The Adaptive Mesh Model:

A New Approach to Efficient Option Pricing

by

Stephen Figlewski*

and

Bin Gao**

DRAFT of May 9, 2023

COMMENTS WELCOME

* New York University Stern School of Business44 West 4th StreetNew York, NY 10012-1126(212)-998-0712sfiglews@stern.nyu.edu

** CB #3490University of North CarolinaChapel Hill, NC 27599(919) 962-7182gaob.bsacd1@mhs.unc.edu

Abstract

Closed-form valuation equations only exist for a small subset of all possible derivative securities. Most have to be priced by some numerical approximation technique, of which Binomial and Trinomial lattice models are probably the most widely used. These methods produce exact theoretical derivative valuation in the limit, but convergence is typically non-monotonic, and frequently requires very large numbers of calculations to achieve an acceptable degree of accuracy. Some valuation problems are theoretically soluble but require so much computation time as to be infeasible in practice. This paper presents a new methodology for constructing derivative valuation lattices that is capable of improving performance enormously. The Adaptive Mesh Model (AMM) essentially grafts one or more small sections of fine high-resolution lattice onto a tree that has coarser time and price steps. This gives more accuracy in critical areas without sacrificing computational efficiency unnecessarily elsewhere.

We describe two types of approximation error, “distribution error” and “truncation error,” that affect convergence in a lattice model, and then demonstrate how to construct an AMM model that sharply reduces truncation error. Three different AMM structures are presented, one for pricing ordinary options, one for barrier options, and one for computing delta and gamma efficiently. The AMM approach can be adapted to a wide variety of problems, yielding efficiency gains that can easily be several orders of magnitude for common problems.

I. IntroductionClosed-form valuation equations only exist for a small subset of all

possible derivative securities. Most have to be priced by some numerical approximation technique, of which Binomial and Trinomial lattice models are probably the most widely used. These methods have considerable appeal because they are intuitive and very flexible. A Binomial model that can be set up relatively easily is capable of dealing with a wide variety of contingencies such as early exercise of American options, caps and barriers that are common in many derivative instruments, and other features of “exotic” option products. Trinomial models have become widely used for pricing interest rate derivatives, since their added flexibility permits the current term structure in the market, as well as mean reversion in interest rates, to be incorporated into the lattice. Moreover, it can be proven that as the lattice is made finer and finer, in the limit these methods converge to the theoretical option values that would be produced by a continuous-time, continuous-state model, like Black-Scholes, that is the solution to the contingent claims fundamental partial differential equation.

Unfortunately, lattice-based models have their own problems. Convergence is normally non-monotonic, and in many cases it can require very large numbers of calculations to achieve an acceptable degree of accuracy. When this occurs, the difficulty is typically that the lattice approximation is not sufficiently accurate in some critical region, such as around the strike price at expiration, or near the price boundary for a barrier option. Greater accuracy can be obtained by making the step size in the lattice smaller, thus increasing its resolution. But the cost of doing so may be a very large increase in the number of computations that must be done, many of them wasted on unimportant regions where greater accuracy is unnecessary. The result is that in some cases, a problem is theoretically soluble but would require so much computation time in practice as to be

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essentially infeasible. In this paper, we present a new approach for constructing a lattice-

based valuation model that allows the user to vary the resolution in different parts of the tree. A relatively coarse grid that is fast to calculate is used for most of the lattice, but a fine grid is constructed where greater accuracy really matters. The Adaptive Mesh Model (AMM) can lead to a significant improvement in accuracy for a relatively small increase in computational effort. For some problems that are commonly found in practice, including calculation of delta and gamma, efficiency can be increased by several orders of magnitude relative to the standard methods.

The value of the AMM procedure comes not just from the fact that it can save computation time by speeding up the calculations required for derivatives pricing and risk management. The potential improvement in performance is so large in some cases that entire classes of problems, which could not reasonably be addressed at all with the standard technology, now become accessible. One example is analyzing the behavior of implied volatilities for a certain type of “exotic” option. Computing an implied volatility requires solving the option valuation model repeatedly with different trial values for the volatility input. Even an efficient search method like Newton-Raphson can involve repricing the option ten times or more to arrive at an acceptable level of accuracy.1 If the valuation algorithm takes 5 minutes, say, to price the option once, constructing an implied volatility data set of a reasonable size for subsequent analysis becomes extremely time-consuming. An AMM procedure that reduces computation time to 5 seconds (a performance improvement that is perfectly possible with the AMM for many problems) greatly expands the possibilities for research in this area.

In the next section, we discuss several numerical solution methods commonly used for derivatives problems, focusing particularly on the

1 This is especially true if numerical derivatives must be computed at each step.

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Binomial and Trinomial models. We describe two sources of approximation error that affect the convergence behavior of these models. The first is distribution error, that comes from the fact that the model is attempting to approximate a lognormal distribution with a binomial or trinomial. Distribution error impacts every part of the tree. The second is truncation error, that arises when the discreteness of the lattice interacts with rapid changes in the option’s delta in particular critical regions. One effect of truncation error in the Binomial model is a peculiar even-odd property in its convergence: The size (and often the sign) of the error alternate between two quite different values as the number of time steps in the tree goes from even to odd and back to even.

Section III illustrates the construction of a simple Adaptive Mesh Model and applies it to value a European put option. This demonstrates both how the AMM procedure works and how it can greatly reduce truncation error with very little increase in the amount of computation required. In Section IV, we examine a different type of valuation problem, pricing a barrier option, in this case a down and out call. Here the truncation error occurs not just at expiration in the region around the strike price, but at every point in time when the asset price approaches the barrier. Because the valuation lattice for this case must allow at least one price step between the initial asset price and the barrier, the number of nodes required in a standard trinomial tree explodes as the initial price approaches the barrier. Use of the AMM methodology permits accurate valuation with a much smaller number of calculations. Section V discusses calculation of the “Greek letter” risk exposures in Binomial and Trinomial lattices. We present a third type of AMM constructed around the initial node of the tree, that markedly improves the accuracy of numerical derivatives. The final section concludes with a brief discussion of further extensions to the AMM approach.

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II. Valuing Options by Numerical ApproximationThe Black-Scholes [1973] (BS) was the first rigorously justifiable closed

form equation for pricing European calls and puts based on observable parameters. Equally importantly, the no-arbitrage principle for obtaining the BS equation pointed the way toward theoretical valuation models for all types of contingent claims.

Closed Form SolutionsUnfortunately, American options presented a problem for the Black-Scholes approach. American calls may be exercised optimally just prior to any ex-dividend date, while exercise of an American put can be optimal at any point during its lifetime. Although the no-arbitrage principle must still hold for American options, the BS equation could not be adapted easily for them, because the optimal exercise strategy depends on the asset price distribution at more dates than just option expiration. While a few modified Black-Scholes-like models were developed 5to deal with early exercise, their complexity prevents extending this line of approach into a usable general valuation technique.

Roll [1977] and Whaley [1981] presented a pricing formula for an American call on a stock that paid one dividend prior to maturity.2 The BS equation involves two terms, each containing the cumulative normal distribution function. Since an American call can be exercised at expiration only if it has not already been exercised on the ex-dividend date, its value depends on the conditional probability distribution for the price at expiration given the price on the ex-dividend date. Thus, the Roll/Whaley formula contains terms involving the cumulative bivariate normal distribution function. Multiple dividends lead to multiple dates on which exercise may occur, and the resulting valuation equation rapidly becomes very complex.

2 Roll is generally credited with developing the approach, but the published version of his paper contains an error in the equation. This was later corrected by Whaley.

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With more than a very small number of dividends, the “closed form” solutions (which are not completely closed-form in practice) become too complex to be usable.3

The American put is the limiting case of the problem just described, since there are an infinite number of potential early exercise dates, independent of dividend payout. Geske and Johnson [1984] developed a model for approximating put values as the limit of a sequence of models like the American call equation. Although Geske and Johnson proposed a clever solution technique using Richardson extrapolation to approximate the put value without solving what amounts to an infinite series of infinitely complicated mathematical expressions, their model can not be considered a closed form valuation equation in the same way that the original BS model is. Rather, it is one way of obtaining an approximate numerical solution to the problem, and it turns out not to be the easiest such method.

Approximating the Fundamental PDEThe Black-Scholes methodology begins with the assumption that the

underlying asset follows the logarithmic diffusion,

where S is the stock price, dS denotes the price change over the infinitesimal time interval dt, μ and σ are the instantaneous mean and volatility, and dz represents Brownian motion, a random infinitesimal increment with mean 0 and variance 1dt. Assuming continuous trading and no transactions costs, the no-arbitrage principle leads to the fundamental partial differential

3 With N dividends, the formula requires evaluating N+1 - variate cumulative normal distributions. While this function can be written easily as an integral of the normal density function, there in no closed form equation for the integral, which must be evaluated by numerical approximation.

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equation (PDE) of contingent claims pricingwhere C is the contingent claim value, r is the riskless interest rate, and T is the time to expiration (T = TEXP - t , where TEXP is expiration and t is the current date).

Any derivative instrument whose payoff depends only on the value of S and t must obey equation (2), subject to a set of boundary conditions that differentiate one contingent claim from another. Solving (2) subject to the appropriate boundary conditions for a European call yields a closed form solution, the Black-Scholes equation. But since equation (2) holds generally, it can be solved approximately by numerical methods to give model values for American puts and other contingent claims for which closed-form solutions are not available. The technique involves replacing equation (2) by a difference equation defined on a discrete lattice of stock prices and dates. Brennan and Schwartz [1977] introduced this “finite difference” approach in their examination of American put pricing.

