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## Properties of Stock Option Prices Chapter 8

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**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.**Risk-free borrowing and lending exists at the unique**risk-free rate. Risk-free borrowing is done by sellingT-bills short and risk-free lending is done by purchasing T-bills. 4. There exist no arbitrage opportunities in the options market**NOTATIONS:**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 the following functions: Ct , ct= c{St , K, T-t, r, , D }, Pt , pt= p{St , K, T-t, r, , D }.**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.**Options Risk-Return Tradeoffs**PROFIT PROFILE OF A STRATEGY A graph of the profit/loss as a function of all possible market values of the underlying asset We will begin with profit profiles at the option’s expiration; I.e., an instant before the option expires.**Options Risk-Return Tradeoffs At Expiration**1. Only at expiry; T. 2. No time value; T-t = 0 CALL is: exercised if S > K expires worthless if S K Cash Flow = Max{0, S – K} PUT is: exercised if S < K expires worthless if S K Cash Flow = Max{0, K – S}**3. All legs of the strategy remain open till expiry.**4. A Table Format Every row is one leg of the strategy. Every row is analyzed separately. The total strategy is the vertical sum of the rows.The profit is the cash flow at expiration plus the initial cash flows of the strategy, disregarding the time value of money.**6. A Graph of the profit/loss profile**The profit/loss from the strategy as a function of all possible prices of the underlying asset at expiration.**The algebraic expressions of profit/loss**at expiration: Cash Flows: Long stock: ST – St Short stock: St - ST Long call: -c + MaX{0, ST -K} Short call: c + Min{0, K- ST } Long put: -p + MaX{0, K- ST} Short put: p + Min{0, ST -K}**Borrowing and Lending:**In many strategies with lending or borrowing capital at the risk-free rate, the amount borrowed or lent is the discounted value of the option’s exercise price: Ke-r(T-t). The strategy’s holder can buy T-bills (lend) or sell short T-bills (borrow) for this amount. It follows that at the option’s expiration time, the lender will receive this amount’s face value, namely, a cash flow of K. If borrowed, the borrower will pay this amount’s face value, namely, a cash flow of – K.**RESULTS FOR CALLS:**1.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.**RESULTS FOR CALLS:**2.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 the NPV[Max{ 0, ST – K }] 0. 3.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.**RESULTS FOR CALLS:**4.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 immediately exercise it for an arbitrage profit of: St – K – Ct > 0; a contradiction of the no arbitrage profits assumption.**RESULTS FOR CALLS:**5.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, 0 < St - Ke-r(T-t) - ct At expiration StrategyI.C.FST < KST > K Sell stock short St -ST -ST Buy call - ct 0 ST - K Lend funds - Ke-r(T-t) K K Total ? K – ST 0**RESULTS FOR CALLS:**6. 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.**RESULTS FOR CALLS:**7.The time value of calls: The longer the time to expiration, the higher is the value of a call. Proof: Let T1 < T2 for two calls on the same underlying asset and the same exercise price. To show that c2 > c1 assume that c1 > c2 or, c1 - c2 > 0. At expiration T1 StrategyI.C.FST1 < KST1 > K Sell c(T1 ) c1 0 -(ST1 –K) Buy c(T2 ) - c2 c c Total ? c ?**RESULTS FOR CALLS:**8.Cash dividends and calls: 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.**RESULTS FOR CALLS:**9. 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 result 8. above.**RESULTS FOR CALLS:**10.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:**RESULTS FOR CALLS:**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. SCUMD SXDIV tA tXDIV tPAYMENT Time line 4. SXDIV = SCDIV - D.