Microsoft PowerPoint - Ch9 Properties of Stock Options.ppt

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1 Notations Ch 9 股票選擇權的特性 Properties of Stock Options C : The value of American call option P : The value of American put option c : The value of European call option p : The value of European put option K : The strike (exercise) price T : The time to maturity σ: The volatility of the stock price r : The risk-free interest rate q : The dividend yield rate of the stock Black-Scholes Pricing Formulas Merton s Pricing Formulas rt c = S N ( d ) Ke N ( d ) where 1 1 rt p = Ke N ( d ) S N ( d ) d d = ln( / ) + ( + σ / ) S K r T σ T S K + r T ln ( / ) ( σ / ) = = d1 σ T 1 σ T qt c= Se Nd ( ) Ke Nd ( ) 1 rt 1 rt p = Ke N( d ) Se N( d ) where d = qt 1 ln( / ) + ( + / ) S K r q σ T σ T ln( S/ K) + ( r q σ / ) T d = = d1 σ T σ T 1

2 影響選擇權價格的因素 Factors Affecting Option Prices Summary 影響選擇權價格的因素有六 : S : The current stock price K : The strike (exercise) price T : The time to maturity σ: The volatility of the stock price r : The risk-free interest rate q : The dividend yield rate of the stock 變數 S 0 K T σ r q c p C P ?? 股價與履約價 Stock Price and Strike price The final payoff of a call option: C(T)= max( S(T) K, 0) S C K C The final payoff of a put option: P(T) =max( K - S(T), 0) S P K P 存續期間 (Time to Maturity) 美式選擇權距到期日越長, 價值越大持有權力越長, 越值錢歐式選擇權則不一定歐式買權 : 如果不發放股利, 距到期日越長, 價值越大若在持有期間內, 發放大量股利, 則距到期日越長, 價值不一定越大歐式賣權 : 如果在價外 (OTM), 距到期日越長, 價值越大如果在深價內 (DITM), 距到期日越長, 價值越小

3 波動度 (Volatility) 詳盡的說明, 於 Ch1 介紹波動度可以想成是標的資產價格的變化程度波動度越大, 表示標的資產價格的變化程度越大波動度越大, 選擇權價值越大若是持有股票, 標的資產價格變化越大, 可能獲利很多, 也可能損失很多若是持有選擇權, 標的資產價格變化越大, 只可能獲利增加, 損失至多為 0 無風險利率 (Risk-free Interest Rate) 利率變動對選擇權價值的影響, 無確定答案利率上升, 其他因素不變 : 後兩者較強勢 r C & P r C & P 折現率上升, 所以 C 與 P 的價值下降 C 可以延遲支付, 收取的利息增加,C 價值上升 P 延遲標的賣出, 利息成本增加,P 價值下降實務上, 利率與股價呈反向變動所以利率上升, 股價會下跌, 因此 : C P 股利 (Dividends) 股利股價股利 C 股利 P 選擇權價格的上下界 Upper and Lower Bounds for Option Prices 3

4 Assumptions 上界 (Upper Bounds) There are no transaction costs. All trading profits (net of trading losses) are subject to the same tax rate. Borrowing and lending are possible at the riskfree interest rate. c S, C S 買權給予投資人購買某標的資產的權利, 故該權利價值不應超過現貨價格, 否則以現貨價格購入即可 C 最好是在 K=0, 該權利允許免費持有 S, 因此該權利最大價值是 S, 若超過, 直接在市場上買 p K, P K 賣權給予投資人出售某標的資產的權利, 故該權利價值不應超過履約價格 P 最好是在 S=0, 可以履約價賣, 收入 K 因此 P 的最大價值就是 K 超過 K, 透過 P 賣收入為負, 不如不要賣 p K exp(-rt) 無現金股利的股票買權價格下界 Lower Bound for Calls on Non-Dividend-Paying Stocks Example 9.1 歐式買權的下界 : Max ( S K exp(-rt), 0 ) 證明 : Portfolio A : 一單位買權的長部位 Portfolio B : 買一單位股票並借入現金 K exp(-rt) 到期時 (time T), 兩個投資組合的到期現金流量為 : 若 S(T) K: A = (S(T) - K) ; B = S(T) K A=B 若 S(T) < K: A= 0 ; B= S(T) K < 0 A > B Payoff A Payoff B 因此在無套利機會下, Price A > Price B. c S - K exp(-rt) c S - K exp(-rt) c Max ( S K exp(-rt), 0 ) S=0, K=18, r=10%, T=1 S K exp(-rt) = 3.71 若 c = 3 ( 太低 ), 有套利機會存在 : 期初 : 賺取 = 0.71 放空一單位股票買一單位買權存入 K exp(-rt) = 16.9 期末 :Payoff 恆大於 0!! 若 S(T) K=18, -S + (S-K) + K = 0 若 S(T) < K=18, 假設 S(T)=17, K = 1 4