But despite the power of the methodology to solve a much broader class of valuation problems, it suffers from several important drawbacks. It is computationally burdensome, and can easily reach the limits of available computer resources when one introduces contingent claims based on more than one stochastic variable. Second, the solution is only approximate and convergence to accurate values may be slow (i.e., requiring a very fine lattice, with a very large number of computations). Finally, it is rather nonintuitive for those who do not have a taste for computational methods.

The Binomial ModelThe approximation procedure that has become the workhorse of

practical contingent claims valuation is the Binomial model, first introduced as a pedagogical device by Sharpe in a textbook [1978] and then developed

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formally into a general valuation procedure by Cox, Ross, and Rubinstein (CRR) [1979].

While the finite difference method begins with the continuous-time and continuous state logarithmic diffusion specified in (1) and uses a discrete approximation to the PDE that results, the Binomial model starts by approximating the asset price process itself, with a discrete-time multiplicative random walk. Over a single time step of length Δt, the stock price is restricted to change from S to one of two values, an up-move to uS or a down-move to dS. Here u (d) represents “1 plus the rate of return if the stock takes an up-move (down-move).” Figure 1 shows the possible stock price movement over one time step and Figure 2 illustrates how the binomial lattice is extended to multiple time steps.

The no-arbitrage principle along with the dynamic hedging strategy introduced by Black and Scholes is also the basis of valuation in the binomial framework. Since there are three securities, the stock, the option and a riskless bond, but only two possible stock prices in the next time period, the option’s value at both stock prices can be replicated by a portfolio of the stock and the bond. If Cu and Cd are the option values in the up and down states (e.g., stock prices uS and dS, respectively, in Figure 1) and R denotes gross interest per period (i.e., R = erΔt where r is the continuously compounded annual rate), then the option value C one period earlier is equal to the value of the replicating portfolio, given by

The variable p is defined by p = (R - d)/(u - d).The binomial tree that describes the future stock price, along with a

known interest rate that determines the bond value, induces a binomial tree for the value of the option. An option pricing problem is solved by filling in all of the nodes at expiration day, where the option value is known for each

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possible stock price, then using (3) to roll backward through the option price tree until the initial value is obtained.

Equation (3) gives the cost of the portfolio of stock and bonds that replicates the option value over the next time step, so p is derived from the no-arbitrage condition and does not depend on the probability distribution for the stock price change. But if p were the probability of an up move, the right hand side of (3) would be the expected value of the option price in the next period, discounted at the riskless interest rate. In that case, the option’s expected return would be the riskless rate, as it would be in a world where investors were indifferent to risk. Since the Binomial model embodies the no-arbitrage condition, it exhibits risk neutral valuation: An option’s fair value is the same as it would be in a risk neutral world, and p is typically known as the “risk neutral probability.”

Under risk neutral valuation, contingent claims can be priced by calculating their expected payoffs under the risk neutral probability distribution for the stock price and discounting them at the riskless interest rate. An important result is that once market completeness is established there is always a risk-neutral distribution under which a unique option value can be obtained by taking a discounted expectation. Any procedure with a distribution that approximates the risk neutral distribution and converges to it in the limit can be used to price the option correctly, even though replication may not be possible in such a procedure. This will allow us to replace the binomial lattice with a trinomial without losing the ability to compute unique option values. Before doing so, however, we examine the convergence of the Binomial option price to its theoretical value.

Convergence in the Binomial ModelTo price an actual option with the Binomial model, the asset price tree

must be set up to incorporate the volatility of the price process. The mean return or drift, however, is not constrained to be any particular value, which

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gives a degree of freedom in constructing the tree. To match an annualized

volatility σ requires that u and d be defined aswhere Q is the drift rate. However, while Q is a free parameter, in that the Binomial model will converge to the same option value as Δt 0, regardless of the value of Q, the speed of convergence is affected by Q. Setting Q equal to r - σ2/2 results in a risk neutral probability p = 1/2 and favorable convergence properties.4

CRR prove that with u and d values defined as in (4), the probability distribution for the terminal stock price becomes lognormal and the Binomial option price converges to the Black-Scholes value as the size of the time step goes to zero.5 Thus the Binomial model offers both intuition and an asymptotically exact approximation to the theoretical option value under Black-Scholes assumptions, for American options and other contingent claims that do not have closed form valuation equations. The model is powerful, flexible, and easy to use.

In practice, however, the Binomial may exhibit several problems in reaching convergence. Using a binomial tree to approximate the value of an option on an asset whose price is actually generated by a logarithmic diffusion introduces two types of approximation error, which we call distributional error and truncation error. Suppose we wanted to value put options using the 4 step binomial tree for the underlying asset price that is shown in Figure 3. There are only five possible stock prices as of the expiration date, while the true stock price will be drawn from a continuous

4 See Courtadon [1990], for a discussion of this point.

5 This also requires that the up and down step sizes both go to zero at about the same rate. A second type of convergence, with one step remaining finite but with a probability that goes to zero, leads to a Poisson model in the limit.

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lognormal distribution. This discrepancy between the true and the approximating distributions will produce distributional error in the option value. Notice that distributional error impacts the model at every time step.

To understand the nature of truncation error, consider two put options with strike prices K1 and K2. In this tree, both options are in-the-money in the bottom two nodes and out-of-the-money in the highest three. The difference in strike prices is reflected in the fact that the payoff for option 1 will be less than for option 2 by the amount (K2 - K1) at the two in-the-money nodes. However, in the tree the probabilities of ending up in-the-money are the same. In other words, the payoffs that option 2 will actually receive if the final stock price falls between K1 and K2 do not translate into additional option value within the binomial tree.

While distributional error contributes random noise to the valuation approximation, truncation error induces a peculiar “even-odd” property to convergence within the Binomial model. Figures 4 and 5, showing 4- and 5-step trees for the same put option illustrate the source of the problem. In Figure 4, we see that the option is in-the-money at the lowest 2 of the 5 possible price nodes, while Figure 5 shows how dividing the option’s lifetime into 5 time steps produces a tree for which it ends up in-the-money in half of the terminal nodes (the lowest 3 out of 6).

Quantitatively, this effect can be surprisingly large. For example, consider a one year 35 strike European put option on an underlying stock whose current price is 40. Assume volatility is 0.40 and the simple interest rate is 5.0%. Valuing the put with a binomial tree gives a price of 3.55 when 4 steps are used, but only 2.81 with 5 steps.

Figure 6 shows the even-odd pattern of convergence to the Black-Scholes value of 3.065 as the number of time steps in the Binomial model increases. Some analysts have suggested that a better estimate of the true value may be obtained by averaging values from N and N+1 step trees. Of

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course this about doubles the number of calculations required. The approximation error of the average is also not necessarily close to zero. Along with the even and odd Binomial values, Figure 6 also plots their average. The average displays much smoother convergence than the simple values, but convergence is still not monotonic.

The Trinomial ModelThe Binomial model is very intuitive and it demonstrates clearly how

option replication and the no-arbitrage principle lead to a uniquely determined option value that is independent of the expected return on the underlying asset. However, the resulting model has little flexibility to deal with problems such as the convergence issue just raised. But once risk neutral valuation is established, it no longer is necessary to be able to replicate an option in order to value it within an approximation framework. Thus the Trinomial model has proven to be more useful and adaptable than the Binomial for many derivatives applications.

A trinomial tree is set up to approximate the risk neutral distribution. Since the asset price is assumed to be lognormal, it is often convenient to construct the tree in terms of the log of S.6 Let X ln(S) denote the log of the asset price, meaning that X is normally distributed. Under risk neutrality, the process for X is given by

where α = r - q - σ2/2 , and q denotes the instantaneous rate of dividend payout.

In a Trinomial tree, over the next time step the underlying asset price is allowed to move to one of three values, designated up (u), down (d) and

6 Note that depending on how the procedures we describe below are actually implemented, this may mean that the option “prices” computed in the tree can be the logs of the dollar prices. We assume the reader will understand when and how the appropriate conversions need to be done.

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middle (m). Associated with these branches are three risk-neutral probabilities, pu, pd, and pm. Adding a third branch at each node allows a considerable increase in flexibility, but it means that in the Trinomial it is no longer possible to replicate an option’s payoff with a combination of the underlying asset and riskless bonds. A binomial tree can be built to embody any expected return on the asset without affecting the (asymptotic) option value. But the size of the binomial up and down price moves then is determined by the volatility of the asset price that the lattice is attempting to approximate.

A Trinomial has three asset prices and three probabilities to be specified. Often the middle price is chosen to be no change and the up and down moves to be of equal magnitude. Let k denote the length of a time step and h be the size of an up or down move. Thus, over one time period the log stock price X goes to X + h with probability pu, X - h with probability pd, and remains unchanged with probability pm. This is illustrated in Figure 7.

The resulting system must obey three constraints: The expected return over the next period must be the riskless interest rate (since the tree is approximating the risk neutralized asset price distribution); the standard deviation must be consistent with the volatility of the underlying; and the probabilities must sum to one.

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In this case we have described how the next period prices may be chosen freely, leaving the probabilities to be determined such that (6) is satisfied. Our choice of possible price changes is convenient and intuitive, but in many cases it does not produce the most rapid convergence. The flexibility of the Trinomial may be exploited further in order to improve the accuracy of the approximation and to speed convergence for certain types of problems, as some authors have proposed. For example, Cho and Lee [1995] suggest constraining the price moves to match higher moments of the lognormal probability distribution. Gao [1996] reviews a number of approaches to tree building, and shows how the available degrees of freedom may best be used to optimize performance. We do not pursue all such possibilities here, however, because that would not allow us to use the same type of tree for both American puts and barrier options in the examples that follow.

