選擇權(Options) 目標： 1 名詞解釋 2 風險、報酬率：Options與Underlying assets之比較 3 如何畫Payoff圖 4 Put-call parity 5 Option valuation方法：何者錯誤、何者正確 6 發行Options與避險 7 Options與underlying assets市場之波動性, Cascade (瀑布) effect

1:名詞解釋 選擇權 : 標的物(underlying asset): 買權 : 在約定之期間內有權力(但不是義務)以 約定之價格購買某一證券。 賣權 : 在約定之期間內有權力以約定之價格 賣出某一證券。 履約價格 : 選擇權持有人購買(或賣出)標的物 (underlying asset)之事先約定價格 Exercise price or strike price

台指選擇權. 台灣加權指數= 5844.76 on 2004/11/30 收盤

其他重要名詞 到期日 : 能履行選擇權之最後一日。 權利金 、 選擇權的價格(price)或價值 (value) 美式選擇權VS.歐式選擇權 與K之關係 ( =標的物之價格): 與K之關係 Call Put S > K S < K S = K 價內 價外 價平 價外 價內 價平

Cont’d “In the money” “Out of the money” “At the money” Call Put S= underlying asset’s price. K= exercise price Call Put S > K S < K S = K in-the-money out-of-the-money at-the-money out-of-the-money in-the-money at-the-money

Options vs. futures Obligation or right ? How to ensure zero default risk ? Risk and return characteristics

2: Profit/loss profile of options: (1) buying a call option vs. ___ Price of stock X= 100. Call option price=2 Strike price=120. Time to expiration=1 year Price of X at expiration Net profit/loss Buy a call Ret. Buy … Ret. $ 150 30-2 140 20-2 120 -2 100 -2 98 -2 90 -2 80 -2 60 -2 Which is more risky on an investment-return basis ? Figure =>

Profit/loss profile: (2) a long put position vs. ____ Price of stock X= 100. Put option price=2 Strike price=100. Time to expiration=1 year Price of X at expiration Net profit/loss Long put Ret. Short stock Ret. $ 150 -2 -50 140 -2 -40 100 -2 0 99 -1 1 98 0 2 90 8 10 80 18 20 60 38 40 Which is more risky on a return basis ? Figure =>

Intrinsic value, time value (時間價值) 1. Components of option price: = Intrinsic value + time value Eg: Eg: If the option is out of the money, 2. Upper bound : stock price, S 3. Lower bound : max (0, S-K) 4. If the stock ( of the company ) is bankrupt, i.e. stock value= 0, the option value = 0 .

3: Payoff圖(profit/loss profile)(暫不考慮option prc) 舉例說明如下 (underlying asset = a stock): 買權的 payoff t=1 1.買權 K=65 to buyer (或holder) = max(0, S -k) 65 標的物價格S,t=1 賣權的 payoff t=1 2.賣權 K=65 to buyer =max(0, k- S ) 65

3. Underlying stock to buyer Your payoff t=1 3. Underlying stock to buyer Your payoff t=1 4.一單位賣權 + 一單位股票 to the buyer 65

7.若考慮成本(即選擇權價格, 但暫不考慮現值之概念) 其他例子 5.選擇權賣方的報酬? 6. Eg: call k=65 to the seller 65 7.若考慮成本(即選擇權價格, 但暫不考慮現值之概念) Eg: Call k=65 cost of the call=2 to buyer 65 67

討論: 1.是否考慮選擇權的價格? =>需一致 2.在推導put-call parity時, 不考慮選擇權的價格 => ∵_____________ 3.如果考慮選擇權的價格,報酬圖如何畫? 4.為什麼報酬圖是重要的? (1)清楚知道自己的部位. 對風險控制很重要. (2) cp. futures (3) to derive put-call parity.

4: Put-call parity P + S = C + PV( k ) 其中 C : 買權的價值 K : 履約價 PV(•) : 無風險利率所估計的現值 P : 賣權的價值 S : 標的資產的價格(Eg:股價) C 與 P: underlying asset相同? K相同? Expiration date相同?

