《衍生品证券derivativeschapter.ppt》由会员分享，可在线阅读，更多相关《衍生品证券derivativeschapter.ppt（56页珍藏版）》请在三一办公上搜索。
1、1,Derivative SecuritiesSAIFDr.Ming Guo,Chapter 10 Binomial Option Pricing I,2,Chapter Outline,How the binomial model works and how to use itA one-period binomial treeTwo or more binomial periodsCall options and put optionsEuropean options and American optionsOptions on other assets:stock index,curre
2、ncies,futures contractsChapter 11:the assumptions underlying the model,3,Introduction to Binomial Option Pricing,Binomial option pricing enables us to determine the price of an option,given the characteristics of the stock or other underlying assetTo perform synthetic replicationA simple yet powerfu
3、l approachAssumption:the price of the underlying asset follows a binomial distribution that is,the asset price in each period can move only up or down by a specified amount,4,A One-Period Binomial Tree,Consider a European call option on the stock of XYZ,with a$40 strike and 1 year to expirationXYZ d
4、oes not pay dividends,and its current price is$41The continuously compounded risk-free interest rate is 8%The following figure depicts possible stock prices over 1 year,i.e.,a binomial tree,5,Computing the Option Price,Next,consider two portfoliosPortfolio A:buy one call optionPortfolio B:buy 2/3 sh
5、ares of XYZ and borrow$18.462 at the risk-free rateCostsPortfolio A:the call premium,which is unknownPortfolio B:2/3$41$18.462=$8.871,6,Computing the Option Price(contd),7,Computing the Option Price(contd),Portfolios A and B have the same payoff,thus,they should have the same cost.Since Portfolio B
6、costs$8.871,the price of one option must be$8.871The payoff to a call is created by borrowing to buy shares.Portfolio B is a synthetic callOne option has the risk of 2/3 shares.The value 2/3 is the delta()of the option:the number of shares that replicates the option payoff,8,Some Concerns,Does the p
7、robability of the up or down movement in stock prices matter?The no-arbitrage argument is independent on the real probabilities.What about the expected return on the stock?There is one-to-one correspondence between return and probability.What is the expected return on the option?The call is equivale
8、nt to a leveraged position in the stock.,9,The Binomial Solution,How do we find a replicating portfolio consisting of shares of stock and a dollar amount B in lending,such that the portfolio imitates the option no matter the stock rises or falls?Suppose that the stock has a continuous dividend yield
9、 of,which is reinvested in the stock,and the risk-free interest rate is r.uS0 denotes the stock price when the price goes up,and dS0 denotes the stock price when the price goes down.,10,The Binomial Solution(contd),Stock price tree:Corresponding tree for the value of option:The value of the replicat
10、ing portfolio at time h,with stock price Sh,is,11,The Binomial Solution(contd),At the prices Sh=uS and Sh=dS,a replicating portfolio will satisfy(uS eh)+(B erh)=Cu(dS eh)+(B erh)=CdSolving for and B gives,12,The Binomial Solution(contd),The cost of creating the option is the cash flow required to bu
11、y the shares and bonds.Thus,the cost of the option is S+B.The no-arbitrage condition is u e(r)h d Verify the numbers in our example using formulas(10.1),(10.2),and(10.3).,13,Arbitraging a Mispriced Option,If the observed option price differs from its theoretical price,arbitrage is possibleIf an opti
12、on is overpriced,we can sell the option.However,the risk is that the option will be in the money at expiration,and we will be required to deliver the stock.To hedge this risk,we can buy a synthetic option(borrow to buy)at the same time.If an option is underpriced,we buy the option and sell a synthet
13、ic option at the same time.,14,A Graphical Interpretation of the Binomial Formula,The portfolio describes a line with the formula Where Ch and Sh are the option and stock value after one binomial period,and supposing=0We can control the slope of a payoff diagram by varying the number of shares,and i
14、ts height by varying the number of bonds,BAny line replicating a call will have a positive slope(0)and negative intercept(B 0),15,A Graphical Interpretation(contd),Figure 10.2 The payoff to an expiring call option is the dark heavy line.The payoff to the option at the points dS and uS are Cd and Cu(
15、at point D).The portfolio consisting of shares and B bonds has intercept erh B and slope,and by construction goes through both points E and D.The slope of the line is calculated as Rise/Run between points E and D,which gives the formula for.,16,Risk-Neutral Pricing,We can interpret the terms(e(r)h d
16、)/(u d)and(u e(r)h)/(u d)as probabilities(adding to 1)Let Then equation(10.3)can then be written as where p*is the risk-neutral probability of an increase in the stock price,17,Constructing a Binomial Tree,Goal:to characterize future uncertainty about the stock price in an economically reasonable wa
