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Properties of Stock Option Prices Chapter 9 1 ASSUMPTIONS: 1. The market is frictionless: No transaction cost nor taxes exist. Trading are executed instantly. There exists no restrictions to short selling. 2. Market prices are synchronous across assets. If a strategy requires the purchase or sale of several assets in different markets, the prices in these markets are simultaneous. Moreover, no bid-ask spread exist; only one trading price. 2 3. Risk-free borrowing and lending exists at the unique risk-free rate. Risk-free borrowing is done by selling T-bills short and risk-free lending is done by purchasing T-bills. 4. There exist no arbitrage opportunities in the options market 3 NOTATIONS: t = The current date. St= The market price of the underlying asset. K = The option’s exercise (strike) price. T = The option’s expiration date. T-t = The time remaining to expiration. r = The annual risk-free rate. = The annual standard deviation of the returns on the underlying asset. D = Cash dividend per share. q = The annual dividend payout ratio. 4 Factors affecting Options Prices (Secs. 9.1 and 9.2) Ct = the market premium of an American call. ct = the market premium of an European call. Pt = the market premium of an American put. pt = the market premium of an European put. In general, we express the premiums as functions of the following variables: Ct , ct = c{St , K, T-t, , r, D }, Pt , pt = p{St , K, T-t, , r, D }. 5 FACTORS AFFECTING OPTIONS PRICES AND THE DIRECTION OF THEIR IMPACT: Factor European European American American call put call put St + - + - K - + - + T-t ? ? + + + + + + r + - + - D - + - + 6 Bounds on options market prices (Sec. 9.3) Call values at expiration: CT = cT = Max{ 0, ST – K }. Proof: At expiration the call is either exercised, in which case CF = ST – K, or it is left to expire worthless, in which case, CF = 0. 7 Minimum call value: A call premium cannot be negative. At any time t, prior to expiration, Ct , ct 0. Proof: The current market price of a call is NPV[Max{ 0, ST – K }] 0. 8 Maximum Call value: Ct St. Proof: The call is a right to buy the stock. Investors will not pay for this right more than the value that the right to buy gives them, I.e., the stock itself. 9 Put values at expiration: PT = pT = Max{ 0, K - ST}. Proof: At expiration the put is either exercised, in which case CF = K - ST, or it is left to expire worthless, in which case CF = 0. 10 Minimum put value: A put premium cannot be negative. At any time t, prior to expiration, Pt , pt 0. Proof: The current market price of a put is NPV[Max{ 0, K - ST}] 0. 11 Maximum American Put value: At any time t < T, Pt K. Proof: The put is a right to sell the stock For K, thus, the put’s price cannot exceed the maximum value it will create: K, which occurs if S drops to zero. 12 Maximum European Put value: pt Ke-r(T-t). Proof: The maximum gain from a European put is K, ( in case S drops to zero). Thus, at any time point before expiration, the European put cannot exceed the NPV{K}. 13 Lower bound: American call value: At any time t, prior to expiration, Ct Max{ 0, St - K}. Proof: Assume to the contrary that Ct < Max{ 0, St - K}. Then, buy the call and simultaneously exercise it for an arbitrage profit of: St – K – Ct > 0. a contradiction. 14 Lower bound: European call value: At any t, t < T, ct Max{ 0, St - Ke-r(T-t)}. Proof: If, to the contrary, ct < Max{ 0, St - Ke-r(T-t)} then, ct < St - Ke-r(T-t) 0 < St - Ke-r(T-t) - ct P/L PROFILE OF AT EXPIRATION STRATEGY: ICF ST < K ST > K SELL THE STOCK St - ST - ST SHORT BUY CALL -ct 0 ST - K LEND NPV OF K -Ke-r(t-t) K K TOTAL ? K - ST 0 15 American vs European Calls The market value of an American call is at least as high as the market value of a European call. Ct ct Max{ 0, St - Ke-r(T-t)}. Proof: An American call may be exercised at any time, t, prior to expiration, t<T, while the European call holder may exercise it only at expiration. 