【導(dǎo)讀】ForwardandFutures. Options. RiskNeutralPricing. (maturitydate,settlementdate).markets.barrel,Deltaincursalossof$20,000.45°Thespotprice,ST,atmaturity.Theprofit(orloss)incurred,F0–ST.thefuturesprice.

　　

【正文】 up at a rate of 12 percent (u = ) or go down at a rate of 10 percent (d = ). What is the value of the structure bond? TwoPeriod Binomial Tree UU UD DD U D $1,275 $1,428 $1, $1, $1, $1, Samp。P 500 [Vuu=$ =+] [Vud=$ =+] [Vdd=$] [Structured Bond] π 1π π 1π π 1π Identify the Risk Neutral Probabilities ?Since both u and d do not change over periods, the risk neutral probabilities at each period should be the same. ?The following formula gives the value of risk neutral probabilities. πu + (1 π)d = 1 + rf π = (1 + rf d)/(u d) = ( – )/( – ) = Find the Value of Structured Bond ? Once we obtain the risk neutral probabilities we can solve the problem backward through the tree diagram to find the value of structure bond, using the risk neutral technique. ? The value of structured bond at node U, ? Vu= [πVuu + (1 π)Vud]/(1 + rf) = (*+*)/=$ ? The value of structured bond at node D, ? Vd= [πVud + (1 π)Vdd ]/(1 + rf) = (*+*)/=$ ? The value of structured bond currently, ? V= [πVu + (1 π)Vd]/(1 + rf) = (*+*)/=$ A General Formula Under Two Period Binomial Process ?The value of derivative at node U, ?The value of derivative at node D, ?The value of derivative at beginning, fuduuu rVVV????1)1( ??fddudd rVVV????1)1( ??222)1()1()1(21)1(fdduduufdurVVVrVVV????????????????Dynamic Tracking Portfolios ? We can also derive the value of the structured bond by tracking its future payoffs backward using the underlying asset and the risk free rate. The tracking portfolios are as the following, ? The tracking portfolio at node U, (Δu, Bu) is (1, $) ? The tracking portfolio at node D, (Δd, Bd) is (, ) ? The tracking portfolio at beginning, (Δ1, B1) is (, $) ? An investor who wants to track the future payoffs of a structured bond need to adjust her tracking portfolio at the beginning of each period. Binomial Valuation of European Options ? Consider a European call option, which will expire in the next period, on a stock without dividend. The current stock price is S0 and the strike price is K. The Stock price follows a one period binomial process which will either increase to uS0 or decrease to dS0. The risk free rate is rf. The binomial tree is as the follows, S0 uS0 U D uS0 Max[uS0K, 0] Max[dS0K, 0] π 1π Value Under One Period Binomial Process ?Identify the risk neutral probabilities ?Value the option dudr f???? 1?duru f????? 11 ?frKdSKuSc??????1]0,ma x [)1(]0,ma x [ 000??Valuation Assuming NPeriod Binomial Process ? If the u and d are constant over periods the risk neutral probabilities will not change. ? Nperiod binomial process will generate N+1 outes at maturity, uNS0, uN1dS0, … , ud N1S0, and dNS0. ? Under risk neutral measure, the probability for the oute uNS0 is πN. The probability associated with ujdNjS0 is N!/j!(Nj)! πj(1π)Nj. ? The value of call option is, ???? ?????NjjNjjNjNfKSdujNj Nrc000 ],0m a x[)1()!(!!)1(1 ?? Binomial Valuation of American Options ?The general procedure for solving American option value through binomial approach is similar to that for European option. ? Work backward from the righthand side of the tree diagram. ? At each node in the tree diagram, discount the expected future payoffs under risk neutral measure to determine the value of the option at that node. ? Compare this value to the value if the option is early exercised at that node. The value of option at the node is the larger one. ? Continue to work backward until the beginning of all periods. American Puts ?Assume that each period, the nondividendpaying stock of Chiron, a drug pany, can either double or halve in value, that is, u = 2, d = . If the initial price of the stock is $20 per share and the risk free rate is 25 percent per period, what is the value of an American put expiring two periods from now with a strike price of $ UU UD DD U D $20 $40 $80 $10 $20 $5 $0 $ $ π = π = π = 1π = 1π = 1π = $3 $12 $ = (3*+*)/ Exercise: $ = 20 Exercise: $ = 10 Exercise: $0 Binomial Tree for American Put Option Solve the Value for American Put ? The risk neutral probabilities π satisfies, 2*π + *(1π)=, which implies π = . ? At node U, the risk neutral price of put is (*0 + *)/ = $3, which is the price if the put option is hold over the last period. It is higher than the immediate exercising value ($0) at node U. ? At node D, the risk neutral price of put is (* + *)/ = $12, which is lower than the exercising value ($$10). Investor will exercise the put at D. ? The beginning risk neutral value is (*3 + *)/ = $, which is higher than the immediate exercising value ($$20). ? Therefore the current value of American put option is $ Tracking Portfolio and Value of Early Exercising ? The tracking portfolio for the American put option over the first period should replicate the option’s payoffs at the end of first period (3, ). Therefore the portfolio (ΔA1, BA1) is (, $). The American put value is *20+= ? The tracking portfolio for the European put option over the first period should replicate the option’s payoffs at the end of first period (3, 12). Therefore the portfolio (ΔE1, BE1) is (, $12) ? The value of European put option is *20+12=$6, which is $ less than the American put option. ? This difference in the value between American and European options can also be obtained by equation: *() American Options on DividendPaying Stocks ? A stock that pays dividends has two values at the exdividend date: ? Cumdividend value: the value of the stock prior to the exdividend date. ? Exdividend value: the stock price after the exdividend date