衍生品证券DerivativesChapter10.ppt

1,Derivative Securities SAIF Dr. Ming Guo,Chapter 10 Binomial Option Pricing I,2,Chapter Outline,How the binomial model works and how to use it A one-period binomial tree Two or more binomial periods Call options and put options European options and American options Options on other assets: stock index, currencies, futures contracts Chapter 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 asset To perform synthetic replication A simple yet powerful approach Assumption: 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 expiration XYZ does not pay dividends, and its current price is $41 The 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 portfolios Portfolio A: buy one call option Portfolio B: buy 2/3 shares of XYZ and borrow $18.462 at the risk-free rate Costs Portfolio A: the call premium, which is unknown Portfolio 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 costs $8.871, the price of one option must be $8.871 The payoff to a call is created by borrowing to buy shares. Portfolio B is a synthetic call One 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 probability 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 equivalent 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 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 replicating 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) = Cd Solving for and B gives,12,The Binomial Solution (contd),The cost of creating the option is the cash flow required to buy 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 possible If an option 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 synthetic 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 = 0 We can control the slope of a payoff diagram by varying the number of shares, , and its height by varying the number of bonds, B Any 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 (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 )/(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 way. 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 have The price next period equals the forward price,18,Constructing a Binomial Tree (contd),With uncertainty, the stock price evolution can be modeled as Where is the annualized standard deviation of the continuously compounded return We 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 have p* is decreasing in h and approaches 0.5 as h approaches 0.,20,Summary,In order to price an option, we need to know Stock price Strike price Standard deviation of returns on the stock Dividend yield Risk-free rate We 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 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 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:,$41 $7.839 =0.738 B=$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 each 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.,26,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 with two binomial periods, we work backward through the tree In year 2, since we are at expiration, the option value is max (0, S K) u = e0.08+0.3 = 1.4623; d = e0.08-0.3 = 0.8025,28,Pricing the Call Option (contd),We compute the option value using equation (10.3), In year 1 In year 0, Stock Price = $41: similarly, the option value is computed to be $10.737,29,Pricing the Call Option (contd),Notice that The option was priced by working backward through the binomial tree The 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 to 1 as we go further into the money Permitting 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 us to generate a more realistic tree with a larger number of different values at expiration Consider the previous example of the 1-year European call option Let 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 the 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 = 1.2212 and d = 0.8637 (as opposed to 1.462 and 0.803 for h = 1) The second-period nodes are computed as follows The remaining nodes are computed similarly Analogous 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 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 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. Difference 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 prices The only difference with a European put option occurs at expiration Instead of computing the price as max (0, S K), we use max (0, K S),36,Put Options (contd),A 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 option 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 put For 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 follows At each node, we check for early exercise If the value of the option is greater when exercised, we assign that value to the node. Otherwise, we assign the value of the option unexercised We 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 = 1.00 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 value of the option at that node ripples back through the tree Thus, 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 stocks The difference for different underlying assets is (i) the construction of the binomial tree, and (ii) the risk-neutral probability We 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 rate The procedure for pricing this option is equivalent to that of the first example, which was used for our derivation. Specifically the 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 by equation (10.5),43,Options on a Stock Index (contd),Consider an American call option on a stock index. Assume Current index value is $110.00 The strike price is $100.00 The volatility of the stock index = 0.30 The risk-free interest rate is 5% The stock index is continuously paying dividend at rate of 3.5% The 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 is where rf is the foreign interest rate Thus, 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 currency Taking into account interest on the foreign-currency denominated obligation, the two equations are The risk-neutral probability of an up move is,47,Options on Currency (contd),Consider a dollar-denominated American put option on the euro, where The 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 expires 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,Assume the forward price is the same as the futures price The nodes are constructed as We need to find the number of futures contracts, , and the lending, B, that replicates the option A replicating portfolio must satisfy,50,Options on Futures Contracts (contd),Solving for and B gives tells us how many futures contracts to hold to hedge the option, and B is simply the value of the option The 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 contract is the ability to earn interest on the mark-to-market proceeds When 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 (contd),Consider an