In quantitative finance, a lattice model is a numerical approach to the valuation of derivatives in situations requiring a discrete time model. For dividend paying equity options, a typical application would correspond to the pricing of an American-style option, where a decision to exercise is allowed at the closing of any calendar day up to the maturity. A continuous model, on the other hand, such as the standard Black–Scholes one, would only allow for the valuation of European options, where exercise is limited to the option's maturity date. For interest rate derivatives lattices are additionally useful in that they address many of the issues encountered with continuous models, such as pull to par.Hull, J. C. (2006). Options, futures, and other derivatives. Pearson Education India. The method is also used for valuing certain exotic options, because of path dependence in the payoff. Traditional Monte Carlo methods for option pricing fail to account for optimal decisions to terminate the derivative by early exercise,Cox, J. C., Ross, S. A., & Rubinstein, M. (1979). "Option pricing: A simplified approach". Journal of Financial Economics, 7(3), 229–263. but some methods now exist for solving this problem.
Equity and commodity derivatives Tree-based equity option valuation: 1. Construct the tree of equity-prices: Either forward-construct, applying an up or down factor ( or ) to the current price, such that in the next period the price will either be or ; or given that the tree is recombining, directly via , where is the number of up ticks and is the number of down ticks.
2. Construct the corresponding option tree: at each final node of the tree—i.e. at expiration of the option—the option value is simply its intrinsic, or exercise, value; at earlier nodes, value is via expectation, , being the probability of an up move; where non-European value is the greater of this and the exercise value given the corresponding equity value. In general the approach is to divide time between now and the option's expiration into N discrete periods. At the specific time n, the model has a finite number of outcomes at time n + 1 such that every possible change in the state of the world between n and n + 1 is captured in a branch. This process is iterated until every possible path between n = 0 and n = N is mapped. Probabilities are then estimated for every n to n + 1 path. The outcomes and probabilities flow backwards through the tree until a fair value of the option today is calculated.
Variants For European equity- and commodity options the application is as follows. The first step is to trace the evolution of the option's key underlying variable(s), starting with today's spot price, such that this process is consistent with its volatility; log-normal Brownian motion with constant volatility is usually assumed.Chance, Don M. (March 2008). "A Synthesis of Binomial Option Pricing Models for Lognormally Distributed Assets". . Journal of Applied Finance, Vol. 18. The next step is to value the option recursively: stepping backwards from the final time-step, where we have exercise value at each node; and applying risk neutral valuation at each earlier node, where option value is the probability-weighted present value of the up- and down-nodes in the later time-step. See for more detail, as well as for logic and formulae derivation.
As stated above, the lattice approach is particularly useful in valuing American options, where the choice whether to exercise the option early, or to hold the option, may be modeled at each discrete time/price combination; this is also true for Bermudan options. For similar reasons, real options and employee stock options are often modeled using a lattice framework, though with modified assumptions. In each of these cases, a third step is to determine whether the option is to be exercised or held, and to then apply this value at the node in question. Some exotic options, such as barrier options, are also easily modeled here; for other Path-Dependent Options, simulation would be preferred. (Although, tree-based methods have been developed. Timothy Klassen (2001). "Simple, Fast and Flexible Pricing of Asian Options". Journal of Computational Finance. 4 (3) 89–124. John Hull and Alan White. "Efficient procedures for valuing European and American path-dependent options". Journal of Derivatives. Fall 1993. 21–31. )
The simplest lattice model is the binomial options pricing model;Ronnie Becker (N.D.). "Pricing in the Binomial Model" , African Institute for Mathematical Sciences. the standard ("canonical"Prof. Markus K. Brunnermeier. "Multi-period Model Options", Princeton University.) method is that proposed by Cox, Ross and Rubinstein (CRR) in 1979; see diagram for formulae. Over 20 other methods have been developed,Mark S. Joshi (2008). "The Convergence of Binomial Trees for Pricing the American Put" with each "derived under a variety of assumptions" as regards the development of the underlying's price. In the limit, as the number of time-steps increases, these converge to the Log-normal distribution, and hence produce the "same" option price as Black-Scholes: to achieve this, these will variously seek to agree with the underlying's central moments, raw moments and / or log-moments at each time-step, as measured discretely. Further enhancements are designed to achieve stability relative to Black-Scholes as the number of time-steps changes. More recent models, in fact, are designed around direct convergence to Black-Scholes.
A variant on the Binomial, is the Trinomial tree,Mark Rubinstein (2000). On the Relation Between Binomial and Trinomial Option Pricing Models. The Journal of Derivatives, Winter 2000, 8 (2) 47-50Zaboronski et al (2010). Pricing Options Using Trinomial Trees. University of Warwick developed by Phelim Boyle in 1986. Here, the share price may remain unchanged over the time-step, and option valuation is then based on the value of the share at the up-, down- and middle-nodes in the later time-step. As for the binomial, a similar (although smaller) range of methods exist. The trinomial model is considered to produce more accurate results than the binomial model when fewer time steps are modelled, and is therefore used when computational speed or resources may be an issue. For vanilla options, as the number of steps increases, the results rapidly converge, and the binomial model is then preferred due to its simpler implementation. For exotic options the trinomial model (or adaptations) is sometimes more stable and accurate, regardless of step-size.