17、y.In the absence of uncertainty,a stock must appreciate at the risk-free rate less the dividend yield.Thus,from time t to time t+h,we haveThe price next period equals the forward price,18,Constructing a Binomial Tree(contd),With uncertainty,the stock price evolution can be modeled asWhere is the ann
18、ualized standard deviation of the continuously compounded returnWe can also rewrite(10.9)as We refer to a tree constructed using equation(10.10)as a“forward tree.”,19,An Expression of p*in Forward Tree,From(10.6),we havep*is decreasing in h and approaches 0.5 as h approaches 0.,20,Summary,In order t
19、o price an option,we need to know Stock priceStrike priceStandard deviation of returns on the stockDividend yieldRisk-free rateWe can approximate the future distribution of the stock by creating a binomial tree using equation(10.10)(forward tree)Once we have the binomial tree,it is possible to price
20、 the option using equation(10.3),21,Another One-Period Example,Reconsider the European call option on the stock of XYZ with K=$40 and h=1 year.XYZ does not pay dividends(=0),and its current price S0=$41.The continuously compounded risk-free interest rate r=8%.Instead of assuming the price next year
21、will be either$60 or$30,here,we assume that the volatility=30%.By construction,22,Another One-Period Example(contd),Accordingly,uS=$59.954,dS=$32.903 and Cu=$19.954,Cd=$0.Using formula(10.3),the option price is=0.738,B=$22.405(not showing here),23,Another One-Period Example(contd),The binomial tree:
22、,$41$7.839=0.738B=$22.405,$59.954$19.954,$32.903$0.000,24,A Two-Period European Call,We can extend the previous example to price a 2-year option,assuming all inputs are the same as before:S=$41.00,K=$40.00,=0.30,r=0.08,T=2.00 years,=0.00,and h=1.000.Steps:calculate the u and d,find stock price at ea
23、ch node,and construct the binomial true.Work backward through the tree.Find option value at each node.Both the option value and the corresponding,and B are indicated at each node.,25,A Two-Period European Call(contd),Figure 10.4,Numbers in bold italic signify that exercise is optimal at that node.,2
24、6,A Two-period European Call(contd),Note that an up move by the stock followed by a down move(Sud)generates the same stock price as a down move followed by an up move(Sdu).This is called a recombining tree.(Otherwise,we would have a nonrecombining tree),27,Pricing the Call Option,To price an option
25、with two binomial periods,we work backward through the treeIn year 2,since we are at expiration,the option value is max(0,S K)u=e0.08+0.3=1.4623;d=e=0.8025,28,Pricing the Call Option(contd),We compute the option value using equation(10.3),In year 1In year 0,Stock Price=$41:similarly,the option value
26、 is computed to be$10.737,29,Pricing the Call Option(contd),Notice thatThe option was priced by working backward through the binomial treeThe option price is greater for the 2-year than for the 1-year option($7.839)The options and B are different at different nodes.At a given point in time,increases
27、 to 1 as we go further into the moneyPermitting early exercise would make no difference.At every node prior to expiration,the option price is greater than S K;thus,we would not exercise even if the option was American,30,Many Binomial Periods,Dividing the time to expiration into more periods allows
28、us to generate a more realistic tree with a larger number of different values at expirationConsider the previous example of the 1-year European call optionLet there be three binomial periods.Since it is a 1-year call,this means that the length of a period is h=1/3.(T=1)Assume that other inputs are t
29、he same as before(so,S=$41.00,K=$40.00,=0.30,r=0.08,T=1.00 year,=0.00,and h=1/3.),31,Many Binomial Periods(contd),The stock price and option price tree for this option,32,Many Binomial Periods(contd),Note that since the length of the binomial period is shorter,u and d are smaller than before:u=1.221
30、2 and d=0.8637(as opposed to 1.462 and 0.803 for h=1)The second-period nodes are computed as followsThe remaining nodes are computed similarlyAnalogous to the procedure for pricing the 2-year option,the price of the three-period option is computed by working backward using equation(10.3).The option
31、price is$7.074.,33,Connection to B-S Model,Holding fixed the time to expiration,we vary the number,n,of the binomial steps to get binomial call prices.(T=1)As n goes to infinite,we have the limiting value for the price.This price is given by the Black-Scholes formula.Thus,the B-S model is a limiting
32、 case of the binomial formula for the price of a European option.,34,Role of Risk-Neutral Probability,Two-period European Call:S=$41.00,K=$40.00,=0.30,r=0.08,T=2.00 years,=0.00,and h=1.000.u=1.4623;d=0.8025(slide#24)Cuu=$47.669,Cud=Cdu=$8.114,Cdd=0.(slide#25)Option price is$10.737(slide#25.Differenc
33、e caused by rounding error).,35,Put Options,We compute put option prices using the same stock price tree and in the same way as call option pricesThe only difference with a European put option occurs at expirationInstead of computing the price as max(0,S K),we use max(0,K S),36,Put Options(contd),A
