金融学概论讲义(北大光华管理学院)lecture07

Principles of FinanceLecture 07 Option Pricing ModelsOptions: Definition∙An American-style call option gives the owner the right (but not obligation) to buy a specified asset at a specified price at any point in time prior to a specified date∙An American-style put option gives the owner the right (but not obligation) to sell a specified asset at a specified price at any point in time prior to a specified date∙For the European-style options, the owner can only exercise on the expiration dateOptions around usGovernment price support: Governments sometimes provide assistance to farmers by offering to purchase agricultural products at a specified support price. If the market price is lower than the support, then a farmer will exercise her right to ‘put’ her crop to the government at the higher price. Insurance: Insurance policy often gives you the right, but not the obligation to do something, it is therefore option-like.Options around usLimited liability: The owners of a limited liability corporation have the right, but not the obligation, to ‘put’ the company to the corporation’s creditors and bondholders. Limited liability is, in effect, a put option.Trading on Commission: You are a trader with a contract giving you a commissi on of 20% of each month’s trading profits. If you make a loss, then you walk away, but if you make a profit, you stay. (You may be tempted to increase your volatility to boost the value of your option).Specifications of Options Contracts ∙ European or American∙ Call or Put (Option Class)∙ Underlying Asset∙ Strike Price∙ Maturity (Expiration) DateSome Terminology∙Long position: The party who buys the options is said to have a long position∙Short position: The party who writes (i.e. sells) the option is said to have a short positionSome Terminology∙In the Money: exercise of the option would be profitable.Call: market price>exercise pricePut: exercise price>market price∙Out of the Money: exercise of the option would not be profitable.Call: market price<exercise pricePut: exercise price<market price∙At he Money: exercise price and asset price are equal.Payoff to Call OptionsPayoff to a long call: []max 0,purchase price T S E --if 0 if T T T S E S E S E -><Payoff to a short call: []m in 0,+purchase price T E S -if 0 if T T T E S S E S E -><Payoff to Call OptionsPayoffLong Call 0Short CallStock PricePayoff to Put OptionsPayoff to a long put: []m ax 0,purchase price T E S --0 if if T T T S E E S S E >-<Payoff to a short put: []m in 0,+purchase price T S E -0 if if T T T S E S E S E >-<Payoff to Put Options PayoffShort Put 0Long PutStock PriceProtective PutUse – to limit lossPosition - long the stock and long the putPayoff T S E < T S E > Stock T S T S Long Put T E S - 0 Total E T SPayoff of a Protective European PutPayoff of a Long European Call and E BondsCovered CallUse - Some downside protection at the expense of giving up gain potential.Position - Own the stock and write a call.Payoff T S E < T S E > Stock T S T S Short Call 0 T E S - Total T S EPayoff of a Covered CallEEuropean Options Put-Call ParityC r EP S t++=+)1( Proof:∙ At time 0=t , construct two portfolios A and B- Portfolio A: One long European call and lending oftr E -+)1((investing in E of pure discount treasury bonds)- Portfolio B: One long European put and buy one share∙ Value of two portfolios at maturityPayoff of Portfolio APayoff of Portfolio B∙ Payoffs of portfolio A and portfolio B are identical at maturity T∙ No arbitrage requires that two portfolios must have identical value at time 0=t .Value of portfolio A at 0=t : C r E T++-)1( Value of portfolio B at 0=t : P S +Therefore:C r E P S T++=+-)1(European Options Put-Call Parity The put-call parity could be written in the form:-=-+C P S E r-(1)TThe left side, which is a portfolio of a long call and a short put, equals the right side, which is called leveraged equity.Payoff of Leveraged EquityPayoffAn Arbitrage Opportunity?∙ Suppose that0.3market =c , 0.310=S , T = 3, 30=Emonthly interest rate is %1=r with monthly compounding. What is the price of put?∙ The theoretical (fair) price for the put: 12.131)01.01(303)1(3f air =-+⨯+=-++=--S r E c p TAn Arbitrage Opportunity?∙ An arbitrage opportunity if 0.2market=p?