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Unformatted text preview: 1 MECHANICS OF OPTIONS MARKETS Options, Futures, and Other Derivatives, 7th Ed., John C. Hull (2008):
Chapter 910 Payoffs from Options What is the Option Position in Each Case? 2 K = Strike price, ST = Price of asset at maturity Payoff K long call Payoff K long put Payoff
K
ST ST short call Payoff
K ST ST short put Lower bounds for option prices 3 Lower Bound for European Call Option Prices; No Dividends: c max(S0 –Ke –rT, 0) Lower Bound for European Put Prices; No Dividends: p max(Ke rT–S0, 0) European options c + Ke ‐rT = p + S0 Types of Strategies 4 Suppose the market is generally expected to be bullish. So the price of a XYZ stock is expected to rise. However, your client is skeptical about the price rising more than a moderate amount and is therefore reluctant to pay for such an eventuality. Devise a strategy for your client that will best meet his needs. Suppose call options on XYZ stock with strike prices $30 and $35 cost $7 and $4, respectively. 4 Butterfly Spread Using Calls 5 Assume K2 is the midpoint of K1 and K3 Profit
K1 K2 K3 ST 5 A Straddle Combination 6 Profit K ST 6 Strangle 7 Payoff K1 K2 ST 7 A Simple Binomial Model 8 A stock price is currently $20 In 3 months it will be either $22 or $18 We set up a portfolio such that: It’s value at the end of three months is certain Arbitrage opportunities do not exist Stock Price = $22
Stock price = $20
Stock Price = $18 8 A Call Option 9 A 3‐month call option on the stock has a strike price of 21. Stock Price = $22
Option Value = $1
Stock price = $20
Option Price=?
Stock Price = $18
Option Value = $0 9 Setting Up a Riskless Portfolio 10 Consider the Portfolio: long shares short 1 call option 22– 1 18 10 Valuing the Portfolio 11 The riskless portfolio is: long 0.25 shares short 1 call option The value of the portfolio in 3 months is 22 0.25 – 1 = 4.50 11 Valuing the Option 12 The portfolio that is long 0.25 shares short 1 option is worth 4.367 The value of the shares is 5.000 (= 0.25 20 ) 12 Generalization 13 A derivative lasts for time T and is dependent on a stock S0
ƒ S 0u
ƒu
S 0d
ƒd
13 Generalization (continued) 14 Consider the portfolio that is long shares and short 1 derivative S0u– ƒu S0d– ƒd The portfolio is riskless when S0u– ƒu = S0d– ƒd or ƒu fd S 0u S 0 d
14 Delta 15 Delta () is the ratio of the change in the price of a stock option to the change in the price of the underlying stock The value of varies from node to node 15 Generalization (continued) 16 Value of the portfolio at time T is S0u– ƒu Value of the portfolio today is (S0u – ƒu)e–rT
Another expression for the portfolio value today is S0– f
Hence ƒ = S0– (S0u– ƒu )e–rT 16 Generalization (continued) 17 Substituting for we obtain ƒ = [ pƒu + (1 – p)ƒd ]e–rT where e rT d
p
ud 17 Irrelevance of Stock’s Expected Return 18 When we are valuing an option in terms of the price of the underlying asset, the probability of up and down movements in the real world are irrelevant This is an example of a more general result stating that the expected return on the underlying asset in the real world is irrelevant 18 p as a Probability 19 It is natural to interpret p and 1‐p as probabilities of up and down movements The value of a derivative is then its expected payoff in a risk‐
neutral world discounted at the risk‐free rate S0
ƒ S0u
ƒu
S0d
ƒd
19 Risk‐Neutral Valuation 20 When the probability of an up and down movements are p and 1‐p the expected stock price at time T is S0erT
This shows that the stock price earns the risk‐free rate Binomial trees illustrate the general result that to value a derivative we can assume that the expected return on the underlying asset is the risk‐free rate and discount at the risk‐free rate This is known as using risk‐neutral valuation 20 Original Example Revisited 21 S0 ƒ S0u = 22
ƒ = 1
u S0d = 18
ƒd = 0 Since p is the probability that gives a return on the stock equal to the risk‐free rate. We can find it from 20e0.12 0.25 = 22p + 18(1 – p )
which gives p = 0.6523 Alternatively, we can use the formula e rT d
e 0.12 0.25 0 .9 0 .6523
p
1 .1 0 .9
ud
21 Valuing the Option Using Risk‐Neutral Valuation 22 S0 ƒ S0u = 22 ƒu = 1 S0d = 18
ƒd = 0 The value of the option is e–0.120.25 (0.65231 + 0.34770) = 0.633 22 Choosing u and d 23 One way of matching the volatility is to set u e t d e t 1 u where is the volatility andt is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein 23 A Two‐Step Example 24 20 2
2 24.2
19.8 18
16.2 Each time step is 3 months A call option with K=21, r=12% 24 A Put Option Example 25 K = 52, time step =1yr r = 5% 50
4.1923 A D 60 B 1.4147
40 72
0
48
4 E C 9.4636
F 32
20 25 What Happens When an Option is American 26 Put option: K = 52, time step =1yr r = 5% 60 D 72
0 B 50
5.0894 A 1.4147
40 48
4 E C 12.0
F 32
20 26 ...
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 Spring '12
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