an option can never be worth less than zero - the holder can always simply let it expire. To derive another lower bound, consider a stock that pays dividend D just before maturity. Compare the payoff from the following two portfolios: A ( option position ): long one European call option on one share of stock; B ( leveraged equity position ): long one share of stock and short a zero bond with face value X + D and maturity T. Table 1 gives the position values at date t and at maturity date T. As we can see, with certainty, portfolio A is at least as valuable as portfolio B at maturity. Thus, by no arbitrage, the value of portfolio A at any date t < T must also be at least that of portfolio B. This provides us with another lower bound on the value of a European call option: the price of the underlying minus the present value of strike price and dividends. Combined with the restriction of nonnegative value, we have: c t max[0, S t P V t (X) P V t (D)]. The maximum payoff from an option at maturity is max[s T X, 0] S T. Therefore, no arbitrage implies that at any date t no one would be willing to pay more than the current price of the underlying asset, S t to purchase the option to buy it at date T for X > 0.
3 Position value value at T if at t S T X S T > X A c t 0 S T X B S t S T + D S T + D X+D (X + D) (X + D) (1+r f ) (T t)/365 S t X+D (1+r f ) (T t)/365 S T X S T X value A value B value A = value B Table 1: Payoffs of option position versus leveraged equity position. Result 1 The value of a European call option lies within the following bounds: S t c t max[0, S t P V t (X) P V t (D)]. (1) 1.3 Bounds on the value of an American call An American call option is at least as valuable as a European one with the same characteristics, since it provides the additional option to exercise early (if no early exercise occurs, the payoff is the same as that for the European call). Thus, S t C t c t. Clearly, the upper bound for the European call option also applies for an American call option: the maximum payoff from exercising the option at any date t = 0,..., T max[s t X, c t ] S t. As we will see, absent dividends, such an option will never be exercised early. No dividends on underlying asset We know from our previous discussion that an American option guarantees at least the payoff from that on a European option, by choosing not to exercise it before maturity: C t c t max[0, S t P V t (X)]. (2) if the American option is exercised at some date t < T, it yields a payoff of: C t (if exercised) = S t X. (3) Now since P V t (X) < X, we conclude that the payoff from holding on to the option [equation (2)] is greater than from exercising it at date t < T [equation (3)]. Therefore, the option is worth more alive than dead, i.e., the option to exercise early has no value.
4 Result 2 An American call option on a non-dividend-paying stock will never be exercised before the maturity date. From this we conclude, that for non-dividend-paying underlying assets, the values of a European and an American call option are identical: C t = c t, t = 0,..., T. Dividends on underlying asset Suppose now that the underlying stock makes a dividend payment D on date T. As we will see, with dividends the lower price bound in (2) can be made tighter, i.e. we can find an even higher value for the lower bound. Consider an American call option which has not been exercised prior to date t. We know that, if it is not exercised on t, it will be worth as much as a European call option: c t max[0, S t P V t (X) P V t (D)]. If the option is immediately exercised (which of course only pays if S t > X), it yields S t X. Suppose that the option is in the money, i.e., S t > X, but that P V t (X + D) > S t. Then, the value of the option if exercised might exceed the value of holding on to it for another day. Result 3 An American call option on stock that pays dividends during its time-to-maturity may be exercised before the maturity date. Thus, with dividend-paying stocks, the right to exercise early potentially has a value. 1.4 Bounds on the value of a European put Since the payoff at maturity of a European put is max[0, X S T ] X, the maximum value that a put can have clearly is X. As with call options, a put option always has nonnegative value. To derive another lower bound for a European put option on non-dividend paying stock, consider the payoff from the following two portfolios: A ( option-stock position ): long one share of stock and one European put option on one share of stock; B ( fixed income position ): long one zero bond with face value X and maturity T. Table 2 gives the position values at date t and at maturity date T. As we can see, with certainty, portfolio A is at least as valuable as portfolio B at maturity. Thus, by no arbitrage, the value of portfolio A at any date t < T must also be at least that of portfolio B:
5 Position value value at T if at t S T X S T > X A S t S T S T p t X S T 0 S t + p t X S T B X (1+r f ) (T t)/365 X X value A = value B value A > value B Table 2: Payoffs of option-stock position versus fixed-income position. Result 4 The value of a European put option lies within the following bounds: X p t max[0, P V t (X) S t ]. (4) 1.5 Bounds on the value of an American put Due to the early exercise feature, we know that the American put option is at least as valuable as its European counterpart: P t p t max[0, P V t (X) S t ]. In the case that the put is exercised early at date t < T (which of course only pays if S t < X), the payoff is X S t. Since P V t (X) < X, early exercise might be profitable. Thus, an American put is more valuable than its European counterpart. Result 5 The value of a American put option lies within the following bounds: X P t max[0, X S t ]. (5) It is straightforward to repeat the above exercise including dividends. What emerges is that, in contrast to the case of American call options, dividend payments make early exercise less likely for put options. The intuition is straightforward, dividends increase the lower bound on the European put options, which is the value of an American option not exercised early. From results 2 and 5, it follows immediately that the put-call parity derived in the last lecture for European options does not hold for American options.