16 Lower bound: American put value: At any time t, prior to expiration, Pt Max{ 0, K - St}. Proof: Assume to the contrary that Pt < Max{ 0, K - St}. Then, buy the put and simultaneously exercise it for an arbitrage profit of: K - St – Pt > 0. A contradiction of the no arbitrage profits assumption. 17 American vs European Puts (Sec. 9.6) Pt pt Max{0, Ke-r(T-t) - St}. Proof: First, An American put may be exercised at any time, t; t < T. A European put may be Exercise only at T. If the price of the underlying asset fall below some price, it becomes optimal to exercise the American put. At that very same moment the European put holder wants to exercise the put but cannot because it is European. 18 Second, the other side of the inequality: At any t, t < T, pt Max{ 0, Ke-r(T-t) - St }. Proof: If, to the contrary, pt < Max{ 0, Ke-r(T-t) - St } then, pt < Ke-r(T-t) – St 0 < Ke-r(T-t) - St - pt P/L PROFILE OF AT EXPIRATION STRATEGY: ICF ST < K ST > K Buy the stock - St ST ST Buy the put - pt K - ST 0 Borrow NPV OF K Ke-r(t-t) -K -K TOTAL ? 0 K ST - 19 American put is always priced higher than its European counterpart. Pt pt P/L For S< S** the European put premium is less than the put’s intrinsic value. For S< S* the K American put premium Ke-r(T-t) coincides with the put’s intrinsic value. P p S* S** K S 20 The put-call parity (Sec. 9.4) European options: The premiums of European calls and puts written on the same non dividend paying stock for the same expiration and the same strike price must satisfy: ct - pt = St - Ke-r(T-t). The parity may be rewritten as: ct + Ke-r(T-t) = St + pt. 21 P/L AT EXPIRATION PROFILE of ICF STRATEGY ST < K ST > K BUY STOCK - St ST ST BUY PUT - pt K - ST 0 TOTAL - [St + pt] K ST P/L AT EXPIRATION PROFILE of STRATEGY ICF ST < K ST > K BUY CALL - ct 0 ST - K – r(T – t) LEND - Ke K K NPV(K) – r(T – t) TOTAL - [ct + Ke ] K ST 22 Synthetic European options: The put-call parity ct + Ke-r(T-t = St + pt can be rewritten as a synthetic call: ct = pt + St - Ke-r(T-t), or as a synthetic put: pt = ct - St + Ke-r(T-t). 23 Synthetic Risk-free rate The put-call parity ct + Ke-r(T-t) = St + pt 1 K r ln T t St p t ct For another synthetic risk-free rate we next analyze the Box Spread strategy: 24 Stock XYZ, TH SEP 21, 2007. All prices $/share. S=61.48 calls puts K OCT NOV JAN APR OCT NOV JAN APR 40 22 23 .56 50 12 .63 55 8.13 11.5 1.25 3.63 60 4.75 8.75 2.88 4.00 5.75 65 2.50 3.98 5.75 8.63 6.00 6.63 8.38 10.00 70 .94 3.88 9.25 11.25 75 .31 5.13 13.38 16.79 80 1.63 90 .81 95 .44 25 1. Stock and options markets P/L of strategy ICF AT EXPIRATION ST < 75 ST > 75 Buy the STOCK -61.48 ST ST Buy APR 75 PUT -16.79 75 - ST 0 Sell APR 75 CALL 5.13 0 75 - ST TOTAL -73.14 75 75 P/L 1.86 1.86 This strategy guarantees its holder a sure profit of $1.86/share for an investment of $73.14/share in a 7 months period. For this strategy to work all the initial prices – the stock the put and the call must be available for the investor at the same instant. If the strategy is possible, it creates a RISK-FREE rate: 26 7 r 73.14e 12 75 1 75 r ln 4.305% 7/12 73.14 27 Suppose that the yield on T-bills that mature on the option’s expiration is r = 5.17%. Then, to make arbitrage profit: P/L of strategy ICF AT EXPIRATION ST < 75 ST > 75 sell the STOCK short 61.48 -ST -ST short APR 75 PUT 16.79 -[75 – ST] 0 long APR 75 CALL -5.13 0 -[75 – ST] Buy T-bills -73.14 75.38 75.38 TOTAL 0 0.38 0.38 0.38 0.38 When the T-bills mature, the GOV pays you: .0517[7/12] 73.14e = 75.38. The above strategy guarantees you an ARBITRAGE PROFIT of 38 cents per share. 