Solving equations (6) for the probabilities yields

The option price is determined by starting at the known asset price

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contingent payoffs at maturity and rolling backward through the tree using equation (8).

The value of k is determined by the option’s maturity, T, and the number of time steps desired for the tree, N.

The price step h remains a free parameter. But once the time step k is set, not every h value will produce nonnegative probabilities for all three nodes in equations (7). In general, both k and h will be far below 1,

so

To guarantee nonnegativity, h should be of the same order as k. Let us define a free parameter λ > 1 connecting the sizes of the time step and the price step, such that

A judicious choice for λ can improve the convergence properties of a tree substantially. For ordinary options, λ = 3 is normally a good choice (see Gao [1996]). Ritchken [1995] shows that for a barrier option, performance can be enhanced significantly by restricting λ to be a value that produces an integer number of price steps between the current asset price and the barrier. Below, we will refer to that approach as the “Restricted Trinomial Model” (RTM).

Truncation error occurs in a lattice approximation model when the option value between two nodes does not change proportionally as the asset price changes, that is, when linear interpolation does not give a good

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estimate of the option value between two nodes. This will happen when gamma is large and the option’s delta changes sharply in that price range. For a European option, this occurs around the strike price at expiration. It turns out that an American option’s truncation error is also almost entirely accounted for by the error in the last time step, for the prices that bracket the strike price.7

The standard technique for reducing the approximation error in a binomial or trinomial tree is to use finer time steps. Unfortunately, this increases the number of nodes geometrically. Designating the initial node as n = 0, the Binomial model has n+1 nodes at time step n, or [(N+1)2+(N+1)]/2 = (N2+3N+2)/2 nodes in total. A trinomial tree has 2n+1 nodes at each step, and (N+1)2 in total. If h is proportional to k, to cut the size of the price step in half requires a tree with four times as many time steps. Thus computation time increases sharply as greater accuracy is required.

One clever technique for minimizing truncation error in an American option pricing lattice is to recognize that by construction “early” exercise is impossible within the final time step before expiration. Thus the option is European at that point in time. The truncation error occurring at maturity can be eliminated by the simple expedient of substituting the Black-Scholes value for the option price at every node in the second to last time step, and then rolling back through the tree to earlier dates using the standard recursion shown in (8). While this does not take into account the possibility of exercising an actual option during that final interval, the improvement in accuracy of the approximation is very striking. Unfortunately, this approach is only viable when a closed form solution exists for the European option with one period to maturity.

7 Note that prior to expiration, for the nodes around an American option’s early exercise boundary, there will be an approximation error with regard to where exercise will occur. But this does not translate into significant truncation error in valuing the option because the delta does not change sharply at the early exercise boundary.

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The problem with increasing the number of time steps to achieve greater accuracy in a lattice is that most of the additional computation is largely wasted, since finer resolution is really only important in one critical area. What we would like is to be able to use a very fine lattice that has high resolution just in the region of the strike price at expiration, while allowing it to remain coarse and fast to calculate everywhere else. We call such a hybrid tree an “Adaptive Mesh Model” (AMM) and will now show how to construct one for the American put problem.

III. The Adaptive Mesh Model for the American PutFigure 8 illustrates the Adaptive Mesh trinomial tree that we wish to

construct to value a put option. The coarse lattice, with price and time steps h and k, is represented by the heavy lines. (This should be thought of as a subtree, that covers the expiration date and the last few periods prior to expiration, for asset prices in the immediate vicinity of the strike price.) The objective is to build a fine mesh tree around the strike price at expiration and join it to the coarse tree so that the valuation information is transmitted properly. By setting the fine price steps to be h/2 and (therefore) the time steps to be k/4, we guarantee that fine mesh nodes will overlap coarse mesh nodes one (coarse) step before expiration. The procedure is to use the fine mesh to obtain the option values for these overlapping nodes, A2 through A5. Option values for the other coarse mesh nodes at this time step are computed in the usual way using the terminal payoffs from the coarse lattice . Once this is done, we have incorporated the more accurate fine mesh valuation for the critical nodes into the coarse tree. Valuation then proceeds by rolling back through the coarse tree to the initial date.

As is clear from Figure 8, the fine mesh is only needed for dates between T-k and T, in the region around the strike price. To be precise, the fine mesh needs to cover all nodes at date T-k from which there are both

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paths that end up in-the-money and paths that end up out-of-the-money. For example, there is no need to use the fine mesh to obtain a value for node A6 at T-k because every possible fine mesh path that would begin at that point would end up out-of-the-money, at a node for which there would be no truncation error. However, from node A5, while the coarse lattice paths all end up out-of-the-money, a fine mesh path with only down moves between T-k and T would end up below K and in-the-money, while paths with one or more up or middle moves in that interval would arrive at asset prices above K on date T. Since it is around K that truncation error occurs, the fine mesh value for node A5 will be more accurate than what the coarse mesh would have produced for that node.

Let us denote the two (coarse mesh) date T asset prices that bracket the strike price as XK- and XK+. As Figure 8 shows, the fine mesh needs to cover all of the prices that can be reached starting at date T-k coarse mesh nodes in the range from XK+ - 2h to XK- +2h. This increases the number of nodes to be calculated in this area from 12 in the coarse mesh lattice to 52 in the fine mesh (4 at date T-k, plus 9, 11, 13, and 15 for dates T-3/4k, ... , T). Thus, even for a trinomial tree with as few as 20 steps, the additional computational burden to use an Adaptive Mesh Model is less than increasing the number of time steps by 1. For a bigger trinomial tree, 100 time steps, say, adding one level of adaptive mesh increases the number of nodes by less than 4%, and with 500 steps, the increase is only 0.02%.

One very attractive feature of the AMM approach is that it is isomorphic at successive levels of refinement. That is, once an AMM tree is constructed as in Figure 8, it is simple to add another level of still finer mesh, with price steps of size h/4, by following exactly the same procedure as we have just described for the period T-k/4 to T. The total “cost” of this further refinement is just an additional 40 node calculations.

Table 1 presents comparative performance statistics for the standard

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Binomial and Trinomial models and the AMM with one and two levels of fine mesh added around the strike price at expiration, as in Figure 8. The models are used to compute theoretical values, deltas and gammas for a test set of 27 European put options. European options are used here so that we may compare the accuracy of the various approximations against an exact benchmark. Obviously, there would be no need to use any of these methods to price European options in practice.

We assume an initial asset price of 40, riskless interest of 5% (corresponding to 4.88% as a continuous rate), and no dividend payout. The 27 combinations of strike prices of 35, 40, and 45; maturities of 1, 4, and 7 months; and volatilities of 0.20, 0.30, and 0.40 are used. The table shows the root mean squared error for the computed values, as well as the approximate execution time on a Sun Sparc 20 computer. (There is a small amount of noise in the very short execution time estimates.) Lattices with 25, 100, 250 and 1000 time steps are examined.

Several features of these results are apparent. First, the ability of the AMM approach to increase the accuracy of the price computation at virtually no increase in execution time is very striking. The AMM-2 model, for example, achieves a level of accuracy with only 25 time steps that is much better than a standard Trinomial with 250 time steps, and only somewhat less accuracy than the Binomial obtains with 1000 steps and an execution time that is about 300 times greater. Comparing across the models for a given number of time steps, we see that computation times for the three trinomial-based models are about the same. The Binomial runs distinctly faster, but accuracy is only about half of that for the standard Trinomial and much worse than the two AMM models. The AMM-1 model is about twice as accurate as the standard Trinomial, and the AMM-2 is about four times as accurate as the AMM-1.8

8 These results indicate a faster rate of convergence as N increases than we anticipated. We had expected mean squared error in the

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Looking at the delta and gamma calculations, we see that the AMM approach does not seem to make much difference to either of them for this case. This seems to be because these computations involve differences between option values at slightly different asset prices, so that the truncation error around the strike price at expiration that affects them in a similar fashion will tend to be eliminated by the differencing. Note that we do not compute the delta and gamma by perturbing the starting asset price, which is a rather inaccurate procedure, as we will explain more fully below, and is affected by the truncation error problem. Instead, we adopt the approach suggested by Pelsser and Vorst[1994], that involves extending the tree back one period prior to the initial date and calculating the “Greek letters” from the values within the tree. In Section V, we will examine the calculation of delta and gamma more carefully, and show how another AMM technique, this time applied to the initial nodes in the tree, can lead to a major improvement in estimating the “Greek letter” risk exposures.

IV. An Adaptive Mesh Model for Barrier OptionsThe previous section demonstrated that significant improvement in the

accuracy of an option valuation lattice could be obtained with very little increase in computation time by adding a small section of fine resolution mesh near the strike price at expiration. This section will show how the AMM approach can be extended for use with barrier options and similar instruments, whose values depend on whether the asset price reaches a specified level at any point during the option’s life.

As the derivatives market has expanded in recent years, large numbers of new instruments have been created, that have a wide variety of early exercise features and other contingencies. Closed-form solutions exist for European versions of many of these, but for those with American exercise features or more complex contingencies, valuation by numerical

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approximation may be the only available technique. And in some cases, convergence is so slow within a standard Binomial or Trinomial model that although valuation is theoretically possible, it is infeasible in practice.

