Unformatted text preview: Last chapter Last Chap 10: Option Markets Outline Outline Option basics Option Option diagrams Option strategies Putcall parity Option valuation Binomial model Black an Scholes model Risk neutral probabiliy I. Option basics I. Derivatives Derivatives A derivative is a financial instrument derivative whose price depends on the price of another underlying asset another Major derivative contracts are:
Futures and forward contracts Call and put options Swaps
4 Definitions Definitions
A call option is the right to buy an asset for a certain call buy price (strike price) at a certain time in the future (expiration date) (expiration A put option is the right to sell an asset for a certain price put sell (strike price) at a certain time in the future (expiration date) date) Exercise style: Exercise European: can be exercised at maturity only American: can be exercised at any time before maturity American: Bermudan: can be exercised only at some predefined times Bermudan: (e.g employee stock options) (e.g 5 Option markets Option Exchange traded options: Stock options (CBOE, Philli, AMEX, NYSE) ForEx options (Philli) Index options (CBOE) Futures options Trading: Market makers, offsetting orders, Trading: margins (for writers and not buyers) margins Options clearing corporation (OCC) Options Option Contracts Option Long position: The option buyer or Long buyer holder pays a premium and receives the right to buy or sell an asset right Short position: The option seller or Short writer receives a premium and has the writer obligation to deliver or purchase an obligation asset whenever the option buyer chooses chooses 7 Option Contracts Option In the money option: if the immediate In exercise is profitable for the holder exercise Out of the money: immediate exercise Out generates losses for the holder generates At the money: Strike is equal to current At underlying price. underlying
8 II. Option diagrams Some notations Some
• T: Expiration or strike date • K: Exercise or strike price • St: Price of the underlying asset at time t • Ct: Price of a call option at time t • Pt: Price of a put option at time t • Sometimes we will use S (resp. C and P) instead of S0 (resp. C0 and P0) Payoff of a Call option with strike price $20 at expiration price
CT =Max(STK,0)=Max(ST20,0) =Max(S Payoff of a Put option with strike price $20 at expiration price Payoff on a short position in a Call Option at expiration
Cash flow is Max(STK,0)=Max(ST20,0) Payoff on a short position in a Put Option at expiration ($20 strike) strike) Option quotations for Amazon.com Amazon.com Problem Problem Assume that you decided to purchase Assume the January call options with exercise price 50 quoted in the Amazon table on November 29, 2005, and you financed the position by shorting a twomonth bond with a yield of 3% EAR. Plot the profit of the position as a function of stock price at expiration. Repeat the exercise for all the other calls. exercise Profit from holding a call option until expiration until Example Example Assume that you decided to purchase Assume each of the January put options quoted in the Amazon table on November 29, 2005, and you financed each position by shorting a twomonth bond with a yield of 3%. Plot the profit of each position as a function of stock price at expiration. expiration. Solution Solution Options and leverage Options Compute the return on holding the Compute Amazon call vs holding the Amazon stock. Consider two scenario of stock returns: +20% and 20%. returns: Simulation of the call return Simulation
t=0 Cash flow (Stock position) Return (Stock position) Cash flow (Call, strike=45) Return (Call, strike=45) Cash flow (Call, strike=47.5) Return (Call, strike=47.55) Cash flow (Call, strike=50) Return (Call, strike=50) Cash flow (Call, strike=55) Return (Call, strike=55) 48.35 T=Jan 06 58.02 20% 5 13.02 160% 3.30 10.52 219% 2.05 8.02 291% 0.70 3.02 T=Jan 06 38.68 20% 0 100% 0 100% 0 100% 0 Economic insight: call is like a leveraged stock
331% 100% Simulation of the put return Simulation