2 The binomial option pricing model Most option pricing models rely on advanced mathematical techniques, which are beyond the scope of this course. The simple binomial option pricing model, developed by Cox, Ross, and Rubinstein (1979) and Rendleman and Bartter (1979), illustrates the logic of pricing by no-arbitrage that underlies these more complex models. Moreover, it provides an easily implementable option pricing tool. The model imposes the following assumptions: frictionless capital markets (no transaction costs, no taxes); price-taking behavior; all market participants have the same information; no short-sales constraints; the risk-free rate of return is r f per period; the stock price at any date t, S t can only move up to S t+1 = u S t or down to S t+1 = d S t, where u > > 1 > d > 0, P rob( up ) = q, and P rob( down ) = 1 q. We will only look at plain vanilla options, i.e. options with standard exercise terms and no special clauses. 2.1 The one-period version Let the risk-free rate of return for one period be r f = 0.08. Suppose that a stock currently sells at S 0 = 100 and can either increase (by a factor of u = 2) to 200 or decrease (by a factor of d = 0.5) to 50 at year-end. Let the probability of an upward movement be q = 0.5. Consider a call option with strike price X = 125 which expires at the end of the year. The payoff of the option depends on the realized end-of-period stock price S 1 (see figure 1): 75 = S 1 X if S 1 = u S 0 = 200 value of call at end of period = 0 if S 1 = d S 0 = 50 The first step in valuing this call option is to construct a riskless hedge portfolio. Consider a portfolio long H shares of stock and short one call option written on the stock. Figure 2 illustrates how this portfolio s value evolves. It is worth H S 0 c 0 at the beginning of the period. The end-of-period value depends on whether the stock price moves up (value: 6
7 Beginning of period End of period Beginning of period End of period q S 1 =us 0 [ 200] q c u =max[0,us 0 -X] [ 75] S 0 [ 100] c 0 1-q 1-q S 1 =ds 0 [ 50] c d =max[0,ds 0 -X] [ 0] Stock price Value of a call option with strike price X Example: S 0 = 100, u=2, d=0.5, X= 125 Figure 1: Payoffs in the one-period binomial model Beginningof- period value Endof-period value q HuS 0 -c u risk-free portfolio if HuS 0 -c u = HdS 0 -c d HS 0 -c 0 1-q HdS 0 -c d hedge ratio: c H = S 0 c u d ( u d). Hedge portfolio Figure 2: Payoffs in the one-period binomial model
8 Position value value at end of period if at t = 0 S 1 = 50 S 1 = 200 long one stock S 0 = 100 S 1 = 50 S 1 = 200 short (written) 2 calls 2 c 0 0 150 sum 100 2 c 0 50 50 Table 3: Hedging with underlying asset. H u S 0 c u ) or down (value: H d S 0 c d ). To obtain a riskless portfolio, we must choose H so that the end-of-period portfolio value is constant across states: H d S 0 c d = H u S 0 c u c u c d range of option values at end of period H = = S 0 (u d) range of stock values at end of period (hedge ratio). H is called the hedge ratio, it gives the number of shares that need to be purchased to offset the risk from one call written (i.e., short position) on the stock. Or, put differently, H is the number of shares you need to sell for every long call position held. Returning to our initial example values, we can compute the hedge ratio to be H = 1/2. That is, two written call options can be perfectly