# Adaptive Mesh Trinomial Tree for Vanilla Option Pricing

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This Demonstration illustrates the application of the recombining adaptive mesh trinomial tree method to numerically approximate the value of the European- or American-type vanilla call/put option, assuming constant volatility and risk-free rate. The adaptive mesh trinomial tree method reduces nonlinearity error and improves pricing efficiency (e.g., reduces CPU time) by overlaying a higher-resolution mesh tree over the conventional coarse tree lattice [1]. The method may be used to approximate the implied volatility parameter for European- or American-type vanilla call/put options. Furthermore, the trinomial adaptive mesh tree method may be extended to estimate the value of other types of options (e.g., barrier options).

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Contributed by: Darius Kirevicius (February 2015)

Open content licensed under CC BY-NC-SA

## Details

The Demonstration illustrates application of the recombining trinomial adaptive mesh tree method to approximate value of the European and American-type call/put option. The recombining trinomial adaptive mesh tree is generated by overlaying the coarse trinomial tree with the finer-mesh trinomial tree around the strike price and close to maturity.

The recombining trinomial coarse tree is generated by allowing only three things to happen to the price of the underlying asset—increase, decrease, or remain unchained—one unit of time later (e.g., one tick, day, week, etc.). Specifically, the underlying asset price is allowed to: (1) increase by the factor ; (2) decrease by the factor ; or (3) remain unchanged, hence scaled by the factor , where is the annualized volatility of the underlying asset and is the unit of time between successive tree nodes. The three possible outcomes , , and are assigned their risk-neutral probabilities , , and , respectively:

,

,

,

where , , is the risk-free rate (annualized), is the dividend yield of the underlying asset (annualized), and and are as defined above. Further, to ensure that probabilities , , and are in the interval and their sum is equal to 1, the following condition must hold: .

The finer-mesh tree is generated in exactly the same way as the coarse tree, except that the unit of time, , between successive mesh tree sites is set to , where is the number of mesh tree steps to the successive coarse tree node. For the purpose of this Demonstration, is set to 4. For a detailed discussion regarding the adaptive model (AMM), please refer to [1].

The adaptive mesh trinomial tree often improves pricing of efficiency of European- and American-type options at a lower computational cost (e.g., CPU time) relative to the standard trinomial tree method (especially for out of the money and longer maturity options). For instance, consider a scenario without any finer-mesh refinement in Snapshot 1, where the European-type call option price is approximated to be 0.566 (to three decimal places). The estimated price under the Black–Scholes–Merton (BSM) method is 0.599 (to three decimal places). Compare this to the adaptive mesh trinomial tree price approximation of 0.585 (to three decimal places) in Snapshot 2.

To explore nonlinearity errors of standard and adaptive mesh trinomial methods, observe what happens to the European-type options prices under scenarios in Snapshots 1 and 2 when the number of tree steps is, for instance, reduced from 10 to 9 to 8.

Reference

[1] S. Figlewski and B. Gao, "The Adaptive Mesh Model: A New Approach to Efficient Option Pricing," *Journal of Financial Economics*, 53(3), 1999 pp. 313–351.doi:10.1016/S0304-405X(99)00024-0

## Snapshots

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