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单击此处编辑母版标题样式,*,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,Lecture 8: Binomial Option Pricing,1,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 bounds 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 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 purpose of this lecture is to study a model of asset price.,2,Basic assumptions,1. Assume that the stock price can take one of two possible values at the end of one period.,2. There exists a risk-free security,3. There are no arbitrage opportunities,4. There is no interest rate uncertainty,3,The importance of binomial price model,1. It yields important insights into the pricing and hedging all derivatives.,2. The basic 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 will converge to the stock price dynamics used to drive Black-Scholes option pricing formula.,4,4. Binomial tree can be used to model stock price dynamics when the volatility is a function of stock price.,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 numerical analysis, binomial tree approach is one of the most important tools.,5,General assets pricing methods,1. General equilibrium approach: it is 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 maximize their objective functions.,6,1. 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 derivative 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 assumption 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.,7,Notations,S: current stock price,r: 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 upward movement in asset price.,8,X: exercise price of option,T: time to maturity of option (current time is 0),t: length of time interval.,: volatility = ln(u)/(,t),0.5,For convenience, we some times use u=1/d and impose the restriction: d e,r,t, u,9,One-period binomial generating process,a. Stock price dynamics,Su(22),S (20),Sd(18),10,a. Option price dynamics,f,u,f,f,d,f can be a call, a put or an other derivative security.,11,The call option dynamics (X=21),c,u,=maxSu-X,0 (1),c,c,d,=maxSd-X,0 (0),12,The put option dynamics,p,u,=maxX-Su,0 (0),p,p,d,=maxX-Sd,0 (3),13,Deriving the binomial option pricing model,a. Consider a portfolio consisting of:,shares of stock S,one short call option on stock S,b. We construct a portfolio in away to ensure that its value at the maturity of the option remains constant irrespective of the stock price.,Su-c,u,=,Sd-c,d,14,c. Solving for, the hedge ratio, to yield,=(c,u,-c,d,)/S(u-d) The hedge portfolios payoff at maturity is known beforehand. Therefore, the portfolios price (today) is equal to the present value of its payoff discounted at the risk free rate of interest:,S-c=(,Su-c,u,)e,-r,t,. This implies c=,S-(,Su-c,u,)e,-r,t,=q,*,c,u,+(1-q,*,)c,d,e,-r,t, where q,*,=(e,r,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-c,u,)=4.5,S-c=4.367, q,*,=0.6523, c=0.633.,15,Risk-neutral valuation:,a. The option price does not depend upon q. It instead relies upon the risk neutral probability (equivalent martingale measure) 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,*,)Sde,-r,t,. This implies that the option price is independent of the expected rate of return of the underlying asset.,16,a. This has important implication. Consider two individuals, one an optimistic and the other a pessimist. The optimistic (pessimist) 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 is 18, then they both will agree that the traded options value today is 0.633.,17,Replication approach,In the above approach, the combination of a stock and a call replicated 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 we will get e,12%*0.25,three months latter. Suppose we buy m,0,shares of stock and invest b,0,dollars in the risk-free asset.,18,The value of the portfolio is V(0) = m,0,20+b,0,. But what must m,0,and b,0,be to mimic the payoffs of the call option?,m,0,22+b,0,e,12%*0.25,=1 (why?),m,0,18+b,0,e,12%*0.25,=0 (why?),Can we design a portfolio to satisfy the above conditions? In general, the answer is yes. (why?),19,m,0,=1/(22-18)=0.25=,b,0,=(1-0.25*22)e,-12%*0.25,=-4.367,V(0) = 0.25*20-4.367=0.633,What 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?,20,Option delta (hedge ratio),The option delta (hedge ratio), represents the slope of the call or put option pricing at point S.,21,Put-Call parity,c-p=S-Xe,-r,t,22,Example,1. Call option,S=$25 u=1.2 d=1/1.2=0.833 X=25 T=1year r=0.10,Stock price dynamics,30=25*1.2,25,20.83=25/1.2,23,Call option price dynamics,max30-25,0=5,3.35,max20.83-25,0=0,q,*,=(e,.1,-0.8333)/(1.2-0.8333)=0.7414 1-q,*,=0.2586,c=0.7414*5+0.2586*0e,-.1,=3.35,hedge ratio = (c,u,-c,d,)/S(u-d) = (5-0)/25(1.2-0.83330=0.5454,24,Payoff structure of the hedge portfolio,State Portfolio Payoff,Up,Su-c,u,0.5454*25*1.2-5=11.36,Down,Sd-c,d,0.5454*25*0.8333-0=11.36,The portfolio is indeed riskless!,25,Two-period binomial model,1. Stock price dynamics,24.2,22,20 19.2,18,16.2,26,2. Call options price dynamics,3.2,2.0257,1.2823 0.0,0.0,0.0,Note 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),27,The system can be generalized. It can be solved recursively starting from the options maturity date. Invoking the risk neutral valuation argument to yield,c,u,=q,*,c,uu,+(1-q,*,)c,ud,e,-r,t,c,d,=q,*,c,ud,+(1-q,*,)c,dd,e,-r,t,c=q,*,c,u,+(1-q,*,)c,d,e,-r,t,Substituting the terms, we obtain,c,u,=q,*2,c,uu,+2q,*,(1-q,*,)c,ud,+(1-q,*,)c,dd,e,-2r,t,28,American put options,1. Stok price dynamics,Today 6-month 12-month,138.11,117.52,100 104.09,88.57,78.45,u=1.1752, d=0.8857, r=6%, p,*,=0.5,29,2.,Put price dynamics (X=110),Today 6-month 12-month,0,D(-7.52) A(2.87),D(10) A(11.79) 5.591,D(21.43) A(18.18),31.55,30,n-period binomial model,The options life is split into n time intervals of equal length (,t=T/n). Applying the one-period operation recursively to yield the n-period generalization for calls:,c=,n,j=0, n!/(n-j)!j!q,*j,(1-q,*,),n-j,maxX-Su,j,d,n-j,0e,-rT,31,A 5-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.,32,Stock price tree,134.2573,125.5465,117.4009 117.4009,109.7837 109.7837,102.6608 102.6608 102.6608,96 96 96,89.7714 89.7714 89.7714,83.9469 83.9469,78.5003 78.5003,73.4071,68.6643,33,European call option price tree,B-S=5.0517 34.2573,26.0453,18.3959 17.4009,12.3829 10.2825,8.0550 5.9835 2.6608,5.1064 3.4398 1.3782,1.9581 0.7139 0,0.3698 0,0 0,0,0,34,European put option price tree,B-S=6.5827 0,0,0 0,1.1104 0,3.4140 2.3277 0,6.6373 5.9510 4.8795,10.2066 9.9475 10.2286,14.9341 15.5544,20.5047 21.4997,26.0942,31.3557,35,A 5-step American 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.,36,Stock price tree,134.2573,125.5465,117.4009 117.4009,109.7837 109.7837,102.6608 102.6608 102.6608,96 96 96,89.7714 89.7714 89.7714,83.9469 83.9469,78.5003 78.5003,73.4071,68.6643,37,American call option price tree,34.2573,26.0453,18.3959 17.4009,12.3829 10.2825,8.0550 5.9835 2.6608,5.1064 3.4398 1.3782,1.9581 0.7139 0,0.3698 0,0 0,0,0,38,American put option price tree,0,0,0 0,1.1104 0,3.4780 2.3277 0,6.9583 6.0851 4.8795,10.8099 10.2286 10.2286,16.0531 16.0531,20.5047 21.4997,26.5929,31.3557,39,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 options:,c=SN(d,1,) - Xe,-r(T-t),N(d,2,),p=Xe,-r(T-t),N(-d,2,)-SN(-d,1,),where d,1,=ln(S/X)+(r+,2,/2)/(T-t)/,(T-t),0.5, d,2,=d,1,-,(T-t),0.5, N(*) is the cumulative normal distribution function.,40,Example: S=96, X=100, T-t= 0.25 year, r=10% (a.c.c), and,=0.3,d,1,=-0.0305, d,2,=-0.1805, c=5.0517, p=6.5827,We will spend another class to discuss B-S formula in detail. We cant say that we have mastered option pricing theory without understanding the B-S formula.,41,Dividend and tree recombination,Think of an example. A one period is divided into four intervals. Just prior to the ex-dividend date, at date 2, there are three possible stock prices. When stock goes to ex-dividend, three new sub-trees