28 Another Synthetic Risk-free rate A Box spread (p. 235): K1 < K 2 . P/L PROFILE of AT EXPIRATION ICF STRATEGY ST < K1 K1<ST < K2 ST > K2 Buy p(K2) - p2 K2 - ST K2 - ST 0 Sell p(K1) p1 ST - K1 0 0 Sell c(K2) c2 0 0 - [ST - K2] Buy c(K1) -c1 0 ST – K1 ST - K1 TOTAL ? K2 - K1 K2 - K1 K2 - K1 29 The initial cost of the box spread is: c1 - c2 + p2 - p1 The certain income from the box spread at the options’ expiration, T, is: K2 - K1 Thus: c1 - c2 + p2 - p1 = (K2-K1)e-r(T-t) 30 Reiterating: For the Box spread strategy: An initial investment of c1 - c2 + p2 - p1 yields a sure income of K2-K1regardless of the underlying asset’s market price. Thus, solving c1 - c2 + p2 - p1 = (K2-K1)e-r(T-t) for r, yields a risk-free rate: 1 K 2 K1 r ln T t c1 c2 p2 p1 31 2. Options markets only. A BOX SPREAD. (p. 235) P/L of strategy ICF AT EXPIRATION ST < 55 55 < ST < 60 ST > 60 Buy the JAN 55 CALL - 11.50 0 ST - 55 ST - 55 Buy the JAN 60 PUT - 5.75 60 - ST 60 - ST 0 Sell the JAN 60 CALL 8.75 0 0 60 - ST Sell the JAN 55 PUT 3.63 ST - 55 0 0 TOTAL - 4.87 5.00 5.00 5.00 This strategy guarantees its holder a sure profit of $.13/share for an investment of $4.87/share in a 4 months period. For this strategy to work all the initial prices – the CALLS and the PUTS must be available for the investor at the same instant. If the strategy is possible, it creates a RISK-FREE rate: 32 4 r 4.87e 12 5 1 5 r ln 7.903% 4/12 4.87 Of course, an annual risk-free rate of 7.903% is large and it indicates that one could NOT have been able to create this strategies with the prices given in the table. 33 Summary We have seen that there are strategies that yield synthetic Risk-free rates. 1. The put-call parity yields a risk-free rate that requires inputs from the options market and the stock market. 2. The Box spread yields a risk-free rate that requires inputs from the options market ONLY. Of course, T-bill rates are risk-free. IN AN EFFICIENT ECONOMIC MARKETS ALL THESE RATES MUST BE EQUAL. 34 Summary If the above rates are not equal arbitrage profit exists. You may use a strategy to create a positive, risk-free cash flow; i.e., borrow at the lower risk-free rate, and invest the proceeds in the strategy that yields the higher risk-free rate. The above is exactly what professional arbitrageurs do, mainly, using the Box Spread strategy. 35 Example Suppose that calculating the risk-free rate from a Box spread on slide 26 yields r = 3%. The options are for .5yrs and the 6-month T-bill yield a risk-free rate of 5%. Arbitrage: Borrow money employing the reverse Box Spread ( effectively borrowing at 3%) and invest it in the 6-month T-bill. At expiration receive 5% from the GOV and repay your 3% Debt for An ARBITRAGE PROFIT of 2%. 36 DIVIDEND FACTS: Firms announce their intent to pay dividends on a specific future day – the X-dividend day. Any investor who holds shares before the stock goes – X-dividend will receive the dividend. The checks go out about one week after the X-dividend day. S CDIV S XDIV tAnnouncement t XDIV t PAYMENT Time line 4. S XDIV =SCDIV - D. 37 DIVIDEND FACTS: 1. The share price drops by $D/share when the stock goes x-dividend. 2. The call value decreases when the price per share falls. 3. The exchanges do not compensate call holders for the loss of value that ensues the price drop on the x-dividend date. S CDIV S XDIV tAnnouncement t XDIV t PAYMENT Time line 4. S XDIV =SCDIV - D. 38 The dividend effect Early exercise of Unprotected American calls on a cash dividend paying stock: Consider an American call on a cash dividend paying stock. It may be optimal to exercise this American call an instant before the stock goes x-dividend. Two condition must hold for the early exercise to be optimal: First, the call must be in-the-money. Second, the $[dividend/share], D, must exceed the time value of the call at the X-dividend instant. To see this result consider: 39 The dividend effect The call holder goal is