Pricing Exotic Options With Binomial Trees

1. Pricing Exotic Options With Binomial Trees By Erik O. Mayer

2. Quick Recap • Binomial Tree Option Pricing is a numerical approximation to the Black Scholes PDE Method (for Vanilla Options) • For a given stock price, volatility, riskless interest rate, a time to expiry on an option, and a number of time intervals, we model the following: • A Stock Price Binomial Tree by multiplying S0 by the up and down factors • An Option Value Binomial Tree by calculating the option payoff at the time of expiry and working towards t0 through backward induction Stock Price Tree (where node * is located m up moves and k down moves from S0) Riskless Probability of an “Up” Move, p

3. “Binomial Value” at V* Option Value Tree V0 = Price of the Option at t0 • For American Options, which can be exercised at any time before expiry, the value of the option at each node is the maximum of the “Binomial Value” at that node and the value of early exercise at that node.

4. Exotic Options, You Say? • Recall the example of a European Call from the previous presentation: S0= $100 K = $100 σ = .30 r = .05 T = 1 n = 4 Δt = .25 u = 1.1618 d = 0.8607 p = 0.5043 • Using this information, we modeled a stock price tree, and then we calculated the Value of the Option at the Time of Expiry for each node at T. Using those Option Values, we calculated the Option Values for the nodes at each preceding time step until we arrived at t0, where we got an option price of $13.53. • Now, let’s change this European Call into an Exotic Option, the Up-and-Out Barrier Call, by setting the Barrier at $150. Our Stock Price Tree will remain the same, except for an indicator of the Barrier’s position: B = $150 By definition, the value of the Up-and-Out Barrier Option becomes worthless if the Stock Price goes above the Barrier at any point during the course of its path. How does this change the way we approach pricing the option? Can we still value it at the Time of Expiry? T

5. Option Value Tree? Why is this not correct? How do we get around this problem of path-dependency? • Here’s the problem: • We can’t value the option at Time of Expiry like we did with Europeanoptions because the payoff at T depends on more than simply the price of the stock at T. And because we don’t have the option values at T, we can’t use backward induction to fill out the rest of the Option Value Tree • Here’s what we know: • Every possible path the stock can take on our binomial tree • The payoff of any individual path the stock price takes • The total probability of each possible path

6. The Paths on Our Tree p For a stock price tree of this form, with n = 1 time interval,the stock price can take two paths: up or down. The probability of the up path is p, while the probability ofthe down path is (1-p). 1-p Δt p For a stock price tree with n = 2 time intervals, thestock price can take four different paths: one with two up moves, one with two down moves, one with an upmove followed by a down move, and one with a down move followed by an up move, with total probabilities p2, (1-p)2, p(1-p), and (1-p)p respectively. p 1-p p 1-p 1-p Δt Δt We’ll say that in general, a stock price tree with n time intervals has 2n differentpaths that the stock price can take, each with a total probability Q = pm(1-p)k,where m is the number of up moves the stock price takes and k is the numberof down moves it takes.

7. The Option Value Formula Generalized • For a one step sub-section of the option value tree, the formula for V looked like this: Let’s generalize this. Instead of using “up” and “down” values of the option, let’s refer to them as Payoff “Outcomes,” O1 and O2. Q1 will refer to the probability attached to Outcome 1, and Q2 will refer to the probability attached to Outcome 2. Now let’s apply this to what we know in terms of paths. There are 2ndifferent paths on an n time interval stock price tree, and each of those paths has a probability attached to it and a payoff outcome for the corresponding option. So there will be 2n possible payoff outcomes for an entire stock price tree. And since we’re now talking about the entire tree, Δt will be replaced with T, the time to expiry, and the arbitrary V will be replaced with V0, the price of the option at t0. Now we have a formula for the price of an option on a stock that’s based on the different possible paths the stock can take on its tree. With this we will be able to price path-dependent options.

8. Pricing the Barrier Option • Now we return to the example of the up-and-out barrier call with the barrier set at $150 and the strike price at $100. Time to expiry was one year, with four subintervals. So there are 24= 16 different paths on this stock price tree. Let’s find all the paths, the payoff of each path, and the probability of each path happening. Payoff of a Path = max(0, ST – K) unless the stock rises above $150, in which case payoff is $0 Total Probability Q = pm(1-p)k, where p = 0.5043 in this example B = $150

9. 2 3 4 1 Payoff = $0 Payoff = $0 Payoff = $34.99 Payoff = $34.99 Q = 0.0647 Q = 0.0636 Q = 0.0636 Q = 0.0636 Payoff = $34.99 Payoff = $0 Payoff = $0 Payoff = $0 Q = 0.0636 Q = 0.0625 Q = 0.0625 Q = 0.0625 Payoff = $0 Payoff = $0 Payoff = $0 Payoff = $0 Q = 0.0625 Q = 0.0625 Q = 0.0625 Q = 0.0614 Payoff = $0 Payoff = $0 Payoff = $0 Payoff = $0 Q = 0.0614 Q = 0.0614 Q = 0.0614 Q = 0.0604 5 6 7 8 9 11 12 10 13 14 15 16

