lecture8(英文)

1、Lecture 8: Binomial Option Pricing,,,We have derived upper and lower bounds for options by using simple no arbitrage arguments. Although these bounds limit the price of the option, the difference of the upper and lower b

2、ounds can be quite large. For example, consider a European call option with strike price of 100, maturity date in six months, and where the underlying asset price is 100. We know the option price must be in the range of

3、2.96 and 100, assuming the interest rate of 6%. To price options more precisely, we must make additional assumptions about the probability distribution describing the possible price changes in the underlying asset. The p

4、urpose of this lecture is to study a model of asset price.,·        Basic assumptions,1.     Assume that the stock price can take one of two possible values at

5、the end of one period.2.     There exists a risk-free security3.     There are no arbitrage opportunities4.     There is no interest rate uncertainty,·&

6、#160;       The importance of binomial price model,1.     It yields important insights into the pricing and hedging all derivatives.2.     The basic

7、logic of this approach is similar to the logic of the majority derivative security models in use today.3.     If short rate is a constant, under some conditions, the binomial model of stock price wil

8、l converge to the stock price dynamics used to drive Black-Scholes option pricing formula.,,4.     Binomial tree can be used to model stock price dynamics when the volatility is a function of stock pr

9、ice.5.     In the practice, a binomial tree may be estimated or constructed based on simple options (implied binomial tree) and then it is used to price exotic options.6.     In

10、numerical analysis, binomial tree approach is one of the most important tools.,·        General assets pricing methods,1.     General equilibrium approach: it i

11、s used to price basic assets. You have to consider investors/consumers' utility functions, producers' production functions. The asset prices are determined by market equilibrium conditions through individuals to

12、maximize their objective functions.,,1.     No-arbitrage approach: It is used to price derivatives. In general, you take the basic assets' prices as given and think that the payoffs from a derivat

13、ive can be duplicated by payoffs of a portfolio of basic assets. Thus the price of derivative is the price of the portfolio of basic asset.2.     No-arbitrage approach commonly is used under assumpti

14、on of complete market. When market is incomplete you may have to use equilibrium approach even for derivatives. An example of this is the stochastic volatility of the underlying asset.,·     

15、;   Notations,S: current stock pricer: risk free rate (a.c.c)u: upward movement factor in asset price over time interval ?t.d: downward movement factor in asset price over time interval ?t.q: probability of

16、 upward movement in asset price.,,X: exercise price of optionT: time to maturity of option (current time is 0)?t: length of time interval.?: volatility = ln(u)/(?t)0.5For convenience, we some times use u=1/d and imp

17、ose the restriction: d < er?t < u,·        One-period binomial generating process,a.      Stock price dynamics 

18、 Su(22) S (20) Sd(18),,,,a.      Option price dynamics  fu f

19、 fdf can be a call, a put or an other derivative security.,,,,The call option dynamics (X=21)  cu=max[Su-X,0] (1) c c

20、d=max[Sd-X,0] (0),,,,The put option dynamics pu=max[X-Su,0] (0) p pd=max[X-Sd,0] (3),,,·       

21、 Deriving the binomial option pricing model,a.      Consider a portfolio consisting of: ? shares of stock S one short call option on stock Sb.     We construct a port

22、folio in away to ensure that its value at the maturity of the option remains constant irrespective of the stock price. ?Su-cu=?Sd-cd,,c.      Solving for ?, the hedge ratio, to yield

23、 ?=(cu-cd)/[S(u-d)] The hedge portfolio's payoff at maturity is known beforehand. Therefore, the portfolio's price (today) is equal to the present value of its payoff discounted at the risk free rate of interest:

24、 ?S-c=(?Su-cu)e-r?t. This implies c=?S-(?Su-cu)e-r?t=[q*cu+(1-q*)cd]e-r?t, where q*=(er?t-d)/(u-d)Ex: S=20, u=1.1, d=0.9, r=12%, ?t=0.25. We have 22?-1=18?, ?=0.25, (?Su-cu)=4.5, ?S-c=4.367, q*=0.6523, c=0.633.,·&

25、#160;       Risk-neutral valuation:,a.      The option price does not depend upon q. It instead relies upon the risk neutral probability (equivalent martingale measu

26、re) q*. b.     Under this setup, the current stock price equals the expected future value discounted at the risk free rate, i.e., S==[q*Su+(1-q*)Sd]e-r?t . This implies that the option price is indep

27、endent of the expected rate of return of the underlying asset.,,a.      This has important implication. Consider two individuals, one an optimistic and the other a pessimist. The optimistic (pess

28、imist) believes that the probability the stock price going up is 90% (10%). Provided these two agrees that the stock price today is 20, and the stock price in the up state is 22 and that the stock price in the down state

29、 is 18, then they both will agree that the traded option's value today is 0.633.,·        Replication approach,In the above approach, the combination of a stock and a call repl

30、icated a risk-free asset. The more natural way is to think that a stock and a bond (the risk-free asset) can replicate the payoffs of options.Based on the above example, if we invest one dollar in risk free asset today

31、we will get e12%*0.25 three months latter. Suppose we buy m0 shares of stock and invest b0 dollars in the risk-free asset.,,The value of the portfolio is V(0) = m020+b0. But what must m0 and b0 be to mimic the payoffs of