to maximize the Cash flow from the call. Thus, at any moment in time, exercising the call is inferior to selling the call. This conclusion may change, however, an instant before the stock goes x-dividend: Exercise Do not exercise Cash flow: SCD – K c{SXD, K, T - tXD} Substitute: SCD = SXD + D. Cash flow: SXD –K + D SXD – K + TV. 40 The dividend effect Early exercise of American calls may be Optimal if: 1. The call is in the money 2. D > TV. In this case, the call should be (optimally) exercised an instant before the stock goes x-dividend and the cash flow will be: SCD – K = SXD –K + D. 41 Early exercise of Unprotected American calls on a cash dividend paying stock: The previous result means that an investor is indifferent to exercising the call an instant before the stock goes x-dividend if the x- dividend stock price S*XD satisfies: S*XD –K + D = c{S*XD , K, T - tXD}. It can be shown that this implies that the Price, S*XD ,exists if: D > K[1 – e-r(T – t)]. 42 Early exercise of Unprotected American calls on a cash dividend paying stock: D > K[1 – e-r(T – t)]. Example: r = .05 T – t = .5yr. K = 30. 30[1 – e-.05(.5)] = $.74. Thus, if the dividend is greater than 74 cent per share, the possibility of early exercise exists. 43 Explanation C XD SXD Ke r(T t) * S XD SXD SXD K D Ke r(Tt) K -r(T-t) K D -K+D> Ke r(T t) -r(T-t) - Ke D> K[1-e ] 44 Early exercise: Non dividend paying stock It is not optimal to exercise an American call prior to its expiration if the underlying stock does not pay any dividend during the life of the option. Proof: If an American call holder wishes to rid of the option at any time prior to its expiration, the market premium is greater than the intrinsic value because the time value is always positive. 45 Early exercise: Non dividend paying stock The American feature is worthless if the underlying stock does not pay out any dividend during the life of the call. Mathematically: Ct = ct. Proof: Follows from the previous result. 46 American put on a non dividend paying stock It may be optimal to exercise a put on a non dividend paying stock prematurely. Proof: There is still time to expiration and the stock price fell to 0. An American put holder will definitely exercise the put. It follows that early exercise of an American put may be optimal if the put is enough in-the money. 47 The put-call parity of European options with dividends: Consider European puts and calls are written on a dividend paying stock. The stock will pay dividend in the amounts Dj on dates tj; j = 1,…,n, and tn < T. rj = the risk-free during tj – t; j=1,…,n. Then: n D je rT (T t) rj (t j t) c t p t St Ke j1 48 The put-call parity with dividends P/L AT EXPIRATION PROFILE of ICF STRATEGY tj ST < K ST < K SELL STOCK St -Dj - ST - ST SHORT SELL PUT pt ST - K 0 BUY CALL - ct 0 ST - K LEND - Ke-rT(T-t) K K NPV(K) LEND - Dje-rj(tj-t) Dj NPV(DJ) TOTAL 0 0 0 0 49 The put-call “parity” for American options on a non dividend paying stock: (p. 220) At any time point, t, the premiums of American options on a non dividend paying stock, must satisfy the following inequalities: -r(T-t) St - K < Ct - Pt < St - Ke 50 Proof: Rewrite the inequality: St - K < Ct - Pt < St - Ke-r(T-t). The RHS of the inequality follows from the parity for European options: Ct = ct because the stock does not pay dividend. Moreover, For the puts Pt > pt. Next, on the LHS, suppose that: St - K > Ct - Pt i.e., St - K - Ct + Pt > 0. It can be easily shown that this is an arbitrage profit making strategy, and hence Cannot hold. 51 The put-call “parity” for American options on dividend paying stock: Let NPV(D) denote the present value of the dividend payments during the life of the options. n NPV(D) D e rj (t j t) j j1 Then: -r(T-t) St – NPV(D) – K < Ct - Pt < St - Ke 52

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