**Early exercise of Unprotected American calls on a cash**dividend paying stock: 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: ExerciseDo not exercise Cash flow: SCD – K c{SXD, K, T - tXD} Substitute: SCD = SXD + D. Cash flow: SXD –K + D SXD – K + TV.**Conclusion:**Early exercise of American calls may be optimal: If the call is in the money and If D > TV, early exercise is optimal. In this case, the call should be (optimally) exercised an instant before the stock goes x-dividend.**Early exercise of Unprotected American calls on a cash**dividend paying stock: This 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)].**RESULTS FOR CALLS:**11.The money value of calls: The higher the exercise price, the lower is the value of a call. Proof: Let K1 < K2 be the exercise prices for two calls on the same underlying asset and the same time to expiration. To show that c2 < c1 assume, to the contrary, that c2 > c1 or, c2 - c1 > 0. Then,**RESULTS FOR CALLS:**At expiration StrategyICFST< K1K1<ST < K2 ST >K2 Sell c(K2 ) c2 0 0 -(ST –K2) Buy c(K1 ) -c1 0 ST –K1 ST–K1 Total ? 0 ST –K1 K2 - K1**RESULTS FOR CALLS:**12.The money value of American calls: Let K1 < K2 then C1 - C2 K2 - K1 Proof: Let K1 < K2 be the exercise prices for two American calls on the same underlying asset and the same time to expiration. Assume that C1 - C2 > K2 - K1 or, equivalently: C1 - C2 – (K2 - K1) > 0. Then,**RESULTS FOR CALLS:**At expiration StrategyICFST< K1K1<ST < K2 ST >K2 Sell C(K1 ) C1 0 -(ST –K1) -(ST –K1) Buy C(K2 )-C2 0 0ST–K2 Lend – (K2 - K1) K2-K1+i K2- K1+i K2- K1+i Total ? K2-K1+i K2-ST+i i i = interest Even if the sold call is exercised before Expiration, the total value in hand is >0.**RESULTS FOR CALLS:**13.The money value of calls: Let K1<K2 then c1-c2 (K2-K1)e-r(T-t) Proof: Let K1 < K2 for two European calls On the same underlying asset and the same time to expiration. Assume that c1-c2 > (K2-K1)e-r(T-t) or, c1-c2 -(K2-K1)e-r(T-t) >0. Then,**RESULTS FOR CALLS:**At expiration StrategyICFST< K1K1<ST < K2 ST >K2 Sell c(K1 ) c1 0 -(ST –K1) -(ST –K1) Buy c(K2 ) -c2 0 0ST–K2 Lend -(K2-K1)e-r(T-t) K2-K1 K2-K1 K2-K1 Total ? K2-K1 K2- ST 0**RESULTS FOR CALLS:**14.The money value of calls: Let K1 < K2 < K3 and K2 = K1 + (1 - )K3 for any : 0 < < 1. The premiums on the three calls must satisfy: c2 c1 + (1 - )c3**RESULTS FOR CALLS:**At Expiration STRATEGY ICFST < K1K1<ST < K2K2 <ST < K3ST > K3 Buy calls K1 -c1 0 (ST – K1) (ST – K1) (ST – K1) Sell one call K2 c2 0 0 -(ST – K2) -(ST – K2) Buy 1- calls K3 -(1-)c3 0 0 0 (1-)(ST–K3) Total c2-c1-(1-)c3 0 (ST–K1) (1-)(K3-ST) 0 All the cash flows at expiration are non negative. Hence, the Initial Cash Flow cannot be positive! c2-c1-(1-)c3 0. Or, c2 c1 + (1 - )c3.**RESULTS FOR CALLS:**Example: Let $80, $95 and $100 be the three exercise prices. $95 = (.25)$80 + (.75)$100. Thus, = ¼. Result 14. Asserts that c(95) ¼c(80) + ¾c(100). Or, 4c(95) c(80) + 3c(100). If the latter inequality does not hold, then 4c(95) > c(80) + 3c(100) or, 4c(95) - c(80) - 3c(100) > 0 and arbitrage profit can be made by the Strategy: Buy the $80 call and sell four $95 Calls and buy three $100 calls.**RESULTS FOR CALLS:**15.Volatility: The higher the price volatility of the underlying asset, the higher is the call value. Proof: The call holder never loses more than the initial premium. The upside gain, however, is unlimited. Thus, higher volatility increases the potential gain while the potential loss remains unchanged.**RESULTS FOR CALLS:**16.The interest rate: The Higher the risk-free rate, the higher is the call value. Proof: The result follows from result 6: Ct ct Max{ 0, St - Ke-r(T-t)}. With increasing risk-free rates, the difference St - Ke-r(T-t) increases and the call value must increase as well.