5 Example 9. 無現金股利的股票賣權價格下界 Lower Bound for Puts on Non-Dividend-Paying Stocks Consider a European call option on a non-dividend-paying stock when the stock price is $51, the strike price is $50, the time to maturity is 6 months, and the risk-free rate of interest is 1% per annum. In this case, S=51, K=50, T=0.5, r=0.1. A lower bound for the option price is S-K exp(-rt) = exp( ) = $3.91 歐式賣權的下界 : Max (K exp(-rt) - S, 0 ) 證明 : Portfolio C : 一單位賣權的長部位 Portfolio D : 現金存款 K exp(-rt) 並放空一單位股票到期時 : Payoff C Payoff D 若 S(T) < K: C = K - S(T) ; D = K S(T) C=D 若 S(T) K: C = 0 ; D= K S(T) < 0 C > D 無套力機會存在下, Price C > Price D. p K exp(-rt) - S p K exp(-rt) - S p Max (K exp(-rt) S, 0 ) Example 9.3 Example 9.4 S=37, K=40, r=5%, T=0.5 K exp(-rt) - S =.01 若 p = 1 ( 太低 ), 有套利機會存在 : 期初 : 賺取 = 1.01 進入一單位賣權長部位買入一單位股票借款 K exp(-rt) =39.01 期末 :Payoff 恆大於 0 若 S(T) > K, 假設 S(T)=43: =3 >0 若 S(T) < K: (K-S(T)) +S(T) K =0 Consider a European put option on a non-dividend-paying stock when the stock price is $38, the strike price is $40, the time to maturity is 3 months, and the risk-free rate of interest is 10% per annum. In this case, S=38, K=40, T=0.5, r=0.1. A lower bound for the option price is K exp(-rt) - S = 40 exp( ) - 38 = $1.01 5

6 買權賣權平價公式 (Put-Call Parity) Put-Call Parity for European Options: 證明 : c(t) + K exp(-r (T-t)) = p(t) + S(t) At time 0: Portfolio A: + c + K exp(-rt) Portfolio B: + p + S At time T: Both are worth max( S(T), K). 兩投資組合在期末有相同收益, 又因為是歐式選擇權, 到期前不能提前履約, 所以在無套利機會下, 兩投資組合期初價值相同買權賣權平價公式 (Put-Call Parity) Put-Call Parity for European Options: c(t) + K exp(-r (T-t)) = p(t) + S(t) Put-call parity 說明了現貨買權賣權三者之間的關係 Put-call parity 若不成立, 則會有套利機會出現 Example S=31, r=10%, T=0.5y,c = 3, K=30 p = 1.6 If p =.5 (Too High!), then 期初 : 賺取 =0.99 一單位買權長部位, 放空一單位股票, 持有存款 K exp(-rt) = 9.6 放空一單位賣權期末 : S(T) > K=30 (S(T) K ) - S(T) +K -0 = 0 S(T) < K =30 0 S(T) +K (K-S(T)) = 0 Example S=31, r=10%, T=0.5y,c = 3, K=30 p = 1.6 If p =1 ( 太低 ) 期初賺取 : =0.6 進入一單位賣權長部位進入一單位買權短部位, 借款 K exp(-rt) = 9.6, 買進一單位股票期末時 : S(T) > K= (K-S(T)) K +S(T) = 0 S(T) < K =30 (K-S(T)) +0 K + S(T) = 0 6

