Option Basics

2 Option Basics It is unlikely that a reader will pick up a book at this level without already having some idea of what options are about. However it is worth establishing a minimum base of knowledge and jargon, without which it is not worth proceeding further. All the material in this chapter was well known before modern option theory was developed. 2.1 PAYOFFS (i) A call option on a commodity is a contract which gives the holder of the option the right to buy a unit of the commodity for a fixed price X (the strike price). The key feature of this contract is that while it confirms the right, it does not impose an obligation. If it were a contract which both allowed and obligated the option holder to buy, we would have a forward contract rather than an option. The difference is that the option holder only exercises his right if it is profitable to do so. For example, suppose an option holder has a call option with X = $10. If the price of the commodity in the market is $12, the option can be exercised for $10 and the underlying commodity sold for $12, to yield a profit of $2; on the other hand, if the market price is $8, the option will not be exercised. The outcome of this type of option contract can be summarized mathematically as follows: Payoff = max [ 0, (S T − X ) ] or (S T − X ) + which means that the payoff equals S T − X , but only if this is positive; otherwise it is zero. The payoff may equally be regarded as the value of the call option at exercise C payoff . Much of this book is dedicated to the following problem: if we know C payoff , how can we calculate the value of the option now? A put option gives the holder the right (but not the obligation) to sell a unit of a commodity for a strike price X. This type of option is completely analogous to the call option. The payoff (option value at exercise) can be written P payoff = max[0, (X − S T )] or (X − S T ) + (ii) The payoff of a call, a put and a forward contract are shown in Figure 2.1. These are the so- called “hockey-stick” diagrams which show the value at exercise or payoff of the instruments as a function of the price of the underlying commodity. (iii) An option is an asset with value greater than or equal to zero. If we buy an option we own an asset; but someone out there has a corresponding liability. He is the option writer and is said to be short an option in the jargon of Section 1.1. An option is only exercised if it yields 2 Option Basics Payoff Payoff Payoff Call Option Put Option Forward Contract T S X T S T S X X Payoff Payoff Payoff T S X T S T S X X Figure 2.1 Payoff diagrams a profit to the holder, i.e. if the option writer incurs a loss. The payoff diagrams of such short positions are shown in Figure 2.2 and are reflections of the long positions in the x-axis. Payoff Payoff Short Call Short Put Short Forward Payoff T S X T S T S X X Payoff Payoff Payoff T S X T S T S X X Figure 2.2 Payoff diagrams for short positions (iv) Put and call options exist in two forms: European and American. A European option has a fixed maturity T and can only be exercised on the maturity date. An American option is more flexible; it also has a fixed expiry date, but it can be exercised at any time beforehand. American options are the more usual in the traded options markets. Looking back at the payoff diagrams of the previous paragraphs, these apply to European options on the maturity dates of the options. On the other hand, the payoff diagrams for Ameri- can options could be achieved whenever the holder of the option decides to exercise. In general, European options are much easier to understand and value, since the holder has no decision to make until the maturity date; then he merely decides whether exercise yields a profit or not. With an American option, the holder must decide not only whether to exercise but also when. 2.2 OPTION PRICES BEFORE MATURITY (i) Put-Call Parity for European Options: Consider the following two portfolios: r A forward contract to buy one share of stock in time T for a price X. r Long one call option and short one put option each on one share of stock, both with strike price X and maturity T. The values of the portfolios now and at maturity are shown in Table 2.1. It is clear that whatever the maturity value of the underlying stock, the two portfolios have the same payoff 16 2.2 OPTION PRICES BEFORE MATURITY Table 2.1 Initial and terminal values of two portfolios Value now Value at t = T Value at t = T S T < XX< S T Forward purchase of stock at X f 0 (T ) = S 0 − X e −rT S T − XS T − X Long call; short put C 0 (X, T ) − P 0 (X, T ) C payoff − P payoff =−P payoff = S T − X C payoff − P payoff = C payoff = S T − X value. Therefore, by the no-arbitrage proposition 1.2(ii), the two portfolios must have the same value now. This important relationship is known as put–call parity and may be expressed as