hedged using one share of stock (see Table 3). The second step exploits the assumption that there should be no arbitrage opportunities in a well-functioning capital market. Since the return on the hedge portfolio is risk-free, it must be equal to the risk-free rate of return: H u S 0 c u H S 0 c 0 = c 0 = H S 0 [( ) u] + c u. (6) Substituting in for the hedge ratio H in (6) yields [ ( ) ( )] (1 + rf ) d u (1 + rf ) c 0 = c u + c d u d u d Note that the terms in brackets sum up to unity. That is, we can denote by 1. (7) p = ( ) d u d and 1 p = u ( ) u d to obtain c 0 = [p c u + (1 p) c d ] 1. (8) Inspecting (8), we can see that the price of a call option is equal to the expected value of the call option at the end of the period (using the probabilities p and 1 p) discounted
9 with the risk-free rate. In other words, the call is priced as if the probability of an upward movement in the stock price was p and all investors were risk-neutral. Therefore, p and 1 p are called risk-neutral probabilities. Note that nothing in our argument required making any assumptions on investors actual risk preferences. Moreover, the objective probability of an upward movement in the stock, q, has no impact on the value we derived for the call option. The option is priced relative to the given parameters S 0, u, d, X, and r f. If investors disagreed on the probabilities for movements in the stock price, this would only indirectly affect the option s value through the market views being reflected in the stock price S 0. In our example, we obtain a risk-neutral probability of p 0.39. Note that this is different from the objective probability q = 0.5 that we assumed to hold. The call option is priced at c 0 = 26.85. Looking again at table 3, we see that indeed the return on the hedge portfolio is equal to the risk-free rate: 50 100 2 26.85 1 = r f = 0.08. The concept used to price the option is simple: from securities for which we know the price construct a risk-free hedge portfolio involving the option to be priced. No arbitrage then implies that the return on this portfolio must be equal to the risk-free rate. The same approach underlies more complex option pricing models as well (they just use more realistic assumptions on the price distribution for the underlying asset). 2.2 The pricing of European call options Using the insights from the previous section, it is easy to use the binomial model to price plain vanilla European call options on underlying assets which involve no payments until the maturity date (e.g. a non-dividend paying stock). We start off, by pricing a two-period option. The two-period model It is simple to generalize the one-period model in the case of European call options. Consider the extension to two periods in figure 3. Working backward through the tree, we can treat each second-period branch as a one-period model, and thus compute the input values for the next lower branch to be worked on. We will illustrate this using the values of our initial example for an option with strike price X = 75 now (see also figure 3). Starting at the upper end branch (2a in figure 3), we see that the option value at date 2 is either c uu = max[0, u u S 0 X] = 325 or c ud = max[0, u d S 0 X] = 25. Thus, we can compute the value of the option at date 1, ensuing an initial increase in the stock price, c u = p c uu + (1 p) c du 130.56.