Lattice-based methods for many such options can be enhanced enormously by an appropriate AMM procedure. In some cases, an AMM lattice incorporating one or more layers of fine mesh near the price barrier has comparable accuracy to a standard Trinomial that requires orders of magnitude more calculations. The AMM allows much more efficient valuation for a large number of important “exotic” derivative instruments and offers the first usable approach for valuing many more, as well as for performing computationally intensive procedures like calculating implied values for

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volatilities and other parameters.9

We will apply the AMM technology to the common but difficult problem of pricing a barrier option when the barrier lies close to the current asset price. A typical barrier option pays off at expiration like an ordinary call or put, except that the payoff is contingent upon whether the price for the underlying asset has reached the specified barrier price at some earlier point during the option’s lifetime. For example, a “down-and-out” call option with strike price K and barrier price H (known as the “out-strike”) has the same payoff at expiration as a European call, conditional upon the stock price having been above H throughout its entire life. But if the price falls to H or below at any point, the option is “knocked-out” and expires worthless, regardless of the underlying asset price at expiration. Such a contract is often known as a “knock-out” option and the out-strike is the “knock-out barrier.”

A barrier option may be a call or a put; it may be knocked-out or knocked-in at the barrier (a knock-in option is worthless at expiration if the barrier was not reached previously); and the barrier may lie above or below the current asset price (“up” and “down” options). Thus a down-and-out call pays off like a regular call if it is still alive at expiration, but it is knocked-out if the asset price ever hits the barrier. An up-and-in put pays off like a regular put only if it has been knocked-in by the asset price rising to the in-strike at some point during its lifetime. There are thus 8 basic types of barrier options, corresponding to the 8 combinations of these three elements. The attraction of such an instrument to the buyer is that it is cheaper than a plain vanilla call or put: It has the same dollar payoff, but the payoff is received in fewer states of the world at expiration (those that the buyer regards as the most probable, however).

Barrier options have become widely used, particularly for foreign currency contracts. There are also a variety of other instruments with similar

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kinds of contingent payoffs, including capped options (that expire early and pay off as soon as they hit the asset price at which they are in-the-money by the prespecified cap amount), ladder options (for which the option’s strike price changes when the underlying asset’s price crosses one or more predetermined barrier levels), interest rate corridors (that accrue interest daily, but only on days in which the rate falls within a specified range), and

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many others.10

Valuing Barrier Options with a Trinomial ModelValuation of these types of instruments can present great difficulty for

standard lattice-based techniques because of truncation error caused by interaction between the discrete prices in the tree and the location of the price barrier. Convergence typically exhibits a choppy pattern similar to the even-odd behavior of the Binomial model for regular options. We will use the example of a European down-and-out call option to illustrate the valuation problems of barrier options and the use of the AMM to deal with them. Notation will be as above, with the addition of the symbol H to denote the option’s knock-out barrier (H < S).

One advantage of this example is that there is a closed-form valuation equation that can serve as our benchmark for evaluating accuracy. Merton

[1973] derives the following solution:where CDO denotes the down-and-out call value and CBS is the Black-Scholes call formula.

Consider using the standard Trinomial model as given by equations (7) and (8) to value a down-and-out call with the following parameter values:

S = 100K = 100T = 1 yearr = 10%σ = 0.25H = 90.

Figure 9 shows the convergence of the approximation produced by the standard Trinomial to this option’s theoretical value, given by Equation (10)

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as 11.323. As the number of time steps increases, the approximation converges toward the correct price and then suddenly jumps away from it. Even with more than 1000 time steps (meaning more than one million node calculations), the typical error is still unacceptably large.

The source of this problem is easily visible in Figures 10 and 11. In Figure 10, the barrier lies just slightly less than two price steps down from the current asset price, so the option is knocked out if the price falls two steps below the initial price at any time prior to expiration. In Figure 11, however, an increase in the number of time steps has decreased the size of the price step just enough that two down moves leaves the asset price still a little short of the barrier. Now three down moves will be needed to knock out the option, which is a distinctly lower probability event. Thus, a small increase in the number of time steps has led to a very small change in the size of the price step, but within the lattice this produces a significant drop in the estimated probability of the option being knocked out and therefore a sharp jump in its approximate value. Because the effective barrier in this case (3 price steps) lies further from the initial price than the true barrier, the down-and-out option will be overvalued. An “in” option, by the same token, will generally be undervalued by a lattice model.

The ideal situation would be for the barrier to coincide with a layer of nodes so that, within the framework of the discrete lattice, the approximate option value incorporates an accurate estimate of the probability of hitting the barrier. The problem has been examined in the literature, and some creative solutions have been offered. One approach is to notice that since the size of the price step depends on the number of time steps, accurate valuation can be obtained simply by choosing the right value for N, as Figure 9 illustrates. For example, Boyle and Lau [1994] show how to compute the sequence of optimal N values for a Binomial tree. However, this uses up the only degree of freedom available in the Binomial, so the method can not be

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extended to deal with any greater complexity, such as multiple or time-varying barriers.

As mentioned above, Ritchken [1995] presents a solution to the problem in a Trinomial model, that works by adjusting the size of the up and down moves to produce an integer number of steps from the current price to the barrier. With more degrees of freedom than the Binomial, this “restricted” Trinomial model (RTM) can be fitted to match a second barrier also. Further generalization typically does not work, however. Even so, Ritchken’s RTM approach is capable of handling many important cases encountered in practice.

The major difficulty these techniques face is when the asset price is close to the barrier. In order for the barrier to coincide with a layer of nodes (and the contract not to be already knocked-out) there must be at least one price step from the initial price to the barrier. But this imposes a maximum value for the price step h, equal to ln(S0 ) - ln(H). The price step, in turn, leads to a maximum time step k, and therefore a minimum value for N that may be very large.

Suppose we needed to value the down-and-out call described above with all of the same parameter values except that the current asset price is much closer to the barrier. For example, if S0 = 90.5 the maximum price step that permits one down-move before hitting the barrier is 0.5. The maximum value for h is h = ln(S0) - ln(H) = ln(90.5) - ln(90) = 0.00554. Setting λ = 3 and plugging into equation (9) gives k = .000164. The resulting tree will therefore have 1/k = 6108 time steps, which will require over 37 million node calculations. If one wanted a more accurate valuation, with two price steps to reach the barrier (which would be needed for an accurate estimate of gamma), the tree would have to have four times as many time steps (24435) and almost 600 million nodes.

Figure 12 illustrates the slow convergence of the standard Trinomial for

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this problem. Even using 1000 time steps, the pricing error is unacceptably large: The estimated option value is 1.102 while the true value from equation (10) is 0.642. Notice that the choppy convergence pattern seen in Figure 9 does not appear here. This is because the option value only jumps when the number of price steps to get to the barrier changes. For this tree, the first jump does not occur until N = 6109, and the second will be at N = 24436. An AMM can resolve this problem by combining a section of high resolution lattice with the necessary fine price step in the relevant region next to the barrier with a coarser lattice that allows efficient computation elsewhere.

Constructing an AMM Lattice for a Barrier OptionWe will now construct an AMM lattice for this case. The flow of pricing

information here is somewhat different than for the American put, where the fine mesh passed information “upward” to the coarse mesh. Here, the information flow is from the coarse mesh downward to the fine mesh, allowing the barrier option to be valued properly at prices that are less than one (coarse) price step away from the barrier.

Figure 13 shows the basic structure of the AMM tree that we wish to construct. In the figure, the heavy lines indicate the coarse mesh lattice, whose nodes will be labeled Aij, with i indicating the number of coarse mesh price steps above the barrier and j the number of the (coarse) time step. Thus, A00 refers to the node falling on the barrier at date t = 0 and A10 is the price that is one coarse mesh step above the barrier. The fine mesh nodes will be denoted by the letter B. We wish to compute the value of the down-and-out call at node B10, which lies one half of a coarse mesh price step above the knock-out price, i.e., at ln(H) + h/2.

The figure indicates how valuation information must flow in the lattice. First the coarse mesh lattice is constructed, and option values are computed at all of the A nodes. This must be done in a particular way to satisfy two

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constraints, as we will discuss below. Next we use the coarse mesh to compute option values at prices of ln(H) and ln(H) + h, for time intervals of k/4 (following the dotted lines). These will be the nodes that anchor the fine mesh to the coarse mesh. Finally, we fill in the remaining fine mesh nodes for the price ln(H) + h/2 and time steps of k/4 (the fine solid lines), culminating in the desired option value at node B10.

The first step of constructing the coarse mesh tree is illustrated in Figure 14. Since we want B10 to correspond to the initial asset price, the value of h must be set to exactly 2(ln(S0) - ln(H)). This determines the initial asset price for the coarse mesh lattice beginning at node A10. Once the time step is set, as described below, the rest of the coarse lattice is filled out in the normal way.

Next we compute the option values for the points where the fine mesh will be connected to the coarse mesh lattice, as shown in Figure 15. These B nodes lie along the barrier and one price step h above the barrier, at time steps of length k/4. In this example, option values at the B0j nodes are all zero, since they fall on the knock-out barrier. In other cases, such as an “in” option or a ladder, it will be necessary to solve for the B0j lattice values on the barrier. This should be done using the same procedure as we describe for the B2j nodes in this tree.