will be generated. The trees may not recombine. This will cause the computing time to increase exponentially, as the number of dividends to be paid over the options life increases. Wed like the trees can be recombined.,Recombination condition: The ex-divided S(u,i,d,j-1,) in its next down move coincides with the ex-dividend S(u,i-1,d,j,) in its next up move.,Some stylized types of dividends,42,1. Proportional dividends,S(u,i,d,j-1,)=Su,i,d,j-1,(1-,) (ex-divided),S(u,i-1,d,j,)=Su,i-1,d,j,(1-,) (ex-divided),Su,i,d,j-1,(1-,)d=Su,i-1,d,j,(1-,)u. So the recombination condition is met.,2. Constant dividend yield,S(u,i,d,j-1,)=Su,i,d,j-1,e,-q(i+j),t,(ex-divided),S(u,i-1,d,j,)=Su,i-1,d,j,e,-q(i+j),t,(ex-divided),Su,i,d,j-1,e,-q(i+j),t,d= Su,i-1,d,j,e,-q(i+j),t,. The condition is met.,43,2. Fixed amount of cash dividends,S(u,i,d,j-1,)=Su,i,d,j-1,-D (ex-divided),S(u,i-1,d,j,)=Su,i-1,d,j,-D (ex-divided),(Su,i,d,j-1,-D)d,(Su,i-1,d,j,-D)u,So the recombination condition is not met.,Solution: Define an adjusted stock price (the cash dividend is assumed to be paid at k,t:,S,*,=S-De,-rk,t,S,*,(u,i,d,j,)=S,*,u,i,d,j,The original stock price becomes,S(u,i,d,j,)=S,*,(u,i,d,j,)+De,-r(k-i-j),t,if i+j,k, and S(u,i,d,j,)=S,*,(u,i,d,j,) if otherwise.,44,A 5-step European option pricing (prop. 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, Div at 3,rd,step (,)=0.0200,45,Stock price tree,131.5722,123.0356,115.0529 115.0529,109.7837 107.5880,102.6608 100.6076 100.6076,96 96 94.0800,89.7714 87.9759 87.9759,83.9469 82.2679,76.9309 76.9309,71.9389,67.2714,46,European call option price tree,31.5722,23.5343,16.0479 15.0529,10.3822 8.0868,6.4869 4.3389 0.6076,3.9538 2.3252 0.3147,1.2447 0.1630 0,0.0844 0,0 0,0,0,47,European put option price tree,0,0,0 0,1.3053 0,3.8992 2.7363 0,7.4048 6.7564 5.7360,11.2886 11.1920 12.0241,16.0531 16.0531,22.0747 23.0697,27.5623,32.7286,48,A 5-step American option pricing (prop. 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, Div at 3,rd,step (,)=0.0200,49,Stock price tree,131.5722,123.0356,115.0529 115.0529,109.7837 107.5880,102.6608 100.6076 100.6076,96 96 94.0800,89.7714 87.9759 87.9759,83.9469 82.2679,76.9309 76.9309,71.9389,67.2714,50,American call option price tree,31.5722,23.5343,17.4009 15.0529,11.0830 8.0868,6.8499 4.3389 0.6076,4.1418 2.3252 0.3147,1.2447 0.1630 0,0.0844 0,0 0,0,0,51,American put option price tree,0,0,0 0,1.3472 0,4.1319 2.8241 0,7.8407 7.1987 5.9200,11.9497 12.0241 12.0241,17.2333 17.7321,23.0697 23.0697,28.0611,32.7286,52,A 5-step European option pricing (prop. 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, Div at 3,rd,step =2.00,53,Adj. stock price tree,131.5020,122.9699,114.9414 114.9914,107.5306 107.5306,100.5539 100.5539 100.5539,94.0298 94.0298 94.0298,87.9290 87.9290 87.9290,82.2240 82.2240,76.8892 76.8892,71.9005,67.2355,54,Stock price tree,ex-dividend 131.5020,122.9699,114.9914 114.9914,109.5206 107.5306,102.5340 100.5539 100.5539,96 96.0198 94.0800,89.9091 87.9290 87.9290,82.2140 82.2240,76.8892 76.8892,71.9905,67.2355,55,European call option price tree,31.5020,23.4687,15.9684 14.9914,10.3298 8.0294,6.4459 4.2958 0.5539,3.9236 2.2960 0.2869,1.2260 0.1486 0,0.0770 0,0 0,0,0,56,A 5-step American option pricing (prop. 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, Div at 3,rd,step =2.00,57,Adj. stock price tree,131.5020,122.9699,114.9414 114.9914,107.5306 107.5306,100.5539 100.5539 100.5539,94.0298 94.0298 94.0298,87.9290 87.9290 87.9290,82.2240 82.2240,76.8892 76.8892,71.9005,67.2355,58,Stock price tree,ex-dividend 131.5020,122.9699,114.9914 114.9914,109.5206 107.5306,102.5340 100.5539 100.5539,96 96.0198 94.0800,89.9091 87.9290 87.9290,82.2140 82.2240,76.8892 76.8892,71.9905,67.2355,59,American call option price tree,31.5020,23.4687,16.9914 14.9914,10.8504 8.0294,6.7155 4.2958 0.5539,4.0633 2.2960 0.2869,1.2260 0.1486 0,0.0770 0,0 0,0,0,60,
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