f 0T = C 0 (X, T ) − P 0 (X, T ) or equivalently P 0 (X, T ) + S 0 = C 0 (X, T ) + X e −rT (2.1) If dividends are taken into account, the last equation may be written P 0 + (S 0 − d e −rτ ) = C 0 + X e −rT discrete dividend at τ P 0 + S 0 e −qT = C 0 + X e −rT continuous dividend rate q (ii) Consider the value of a put option prior to expiry, if the stock price is much larger than the strike price. Clearly the value of this asset cannot be less than zero since it involves no obligation; on the other hand, its value must be very small if S 0 →∞, since the chance of its being exercised is small. The same reasoning applies to a call option for which S 0 → 0. These can be summarized as lim S 0 →∞ P 0 → 0; lim S 0 →0 C 0 → 0 Using both these results in the put–call parity relationship of equation (2.1) gives the following general result for European options without dividends: lim S 0 →∞ C 0 → f 0T = S 0 − X e −rT ; lim S 0 →0 P 0 →−f 0T = X e −rT − S 0 (2.2) These results are illustrated in Figure 2.3. The dotted lines and the x -axes provide the asymp- totes for the graphs of C 0 and P 0 against S 0 , for European options. The third graph illustrates Call Option Put Option Put – Call Parity --rT Xe 0 S 0 S 0 S 0 C 0 P 0 f rT Xe -rT Xe 0 S Figure 2.3 Option values before maturity 17 2 Option Basics the put–call parity relationships with the dotted line representing the value of the forward contract. One feature should be noted. The dotted lines in the first two graphs look very much like payoff diagrams; but they are not the same. Payoff hockey sticks have a fixed position while these asymptotes drift towards the right over time. They only correspond to the payoff diagrams at maturity. 2.3 AMERICAN OPTIONS In the last section it was seen that the curve of the value of a European option always lies above the asymptotic lines. What of an American option which can be exercised at any time before maturity? Some very general and important conclusions can be reached using simple arbitrage arguments. (i) First, we establish three almost trivial looking results: r The prices of otherwise identical European and American options must obey the relationship Price American ≥ Price European This is because an American option has all the benefits of a European option plus the right of early exercise. r An American option will always be worth at least its payoff value: if it were worth less, we would simply buy the options and exercise them. Conversely, an American option will not be exercised if its value is greater than the payoff, as this constitutes the purposeless destruction of value. r The price of a stock falls on an ex-dividend date by the amount of the dividend which is paid. The holder of an option does not receive the benefit of a dividend, so the potential payoff of an American call drops by the value of the dividend as the ex-dividend date is crossed. If an American call is exercised, this will therefore always occur shortly before an ex-dividend date. By the same reasoning, an American put is always exercised shortly after an ex-dividend date. (ii) American Calls: In Section 2.2(ii) we saw that the graph of a call option against price must al- ways lie above the line representing the value of a forward, i.e. C European ≥ f 0T = S 0 − X e −rT . The first point of the last subsection then implies that C American ≥ f 0T = S 0 − X e −rT and if r and T are always positive (i.e. e −rT ≤ 1) then we must also have C American ≥ S 0 − X If this is true, then by the second point of the last subsection, it can never pay to exercise an American call before maturity; but if an American call is never exercised early, this feature has no value and the price of an American call must be the same as the price of a European call. (iii) Dividends: The last conclusion is summed up by the first of the three graphs in Figure 2.4. However if dividends are introduced, the picture changes. Using the discrete dividend model, the line representing the value of the forward becomes S 0 − d e −rτ − X e −rT ; this line may lie to the right of the payoff line S 0 − X , in which case the curve for the American call would cut 18 2.3 AMERICAN OPTIONS the payoff line at some point. It would then pay to exercise the American call, i.e. it may pay to exercise if S 0 − d e −rτ − X e −rT < S 0 − X or if d e −rτ > X(1 − e −rT ) This is a condition that the present value of the dividend is greater than the interest earned on the cash that would be used to exercise the option. This clearly makes sense if an extreme example is considered: suppose a company is about to dividend away three quarters of its value; if S > X it makes sense to exercise just before the dividend. S 0 - X e – r T X S 0 No Dividend Discrete Dividend Continuous Dividend X S 0 X S 0 S 0 - X S 0 e q t - X e - - r t S 0 d e X e r r t - - - t S 0 - X e – r T S 0 - X e r T forward forward