For multiple underlyers – e.g. Rainbow- and Basket options – "multinomial lattices" can be built. Each underlyer will have its own tree, and the per node option-value will be a function of the corresponding nodes on all underlying trees. In the case of two assets, the tree will then be referred to as a "binomial pyramid". Two additional complications exist here.Staff (2024). The Multi-Asset Binomial-Model, fastercapital.com First, the number of nodes increases exponentially with the number of underlyers. Second, in these products, correlations between assets play a significant role, and these must also inhere in the modelling.
Risk measurement and management Various of the Greeks can be estimated directly on the lattice, where the sensitivities are calculated using finite differences.Don Chance. (2010) Calculating the Greeks in the Binomial Model. Delta and gamma, being sensitivities of option value w.r.t. price, are approximated given differences between option prices – with their related spot – in the same time step. Theta, sensitivity to time, is likewise estimated given the option price at the first node in the tree and the option price for the same spot in a later time step. (Second time step for trinomial, third for binomial. Depending on method, if the "down factor" is not the inverse of the "up factor", this method will not be precise.) For rho, sensitivity to interest rates, and vega, sensitivity to input volatility, the measurement is indirect, as the value must be calculated a second time on a new lattice built with these inputs slightly altered – and the sensitivity here is likewise returned via finite difference. See also Fugit, the estimated time to exercise for a non-European option, which is typically calculated using a lattice.
Beginning with the 1987 crash, and especially since the 2008 financial crisis, it has become important to incorporate the volatility smile / surface into pricing models. This recognizes the fact that the underlying price-change distribution displays a term structure and is non-normal, unlike that assumed by Black-Scholes; see and . To do so, banks typically apply stochastic- or local volatility models. In the Lattice framework, implied trees can be constructed; these essentially discretize the latter. Here, the tree is solved such that it successfully reproduces selected (all) market prices, across various strikes and expirations. These trees thus "ensure that all European standard options (with strikes and maturities coinciding with the tree nodes) will have theoretical values which match their market prices".Emanuel Derman, Iraj Kani, and Neil Chriss (1996). Implied Trinomial Trees of the Volatility Smile. Goldman Sachs, Quantitative Strategies Research Notes Using the calibrated lattice one can then price options with strike / maturity combinations not quoted in the market, such that these prices are consistent with observed volatility patterns. For risk management, the Greeks returned here will reflect sensitivities more appropriately.
There exist both implied binomial trees, often Rubinstein IBTs (R-IBT),Mark Rubinstein (1994). Implied Binomial Trees. Journal of Finance. July 1994. and implied trinomial trees, often Derman-Kani-Chriss (DKC; superseding the DK-IBTEmanuel Derman and Iraj Kani (1994). The Volatility Smile and Its Implied Tree. Research Note, Goldman Sachs.). The former is easier built, but is consistent with one maturity only; the latter will be consistent with, but at the same time requires, known (or interpolated) prices at all time-steps and nodes. As regards the construction, for an R-IBT the first step is to recover the "Implied Ending Risk-Neutral Probabilities" of spot prices. Then by the assumption that all paths which lead to the same ending node have the same risk-neutral probability, a "path probability" is attached to each ending node. Thereafter "it's as simple as One-Two-Three", and a three step backwards recursion allows for the node probabilities to be recovered for each time step. Option valuation then proceeds as standard, with these probabilities replacing above. For DKC, the first step is to recover the state prices corresponding to each node in the tree, such that these are consistent with observed option prices (i.e. with the volatility surface). Thereafter the up-, down- and middle-probabilities are found for each node such that: these sum to 1; spot prices adjacent time-step-wise evolve risk neutrally, incorporating dividend yield; state prices similarly "grow" at the risk free rate.Jim Clark, Les Clewlow and Chris Strickland (2008). Calibrating trees to the market prices of options. Energy Risk, August 2008. (Archived, 2015-06-30) (The solution here is iterative per time step as opposed to simultaneous.) As for R-IBTs, option valuation is then by standard backward recursion.
As an alternative, Edgeworth binomial trees Mark Rubinstein (1998). Edgeworth Binomial Trees. Journal of Derivatives, Spring 1998. may be employed, as these allow for an analyst-specified skew and kurtosis in spot-price returns (see Edgeworth series). Here, options with differing strikes will return differing implied volatilities, and the tree may then be calibrated to the volatility smile, by a "judicious choice" of parameter values.