The B2j nodes that overlap A nodes, like B24 and B28, are also known from the coarse lattice. However, the B2j nodes that lie between coarse mesh time steps, such as B27, require special handling. As Figure 15 indicates, these are calculated from the nearest A nodes in the immediately subsequent time step. Thus B21, B22, and B23 are all obtained from A01, A11 and A21, while B25, B26, and B27 are connected to A02, A12, and A22. Computing the B node values simply involves rolling backward through a trinomial tree using equation (8). The only difference is that each of the three B2j nodes, falling on dates that are k/4, 2k/4, and 3k/4 after a coarse lattice time step,

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will have its own set of up, middle, and down probabilities. These are obtained by replacing k in equation (7) by the appropriate effective time step size for that node, i.e., 3k/4 for a B node like B25, 2k/4 for B26 and k/4 for B27.

For example, consider B27. From equation (7), we have

Plugging into (8) gives

where for convenience the notation has been adjusted to indicate explicitly the node to which the call value function C(.) applies. Option values at the other B2j nodes are computed in similar fashion using equation (7) with the appropriate multiple of k/4 for the time step.

Finally, when the B0j and B2j node values have been obtained for all j, we fill in the B1j values as in Figure 16, by defining a third set of probabilities, for price steps of size h/2 and time steps of k/4. Since the dominant term in equations (7) is the one containing k/h2, these probabilities will be almost the same as for the coarse mesh. The up, middle, and down probabilities used in calculating a B1j node are given by

To fill out these middle nodes, one must start from the expiration date and roll back through the tree with the normal recursion. This eventually gives the node B10 option value as

Setting the Coarse Mesh Time StepThis discussion has shown how to construct the AMM lattice given the

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coarse mesh tree. We now return to the coarse tree, that must be set up carefully in order to satisfy two conditions. First, as discussed above, the fine mesh must place the current asset price exactly one price step above the barrier, meaning h/2 = ln(S0 ) - ln(H). Second, the number of coarse mesh time steps in the tree, N = T/k, must be an integer. Note that by setting the coarse time step correctly, all finer mesh sub-lattices will also have integer numbers of time steps.

Before, the values of h and k were linked through equation (9) to the parameter λ, which we set equal to 3. Here, since h and k must obey separate constraints, the choice of λ is no longer free. Referring to it as the “stretch” parameter, Ritchken [1995] discusses how to choose λ in order to create a tree with an integral number of time steps and with a layer of price

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nodes that lies exactly on the out-barrier.11 In this case, we need a variant of the procedure, because the

constraints are on the fine mesh price step and the coarse mesh time step.

Beginning with the price step, we setIf λ is fixed at 3 (or at any other constant value), the price step from equation (15) will normally not produce an integral number of time periods in the option’s life, so the tree must be stretched a little to fit properly. One simple procedure that leaves us with a λ very close to the desired value is to calculate the noninteger number of steps the target λ would yield and then lengthen the time step just enough to eliminate the rounding error.

For example, setting λ = 3, we compute k0 = h2 / 3σ2. Let N0 = T / k0, be the (non-integer) number of time steps in the option’s lifetime. Eliminating the fractional step, the adjusted tree will be built to have N = int(N0) steps of length k = T / N . The resulting time step will be marginally longer than that produced by λ = 3, but the difference will be negligible for a typical lattice (since the step length increases by less than 1 Nth, k - k0 < k0/N).

Expressing the time step calculation in a single general equation, we

haveThe General AMM for a Barrier Option

This procedure has shown how to construct an AMM lattice with one level of finer mesh at the knock-out barrier. One of the significant advantages of the AMM is that further refinement is easily introduced by using the identical approach to add additional levels of progressively finer

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mesh, with the price step at each successive level being half as large as at the previous one. The fine dotted line in Figure 16 indicates where the middle nodes of the next level of mesh would be placed, in order to compute the option value at the asset price C10. Note that since the problem illustrated in Figure 12 occurs when the initial asset price is less than one whole price step above the boundary, adding finer mesh will always be done so that the finest price step is exactly equal to ln(S0 ) - ln(H).

The general AMM methodology for a barrier option is therefore the following. First choose M, the number of levels of fine mesh to be constructed. For the above example, M = 1. Computing the option value at C10 would involve a second level of fine mesh, making M = 2. The finest mesh price step is set to ln(S0) - ln(H), making the coarse mesh step h

The coarsest mesh time step k is given by equation (16). Now the coarsest (A level) lattice is constructed starting at the initial asset price X = ln(S0 ) = ln(H) + h. Then the finer levels (B, C, etc.) are added one at a time, following the procedure outlined above.

It is easy to see from Figure 16 how many additional calculations are required to add a layer of fine mesh to a trinomial lattice. Consider the three layers in the B level. Along the upper and lower layers, at each coarse time step there are three new values to be computed for the B nodes like B01, B02, B21 and B22 that do not overlap coarse mesh nodes. The middle B layer is entirely new, so there are 4 nodes per coarse time step. The total for the first layer of finer mesh is therefore 3 + 3 + 4 = 10 new nodes per time step, or 10N in total. Thus for a 100-step lattice, that contains 1012 = 10,201 nodes, an AMM with a single level of fine mesh would add 1000 new nodes, for a total of 11,201. By contrast, increasing the resolution of a standard Trinomial model by halving the price step (and multiplying N by 4) would

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require increasing the total number of nodes to 4012 = 160,801.Adding further levels of fine mesh requires 10 new nodes for each time

step in the next higher level. Thus the C level mesh in Figure 16 will require 40 new nodes per coarse mesh time step, and in general, the m’th level will add 10 x 4m-1 x N new nodes. Table 2 shows the number of nodes in an AMM lattice as finer levels of mesh are added and compares it to a standard Trinomial model with the same mesh size everywhere. It is obvious that refining the standard Trinomial in order to price barrier options near the boundary rapidly requires unmanageably large numbers of calculations. Of course, the higher resolution throughout the lattice will produce smaller distribution approximation error for the standard Trinomial than the AMM, but there will be no difference between them in the truncation error, since that is determined by the size of the price step immediately adjacent to the barrier, which is the same for both models.

Table 3 shows how this difference translates into computation time for the RTM and the AMM. We value a one year down-and-out call with a strike price of 100 and out-strike H = 90 for a sequence of initial stock prices that approach the barrier. The AMM always uses a coarsest price step of about 2 points, but adds successive layers of fine mesh for asset prices closer to the barrier, while the RTM is constructed to have a price step equal to ln(S0 ) -

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ln(H) in each case.12 (Computation times are for a personal computer with a 90MHz Pentium processor and 32 megabytes of RAM.) The RTM computation became unmanageable at an asset price of 90 1/8 because of the size of the lattice, while the AMM reached accurate answers almost instantaneously in every case. Note that the general rule that halving the price step increases the number of time steps by a factor of 4 and the total number of node calculations by a factor of 16 does not allow us to assume computation times will increase proportionally. As the size of the lattice increases, it becomes impossible to keep the whole tree in fast computer memory at one time and computation speed slows sharply.

V. An AMM for Calculating Delta and GammaWhile academics tend to focus on derivatives valuation, in practical

use, option pricing models are needed more for risk management than for valuation. Practitioners tend to begin by using their model to compute implied volatilities from option prices observed in the market. The “market” volatility then becomes the input parameter for subsequent model calculations. Thus, by construction, any reasonable model must be generally consistent with current market option prices. Its real value to the trader is in indicating how an option’s value will respond when the price of the underlying changes, and how this risk exposure can be hedged.

Two of the most important risk exposures are measured by delta, the change in the option value for a small change in the asset price, and gamma, the change in delta. In a closed-form valuation model, these “Greek letter”

risks can often be obtained in closed-form also, asThere are a variety of ways to estimate delta and gamma in a lattice

model, some of which are distinctly better than others in terms of accuracy

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and speed of convergence. In this section we will discuss the different standard approaches, all of which are subject to the effects of truncation error, and then show how accuracy can be enhanced by an AMM technique.

Delta and Gamma in the Standard Binomial ModelThe standard Binomial model is based upon a strategy of replicating

the next period option payoffs in the up and the down states with a portfolio containing just the underlying asset and riskless bonds. Calling the number of units of the asset in this replicating portfolio nS and the investment in bonds B, the portfolio costs nSS + B, which is therefore the theoretical option value. A unit change in the asset price changes the value of this portfolio by nS, so nS is frequently taken as the estimate for the continuous-time continuous-state delta, in the same way as the option price from the Binomial approximation is the estimate of the Black-Scholes option value.

But while the option replication delta does converge to the correct value in the limit as the length of a time step goes to zero, it actually represents a kind of average delta over the period from date 0 to date 1. Evaluating gamma requires computing the change in delta for different asset prices, meaning that it will be based on the deltas at the two period 1 nodes and involve asset prices at date 2.

The delta from the replicating portfolio will be different from the change in the Binomial option value that would be obtained by perturbing the initial asset price by some small value ε and building a new tree starting from the price S0 + ε. Since the latter more closely corresponds to the partial derivative expression for delta in (18), delta and gamma are often computed as numerical derivatives by perturbing the initial asset price. Unfortunately, this runs into the now-familiar problem of truncation error: the estimates are very noisy and, like the option price, they converge to the true values non-monotonically as the number of time steps goes to infinity.

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Another major shortcoming is that the entire tree must be computed a second time to get delta and a third time to get gamma (or to get a delta centered on the initial asset price).

Pelsser and Vorst [1994] discuss the problem of obtaining delta and gamma for a Binomial model and propose an alternative technique that increases efficiency markedly. Rather than starting from the initial asset price at time 0, they begin two time steps earlier and build the tree so that the middle node at the end of the second period (time 0) falls at the desired starting asset price. With this extended tree, there will now be three nodes at time 0, which permits calculating both delta and gamma centered on the right point in time. This method improves model performance considerably, but its shortcoming is that the perturbation of the asset price used in the numerical derivative need not be small: it is one full price step in the lattice.