32、 the call option?m022+b0e12%*0.25=1 (why?)m018+b0e12%*0.25=0 (why?)Can we design a portfolio to satisfy the above conditions? In general, the answer is yes. (why?),,m0=1/(22-18)=0.25=?b0=(1-0.25*22)e-12%*0.25=-4.367

33、V(0) = 0.25*20-4.367=0.633What should be the value of the traded call option? Suppose traded option is priced at 0.7 what can we do? Suppose traded option is priced at 0.6 what should we do?,·   

34、60;    Option delta (hedge ratio),The option delta (hedge ratio), ?, represents the slope of the call or put option pricing at point S.,·        Put-Call parity,c-p=

35、S-Xe-r?t,·        Example,1.     Call optionS=$25 u=1.2 d=1/1.2=0.833 X=25 T=1year r=0.10Stock price dynamics 30=25*1.2

36、25 20.83=25/1.2,,,,Call option price dynamics max[30-25,0]=53.35 max[20.83-25,0]=0q*=(e.1-0.8333)/(1.2-0.

37、8333)=0.7414 1-q*=0.2586c=[0.7414*5+0.2586*0]e-.1=3.35hedge ratio = (cu-cd)/[S(u-d)] = (5-0)/[25(1.2-0.83330]=0.5454,,,,Payoff structure of the hedge portfolioState Portfolio PayoffUp ?Su-cu

38、 0.5454*25*1.2-5=11.36 Down ?Sd-cd 0.5454*25*0.8333-0=11.36The portfolio is indeed riskless!,·        Two-period binomial model,1. Stock price dynamics

39、 24.2 2220 19.2 18 16.2,,,,,,,,2. Call op

40、tion's price dynamics 3.2 2.02571.2823 0.0

41、 0.0 0.0Note p*=0.6523, we have 2.0257=e-0.12*0.25(0.6523*3.2+0.3477*0) and 1.2823=e-0.12*0.25(0.6523*2.0257+0.3477*0),,,,,,,,The system can be genera

42、lized. It can be solved recursively starting from the option's maturity date. Invoking the risk neutral valuation argument to yieldcu=[q*cuu+(1-q*)cud]e-r?t cd=[q*cud+(1-q*)cdd]e-r?t c=[q*cu+(1-q*)cd]e-r?tSubstit

43、uting the terms, we obtain cu=[q*2cuu+2q*(1-q*)cud+(1-q*)cdd]e-2r?t,·        American put options,1. Stok price dynamics Today 6-month 12-month

44、 138.11 117.52 100 104.09 88.57

45、 78.45 u=1.1752, d=0.8857, r=6%, p*=0.5,,2.     Put price dynamics (X=110) Today 6-month 12-month

46、 0 [D(-7.52) A(2.87)] [D(10) A(11.79)] 5.591 [D(21.43)

47、 A(18.18)] 31.55,·        n-period binomial model,The option's life is split into n time intervals of equal

48、length (?t=T/n). Applying the one-period operation recursively to yield the n-period generalization for calls:c=?nj=0{ n!/[(n-j)!j!]q*j(1-q*)n-jmax[X-Sujdn-j,0]}e-rT,·        A 5-

49、step European option pricing (without dividends),S=96, u=1.0694, X=100, d=1/u=0.9351, T=0.25, q*=0.5206, r=10%, s.d.=0.3 , # of steps =5.,,Stock price tree

50、 134.2573 125.5465 117.4009 117.4009 109.7837

51、 109.7837 102.6608 102.6608 102.660896 96 96 89.7714 89.7714

52、 89.7714 83.9469      83.9469 78.5003    

53、 78.5003 73.4071 68.6643,,European call option price treeB-S=5.0517

54、 34.2573 26.0453 18.3959 17.4009

55、 12.3829 10.2825 8.0550 5.9835 2.66085.1064 3.4398 1.3782 1.9581   

56、  0.7139 0 0.3698 0 0 0

57、 0 0,,European put option price treeB-S=6.5827

58、 0 0 0 0 1.1104

59、 0 3.4140 2.3277 06.6373 5.9510 4.8795 10.2066

60、 9.9475 10.2286 14.9341 15.5544 20.5047 21.4997

61、 26.0942 31.3557,·        A 5-step American option pricing (without dividends),S=96

62、, u=1.0694, X=100, d=1/u=0.9351, T=0.25, q*=0.5206, r=10%, s.d.=0.3 , # of steps =5.,,Stock price tree 134.2573

63、 125.5465 117.4009 117.4009 109.7837 109.7837 102.6608 102.6608

64、 102.660896 96 96 89.7714 89.7714 89.7714 83.9469     

65、 83.9469 78.5003      78.5003 73.4071

66、 68.6643,,American call option price tree 34.2573

67、 26.0453 18.3959 17.4009 12.3829 10.2825 8.0550 5.9835 2.6

68、6085.1064 3.4398 1.3782 1.9581     0.7139 0 0.3698

69、 0 0 0 0

70、 0,,American put option price tree 0 0

71、 0 0 1.1104 0 3.4780 2.3277 06.9583

72、 6.0851 4.8795 10.8099 10.2286 10.2286 16.0531 16.

73、0531 20.5047 21.4997 26.5929

74、 31.3557,·        The continuous-time limit of the BOPM,Let u=exp[?(?t)0.5] and d=exp[-?(?t)0.5]. If ?t?0, it gives rise to the Black & Scholes formula for European opti