© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-1 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 put, the value of the option at a node is given by

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-2 American Options (cont’d) 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 the exercised value to the node. Otherwise, we assign the value of the option unexercised. –We work backward through the tree as usual.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-3 American Options (cont’d) Consider an American version of the put option valued in the previous example.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-4

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-5 American Options (cont’d) The only difference in the binomial tree occurs at the S dd node, where the stock price is $ The American option at that point is worth $40 – $ = $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.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-6 Options on Other Assets The binomial model developed thus far can be modified easily to price options on underlying assets other than non-dividend-paying stocks. The difference for different underlying assets is the construction of the binomial tree and the risk- neutral probability. We examine options on –Stock indexes –Currencies –Futures contracts –Commodities –Bonds

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-7 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 (10.9). –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).

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-8 Options on a Stock Index Given –S = $110; –K = $100; – = 0.30; –r = 0.05 –T = 1 year – = –h = 0.333

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-9 Options on a Stock Index (cont’d) A binomial tree for an American call option on a stock index:

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Options on Currencies With a currency with spot price x 0, the forward price is where r f is the foreign interest rate. Thus, we construct the binomial tree using

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Options on Currencies (cont’d) 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 (10.20)

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Options on Currencies (cont’d) Consider a dollar-denominated American put option on the euro, where –The current exchange rate is $1.05/€ (S); –The strike is $1.10/€ (K); – = 0.10; –The euro-denominated interest rate is 3.1% (); –The dollar-denominated rate is 5.5% (r).

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Options on Currencies (cont’d) The binomial tree for the American put option on the euro

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved 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

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Options on Futures Contracts (cont’d) 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. We can again price the option using equation (10.3). The risk-neutral probability of an up move is given by (10.21)

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Options on Futures Contracts (cont’d) 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.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Options on Futures Contracts (cont’d) A tree for an American call option on a futures contract

Chapter 11 Binomial Option Pricing: Selected Topics

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved The Binomial Tree and Lognormality The usefulness of the binomial pricing model hinges on the binomial tree providing a reasonable representation of the stock price distribution. The binomial tree approximates a lognormal distribution.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved The Random Walk Model It is sometimes said that stock prices follow a random walk. Imagine that we flip a coin repeatedly –Let the random variable Y denote the outcome of the flip. –If the coin lands displaying a head, Y = 1; otherwise, Y = – 1. –If the probability of a head is 50%, we say the coin is fair. –After n flips, with the i th flip denoted Y i, the cumulative total, Z n, is (11.8) It turns out that the more times we flip, on average the farther we will move from where we start.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved The Random Walk Model (cont’d) We can represent the process followed by Z n in term of the change in Z n Z n – Z n-1 = Y n or Heads: Z n – Z n-1 = +1 Tails: Z n – Z n-1 = –1

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved The Random Walk Model (cont’d) A random walk, where with heads, the change in Z is 1, and with tails, the change in Z is – 1

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved The Random Walk Model (cont’d) The idea that asset prices should follow a random walk was articulated in Samuelson (1965). In efficient markets, an asset price should reflect all available information. In response to new information the price is equally likely to move up or down, as with the coin flip. The price after a period of time is the initial price plus the cumulative up and down movements due to informational surprises.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Modeling Stock Prices As a Random Walk The above description of a random walk is not a satisfactory description of stock price movements. There are at least three problems with this model –If by chance we get enough cumulative down movements, the stock price will become negative. –The magnitude of the move ($1) should depend upon the level of the stock price. –The stock, on average, should have a positive return. The random walk model taken literally does not permit this. The binomial model is a variant of the random walk model that solves all of these problems at once. The binomial model assumes that continuously compounded returns are a random walk with drift.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved The Binomial Model The binomial model is Taking logs, we obtain (11.11) –Since ln(S t+h /S t ) is the continuously compounded return from t to t + h, the binomial model is simply a particular way to model the continuously compounded return. –That return has two parts, one of which is certain, (r–)h, and the other of which is uncertain and generates the up and down stock prices moves (plus or minus h).

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved The Binomial Model (cont’d) Equation (11.11) solves the three problems in the random walk: –The stock price cannot become negative. –As h gets smaller, up and down moves get smaller. –There is a (r – )h term, and we can choose the probability of an up move, so we can guarantee that the expected change in the stock price is positive.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Lognormality and the Binomial Model The binomial tree approximates a lognormal distribution, which is commonly used to model stock prices. The lognormal distribution is the probability distribution that arises from the assumption that continuously compounded returns on the stock are normally distributed. With the lognormal distribution, the stock price is positive, and the distribution is skewed to the right, that is, there is a chance that extremely high stock prices will occur.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Lognormality and the Binomial Model (cont’d) The binomial model implicitly assigns probabilities to the various nodes.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Lognormality and the Binomial Model (cont’d) The binomial probability of reaching the final period node, Su n–i d i, is given by where p * is the risk-neutral probability of an up move:

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Lognormality and the Binomial Model (cont’d) The following graph compares the probability distribution for a 25-period binomial tree with the corresponding lognormal distribution

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Is the Binomial Model Realistic? The binomial model is a form of the random walk model, adapted to modeling stock prices. The lognormal random walk model in this section assumes among other things, that: –Volatility is constant. –“Large” stock price movements do not occur. –Returns are independent over time. All of these assumptions appear to be violated in the data.