Delta and Gamma in the TrinomialThe same problems affect calculation of delta and gamma in a

Trinomial model. One difference, however, is that since the Trinomial is not based on option replication, there is no delta estimate comparable to nS from the Binomial. Perturbing the initial asset price either directly or by extending the tree is the standard solution technique. Figures 17 through 19 illustrate the how points just discussed for the Binomial model impact the Trinomial. Figure 17 shows the initial portion of a basic trinomial tree, starting at the initial asset price X0 = ln(S0). The price and time steps are h and k, as before.

Figure 18A shows the process of computing a (one-sided) delta by perturbing the initial price and building a second tree starting from X0 + ε. In Figure 18B, a third tree starting from X0 - ε is constructed, for the calculation of gamma, as well as a centered delta value, as follows:

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It is clear from Figure 18 that computing even a one-sided delta requires twice as many calculations as just pricing the option, and gamma requires three times as many.

Figure 19 illustrates economizing on extra calculations by extending the tree backwards in time. To compute gamma, three asset prices are needed for date 0, which means that the Binomial has to be extended backward for two periods, but the Trinomial only needs to be extended one. As in Figure 18, the dashed lines indicate the new sections of lattice that must be added to the original tree. It is apparent why extending the tree can be much more efficient than simply perturbing the initial asset price. In Figure 18, two new trees were constructed starting from the perturbed time zero asset prices, but in Figure 19, these trees overlap the existing lattice almost entirely. Only two new nodes are required at each time step from 0 to T, a total of 2N + 2. (Since our real interest is in time 0, there is no need actually to compute the time 0 - k option price.) Delta and gamma values are obtained from equation (19) with the price step h substituted for ε.

Relative Performance of the Standard Methods in Estimating Greek Letter Risks

Table 4 compares the performance of these alternative approaches for estimating delta and gamma in standard Binomial and Trinomial models. The table is set up like Table 1 and uses the same test set of 27 European put options. We report root mean squared errors relative to the exact values from the Black-Scholes formula, as well as execution time on a Pentium Pro 200 MHZ personal computer (averaged over 10 runs for each set of parameters). Results are shown for 25, 100, 250 and 1000 step trees.

The first four lines for each value of N relate to the Binomial model. Since the procedure for calculating Greek letter exposures does not change the original valuation tree, the RMSE for the option price is the same for all

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Binomial and all Trinomial variants. In line 1 the delta and gamma are obtained from the Binomial model replicating portfolio. Performance is remarkably good with this data set. Given that delta and gamma are being computed from prices on the wrong date (i.e., not at time 0), this suggests that they are not changing much over time. Since the calculation does not require any alteration to the tree, this is the most computationally efficient method.

The second and third lines use perturbation to compute numerical derivatives. This requires building two additional complete trees, so computation time should be at least three times as long as the previous approach. In fact, with the extra computational overhead the multiple appears to be closer to 5 times as long. Moreover, the perturbation method is much less accurate than using the replication deltas and gammas. We also see an interesting effect with regard to the size of the perturbation. Using a very small value, ε = 0.001, produces much less accurate deltas and, especially, gammas than those obtained with a larger ε = 0.01. The problem appears to be that dividing the change in estimated option value by a very small ε magnifies the effect of the truncation error. Thus, it can be counterproductive to use very small perturbations in estimating the Greek letter risk exposures. The fourth line gives results for the method of extending the tree backwards in time, as in Figure 19. This is considerably more efficient in terms of computation time than the perturbation approaches and it also gives much more accurate results, especially for gamma.

The last three lines for each N value give the results from the standard Trinomial model. We a see similar effect as for the Binomial with respect to the choice of ε, and again we observe that extending the tree is both much more accurate than perturbation and faster in computation time. One surprise is that with a given N the Trinomial is about twice as accurate as the

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Binomial for pricing the options (in twice the execution time), but that does not carry through for the estimates of delta and gamma, where the results are fairly similar, and in a few cases, the Binomial beats the Trinomial.

Unfortunately, the problem of truncation error still remains. This can partly be mitigated by the use of an AMM model to add a section of fine mesh around the strike price at expiration, as demonstrated in Section III. But there is still a difficulty at time 0, because to compute delta and gamma one must either solve three complete lattices with a small ε perturbation as in Figure 18, or else extend the tree as in Figure 19 and compute them from nonmarginal asset price changes. While the results in Table 4 might suggest using the Binomial model and taking delta and gamma from the replicating portfolio, as discussed above the Binomial lacks the flexibility of the Trinomial for pricing non-vanilla options and is therefore difficult to use as an all-purpose valuation technique.

Building an AMM Model to Compute Delta and GammaWe will now show how the problem of estimating the Greek letters can

be solved by an AMM model with a region of fine mesh around the initial asset price. This will allow numerical derivatives to be computed using as small an ε perturbation as we like with only a small increase in the total number of node calculations.

Figure 20 shows a lattice set up much like the one in Figure 19, with perturbation trees that overlap the original one and only add new nodes at the highest and lowest prices in each period. The difference here is that these new sections of lattice begin at time 0 at asset prices of X0 + h/2 and X0 - h/2, that is, at deviations from the original price X0 that are half as large as in the extended tree of Figure 19. To do this, we introduce quadrinomial branching for these two nodes. In the original lattice, starting from X0 there are paths going up to X0 + h, down to X0 - h, or in between to remain at X0 in the next period. The

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extended tree would add a time 0 node at the price X0 + h, from which branches would lead to X0 + 2h, X0 + h, and X0. The quadrinomial branching for the new node we are placing at X0 + h/2 allows for price changes to any of the four second period nodes that could be reached from either X0 or X0 + h. Thus we cut the size of the perturbation used in the delta and gamma calculation in half, but the modified tree requires exactly the same number of new nodes as the extended tree of Figure 19, because it connects to the original lattice in exactly the same way at the end of the first time step.

Quadrinomial branching introduces another degree of freedom into the choice of the probabilities. Consider the new node at X0 + h/2. There are four probabilities attached to the four possible next period nodes that can be reached, which we will designate as puu, pu, pd, and pdd. These correspond to price changes of +3/2h, +h/2, -h/2, and -3/2h, respectively. The trinomial branch probabilities had to satisfy the three conditions that the expected return over the next time step was the riskless rate, the second moment was consistent with the volatility σ of the underlying, and the probabilities summed to 1.0. With a fourth branch, another condition is needed to produce a unique solution for the probabilities. Our choice here is to require that the third moment of the price change over the next time step be equal

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to that of the lognormal distribution that we are trying to approximate.13 With α = r - q - σ2/2 and q denoting the dividend yield as before, we have

Solving these four equations in four unknowns and simplifying yields

where,Adding Finer AMM Levels to the Lattice

Like the other AMM lattice structures we have described, this model is isomorphic, so that finer layers of mesh can be added using the same procedure at each level. Figure 21 illustrates how the second level AMM is constructed, and Figure 21A shows a magnified view so the fine mesh at time 0 can be clearly seen. The first level of the AMM, shown in Figure 19, began with the extended tree lattice, with a price step of h and a time step of k, and added time 0 nodes one half of a price step above and below X0. The time 0 nodes were connected to the coarse lattice at time 1, using trinomial branching from X0 , since it is a node in the coarse lattice, and quadrinomial branching from the two new nodes that fall in between two coarse nodes. The end result is a tree with three nodes at time 0, whose prices differ by the amount h/2. From these nodes there are branches connecting at time 1 to five nodes in a coarse tree with price step h.

To go to the next level of AMM, we must add a new section of lattice with a price step of h/2 and a time step of k/4 and connect it to five nodes in the first level AMM lattice. The initial asset price X0 is the center node at the far left. From it, three branches lead to nodes one (fine mesh) time step

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later, with prices X0, X0 + h/2, and X0 - h/2. As in the level 1 AMM, we also place time 0 nodes one half of a (fine mesh) price step above and below X0, at X0 + h/4 and X0 - h/4. From these prices, a middle “no change” branch would not connect to the next higher level lattice, so quadrinomial branching is required. As before, branches from the three initial price nodes will lead to five nodes of the next higher level mesh in the next (fine) time step. (This means that the coarse tree nodes at X0 + h and X0 - h, which were not necessary for the first level AMM will be needed now).

In Figures 21 and 21A, the heavy lines represent the coarse lattice with step sizes h and k; the dashed lines indicate the section of lattice that is added in the first level AMM; and the fine solid lines show the level 2 AMM section. Adding a third level AMM would follow the same process, with the starting node one finest time step (k/16) earlier and three nodes at X0, X0 + h/8 and X0 - h/8, that branch to five nodes in the next step. In all cases, if there is a node with the same price in the next higher AMM level, trinomial branching is used, but for nodes falling in between prices in the next level, branching is quadrinomial.