t=0 Cash flow (Stock position) Return (Stock position) Cash flow (Put, strike=45) Return (Put, strike=45) Cash flow (Put, strike=47.5) Return (Put, strike=47.55) Cash flow (Put, strike=50) Return (Put, strike=50) Cash flow (Put, strike=55) Return (Put, strike=55) 7.1 2.15 48.35 T=Jan 06 58.02 20% 1.3 0 100% 0 100% 3.5 0 100% 0 100% T=Jan 06 38.68 20% 6.32 386% 8.82 310% 11.32 223% 16.32 130% Economic insight: a put is like a short position on the stock Options and leverage Options Options in the SML Options III. Options strategies III. Options strategies Options Straddle:
• Buy one call and put with the same exercise price Strangle: Buy 1 call (strike $40) and 1 put (strike $30) Buy Butterfly spread : Butterfly
• sell 2 calls (strike $30) and buy 1 call ($20) and 1 call ($40). sell Make money when the stock price is close to the strike M ake Protective put (also called portfolio Protective insurance):
• Buy stock + Put on a stock Portfolio insurance Portfolio IV. Put call parity IV. Put call parity Put
Without dividends With dividends Problem Problem
European call and put options on Exxon are selling for European $3.10 and $0.40 respectively. The exercise price of both options is $45 and have one month to maturity. The current stock price is $46.60 and the onemonth riskless rate is 5% p.a (continuous compounding). Is there an arbitrage opportunity in this market? If so, demonstrate how to exploit it and construct an arbitrage table. how V. Option Valuation V. PFIZER INC COM, April call $20 (CBOE) on March 31, 2008 (CBOE)
C K S Premium of the call=C=$1.01 Intrinsic value=Max(20.8020,0)=$0.80 Time value of the call: Time C – intrinsic value=1.010.80=$0.21 intrinsic Factors affecting the call price Factors
Strike price
If K↓, the call either becomes deeper in the the money or comes closer to being in the money. money Stock price
A S ↑, the call either goes deeper in the money the or comes closer to being in the money or Time to maturity
As T↑, there is a larger likelihood that the stock there price will at some point be higher than it is now. price Factors affecting the call price Factors Volatility
Total risk not market risk Total not High volatility increases the likelihood of a High large price swing that would push the option into the money: if volatility ↑, C ↑. ↑, Interest rate
As r ↑, some people who were buying on some margin decide instead to get their leverage by buying call options: C ↑. Option value and volatility Option VI. The binomial option pricing model model One period binomial tree Stock
60 Bond
1.06 Call Max(6050,0)=10 Stock 50 Bond 1 40 1.06 Max(4050,0)=0 This is our “binomial tree”
38 How can anything this simple be useful? How The binomial model:
gives a value for the call option shows how the call option can be shows hedged. hedged.
• “ Hedge” = reduce/eliminate the risk of the Hedge” option position option Law of one price Law Replicate the call’s payoff with a Replicate porfolio of stocks and bonds porfolio Hold Δ shares of stock and B the initial Hold investment in the bond investment Two equations and two unknowns: Δ and Two B. B. Law of one price Law Solution: Δ=0.5 and B=18.8679 Borrow $18.8679 (at 6%) and buy 0.5 shares Call = levered position on the stock premium of the call = Cost of the levered premium position position C=6.13 and we do not need to know the C=6.13 probability of return (or the expected return) return) Options vs replicating portfolio Options Hedge ratio (∆ ) Hedge
Delta ∆ is called the hedge ratio Delta
How many shares to hold in order to How replicate the call’s payoff replicate Delta is also the sensitivity of price of a Delta derivative asset to the price of the underlying asset underlying • A $1 increase in the stock price corresponds to a $1 $∆ i ncrease in the price of the derivative
43 General formulation
Stock
Su Bond
1+ rf Derivative Cu Stock S Bond 1 Sd 1+ rf Cd S u ∆ + (1 + r f ) B = C u and S d ∆ + (1 + r f ) B = C d 44 Exercise Exercise Multiperiod binomial tree
0 1 50 40 30 2 60 40
Let’s see how to value a call option with K= 50 using this tree and rf = 6 %
46 20 Stock Price Tree Backward induction Backward Price at the node “u”:
• Δu=0.5 and Bu=18.8679 • Cu = $6.13 Price at the node “d”:
• Δd=0 and Bd=0 • Cd = $0 Price at time 0