forward S e - S 0 S 0 - ee Figure 2.4 American calls with dividends The last of the graphs in Figure 2.4 shows the same issue expressed in terms of the continuous dividend model. The value of the forward is now represented by S 0 e −qT − X e −rT . The slope of this line is less than that of the payoff line, so the two lines cross at some point. This happens if S 0 e −qT − X e −rT < S 0 − X or S 0 (1 − e −qT ) > X(1 − e −rT ) Once again, the condition is that the dividends earned are greater than the interest on the exercise price. If it might pay to exercise a call before maturity, then clearly the value of the American option must be greater than its European equivalent. (iv) American Puts: The divergence between the values of American and European options is much starker for puts than for calls. By the same reasoning as in Section 2.3(ii), we may conclude that the value of an American put must lie above and to the right of the diagonal line depicting the value of a short position in a forward contract, i.e. P American ≥−f 0T = X e −rT − S 0 From Figure 2.5 for a non-dividend-paying put, it can be seen that the short-forward diagonal is to the left of the payoff diagonal. The curve for the put option, which is asymptotic to the short forward line, will cut across the payoff line. In the terms of the last couple of subsections, the payoff will be greater than the option price over a substantial region so that the precondition exists for exercise and the American put has a higher price than the European put. 19 2 Option Basics S 0 X - S 0 X e -r t - S 0 P American Short forward Figure 2.5 American put 2.4 PUT–CALL PARITY FOR AMERICAN OPTIONS (i) It will be apparent to the reader that given the more complex behavior of American options, there is no slick formula for put–call parity as there is for European options. However for short-term options, fairly narrow bounds can be established on the difference between American put and call prices. exercisenow maturity τ ττ τ t=0 t= t=T Consider American options with maturity T which may be exercised at a time τ . The value of the proceeds of each option depends not only on the price S T at maturity, but also on whether and when it is exercised. If the option is exercised early, the strike price is paid and the time value of this cash has to be taken into account. For example, an American call option might be exercised at any time τ between now and T . After exercise, the stock that we buy under the option will continue to vary stochastically, achieving value S T at time T; but the exercise price would have been paid earlier than final maturity, so that the time T value of the strike price is X e r(T −τ ) where 0 ≤ τ ≤ T . The generalized payoff value of an American call option assessed at time T may therefore be written as S T − X e r(T −τ ) ; the corresponding value for an American put option is X e r(T −τ ) − S T . Put–call parity relations for American options may be obtained using arbitrage arguments analogous to those for European options. In the analysis that follows, we make the decision ahead of time to hold any American option to maturity. Any short option position may be exercised against us at time τ (0 ≤ τ ≤ T ) and we then maintain the resultant stock position until maturity. (ii) Let us now compare the following two portfolios: r A forward contract to sell one share of stock in time T for a price X. r Long one put option and short one call option each on one share of stock, both with strike price X and maturity T. Our strategy in running this portfolio is only to exercise the put options on their expiry date. Our counterparty may choose to exercise the call against us before maturity, in which case we invest the cash and hang on to the short stock position until maturity. Initial and terminal values of these two portfolios are given in Table 2.2. The notation { Q, 0 } signifies a quantity which could have value Q or 0, depending on whether our counterparty has exercised the call option or not. A few seconds reflection will convince the reader that the 20 2.4 PUT–CALL PARITY FOR AMERICAN OPTIONS Table 2.2 Initial and terminal values of two portfolios Value now Value at t = T Value at t = T S T < XX< S T Forward sale − f 0 (X, T ) = X e −rT − S 0 X − S T X − S T of stock at X Long put; short call P 0 (X, T ) − C 0 (X, T ) P payoff − C payoff = (X − S T ) −  S T − X e r(T −τ ) , 0  P payoff − C payoff = 0 −  S T − X e r(T −τ ) , 0  value of the option portfolio is always equal to or less than the proceeds of the forward share sale, whatever the value of S T . In terms of the present value of the two portfolios, this may be written C 0 (X, T ) − P 0 (X, T ) ≤ S 0 − X e −rT (iii) A very similar argument to that given in the last subsection allows us to establish a different bound. This time we compare the following two portfolios: r A forward contract