10 Strike price: X= 75 1 q c u 2a S 1 =us 0 [ 200] pcuu + ( 1 p) c = 1+ rf [ 130.56] ud q 1-q S 2 =uus 0 [ 400] c uu =max[0,uus 0 -X] [ 325] S 2 =uds 0 [ 100] c ud =max[0,uds 0 -X] [ 25] S 0 [ 100] 1-q 2b q S 2 =dus 0 [ 100] c du =max[0,dus 0 -X] [ 25] ( 1 ) pcu + p cd c0 = 1+ r [ 51.83] f S 1 =ds 0 [ 50] pcdu + ( 1 p) c cd = 1+ rf [ 8.95] dd 1-q S 2 =dds 0 [ 25] c dd =max[0,dds 0 -X] [ 0] Figure 3: Payoffs in the two-period binomial model Working back the lower end branch (2b in figure 3), we see that c du = c ud and c dd = max[0, d d S 0 X] = 0. Thus, c d = p c du + (1 p) c dd 8.95. Using these results as inputs in the starting branch (1 in figure 3), c 0 = p c u + (1 p) c d 51.83. The T-period model The model can be extended to T periods. Note that c 0 = p c u + (1 p) c d = p2 c uu + p (1 p) c ud + (1 p) p c du + (1 p) 2 c dd ( ) 2. Since c ud = c du, all that counts are the number of upward and downward movements and not their order. Thus, we can use the binomial distribution to compute the probabilities of these events. After T -periods with n upward movements and T n downward movements, the option value is: max[0, u n d T n S 0 X]. (9)
11 The probability of ending up in this node of the branch is given by the binomial distribution: 1 T n p n (1 p) T n. Hence, T c 0 = T n n=0 p n (1 p) T n max[0, u n d T n S 0 X] 1 ( ) T. (10) Note that the payoff of a call option is positive only if it finishes off in-the-money, i.e. S T > X. Therefore, the option payoff given in (9) is zero for all n for which S T = u n d T n S 0 X. This helps simplify the formula in (10): T c 0 = S 0 T p n (1 p) T n n n n un d T n ( ) T X ( ) T T T p n (1 p) T n (11) n n n } {{ } = P rob(s T > X) under riskneutral probabilities where n is the smallest integer for which the option finishes in the money, i.e. n = min{n N : u n d T n S 0 > X}. 2.3 The pricing of European put options To price a plain vanilla European put option on a non-dividend paying stock, we can simply invoke the put-call parity relation (see last lecture) and use the price of the European call option as an input: p 0 = c 0 + X ( ) T/365 S 0. 2.4 The pricing of American call options We can invoke result 2, which tells us that an American call option on a non-dividend-paying stock will never be exercised early. Thus, the American call option has the same value as its European counterpart, for which we know how to price it: 1 Recall that for a binomial coefficient @ T n 0 C 0 = c 0. 1 A = T! n! (T n)!.
12 2.5 The pricing of American put options The only problematic case is the pricing of an American put option. Recall that the put-call parity relation does not hold for American options because it may be optimal to exercise early an American put on a non-dividend-paying stock (in contrast to an American call). The advantage of the binomial model is that we can easily modify it, to account for the possibility of early exercise. 2 We will illustrate the approach in a two-period example. The share price on a stock that pays no dividend over the next two periods currently is S 0 = 50. In each period, the stock price can either increase (by a factor u = 1.2) or decrease (by a factor d = 0.6). The yield curve is flat and the risk-free rate of return r f = 0.1 per period. What is the fair price of an American put option, P 0, with exercise price X = 50? Again, we can start with a one-period version of the model and form a hedge portfolio. Recall that we defined the hedge ratio as the number of shares that you need to sell for a long position in options on that stock. As before, the hedge ratio H is chosen so that the portfolio is riskless, i.e. it has a constant end-of-period value: H u S 0 + P u = H d S 0 + P d H = P u P d range of option values at end of period = S 0 (u d) range of stock values at end of period (hedge ratio). Note that since P u < P d, the hedge ratio will be negative, in contrast to the call option case. That is, we can offset the risk of a long put option by going long in shares of the stock. By no-arbitrage, the one-period portfolio return must be equal to the risk-free rate. Thus, given the end-of-period payoffs P u and P d we can price the option at the beginning of the period: ( ) (H S 0 P 0 ) = H u S 0 P u P 0 = u ( ) u d } {{ } =p P d + ( ) d u d } {{ } =1 p P u 1 (substituting in H). To analyze the two-period model, of course, we need to know the option s value ensuing an up or down move in the stock price at each period. As above, we start from the last period: at maturity the payoff of the American put is equal to that of the European counterpart (using 2 All option pricing models for American put options do not have a closed-form solutions and therefore need to rely on numerical methods for deriving the option price.