∙ When 0.2market=p, there is an arbitrage opportunity: (1)TC P S E r --<-+, the leveraged equity is overpriced. So we take a short position in the leveraged equity.An Arbitrage Opportunity?At 0=t :Sell a put option on one underlying share: P Buy a call option on one underlying share: C -Sell short one underlying share: SBuy E pure discount treasury bonds maturing at T :Tr E -+-)1(Net cash flow:88.0)01.01(303312)1(3=+⨯--+=+--+-T r E C S PAt maturity:Get E from the treasury bond, and close your short position on the share. Cash flow: T S E -If E S T ≥: exercise the call option, make a profit. Cash flow: E S T -If E S T ≤: the buyer of the put option exercises the put option, you make a loss. Cash flow: E S T -Net cash flow at maturity: 0Volatility and Option PricesWhat happens to the value of an option when the volatility of the underlying stock increases?We assume a world in which the stock price moves during the year from $100 to one of two new values at the end of the year when the option matures. The strike price for both the call and the put option is also $100.Volatility and Option Prices, P0 = $100, Strike = $100 Stock Price Call Payoff Put Payoff Low Volatility CaseRise120200 Fall80020 Expectation1001010 High Volatility CaseRise140400 Fall60040 Expectation1002020Volatility and Option PricesThe stock volatility in the second scenario is higher, and the expected payoffs for both the put and the call are also higher.Conclusion: V olatility increases all option pricesDifficulty in Pricing Options∙Bond and share valuation: Value of bond and share is given by discounted cash flows∙Options Valuation-DCF does not work-Easy to calculate expected payoff of the options at maturity for an assumed distribution of asset price-Difficult to estimate the discount rateBinomial Option Pricing Model∙To price options we need to know asset price dynamics∙The simple but powerful model is the binomial model-The stock price can take only one of two possible values at the end of each (short) interval-Very good approximation when time intervals are small-Useful to explain the argument underlying options valuation models-Can be used to value derivatives like American optionsSingle Period Binomial ModelExpected return = 1.1Expected variance = 0.09Two Period Binomial ModelPricing Options via Synthetic Replication∙To price a derivative contract, a portfolio of the stock and riskless investment is constructed to mimic or replicate the payoffs of the derivative at maturity∙The portfolio is called a synthetic derivative∙With no arbitrage, the value of the portfolio must equal the price ofa traded derivative∙Synthetic construction is equivalent to hedgingNo Arbitrage Principle for Pricing DerivativesSingle Period: Valuing European Call∙ European call:10%r 0.8,d ,2.1 ,40 ,40 ,10======u E S T∙ Portfolio (Synthetic Call) = Shares + Riskless Asset 000B S V +∆=∙ Portfolio which replicates the payoffs of the call option at maturity 810.1480=⨯+⨯∆B010.1320=⨯+⨯∆B∙ 1.10 = End-of-year wealth for $1 investment today∙ Solving the equations gives 55.14 ,5.00-==∆B∙ The minus sign for 0B implies borrowing∙ 45.555.14405.00=-⨯=V∙ No arbitrage requires 45.50==V c∙ Implication:Value of p is never used: Expected return is irrelevant!Single Period: Valuing European Put∙ Portfolio (Synthetic Call) = Shares + Riskless Asset 000B S V +∆=∙ Portfolio which replicates the payoffs of the call option at maturity 048 1.100B ∆⨯+⨯=032 1.108B ∆⨯+⨯=∙ Solving the equations gives 00.5, 21.82B ∆=-=∙ 00.54021.82 1.82V =-⨯+=Single Period: Valuing European Put∙ No arbitrage requires 0 1.82p V ==∙ The put-call parity relationship implies that:(1)TC P S E r --=-+Here we have 5.45 1.82 3.63C P -=-=1(1)4040 1.1 3.63T S E r ---+=-⨯=So the parity holds. If we know the call price, the put price could be derived using the put-call parity.Generalization ofOne Period Binomial Option Pricing∙ The portfolio replicates the payoff of the call at maturityuc r B u S =++∆)1(00dc r Bd S =++∆)1(00∙ Solving the equations givesdS u S cc du00--=∆, and )1/(0r d u dc uc B u d +⎪⎪⎭⎫⎝⎛--=∙ No arbitrage requires()[]d u dr r c c V c d u --+=+-+==1 where ),1/(10πππMulti-Period Binomial ModelSS +S + +S -S - -S + -S + + +S + + -S + - -S - - -Black-Scholes Option Pricing Model∙The option price and the stock price depend on the same underlying source of uncertainty∙ A riskless hedge portfolio can be constructed using the stock and the option∙The riskless portfolio must earn the riskfree rate of return∙This leads to the Black-Scholes partial differential equation (PDE)The Black-Scholes Formulas()()21d N Ee d SN c r τ--=()()12d SN d N Ee p r ---=-τwhere()()τστσ212/ln ++=r E S d τσ-=12d d().N is the cumulative probability distribution function fora standardised normal variableDeterminants of Options Prices Increase in Call Put Stock Price, S↑↓Strike Price, E↓↑V olatility, σ↑↑Time to Maturity, τ↑↑Interest Rate, r↑↓Cash Dividend, d↓↑Applying the Black-Scholes Formulas50=S , 45=X , () year 0.25i.e.=-t T τ, pa %20=σ %6=r with continuous compounding()()()()2536.125.02.025.006.045/50ln /ln 22.02122=⨯⨯++=++=τστσr X S dand 1536.112=-=τσd dThen ()8950.01=d N , ()8757.02=d N()()93.58757.0458950.05025.006.021=⨯⨯-⨯=-=⨯--e d N Xe d SN c r τand26.0504593.525.006.0=-⨯+=-+=⨯--e SXec p r τDebt and ShareA share in a company is like a long position in a call option on the company whose exercise price equals the value of debt: shareholders get what remains with the company after debt is paid.V SVD。