to buy one share of stock in time T for a price X e rT . r Long one call option and short one put option each on one share of stock, both with strike price X and maturity T. Our strategy in running this portfolio is only to exercise the call options on their expiry date. Our counterparty may choose to exercise the put early. Table 2.3 Initial and terminal values of two portfolios Value now Value at t = T Value at t = T S T < XX< S T Forward f 0 (X e rT , T ) = S 0 − X e r S T − X e rT S T − X e rT purchase of stock at X e rT Long call; short put C 0 (X, T ) − P 0 (X, T ) C payoff − P payoff = 0 −  X e r(T −τ ) − S T , 0  C payoff − P payoff = ( S T − X ) −  X e r(T −τ ) − S T , 0  This time it is obvious that the terminal values of the option portfolio are always greater than or equal to the forward contract proceeds. The inequality may therefore be written C 0 (X, T ) − P 0 (X, T ) ≥ S 0 − X The results of this section can be summarized to give a put–call parity relationship for American options as follows: S 0 − X ≤ C American − P American ≤ S 0 − X e −rT (2.3) This relationship can be generalized to include the effects of dividends by making the normal substitutions S → S e −qt or S → S−PV[D]. 21 2 Option Basics 2.5 COMBINATIONS OF OPTIONS This is a book on option theory and many “how to” books are available giving very full descriptions of trading strategies using combinations of options. There is no point repeating all that stuff here. However, even the most theoretical reader needs a knowledge of how the more common combinations work, and why they are used; also, some useful intuitive pointers to the nature of time values are examined, before being more rigorously developed in later chapters. Most of the comments will be confined to combinations of European options. (i) Call Spread (bull spread, capped call): This is the simplest modification of the call option. The payoff is similar to that of a call option except that it only increases to a certain level and then stops. It is used because option writers are often unwilling to accept the unlimited liability incurred in writing straight calls. The payoff diagram is shown in the first graph of Figure 2.6. It is important to understand that a European call spread (and indeed any of the combinations described below) can be created by combining simple options. The second graph of Figure 2.6 shows how a call spread is merely a combination of a long call (strike X 1 ) with a short call (strike X 2 ). The third graph is the payoff diagram of a short call spread; it is just the mirror image in the x-axis of the long call spread. Call Spread Long and Short Calls Short Call Spread PayoffPayoff Payoff T S 1 X 2 X T S 1 X 2 X T S 1 X 2 X Figure 2.6 Call spreads (ii) Put Spread (bear spread, capped put): This is completely analogous to the call spread just described. The corresponding diagrams are displayed in Figure 2.7. Put Spread Long and Short Puts Short Put Spread Payoff Payoff Payoff T S 1 X 2 X T S 1 X 2 X T S 1 X 2 X Figure 2.7 Put spreads (iii) In glancing over the last two sets of graphs, the reader will notice that the short call spread and the put spread are very similar in form; so are the call spread and short put spread. How are they related? 22 2.5 COMBINATIONS OF OPTIONS All the payoff diagrams used so far have been graphs plotting the value of the option position at maturity against the price of the underlying stock or commodity. But the holder of an option would have had to pay a premium for this position (the price of the option). To get a “total profits” diagram, we need to subtract the future value (at maturity) of the option premium from the payoff value, i.e. the previous payoff diagrams have to be shifted down through the x-axis by the future value of the premium. Similarly, short positions would be shifted up through the x-axis. Call Spread Short Put Spread Box Spread 1 X 2 X rT 12 (C - C)e 12 (X X ) 1 X 2 X rT 12 (P - P ) e 1 XX- 12 Figure 2.8 Equivalent spreads The effects of including the initial premium on the final profits diagram of a call spread and a short put spread are shown in the first two graphs of Figure 2.8. The notation C 1 , C 2 , P 1 , P 2 is used for the prices of call and put options with strikes X 1 , X 2 . The diagonal put and call payoffs are 45 ◦ lines, so that the distance from base to cap must be X 2 − X 1 as shown. Recall the put–call parity relationship for European options C + X e −rT = P + S, from which (C 1 − C 2 )e rt + (P 2 − P 1 )e rt = X 2 − X 1 It follows immediately that these two final profit diagrams are identical. All of these payoffs could be generated using just puts or just calls, and the costs would be the same. This theme is developed further below. Although it is possible to create spreads with American options, re- member that the put–call parity equality no longer holds; American puts