7 Put-Call Parity for American Options S K C - P S- K exp(-rt) See Problem Business Snapshot 9.1 Put-Call Parity and Capital Structure 某公司以零息債券與股票來做為資產的融資來源假設零息債券面額為 K,5 年後到期假設公司不發行股利假設公司 5 年後資產價值為 A 5 年後, 股東權益為 Max( A(T) -K, 0), 等於股東握有一個買權, 標的為公司資產價值 A(T), 履約價為負債價值 K: 若公司資產價值 A(T) 大於 K, 則股東會償還債務若公司資產價值 A(T) 小於 K, 則股東會宣布破產, 而不清償債務 5 年後, 債券價值為 Min (A(T), K) = K- Max( K-A(T), 0): 債權人到期時, 除了債權 K, 還給予股東一個賣權, 此賣權標的為公司資產, 履約價為公司負債 K : 期初, 公司資產負債關係為 : 股東權益 = c 負債 = PV(K) p 資產 A(0) = c+ PV(K) p Put-Call Parity : c + PV(K) = p + A(0) Early Exercise: Calls on a Non-Dividend-Paying Stock Variation of an American or European Call on a Non-dividend-paying Stock with the Stock Price S 若標的是沒有發放股利的股票, 則美式買權的最適履約時點為到期日, 即不提前履約理論上, 若標的是沒有發放股利的股票, 則美式買權等於歐式買權因為不發放股利, 所以股價不會因發放股利大幅下降, 所以延後履約可以賺取利息選擇權價值 = 內含價值 + 時間價值, 馬上履約損失了時間價值若一定要履約, 以持有股票則最佳策略是將買權在市場賣掉, 然後在於市場上購買 c S K exp(-rt), C c, Call option price As r or T or the volatility increases, the line relating the call price to the stock price moves in the direction indicated by the arrows. C S K exp(-rt) C S K ( 內含價值 ) K Stock price, S 7

8 Early Exercise: Puts on a Non-Dividend-Paying Stock 若標的是沒有發放股利的股票, 則美式賣權提前履約是可行的事實上, 當賣權是深價內時, 提前履約才是明智之舉極端例子 : 假設股價為 0, 則賣權最大利潤為 K, 馬上履約即可賺取最大利潤美式賣權價值大於歐式賣權價值, 即 P > p Put option price Variation of Price of an American Put with the Stock Price S 當股價很低時, 提前履約美式賣權是有利的假設 A 為臨界點, 當股價低於 A 時, 會提前履約因此, 當 S < A, 賣權提前履約, 所以賣權價值等於內含價值 = K S When r decreases, when T or the volatility increases, the line relating the call price to the stock price moves in the direction indicated by the arrows. A K Stock price, S Variation of Price of an European Put with the Stock Price S Put option price E 因為美式賣權價值大於歐式賣權, 又因為美式賣權在 S < A 時, 價值等於內含價值, 因此歐式賣權價值可能低於內含價值亦即, 時間價值可能為負 A < B E=K exp(-rt) 股利的影響 Effect of Dividends A B K Stock price, S 8

9 股利的影響 (Effect of Dividends) 之前都是在無股利發放的情形下討論, 以下討論有股利發放的情形在美國, 交易所交易的股票選擇權, 到期日一般皆小於 1 年期間, 股票所發放的股利可以準確的預估以下用 D 來表示選擇權存續期間, 所發股利的現值歐式買權與賣權的下界買賣權的下界 : c S D K exp(-rt) p D + K exp(-rt) S 證明如前 : European Call Portfolio A: + c + D + K exp(-rt) Portfolio B: + S European Put Portfolio C: + p + S Portfolio D: + D + K exp(-rt) 提前履約 (Early Exercise) Put-Call Parity 在有發放股利的情形下, 美式買權提前履約可能是有利的因為在除息日時 (Ex-Dividend Date), 股價會因股利發放而突然下降, 造成買權價值下降, 因此在除息日前提前履約, 可能是有利的美式買權的最適履約時點, 僅有可能在除息日前 c + D + K exp(-rt) = p + S S D K C P S K exp(-rt) 9

10 Exercises,3,7,9,10,11,1,14,16, 10

Microsoft PowerPoint - Ch10 Trading Strategies Involving Options.ppt

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