13 value of European put (no early exercise) early exercise q P uu =p uu =max[0,x-uus 0 ] [ 0] P u =max[p u,x-us 0 ] q [ 2.12=max[ 2.12, - 10] ] [no early exercise] 1-q P 0 1-q value of European put (no early exercise) early exercise q P ud =p ud =max[0,x-uds 0 ] [ 14] P du =p du =max[0,x-dus 0 ] [ 14] [P 0 = 4.64] [p 0 = 3.95] P d =max[p d,x-ds 0 ] [ 20=max[ 15.45, 20] ] [early exercise] 1-q P dd =p dd =max[0,x-dds 0 ] [ 32] Figure 4: Valuing a two-period American put option our example s values). P uu = p uu = max[0, X u 2 S 0 ] = 0 P du = P ud = p ud = max[0, X u d S 0 ] = 14 P dd = p dd = max[0, X d 2 S 0 ] = 32 Using these interim results, we can price the European put option (which cannot be exercised before maturity) at date 1 as follows in our example: 1 p u = [p p ud + (1 p) p uu ] 2.12 pd = [p p dd + (1 p) p du ] 1 15.45. To determine the value of the American put, we need to compare this payoff from not exercising early to that from early exercise at date 1: stock price early exercise no exercise value of American put u S 0 = 60 X u S 0 = 10 p u = 2.12 P u = max[x u S 0, p u ] = 2.12 d S 0 = 30 X d S 0 = 20 p d = 15.45 P d = max[x d S 0, p d ] = 20 As we can see, the American option will be exercised early if the stock moves down in the first period. Hence, we plug in P u = 2.12 and P d = 20 into (12) to obtain: P 0 4.64. In
contrast, the European put would be valued at p 0 3.95. Thus, the early exercise option has a premium of 0.69. The procedure is summarized in figure 4. 3 The Black-Scholes option pricing model The binomial model had the advantage of being tractable and extremely flexible (e.g., it is easy to price American put options). Clearly, the assumption that the stock price can only move up or down by a fixed amount in each time period is restrictive. However, one can add realism by simply splitting a given unit of time (e.g. a month) into more and more time periods. Cox, Ross, and Rubinstein (1979) prove that the limit of the binomial call pricing formula is the Black and Scholes (1973) continuous time pricing formula. The Black and Scholes (1973) formula for pricing call options on non-dividend paying stocks is widely used in practice. In recognition of the significant impact it had on finance theory and practice, Myron Scholes and Robert C. Merton, who also contributed to developing option pricing, were awarded the 1997 Nobel prize (Fisher Black died in 1995). The Black-Scholes option pricing formula for a non-dividend paying call option is: c 0 = S 0 N(d 1 ) X e r f T N(d 2 ) (12) d 1 = ln(s 0/X) + (r f + σ 2 /2) T σ T d 2 = d 1 σ T, (13) N( ) is the cdf of the standard normal distribution. That is, the current option value c 0 is a function of the current stock price S 0, the strike price X, the risk-free rate r f (annualized rate with continuous compounding), the instantaneous variance of the annualized stock return, σ 2, and time to maturity T (in years). Note: the price of the option does not depend on the expected rate of return on the stock. The formula is easy to use given tabulated values for the standard normal cdf or using a computer program. 3 3 An example is the Excel spreadsheet available from the website: http://highered.mcgraw-hill.com/ sites/dl/free/0072861789/116763/ch21_bs_stu.xls. 14
15 Interpretation of the formula To interpret the Black-Scholes formula, it is useful at this point to compare it with the T-period binomial model. We restate here the pricing formula for a T-period European call option (11): T c 0 = S 0 T p n (1 p) T n n n n un d T n ( ) T X ( ) T T T p n (1 p) T n n n n } {{ } = P rob(s T > X) under riskneutral probabilities where n is the smallest integer for which the option finishes in the money, i.e. n = min{n N : u n d T n S 0 > X}. The pricing equation from the Binomial model and that from the Black- Scholes model (12) have a similar structure: the stock price S 0 is multiplied with a factor to compute the expected value of the stock (using the risk neutral probabilities), conditional on finishing in the money; from this we subtract the discounted value of the bond, conditional on finishing in the money. So one can view the second N(d) term in the Black-Scholes formula (12) as the probability that the option will finish in the money and the first N(d) term has an analogous interpretation to the