**RESULTS FOR PUTS:**17.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.**RESULTS FOR PUTS:**18.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 The NPV[Max{ 0, K - ST}] 0.**RESULTS FOR PUTS:**19a.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. 19b.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}.**RESULTS FOR PUTS:**20.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 immediately exercise it for an arbitrage profit of: K - St – Pt > 0. A contradiction of the no arbitrage profits assumption.**RESULTS FOR PUTS:**21.Lower bound: European put value: At any time 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, Ke-r(T-t) - St - pt > 0. At expiration Strategy I.C.FST < KST > K Buy stock -StSTST Buy put - pt K - ST 0 Borrow Ke-r(T-t) - K - K Total ? 0 ST - K**RESULTS FOR PUTS:**22. An American put is always priced higher than an European put. Pt pt Max{0, Ke-r(T-t) - St}. Proof: An American put may be exercised at any time, t, prior to expiration, t < T, while the European put holder may exercise it only at expiration. 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 (optimally) exercise the put but cannot because it is a European put.**For S< S** the European put premium is less than the**put’s intrinsic value. For S< S* the American put premium coincides with the put’s intrinsic value. P/L RESULTS FOR PUTS: 23. American put is always priced higher than its European counterpart. Pt pt K Ke-r(T-t) P p S* S** K S**RESULTS FOR PUTS:**24.The time value of puts: The longer the time to expiration, the higher is the value of an American put. Proof: Let T1 < T2 for two American puts on the same underlying asset and the same exercise price. To show that P2 > P1 assume, to the contrary, that P2 < P1 or, P1 - P2 > 0. At expiration T1 StrategyI.C.FST1 < KST1 < K Sell P(T1 ) P1ST1 –K 0 Buy P(T2 ) -P2 P P Total ? ? P**RESULTS FOR PUTS:**25.The money value of puts: The higher the exercise price, the higher is the value of a put. Proof: Let K1 < K2 for two puts on the same underlying asset and the same time to expiration. To show that p2 > p1 assume, to the contrary, that p2 < p1 or, p1 - p2 > 0. Then,**RESULTS FOR PUTS:**At expiration StrategyICFST< K1K1<ST < K2 ST >K2 Sell p(K1 ) p1ST –K1 0 0 Buy p(K2 ) -p2 K2 –ST K2 - ST 0 Total ? K2 - K1 K2 - ST 0**RESULTS FOR PUTS:**26.The money value of puts: Let K1 < K2 then P2 - P1 K2 - K1 Proof: Let K1 < K2 be the exercise prices or two American puts on the same underlying asset and the same time to expiration. Assume, contrary to the result’s assertion, that P2 - P1 > K2 - K1 or, equivalently: P2 - P1 – (K2 - K1) > 0. Then,**RESULTS FOR PUTS:**At expiration StrategyICFST< K1K1<ST < K2 ST >K2 Sell P(K2 ) P2ST –K2ST –K2) 0 Buy P(K1 )-P1 K1 – ST 0 0 Lend – (K2 - K1) K2-K1+i K2- K1+i K2- K1+i Total ? i ST-K1 +i K2- K1+i i = interest Even if the sold put is exercised before expiration, the total value in hand is >0.**RESULTS FOR PUTS:**27.The money value of puts: Let K1<K2 then p2-p1 (K2-K1)e-r(T-t) Proof: Let K1 < K2 for two European puts on the same underlying asset and the same time to expiration. Assume, contrary to the result’s assertion, that p2-p1 > (K2-K1)e-r(T-t) or, p2- p1 -(K2-K1)e-r(T-t) >0. Then,**RESULTS FOR PUTS:**At expiration StrategyICFST< K1K1<ST < K2 ST > K2 Sell p(K2 ) p2ST –K2ST –K2 0 Buy p(K1 ) -p1 K1 – ST 0 0 Lend -(K2-K1)e-r(T-t) K2-K1 K2-K1 K2-K1 Total ? 0 K2- ST K2-K1 In the absence of arbitrage, the initial cash flow cannot be positive.**RESULTS FOR PUTS:**28.Volatility: The higher the price volatility of the underlying asset, the higher is the put value. Proof: The put holder never loses more than the initial premium. The upside gain, however, is increasing from zero to K. Thus, higher volatility increases the potential gain while the potential loss remains unchanged.