高级公司金融Pricing Derivatives.ppt

1、DRAFT,CONFIDENTIAL,CHAPTER 4,Pricing Derivatives,Chapter Outline,4.1 Introduction to Derivatives 4.2 Forward and Futures 4.3 Options 4.4 The Basics of Derivatives Pricing 4.5 Perfect Tracking Portfolios and Valuation 4.6 Perfect Tracking and Put-Call Parity 4.7 Binomial Pricing Model 4.8 Risk Neutra

2、l Pricing 4.9 Multiperiod Binomial Valuation 4.10 Binomial Valuation of European Options 4.11 Binomial Valuation of American Options 4.12 Black-Scholes Valuation 4.13 Simulation and Derivatives Valuation,4.1 Introduction to Derivatives,A derivative is a financial instrument whose payoffs and values

3、are derived from, or depend on, something else, that are known as the underlying or the primitive. Derivatives are tools to change the firms risk exposure. When the firm reduces its risk exposure with the use of derivatives, it is said to be hedging. When derivative is merely used to change or even

4、increase the firms risk exposure, it is said to be speculating.,4.2 Forward and Futures,A forward contract represents the obligation to buy (sell) the underlying security or commodity at a specified price, known as the forward price, at some future date (maturity date, settlement date). At maturity

5、the firm or person with the long position pays the forward price to the person with the short position, who in turn delivers the underlying asset. Futures contracts are a special type of forward contract that trade on organized exchanges know as futures markets.,An Example of Forward Contract,Suppos

6、e on May 31st Delta Airline wrote an forward with ExxonMobil to buy 1,000 barrels on September 30th at the forward price, $60 per barrel. On September 30th Delta Airline is obliged to buy and ExxonMobil is obliged to sell this amount of oil at the forward price. On September 30th, if the market pric

7、e of oil rises to $80 per barrel, Delta earns a profit of $20,000 from the forward contract. However if the market price of oil is $40 per barrel, Delta incurs a loss of $20,000.,Value of a Forward Contract at Expiration,Value of Forward Contract,ST,Forward Price, F0,45,The payoff of a forward at T:

8、 ST-F0,Distinguishing Forwards from Futures,The essential distinction between a forward and a futures contract lies in the timing of cash flows. With a forward contract, cash is paid for the underlying asset only at the maturity date, T, of the contract. This amount, F0, which is the forward price s

9、et at date 0, is the sum of the following two components The spot price, ST, at maturity. The profit (or loss) incurred, F0 ST.,Marking to Market,With a futures contract the price paid at maturity is the spot price, ST , which is identical to the final days futures price, FT. The profit (or loss), F

10、0-ST is received (paid) on a daily basis before the maturity date instead of in one large amount at the maturity date. At any date before maturity the amount the buyer receives from (pays to) the futures contract is the one-day appreciation (depreciation) of the futures price. The marking to the mar

11、ket takes place automatically, by transferring funds between the margin accounts of the two investors agreeing to the contract. The margin accounts are simply deposits placed with brokers as security against default.,Example: Marking to the Market with S while Black-Scholes model assumes that stock

12、price follows continuous time process. The continuous time process allows for an infinite number of stock price outcomes and it can describe the distribution of stock and option prices at any time. A binomial process approximate the continuous time process if time is divided into large numbers of sh

13、ort periods.,Black-Scholes Differential Equation,Assume the price of a stock paying no dividends follows a Geometric Brownian Motion with annual mean, and annual standard deviation, , The value of a derivative, c(S, t) is a function of the underlying stock price and time t, therefore Itos lemma show

14、s, The uncertainty of the option and the stock is driven by the same underlying stochastic process.,dS= Sdt + Sdw,(3),(4),Black-Scholes Differential Equation,Equations (3) and (4) implies that we can construct a portfolio consisting of longing shares of underlying stock and shorting 1 call option to

15、 eliminate the uncertainty. The current value of this portfolio is, (5) No arbitrage dictates that (7) Insert (5) and (6) into (7) we have the Black-Scholes Differential Equation,(6),(8),Black-Scholes Differential Equation,Since c(S, t) is a general function of underlying asset price and time t, all

16、 derivatives on the underlying assets without cash income should satisfy the Black-Scholes differential equation (8). Different derivatives have specific function forms of c(S, t) and boundary conditions. For a European call option, c(S, T) = max(ST-K, 0) (when t = T), and for a European put option,

17、 c(S, T) = max(K-ST, 0) (when t = T). Given the Black-Scholes differential equation and the boundary condition, the value of a derivative could be solved.,Black Scholes Formula,Given the Black-Scholes differential equation (8) and the boundary condition for European call and put options, the option

18、prices are as the follows,where,and,N(x): the cumulative probability distribution function for a standardized normal distribution.,S0: stock price at time 0.,K: Strike price.,rf: risk free rate.,(9),(10),Black-Scholes Price Sensitivities (S0),Stock price The higher the stock price, the more valuable