first term in the pricing equation above from the Binomial model. Thus, if the option is almost certain to be in the money at maturity, the price is approximately S 0 X e r f T, which is the discounted value of the option payoff (often called the intrinsic value of an option): the claim on the stock is worth S 0 today and the exercise price is discounted. In contrast, if the probability of finishing in the money is almost zero, the option value is roughly zero, as we would expect. For intermediate cases, the N(d) terms adjust the price of the option to reflect the probability of profitable exercise at maturity. Option Delta and other Greeks Another interesting feature of this formula, is that the derivative of the option value with respect to the underlying stock s price is simply: 4 delta = c 0 S 0 = N(d 1 ). This partial derivative is called the delta of the option. It measures the first-order effect of a change in the underlying asset s price on the call option s price. Thus, one can approximately insure a portfolio long one call option against (small) movements in the underlying asset s 4 Note that when taking the derivative one needs to take into account that N(d 1) and N(d 2) also depend on S 0. The effects through them happen to cancel out.
16 price by establishing a short position of H = delta stocks. Then, following a small change in S 0 the change in portfolio value is approximately c 0 H S 0 S 0 S }{{} 0 =1 = 0 if H = c 0 S 0. Such a portfolio would be called delta neutral. We can interpret delta as the hedge ratio at which we must continuously keep our hedge portfolio to maintain a riskless position (if the other factors affecting option value remain constant). Note, that the concept of delta hedging relies on a first-order approximation. The option value is a convex function in the underlying asset s price (see panel (a) in figure 5). Therefore, the approximation is not exact. To correct for this, one can use a second-order term as well (the so-called option gamma ). Other such first-order approximations are commonly used for the impact of changes in the other factors that affect the option value. Because these derivatives are (almost all) denoted by Greek letters, they are often called the Greeks : rho measures interest rate sensitivity, theta measures the sensitivity to changes in time to maturity, vega measures the sensitivity to changes in volatility. As one can see from figure 5, the price of a call option increases with longer time to maturity (panel (b)), with higher interest rate (panel (c)), and with higher variance (panel (d)). The price is roughly linear in these factors so that a first-order approximation can be reasonable. The impact of time to maturity is intuitive: more time left means more chances of the option to move very far into the money. More volatility in the underlying asset has the same impact on the value of the option: since the downside of the call value is limited by the zero floor but the upside is unlimited more variance increases the value of the option. If you are interested in exploring these relations further, use the Excel application referred to in footnote 3 or have a look at the website: http://www.hoadley.net/options/optiongraphs. aspx? References Black, Fisher, and Myron Scholes, 1973, The pricing of options and corporate liabilities, Journal of Political Economy 81, 637 654. Bodie, Zvi, Alex Kane, and Alan J. Marcus, 2008, Investments (Irwin McGraw-Hill: Chicago). Cox, J., S. Ross, and M. Rubinstein, 1979, Option pricing: A simplified approach, Journal of Financial Economics 7, 229 264.
17 Elton, Edwin J., and Martin J. Gruber, 1995, Modern Portfolio Theory and Investment Analysis (Wiley: New York). Rendleman, Richard J., and Brit J. Bartter, 1979, Two-state option pricing, Journal of Finance 34, 1093 1110.
18 call price call price S-PV(X) ('intrinsic value') price of underlying asset S 0 time to maturity T (a) Price of call option as a function of underlying stock price (b) Price of call option as a function of time to maturity call price call price risk-free rate r f return variance of underlying asset (c) Price of call option as a function of risk-free rate (d) Price of call option as a function of underlying asset s Figure 5: Black-Scholes prices for a non-dividend paying call option return variance