10. To verify that the we found all the right paths (ie. we didn’t miss a path or accidentally repeat a path), we can add up all the Q probabilities, and the sum should be 1 (because the probability of the stock taking one of these paths on our tree is 100%). • Now we can use the formula to find the price of the barrier option at t0. = $6.35 The actual price of this barrier option is $5.13 (More on this in a moment) For this option, the barrier “knocked out” two of the paths. Had it been a regular European Call, without a barrier, the payoff of those two paths would have been $82.21 and $34.99 respectively, and the formula would have yielded a result of $13.52, which aside from a penny’s difference due to rounding issues, is precisely the answer we got for this European Call when we used backward induction. V0 = ) (

11. Convergence? • The four time interval binomial tree gave us a price of $6.35, while the online option calculator yielded a price of $5.13. Theoretically, the binomial tree price should converge to the actual price as n ∞. Let’s take a look at how this converges using a Maple worksheet. • That’s not too good of an outlook. It’s not clear how many time intervals it would take for the price to converge, but judging from the first 18 time intervals, it looks like it will be quite slow. • Why is it converging so slowly? Part of the problem with using binomial trees for barrier options is the price’s sensitivity to a barrier slightly above or below a “line” of stock prices.

12. B = $182.20 V0 = $8.46 Actual Price = $10.45 B = $182.22 V0 = $13.52 Actual Price = $10.45 B = $156.84 V0 = $8.46 Actual Price = $6.46 B = $156.82 V0 = $6.35 Actual Price = $6.46 B = $150 V0 = $6.35 Actual Price = $5.13 B = $135.00 V0 = $6.35 Actual Price = $2.28 B = $134.98 V0 = $0 Actual Price = $2.28 Any barrier level on the interval (134.99, 156.83) for this example results in the same option price of $6.35, while the actual prices on that interval range from $2.28 to $6.46. This happens because on this binomial tree, barrier levels on that interval knock out the same two paths. However, moving the barrier slightly above that interval jumps the price to $8.46, while moving it slightly below that interval drops the price to $0, since suddenly every non-zero path payout is excluded by the barrier. These drastic jumps in price will cause problems when increasing the number of time intervals if the barrier changes sides with these “interval endpoints” in the process.

13. n = 5 25= 32 pathsStill only two paths are knocked out, and more paths have positive payouts, so the price goes way up.V0 = $10.33 n = 6 26 = 64 pathsMany more paths are knocked out becausethe barrier is below more nodes now. Onlyone node at time of expiry has can yield apositive payoff now. The price goes way down.V0 = $5.86 If the barrier keeps hopping above and below the “lines” of nodes (for these amounts of time intervals, indicated by the green line), the priceof the option will keep hopping up and down. Convergence will be very slow. Is there any way to adjust for this?

14. White-Hull Up and Down Factors • The CRR Up and Down Factors have the attribute that u multiplied by d equals 1. What this does is create a horizontal “line” of nodes across the binomial tree. If the barrier is above one of those nodes, it’s above all of those nodes, which contributes to the large jump in prices. • An alternative to the CRR model, the White-Hull Up and Down Factors do not have the attribute that u*d = 1, meaning that an up move followed by a down move does not return the stock price to what it was before any moves. This scatters the nodes a little. Here’s how White and Hull defined these up and down factors: • Let’s use the Maple worksheet to see if this helps the convergence. • That’s still not great. In fact, it follows the CRR model pretty closely, at least out to 18 time intervals. It’s still pretty far away from converging to $5.13.

15. What Can Be Said? • It would be great if we could go out to, say, 100 time intervals with our Maple procedure and see how much longer it takes to converge. Unfortunately, since we must keep track of every path on these binomial trees in order to price path-dependent options, Maple is keeping track of a ton of information and becomes very slow. It starts taking several minutes to calculate a price if you go much further than 15 time intervals. If we had more powerful computers with better processing speed, we would be able to say a lot more. • Not all is lost, however. Computers are constantly getting better and faster. Perhaps binomial trees could become a viable method for calculating prices for path-dependent exotic options. What we can take away from this is that binomial trees can be adapted to a very large assortment of options. Even if it isn’t practical to do it this way, it theoretically works. • For now and for the foreseeable future, though, Monte Carlo Simulations are better for calculating prices onexotic options.

16. On a Positive Note:Applying this Framework to Other Options • Once we know all the paths on a binomial tree (and their corresponding probabilities), we can price other path-dependent options, like Asian Options and Lookback Options. Payoff for a Fixed Strike Asian Option = The difference between the Strike and the average stock price over the time period (for each path, we know the stock price for every time step, so we can find its average easily). Payoff for a Fixed Strike Lookback Option = The difference between the Strike and maximum or minimum stock price over the time period (again, we know the stock price at every time step for every path, so we can determine the highest or lowest of those prices). • And there are many, many more path-dependent options, and binomial trees can deal with a great many of them to an extent.

17. Sources • Black-Scholes and Beyond: Option Pricing Models by Neil A. Chriss • Valuing Derivatives via the Binomial Tree Method (Cox-Ross-Rubinstein) by Klaus Volpert • Exotic Options by Chris DiBella • www3.sitmo.com (Online Option Calculator) • Special Thanks: Title Slide Picture – In the Forest by deviantART user Hartmut-Lerch