= Why? +) Stock k Payoff of call at t=1 Payoff of put at t=1 Payoff of gov. bond at t=1 k Payoff of put at t=1 Stock price t=1 Stock = Why?

∴ PV( Call + k ) = PV( put + stock ) ∴ C + PV(k) = P + S Q.1: 假設你有股票,買權,T-bills, 你可以創造出賣權嗎? Q2.Eg: If S=K , (股價=履約價), 買權價格 > 賣權價格 為什麼? Sol: C-P = S-PV(k). When S=K, S-PV(k)>0 ∴C-P >0

Time value, intrinsic value Value of Call 股票的報酬 : 上限 K 下限 max (0, S-K ) (time value) 時間價值 S 選擇權價格= intrinsic value + 時間價值

2 components of option price Option price= intrinsic value + time value Intrinsic value =標的資產 之當期價格與 履約價之價差所產生之價值 if 履約是划算的 Time value = option price – intrinsic value How to determine option price (premium) ?

5: 影響選擇權價格的五個因素 買權價格 賣權價格 風險 () + + 履約價 - + 到期日 + + 標的資產的價格 + - 無風險利率

5: Option price (or premium) 例: t=0 (now) t=1 (1 year later) . up ABC stock $65/share $81.25 (機率=0.6) $52 (機率=0.4) down 買權,履約價格 = $70. 無風險利率= 2.5% (ABC股票的報酬 = 7%), Call option on TSM stock. Expiration: 1 year later. 買權的價格是多少 ? Sol: Several possible methods: Approach (1) :

Approach (1): NPV (淨現值) Step 1: Calculate E(r) of ABC stock (=7%) Step 2: Determine E( payoff) of the call (=6.75) Step 3: Call’s price= PV of E(payoff) of call using E(r) of ABC stock Step 3 assumes that call’s risk = ABC stock’s risk. Correct ?

Approach (2): HEDGE portf. or replication port. Call’s payoff = a function of underlying asset’s payoff = max[0, X-k] where X= underlying asset’s value at that time, k= exercise price So, call’s payoff depends on underlying asset’s payoff. Call’s price= C (a random variable) Underlying asset’s value = X (a random variable) Thus, we use X to eliminate uncertainty of C and get a portfolio with a certain [i.e., risk-free] payoff.

No套利 (no free lunch, market efficiency) : Approach (2) No套利 (no free lunch, market efficiency) : 例: t=0 t=1 . ABC stock $65/share $81.25 (機率=0.6) $52 (機率=0.4) up down 買權,履約價格 = $70. 無風險利率= 2.5% (ABC股票的報酬 = 7%) 買權的價格是多少 ?

SOL: Sol: t=1時買權的payoff: Call 81.25-70 = 11.25 up down 需 相 同 複製法:找一組投資組合,使其payoff = call’s payoff t = 0 t = 1 stock up stock down 買X 股 ABC股票 賣PV(52X)T-bill +81.25x -52x +52x 11.25 (國庫帣)

(continued) ∴ 81.25x-52x=11.25 x = 0.385= Spread of call payoff/spread of ABC payoff = Call option’s delta (意義?) ∴ 複製之投資組合(replication portfolio): 買入 0.385 股 ABC股票並且賣出gov’t bond PV(520.385) = 520.385(1+2.5%) = 19.53 1 ∴買權的價值= 0.385 65-19.53=5.495

Q. Q: 例子中, 股票價格怎麼只有2個可能之價格 ? Q: 如果市場上option 之交易價格不等於理論所算出之價格呢?

(continued) 81.25X-52X=18 X = 0.615 = Spread of Put payoff/ Spread of ABC payoff = Put option’s delta ∴ 複製投資組合: 賣出 0.615 股ABC股票 而且 買入 T-bill 81.250.615 (1.025) = $ 48.78 ∴ 賣權的價格 = -0.615  65+48.78 = 8.805 練習: 如上例, What is the option price when the exercise price of call option = $60 ?