The General Procedure for Constructing an AMM Lattice for Delta and Gamma

In the foregoing description, it was convenient to start with the coarsest mesh lattice and consider adding progressively finer sections. But in constructing this kind of AMM from scratch, it is actually more intuitive to think about it starting from time 0. Suppose we have decided to build a tree with N coarse steps and M AMM levels. Note that the first level does not require adding any mesh with a shorter time step, as is clear from Figure 20. For each AMM level m > 1, there will be one time step of length k / 4m-1. Thus the length of the entire tree will be (N + 1/4 + ... + 1/4M-1 ) coarse time steps. Given option maturity T, volatility σ, and a choice of λ, the values for h and k are given by

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Now, let us denote the price and time steps for the mth level of the AMM as hm and km (hm = h / 2m-1 , km = k / 4m-1 ). The first node is placed at X0

at time 0 and the nodes to be used in the calculation of delta and gamma are placed at X0 + hm / 2 and X0 - hm / 2. These three nodes must branch one km time step later to five nodes in the next highest level AMM lattice, located at X0 + 2hm-1 , X0 + hm-1 , X0 , X0 - hm-1 , and X0 - 2hm-1. The branching from X0 is trinomial, because there is a node in the higher level lattice at X0 . Branching from a node lying between two higher level lattice prices will be quadrinomial. Once the nodes and branching for the first km time step is completed, the price step is doubled and the time step multiplied by four, and the process of constructing the nodes and branching to the next higher level is repeated until the coarsest tree with price and time steps h and k is reached. This coarse lattice is then extended out to option maturity.

Performance of the AMM in Estimating Delta and GammaNow let us see what this process gains us. Table 5 reports the

performance statistics for the standard Trinomial and several versions of AMM models. We examined adding fine mesh at time 0 to improve the estimates of delta and gamma, but we found that these numerical derivatives were affected by the truncation error in the option prices, so that accuracy could be enhanced considerably by using an AMM at expiration (t = T) as well. For each N, the first line of results is for the standard Trinomial, with the Greek letters computed by extending the tree. The next two lines incorporate one and two levels of AMM at time 0 (as in Figures 20 and 21). The last three lines add AMM levels both at the beginning and at expiration.

In principle, adding finer lattice only at the beginning should not affect the calculation of the option value, but we see a very small effect going from AMM 1 to AMM 2, due to the slight change in the time step (e.g., changing

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from 25 to 25 1/4 coarse steps in the tree). The big difference is in the accuracy of the estimates of delta and gamma, which are cut by about 2/3 for delta and 1/2 for gamma in the AMM 1 model, with no increase in execution time. Going from AMM 1 to AMM 2 improves the delta calculation a little, but not gamma.

When we add finer mesh both at the beginning and the end, there is a sharp improvement in both the option value and the delta. Gamma also becomes much more accurate when the first level of AMM mesh is added at expiration, but further refinement does not seem to help; if anything, using a single level of fine mesh is better than using more. (This is a result that we do not yet fully understand.) It is important to notice that while adding finer mesh leads to a large improvement in accuracy, execution time is virtually the same for the most refined AMM model as for the basic Trinomial. Finally, in comparing the AMM approach to the other methods displayed in Table 4, it is apparent that huge gains in performance are possible. For example, a 25 time step AMM model with 3 levels of fine mesh at the beginning and end achieves substantially greater accuracy in pricing these options and computing their deltas than does a 1000 time step standard Trinomial that takes more than 500 times longer to execute. Only in estimating gamma does the AMM seem to have any difficulty: it improves accuracy by only about a factor of 5 in the same amount of time as a standard model.

VI. ConclusionThe significant advance in the technology of option valuation achieved

by the Black-Scholes pricing model quickly reached a theoretical impass with American options, and then other derivative instruments with more varied contingencies. Although the no-arbitrage condition continued to hold so that the valuation principles remained the same, closed-form valuation equations could not be derived. The development of lattice-based models starting with

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the Binomial, and followed by the Trinomial, was the breakthrough that permitted at least an approximate solutions.

Numerous enhancements to tree-based models over the years have led to some improvement in performance, but whole classes of problems, including many that are important for valuing common real-world derivative instruments, remain theoretically soluble, but practically infeasible with the standard methods because they require too many calculations to achieve an acceptable degree of accuracy.

The Adaptive Mesh Model described in this paper offers a way to increase accuracy and reduce computation time enormously for many sorts of valuation problems. We have described three types of AMM, one that involves constructing a section of high resolution mesh around the strike price at expiration, a second that creates whole layers of fine mesh near the barrier of a barrier option, and finally one that adds fine mesh at time 0 to improve the estimates of delta and gamma. All three AMM models produced a major improvement in performance.

The AMM approach can also be adapted relatively easily to higher dimensions, with even greater performance increases relative to constant mesh size lattice methods. For example, we have obtained some results using an AMM procedure to value an “outside barrier option.” This is a barrier option for which the payoff is determined by one asset price while the barrier depends on a different one. An example is an option on the British FT-SE stock index, that is knocked out if the exchange rate on the pound falls below a specified level. These valuation problems are practically insoluble with ordinary lattice methods (or any other technique currently in use). The AMM technology therefore greatly extends the range of derivatives valuation

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problems that may be addressed.14

Along with developing AMM models for use with derivatives that are contingent upon multiple stochastic factors, we are also exploring other “variations on the theme,” including models that allow the price step to shrink more slowly than by a factor of 2 at each level, models that add larger regions of fine mesh with the same step size, and general criteria for determining in which regions of an option’s state space an AMM approach will be most valuable.

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References

Black, F., and M. Scholes. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy 81 (1973), pp. 637-654.

Boyle, P. P., and S. H. Lau. “Bumping Up against the Barrier with the Binomial Method.” The Journal of Derivatives 1, (Summer 1995), pp. 6-14.

Brennan, M. J., and E. S. Schwartz. “The Valuation of American Put Options.”

approximation to be proportional to the length of a time step, so that RMSE would fall with the square root of N. But in these results, the drop in RMSE appears to be proportional to N. We are currently investigating the reason for this.

9 Calculating implied volatilities even for ordinary American calls and puts requires extensive computation to solve the valuation model repeatedly for each option. For example, when Canina and Figlewski [1993] used a 500 time step Binomial model to obtain implied volatilities for about 17,000 stock index calls the required CPU time on a VAX mainframe computer was approximately one week.

10 Rubinstein[1991] describes a wide variety of exotic option contracts, many of which have barriers. Brief explanations of a vast array of traded derivative products, including a large number with various barrier features are given in Gastineau and Kritzman [1996].

11 Ritchken actually defines λ slightly differently than we do: His λ is the square root of the λ given in equation (9).

12 Since the price step is defined in terms of the log of the price, the dollar values mentioned here are exact only at the barrier: it is h that is constant throughout the tree.

13 We have not yet explored alternative choices for the fourth condition. One possibility would be to match the kurtosis of the lognormal rather than its skewness, in view of the importance of tail fatness of the probability distribution in option pricing. In any case, we do not expect this choice to have much overall impact, since it affects only the handful of quadrinomial branching nodes that will be introduced.

14 Gao[1996] discusses an outside barrier option example in which the RTM approach would require over 100 days of CPU time (it is estimated), while an AMM model obtains the correct price to 3 decimal places in under 1 second.

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Journal of Finance 32, (1977), pp. 449-62.

Canina, L. and S. Figlewski. “The Informational Content of Implied Volatility.” Review of Financial Studies 6, (1993), pp. 659-681.

Cho, H.Y. and K.S. Lee. “An Extension of the Three-jump Process Model for Contingent Claim Valuation.” The Journal of Derivatives 3, (Fall 1995), pp. 102-108.

Courtadon, G. “An Introduction to Numerical Methods in Option Pricing.” in Figlewski, et al, eds. Financial Options: From Theory to Practice. Business One Irwin, Homewood, IL, 1990, pp. 538-572.

Cox, J., S. Ross, and M. Rubinstein. “Option Pricing: A Simplified Approach.” Journal of Financial Economics 7, (1979), pp. 229-264.

Gao, Bin. “Essays in Efficient Option Pricing.” Ph.D. Dissertation, New York University Stern School of Business, 1996.

Gastineau, G. and M. Kritzman. Dictionary of Financial Risk Management. Frank J. Fabozzi Associates, 1996.

Geske, R., and H. E. Johnson. “The American Put Option Valued Analytically.” Journal of Finance 39, (1984), pp. 1511-1524.

Merton, R. C. “Theory of Rational Option Pricing.” Bell Journal of Economics and Management Science 4 (Spring 1973), pp. 141-183.

Pelsser, A., and T. C. F. Vorst. “The Binomial Model and the Greeks.” The Journal of Derivatives 1, (Spring 1994), pp. 45-49.

Ritchken, P. “On Pricing Barrier Options.” The Journal of Derivatives 3, (Winter 1995), pp. 19-28.

Roll, R. “An Analytical Formula for Unprotected American Call Options on Stocks with Known Dividends.” Journal of Financial Economics 5, (1977), pp. 251-258.

Rubinstein, M. “Exotic Options.” University of California at Berkeley Research Program in Finance Working Paper No. 220, 1991.

Sharpe, W. F. Investments. Prentice-Hall, Englewood Cliffs,NJ., 1978.

Whaley, R. “On the Valuation of American Call Options on Stocks with Known

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Dividends.” Journal of Financial Economics 9, (1981), pp. 207-211.