and calls are therefore not interchangeable as are their European counterparts. (iv) Box Spread: The third graph of Figure 2.8 shows an interesting application of the concepts just discussed. By definition, a put spread is perfectly hedged by a short put spread; but we have just seen that a European short put spread is identical to a European call spread. Thus a put spread is exactly hedged by a call spread. The combination of the two is called a box spread. Suppose we buy a call spread for C 1 − C 2 and a put spread for P 1 − P 2 ; the put–call parity equality of the last paragraph shows that this will cost (X 1 − X 2 )e −rT . Since a box spread is completely hedged, this structure will yield precisely X 1 − X 2 at maturity. In other words, a combination of puts and calls with individually stochastic prices yields precisely the interest rate. There are two purposes for which this structure is used. First, if one (or more) of the four options, bought in the market to make the box spread, is mispriced, the yields on the cash investment may be considerably more than the interest rate. This is quite a neat way of squeezing the value out of mispriced options. Second, gains on options sometimes receive different tax treatment from interest income, so that this technique has been used for converting between capital gains and normal income. 23 2 Option Basics (v) Straddle: This is another popular combination of options with the payoff shown in the first graphs of Figure 2.9 . This consists of a put and a call with the same strike price. People invest in this instrument when they think the price of the underlying stock or commodity will move sharply, but they are not sure in which direction. Clearly, this is tantamount to betting on the future volatility of the stock. Payoff Payoff Straddle Strangle T S X T S 1 X 2 X Figure 2.9 Straddle and strangle Strangle: A slightly modified version of the straddle is shown in the second graph of Figure 2.9. A straddle is quite an expensive instrument, but by separating the strike prices of the put and the call, the cost can be reduced. (vi) Collar: One of the most important uses of an option is as a hedge against movement in the underlying price. Typically, the owner of a commodity can buy an at-the-money put option; for each $1 drop in the commodity price, there is a $1 gain in the payoff of the put. The put option acts as an insurance policy on the price of the commodity. If an insurance premium is too expensive, it can be reduced by introducing an “excess” or “deductible”. For example, the owner of the commodity bears the first $5 of loss and the insurance covers any further loss. This would be achieved by buying a put whose strike price is $5 below the current market price. Another way in which the insurance cost can be decreased is by means of a collar. In addition to buying a put, the commodity holder sells a call with strike somewhere above the current commodity price. The first graph of Figure 2.10 shows the payoff for a collar. Below X 1 , the Collar Long Commodity Net Exposure Payoff Payoff Payoff T S 1 X 2 X T S T S 1 X 2 X 00 S Figure 2.10 Collars 24 [...]... structures, put–call parity always assures that the cost is the same, whatever elements are used to build them 25 2 Option Basics 2.6 COMBINATIONS BEFORE MATURITY (i) The value of a combination of options before maturity is just equal to the sum of the values of the constituent simple options The evolution of the value of a butterfly at times T1 and T2 before and at maturity is shown in Figure 2.12 Long...2.5 COMBINATIONS OF OPTIONS commodity holder receives $1 from the put for each drop of $1 in the price Above X 2 , he pays away $1 under the call option for each $1 rise in the price If the option positions are combined with his position in the underlying commodity (second graph), the result is his net exposure... particularly popular variety is the zero-cost collar where the strike prices are arranged so that the receipt from the call exactly equals the cost of the put (vii) Butterfly: As with simple put and call options, the writer of a straddle accepts unlimited liability This can be avoided by using a butterfly, which is just a put spread plus a call spread with the upper strike of the first equal to the lower... always come to the same conclusion The relationship between gamma and theta will be rigorously analyzed later in the course; but it is comforting to know that one of the most important conclusions of option theory can be confirmed by a casual glance at the payoff diagrams 27  . liability. He is the option writer and is said to be short an option in the jargon of Section 1.1. An option is only exercised if it yields 2 Option Basics Payoff. European options. The third graph illustrates Call Option Put Option Put – Call Parity --rT Xe 0 S 0 S 0 S 0 C 0 P 0 f rT Xe -rT Xe 0 S Figure 2.3 Option