Approach 3: risk-neutral approach (風險中立法) 將真實機率系統(probability measure)轉換成另一個機率系統. 步驟: 假設所有的投資人為風險中立者. 找到假定之機率 p* 使得Ep*(rstock) = rf. 使用假定機率計算選擇權的期望值at t=1 . 使用 rf 來計算選擇權期望值的現值 = 選擇權的價格

How about put option’s price? Eg:如上例ABC股票,履約價為$70的賣權之價格= ? Sol: put 70-52=18 up down ∴ 複製之投資組合: t=0 買入 x 股的股票 賣出 PV(81.25x) T-bill t=1 ABC up ABC down +81.25x -81.25x +52x 0 18 - + 相同 賣出 買入

Eg. 如上例: ABC Stock $65 $81.25 up $52 rf=2.5% down call option ,k=$70, 買權的價格? Sol: Find p* such that: ∴ p* = 0.5 1-p* = 1-0.5 = 0.5 這並非ABC股票的真實機率.

(continued) 買權的期望值at time 1 = 上漲機率 11.25 + 下跌機率  0 = 0.5 11.25 = $5.625 ∴ 買權的價值 (at t=0) = Eg: 用風險中立法算put option put option k=$70 70-52=18 ABC up ABC down

Sol: ∴ Based on p* , 賣權的期望值= 0×0.5 + 18×0.5= 9 Put option price= 9/1.0251=8.78

討論 我們可以使用 P**, r** , e.g. r**=0.07 來計算選擇權的價格嗎?

Black-Scholes Model: 假設: 1. 交易成本≒0. 稅≒0 2. 允許賣空(shortselling). 3. rf 是常數(constant) ( 當 rf 是隨機 ?) 4. 標的資產不配發現金股利. 5. 歐式選擇權 6. S~指數常態 (i.e. log(S)~N(·, ·))

買權之價格 = N(d1)  S - N(d2)  PV(K) 公式與delta 買權之價格 = N(d1)  S - N(d2)  PV(K) S = 標的證券之現價(current price) k = 履約價格 N(d1) , N(d2)為標準常態分配的累積機率函數(c.d.f.) PV(K) = K的現值 (discount rate= rf) (t = time to maturity)

Eg. Eg: Intel 股票的 σ=32% (年報酬率之標準差) 履約價 =$65, 股價=$65, 到期日=6 個月 rf=2.5% for 6 個月 買權的價格? 賣權的價格? => Excel : N(d1), N(d2)： ∴ Buying a call = Buying  shares,and borrowing the balance EXCEL: 例如：=@NORMSDIST(1.645)

Exotic options Either-or option Out-performance option For example: an outperformance option: Investor will gain if S&P 500 return > FTSE 100 return over a six-month period If S&P return < = FTSE 100 return over this time period, this option will expire worthless.

6: 發行Options與避險 討論: Buying a call option = buying N(d1)(i.e所謂之 call option’s delta, 券商如何 hedge ? ) 同時 借N(d2) of PV(K) 之金額 (at rf ) Q: 某一證券發行ABC stk的買權( i.e. 賣給投資 人買權) 假如ABC股價, 如何部份避險 =>根據 call option delta買入____.

7: Delta hedging, portfolio insurance, cascade (瀑布) effect If you sell call option and do delta-hedging, When stock price , you … When stock price , you … If you sell call on gov’t bonds and do delta-hedging, When bond price , you … When bond price , you … This is why options can be blamed for increases in volatility in financial markets. => Cascade (瀑布) effect

Empirical evidence on option pricing B/S formula is generally OK. B/S formula tends to undervalue deep in-the-money calls and overvalue deep out-of-the-money calls. Why ? B/S assumes no cash dividends. B/S formula performs worst for high div. Modified formula performs well for high div.