• Δ=0.3065 and B=8.67 • C=$3.59 VII The Black and Scholes model VII The geometric Brownian motion The The BlackScholes formula The Premium of a European Call on a non Premium dividend paying stock dividend Black and Scholes call premium Black Time value European Put premium European Using the put call parity gives the Using European put premium Oracle corp. Oracle Exercise Exercise Problem: Oracle Corporation does not pay dividends. Using the above quotations, compare the price on Dec 6, 2005, for the Jan 2006 American call option on Oracle with a strike price of $12.50 to the price predicted by the BlackScholes formula (T=45 days). Assume that the volatility of Oracle is 25% per year and that the short term riskfree rate of interest on Dec 2005, is 4.38% per year (use continuous compounding). Exercise Exercise The present value of the strike The PV(K)=12.5*exp(45/365)=$12.4326 PV(K)=12.5*exp(45/365)=$12.4326 C=0.52 (see the file Oracle.xls) Oracle call bid and ask are $0.5 and $0.6 The replicating portfolio for a call The We know from the binomial model that Compare with the BS formula gives For a call 0<Δ<1 and K<B<0 Δ=dC/dS is a sensitivity or a slope is Computing the Replicating Portfolio Portfolio The hedge ratio as a slope The Replicating portfolio for a put option option Put = Short the stock and invest in the Put risk free asset risk 1<∆ <0 and K>B>0 Options and volatility Options C=C(T,K,S,r,σ) σ is the stdv of the stock annual return: hard to observe hard A common approach is to estimate from common past data (10, 20 or 30 days) past Implied volatility: Implied C(T,K,S,r,?)=observed C C(T,K,S,r,?)=observed Next pages from ivolatility.com S&P500 volatility: March 18th , 2011 S&P500 Set the BlackScholes price equal to the current market price. Work backwards to find the that “the market” is using. S&P500 IV vs HV S&P500 VIII. Risk neutral probability VIII. Back to the binomial model Back A RiskNeutral TwoState Model RiskNeutral Implications of the RiskNeutral World Implications RiskNeutral Probabilities and Option RiskNeutral Pricing Risk neutral world Stock
60 Bond
1.06 Call Max(6050,0)=10 Stock 50 Bond 1 40 1.06 Max(4050,0)=0 What if investors were risk neutral?
65 Pricing in a risk neutral world Pricing Denote by q the (risk neutral) Denote probability of the “up state” probability Pricing the stock 60q + 40(1 − q ) 50 = 1.06 Thus q =0.65 =0.65 Option premium is Option 10 * 0.65 + 0 * (1 − 0.65) = 6.13 1.06 General binomial formulation General
qS u + (1 − q ) S d 1 = R f S
Solving this equation for the riskneutral probability q we get q= (1 + R f ) * S − S d Su − S d Problem Problem Suppose the current price of Narver Network Suppose system stock is $50 per share. In each of the next two years the stock will either increase by 20% or decrease by 10%. The 3% one year risk free rate on interest will remain constant. Imagine all investors are risk neutral and calculate the probability of every state in the next two years. Use these probabilities to calculate the price of a two year European call option on Narver stock with a strike price $60. Then price a twoyear European put option with the same strike. with Questions Questions 69 Supplementary material (optional) (optional) Straddle: Buy one call and put with the same exercise price with Strangle Strangle Butterfly spread Butterfly Solution Solution Option value and volatility Option V. Exercising options early V. Nondividendpaying calls Nondividendpaying
European call 2 benefits of delaying: benefits
• • Earn interest on K Retain the right to not exercise (insurance against sharp Retain decline in S) In absence of dividend, American right is worthless right Nondividendpaying puts Nondividendpaying European put What happens when S=0? Early exercise can be optimal Early exercise Early The following table lists the quote fron the CBOE on The December 5, 2005, for options on Microsoft stock expiring in January 2006. Microsoft will not pay a dividend during this period. Identify any option for which exercising early in better than selling it Early exercise Early Early exercise with dividend Early Call: tradeoff between the benefit of Call: waiting and the loss of dividend waiting It may be optimal to exercise a call just It before the dividend is delivered ...
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 Spring '08
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 Options, Strike price

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