19、 the call option will be. The value of call option before T is a convex function of the price of the underlying stock. When stock price is higher, a unit change in the underlying stock price generates a larger increase in the value of call option than when stock price is lower.,The Value of Call Opt

20、ion Before T,Value of Call at T,Value of Call before T,Black-Scholes Price Sensitivities (S0),Two special points on the above figure. When the stock is worthless the call option is also worthless. When the stock price is extremely large, the call will be almost surely exercised. In this case an opti

21、on is like a forward contract. The option value is S0 - Ke-rT. The Black-Scholes formula shows that when S0 goes to infinity, N(d1) and N(d2) are close to 1. We get the above result.,Black-Scholes Price Sensitivities (S0),Delta: The sensitivity of option value to stock price changes The change in th

22、e value of the derivative security with respect to movements in the stock price, ceteris paribus. The delta of a European option is the derivative of the options price with respect to stock price, , for a call and , , for a put. The Black-Scholes formula (9) and (10) shows that the delta of a call o

23、ption is N(d1).,Black-Scholes Price Sensitivities (S0),Delta as the number of shares of stock in a tracking portfolio Delta can be viewed as the number of shares of stock needed in the tracking portfolio. If in a tracking portfolio investors hold shares of stock, the change in the price of this amou

24、nt of shares is , which exactly offsets the change in the price of call, dc. N(d1) is probability, thus the number of shares needed to track the option lies between zero and one. As time elapses, stock price and t changes, therefore delta changes over time. But these changes do not require additiona

25、l financing,Black-Scholes Price Sensitivities (S0),Delta and the interpretation of the Black-Scholes formula Viewing N(d1) as the number of shares of stock in the tracking portfolio, the Black-Scholes formula is just the value of the tracking portfolio. S0N(d1) is the cost of the shares needed in th

26、e tracking portfolio. represents the number of dollars borrowed at the risk free rate. Call option on stock is equivalent to leveraged positions in stock. Investing on call option is therefore riskier than on stock.,Black-Scholes Price Sensitivities (),The key factor: the variability of the underlyi

27、ng asset The greater the variability of the underlying asset, the more valuable the call option will be. Example: consider two stocks A: 0.5 probability to be 100 and 0.5 probability to be 80 at maturity. B: 0.5 probability to be 120 and 0.5 probability to be 60 at maturity. A call option with strik

28、e price 110 will be more valuable if the underlying stock is B. This is the fundamental distinction between holding a stock and holding a call option.,Black-Scholes Price Sensitivities (),Black-Scholes Price Sensitivities (T),Expiration date: European calls on stocks that pay no dividends are more v

29、aluable the longer the time to expiration for two reasons. The terminal stock price is more uncertain the longer the time to expirations. Given the same strike price, the longer the time to maturity , the lower is the present value of the strike price paid. For European puts, the longer the time to

30、maturity the less valuable the present value of the received payment from strike price. The volatility effect and discounting effect make the price sensitivity to T ambiguous. For American puts and calls, longer maturity leads to higher option value.,Exercise price: An increase in exercise price red

31、uce the call option value but increase the put option value. Risk free rate: For a call option, if an investor exercise the option she pays the fixed strike price K in a future time. The higher the risk free rate the lower the present value of the payment of strike. Call option is more valuable. For

32、 put option, an investor receives the fixed strike price if it is exercised. The higher the risk free rate the lower the present value of this payment. The put option is less valuable.,Black-Scholes Price Sensitivities (K, rf),Estimating Volatility,The only parameter that requires estimation in the

33、Black-Scholes Model is the volatility . There are two approaches to estimate volatility Using historical data Implied volatility If market values for the options exist, there is a unique implied volatility that makes the Black-Scholes model consistent with the market price for the option. Average th

34、e implied volatilities of other options on the same security is a common approach to obtaining the volatility necessary for obtaining the Black-Scholes valuation of an option.,Estimating u and d in Binomial Process,The binomial method is often used to approximate many kinds of continuous distributio

35、ns, for example the lognormal distribution. Once the annualized standard deviation, , of this normal distribution is know, u and d are estimated as follows,T = number of years to expiration N = number of binomial periods e = exponential constant,square root of the number of years per binomial period

36、,=,4.13 Simulation and Derivatives Valuation,Risk neutral pricing formula is as the following, Two approaches to obtain the expectation of the future cash flow under risk neutral probability measure, ERN(CF). Derive the formula for the expectation of future cash flow under risk neutral probability m

37、easure. Given the distribution of future cash flow under risk neutral probability, we can randomly generate N possible future outcomes cfsim1, cfsim2, , cfsimN and then take the average. The law of large number says that when N approach infinite, the average approaches the expectation of the cash fl

38、ow under risk neutral measure,Simulating Binomial Process,To price a derivative, it will be easier to simulate the price processes for the underlying asset (for example, stock) first. If we assume an N-period binomial process for the stock price, we need to estimate the volatility, ; and then its easy to obtain u, d and risk neutral probabilities, for up state and 1- for down state. At each node the stock price follows a Bernoulli distribution with probabil