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Table 1

Performance of Binomial, Trinomial, and Adaptive Mesh Models in Valuing European Puts

Time Approximation RMSE Nodes ExecutionModel steps Price Delta Gamma Time (sec)

Binomial 25 0.020841 0.005805 0.001594 351 0.0183Trinomial 25 0.012025 0.003337 0.000428 676 0.0117AMM-1 25 0.002812 0.003345 0.000539 716 0.0117AMM-2 25 0.000615 0.003359 0.000548 756 0.0150

Binomial 100 0.004929 0.001470 0.000197 5151 0.1050Trinomial 100 0.002770 0.000846 0.000144 10201 0.1600AMM-1 100 0.000600 0.000845 0.000140 10241 0.1600AMM-2 100 0.000151 0.000854 0.000138 10281 0.1750

Binomial 250 0.002214 0.000534 0.000073 31626 0.3967Trinomial 250 0.001360 0.000346 0.000061 63001 1.0133AMM-1 250 0.000245 0.000334 0.000056 63041 1.0133AMM-2 250 0.000057 0.000342 0.000056 63081 1.0133

Binomial 1000 0.000448 0.000145 0.000020 501501 4.4232Trinomial 1000 0.000244 0.000079 0.000015 1002001 15.5344AMM-1 1000 0.000056 0.000086 0.000014 1002041 15.5977AMM-2 1000 0.000016 0.000085 0.000014 1002081 15.7266

The table compares the performance of different lattice-based approximation methods in computing the theoretical price, delta and gamma for a set of 27 European put options. Root mean squared errors relative to the exact Black-Scholes values and computation times on a Sun Sparc 20 are displayed for the CRR version of the Binomial model, the standard Trinomial model, and the Adaptive Mesh Model with 1 and 2 levels of fine mesh around the strike price at maturity (denoted AMM-1 and AMM-2, respectively).

Parameter values: Initial stock price = 40Riskless interest rate = 5.0% simple interest

= 4.88% continuously compoundedNo dividends.

Option test set: European puts with all 27 combinations of: Strike prices = 35, 40, and 45; Maturities = 1, 4, and 7 months;

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Volatilities = 0.20, 0.30, 0.40

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Table 2

Comparison of the Number of Nodes in a Barrier Option Adaptive Mesh Model versus a Standard Trinomial Model with the Same Sized Price Step as the Finest Level AMM

Mesh

Model N = 25 N = 100 N = 400

Standard Trinomial 676 10,201 160,801

Adaptive Mesh, M = 1 926 11,201 164,801Equivalent Trinomial 10,816 163,216 2,572,816

Adaptive Mesh, M = 2 1,926 15,201 180,801Equivalent Trinomial 173,056 2,611,456 41,165,056

Adaptive Mesh, M = 3 5,926 31,201 244,801Equivalent Trinomial 2,768,896 41,783,296

658,640,896

Adaptive Mesh, M = 4 21,926 95,201 500,801Equivalent Trinomial 44,302,336 668,532,736 10,538,254,336

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Table 3

Pricing a Down and Out Call Option with The AMM and the Restricted Trinomial Model when the

Asset Price is Near the Boundary

S0 Analytic RTM AMM Value Value CPU Time Value CPU Time M

92 2.506 2.507 0.033 2.507 0.033 0

91 1.274 1.274 0.750 1.274 0.050 1

90 1/2 0.642 0.642 12.35 0.643 0.059 2

90 1/4 0.323 0.323 364.3 0.323 0.117 3

90 1/8 0.162 N/A N/A 0.162 0.317 4

The table shows the analytic down and out call value from equation (10) and approximate values from the Restricted Trinomial Model (RTM). The RTM constructs a lattice with the same price step size everywhere and with a layer of nodes exactly on the knock-out boundary, one price step below the initial asset price. The Adaptive Mesh Model maintains the same size price step for the coarsest mesh but adds layers of finer mesh as the initial asset price approaches the barrier.

Parameter values: K = 100, r = 10%, T - t = 1 year, σ = 0.25, H (barrier) = 90.

Computation time in seconds is for a PC with a Pentium 90 MHz processor and 32 megabytes of RAM. N/A indicates that the problem became unmanageable for the computer.

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Table 4

Performance of Binomial and Trinomial Models inEstimating Delta and Gamma for European Puts

Approximation RMSE Execution Model Price Delta Gamma Time (sec)

N = 25Binomial Replication 0.020841 0.001978 0.000765 0.0050Binomial Perturbation (ε=.001) 0.020841 0.025559 0.060454 0.0200Binomial Perturbation (ε=.01) 0.020841 0.016775 0.079445 0.0200Binomial Tree Extension 0.020841 0.005805 0.001594

0.0070Trinomial Perturbation (ε=.001)0.012025 0.034958 0.499567 0.0200Trinomial Perturbation (ε=.01) 0.012025 0.020866 0.101378 0.0200Trinomial Tree Extension 0.012025 0.003337 0.000428 0.0100

N = 100Binomial Replication 0.004929 0.000453 0.000417 0.0320Binomial Perturbation (ε=.001) 0.004929 0.013722 1.084769 0.1400Binomial Perturbation (ε=.01) 0.004929 0.005734 0.078385 0.1400Binomial Tree Extension 0.004929 0.001470 0.000197

0.0460Trinomial Perturbation (ε=.001)0.002770 0.015642 0.242340 0.2810Trinomial Perturbation (ε=.01) 0.002770 0.006151 0.044298 0.2800Trinomial Tree Extension 0.002770 0.000846 0.000144 0.0920

N = 250Binomial Replication 0.002214 0.000227 0.000169 0.1472Binomial Perturbation (ε=.001) 0.002214 0.009741 0.671262 0.7010Binomial Perturbation (ε=.01) 0.002214 0.001215 0.044008 0.6910Binomial Tree Extension 0.002214 0.000534 0.000073

0.2123Trinomial Perturbation (ε=.001)0.001360 0.009286 0.242539 1.5820Trinomial Perturbation (ε=.01) 0.001360 0.001689 0.026351 1.5830Trinomial Tree Extension 0.001360 0.000346 0.000061 0.5298

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Table 4 cont.

Approximation RMSE Execution Model Price Delta Gamma Time (sec)

N = 1000Binomial Replication 0.000448 0.000043 0.000041 1.9729Binomial Perturbation (ε=.001) 0.000448 0.004774 0.315333 9.9840Binomial Perturbation (ε=.01) 0.000448 0.000436 0.010133 9.9850Binomial Tree Extension 0.000448 0.000145 0.000020

2.9893Trinomial Perturbation (ε=.001)0.000244 0.004631 0.120938 24.6160Trinomial Perturbation (ε=.01) 0.000244 0.000656 0.004602 24.6650Trinomial Tree Extension 0.000244 0.000079 0.000015 8.1918

The table compares the performance of different approaches for estimating delta and gamma in the CRR version of the Binomial model and the standard Trinomial model. “Replication” refers to the delta and gamma obtained from the option replicating portfolio. “Perturbation” involves constructing new binomial or trinomial trees beginning from time 0 asset prices above and below the original (log) price X0 by an amount ε (for ε=.001 or .01) as shown in Figure 18. Delta and gamma are estimated as numerical derivatives using the option values obtained from the new trees. “Tree extension” involves building the valuation tree starting one (trinomial) or two (binomial) time steps before time 0, so that it produces option values at time 0 for asset prices X0 , X0 + h , and X0 - h, as shown in Figure 19. These are then used to calculate numerical derivatives.

Root mean squared errors relative to the exact Black-Scholes values and computation times on a Dell Pentium Pro 200 MHZ computer are displayed for the same test set of 27 European put options as in Table 1. The execution time is the average from 10 identical runs for each set of parameters.

Parameter values: Initial stock price = 40Riskless interest rate = 5.0% simple interest

= 4.88% continuously compoundedNo dividends.

Option test set: European puts with all 27 combinations of: Strike prices = 35, 40, and 45; Maturities = 1, 4, and 7 months;

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Volatilities = 0.20, 0.30, 0.40

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Table 5

Performance of Trinomial AMM Models inEstimating Delta and Gamma for European Puts

AMM Level Approximation RMSE Execution t = 0 t = T Price Delta Gamma Time (sec)

N = 25Trinomial 0 0 0.012025 0.003337 0.000428 0.0090AMM 1 0 0.012025 0.001087 0.000333 0.0090

2 0 0.011909 0.000810 0.000400 0.00901 1 0.002812 0.000845 0.000080 0.01202 2 0.000596 0.000205 0.000113 0.01313 3 0.000193 0.000053 0.000120 0.0140

N = 100Trinomial 0 0 0.002770 0.000846 0.000144 0.0931AMM 1 0 0.002770 0.000265 0.000068 0.0922

2 0 0.002764 0.000188 0.000077 0.09311 1 0.000600 0.000210 0.000020 0.09422 2 0.000153 0.000056 0.000028 0.09713 3 0.000043 0.000014 0.000027 0.0991

N = 250Trinomial 0 0 0.001360 0.000346 0.000061 0.5358AMM 1 0 0.001360 0.000115 0.000029 0.5288

2 0 0.001350 0.000085 0.000031 0.52871 1 0.000245 0.000079 0.000006 0.53082 2 0.000057 0.000023 0.000010 0.58783 3 0.000019 0.000005 0.000011 0.5738

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Table 5 cont.

AMM Level Approximation RMSE Execution t = 0 t = T Price Delta Gamma Time (sec)

N = 1000Trinomial 0 0 0.000244 0.000079 0.000015 8.5072AMM 1 0 0.000244 0.000021 0.000006 8.3711

2 0 0.000243 0.000016 0.000007 8.36401 1 0.000056 0.000023 0.000002 8.31702 2 0.000016 0.000005 0.000003 8.31303 3 0.000006 0.000001 0.000003 8.3751

The table compares the performance of Trinomial AMM models with different levels of fine mesh added at t = 0 around the initial asset price, and at t = T around the strike price. The first line in each panel gives the results for the standard Trinomial model. Delta and gamma are estimated as numerical derivatives using the option values obtained from the finest level of mesh at t = 0.

Root mean squared errors relative to the exact Black-Scholes values and computation times on a Dell Pentium Pro 200 MHZ computer are displayed for the same test set of 27 European put options as in Table 1. The execution time is the average from 10 identical runs for each set of parameters.

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