34、binomial tree for a European put option with 1-year to expiration,S=$41.00,K=$40.00,=0.30,r=0.08,T=1.00 year,=0.00,and h=0.333.,37,American Options,The value of the option if it is left“alive”(i.e.,unexercised)is given by the value of holding it for another period,equation(10.3)The value of the opti
35、on if it is exercised is given by max(0,S K)if it is a call and max(0,K S)if it is a putFor an American call,the value of the option at a node is given by,38,American Options(contd),The valuation of American options proceeds as followsAt each node,we check for early exerciseIf the value of the optio
36、n is greater when exercised,we assign that value to the node.Otherwise,we assign the value of the option unexercisedWe work backward through the tree as usual,39,American Options(contd),Consider an American version of the put option valued in the previous example,S=$41.00,K=$40.00,=0.30,r=0.08,T=1.0
37、0 year,=0.00,and h=0.333.,$8.363 if holding,40,American Options(contd),The only difference in the binomial tree occurs at the Sdd node,where the stock price is$30.585.The American option at that point is worth$40$30.585=$9.415,its early-exercise value(as opposed to$8.363 if unexercised).The greater
38、value of the option at that node ripples back through the treeThus,an American option is at least as valuable as the otherwise equivalent European option,41,Options on Other Assets,The model developed thus far can be modified easily to price options on underlying assets other than nondividend-paying
39、 stocksThe difference for different underlying assets is(i)the construction of the binomial tree,and(ii)the risk-neutral probabilityWe examine options on-Stock indexes-Commodities-Currencies-Bonds-Futures contracts,42,Options on a Stock Index,Suppose a stock index pays continuous dividends at the ra
40、te The procedure for pricing this option is equivalent to that of the first example,which was used for our derivation.Specificallythe up and down index moves are given by equation(10.10)the replicating portfolio by equation(10.1)and(10.2)the option price by equation(10.3)the risk-neutral probability
41、 by equation(10.5),43,Options on a Stock Index(contd),Consider an American call option on a stock index.AssumeCurrent index value is$110.00The strike price is$100.00The volatility of the stock index=0.30The risk-free interest rate is 5%The stock index is continuously paying dividend at rate of 3.5%T
42、he option expires in one year and there are three periods.,44,Options on a Stock Index(contd),The binomial tree for the American call option on a stock index:,The holding value is$56.942.,45,Options on Currency,With a currency with spot price x0,the forward price iswhere rf is the foreign interest r
43、ateThus,we construct the binomial tree using,46,Options on Currency(contd),Investing in a“currency”means investing in a money-market fund or fixed income obligation denominated in that currencyTaking into account interest on the foreign-currency denominated obligation,the two equations areThe risk-n
44、eutral probability of an up move is,47,Options on Currency(contd),Consider a dollar-denominated American put option on the euro,whereThe current exchange rate is$1.05/The strike is$1.10/The euro-denominated interest rate is 3.1%The dollar-denominated rate is 5.5%The volatility is 10%The option expir
45、es in 6 months and there are 3 periods.(T=0.50 and h=0.167),48,Options on Currency(contd),The binomial tree for the American put option on the euro,Because volatility is low and the option is in-the-money,early exercise is optimal at three nodes prior to expiration.,49,Options on Futures Contracts,A
46、ssume the forward price is the same as the futures priceThe nodes are constructed asWe need to find the number of futures contracts,and the lending,B,that replicates the optionA replicating portfolio must satisfy,50,Options on Futures Contracts(contd),Solving for and B gives tells us how many future
47、s contracts to hold to hedge the option,and B is simply the value of the optionThe option can be again priced using equation(10.3)or simply(0)+B=B.The risk-neutral probability of an up move is given by,51,Options on Futures Contracts(contd),The motive for early-exercise of an option on a futures con
48、tract is the ability to earn interest on the mark-to-market proceedsWhen an option is exercised,the option holder pays nothing,is entered into a futures contract,and receives mark-to-market proceeds of the difference between the strike price and the futures price,52,Options on Futures Contracts(cont
49、d),Consider an American call option on a gold futures contract.AssumeThe price of the futures contract expiring in one year is$300.00 per ounce.(The value of futures contract at time zero is zero.Hence the strike price can be considered as$300.00 per ounce as well.)The risk-free interest rate is 5%a
50、nd the volatility of the futures price is 10%.Assume there are three periods.,53,Options on Futures Contracts(contd),A tree for an American call option on a gold futures contract,B should be$36.113,54,Options on Commodities,It is possible to have options on a physical commodityIf there is a market f
