Edgeworth binomial tree

In finance, a lattice model is a technique applied to the valuation of derivatives, where a discrete time model is required. For equity options, a typical example would be pricing an American option, where a decision as to option exercise is required at "all" times (any time) before and including maturity. A continuous model, on the other hand, such as Black–Scholes, would only allow for the valuation of European options, where exercise is on 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. The method is also used for valuing certain exotic options, where because of path dependence in the payoff, Monte Carlo methods for option pricing fail to account for optimal decisions to terminate the derivative by early exercise, though methods now exist for solving this problem.

Equity and commodity derivatives

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.

For equity and commodities 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 200
''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.

)

The simplest lattice model is the binomial options pricing model; the standard ("canonical") 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
/ref> 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, 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.

Various of the Greeks can be estimated directly on the lattice, where the sensitivities are calculated using finite differences. 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, which is typically calculated using a lattice.

When it is important to incorporate the volatility smile, or surface, implied trees can be constructed. 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. There exist both implied binomial trees, often Rubinstein IBTs (R-IBT), and implied trinomial trees, often Derman-Kani- Chriss (DKC; superseding the DK-IBT). 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. (DKC is effectively a discretized local volatility model.)

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 substituted for ''p''. 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. (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

allow for an analyst-specified skew and kurtosis in spot price returns; see Edgeworth series. This approach is useful when the underlying's behavior departs (markedly) from normality. A related use is to calibrate the tree to the volatility smile (or surface), by a "judicious choice" of parameter values—priced here, options with differing strikes will return differing implied volatilities. For pricing American options, an Edgeworth-generated ending distribution may be combined with an R-IBT. This approach is limited as to the set of skewness and kurtosis pairs for which valid distributions are available. The more recent Johnson binomial trees
use the Johnson "family" of distributions, as this is capable of accommodating all possible pairs.

For multiple underlyers, multinomial lattices can be built, although the number of nodes increases exponentially with the number of underlyers. As an alternative, Basket options, for example, can be priced using an "approximating distribution" via an Edgeworth (or Johnson) tree.

Interest rate derivatives

Lattices are commonly used in valuing bond options, Swaptions, and other interest rate derivatives In these cases the valuation is largely as above, but requires an additional, zeroeth, step of constructing an interest rate tree, on which the price of the underlying is then based. The next step also differs: the underlying price here is built via "backward induction" i.e. flows backwards from maturity, accumulating the present value of scheduled cash flows at each node, as opposed to flowing forwards from valuation date as above. The final step, option valuation, then proceeds as standard. See top for graphic, and aside for description.

The initial lattice is built by discretizing either a short-rate model, such as Hull–White or Black Derman Toy, or a forward rate-based model, such as the LIBOR market model or HJM. As for equity, trinomial trees may also be employed for these models; this is usually the case for Hull-White trees.

Under HJM,Pricing Interest Rate-dependent Financial Claims with Option Features
Ch 11. in Rendleman (2002), per Bibliography. the condition of no arbitrage implies that there exists a martingale probability measure, as well as a corresponding restriction on the "drift coefficients" of the forward rates. These, in turn, are functions of the volatility(s) of the forward rates. A "simple" discretized expression for the drift then allows for forward rates to be expressed in a binomial lattice. For these forward rate-based models, dependent on volatility assumptions, the lattice might not recombine. (This means that an "up-move" followed by a "down-move" will not give the same result as a "down-move" followed by an "up-move".) In this case, the Lattice is sometimes referred to as a "bush", and the number of nodes grows exponentially as a function of number of time-steps. A recombining binomial tree methodology is also available for the Libor Market Model.

As regards the short-rate models, these are, in turn, further categorized: these will be either equilibrium-based ( Vasicek and CIR) or arbitrage-free ( Ho–Lee and subsequent