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Lattice model (finance)

For other meanings, see lattice model (disambiguation)

In finance, a lattice model is a discretization used to find the fair value of a stock option; variants also exist for interest rate derivatives. [1]

The model divides 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.

Equity and commodity derivatives

The simplest lattice model for options on equity and commodities is the binomial options pricing model,[2] while a more sophisticated variant [3] is the Trinomial tree. As above, these models trace the evolution of the option's key underlying variable(s) in discrete-time, starting with today's spot price, and consistent with its volatility; log-normal Brownian motion with constant volatility is usually assumed.[4]

Lattice models are 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 true also for Bermudan options. See Binomial options pricing model#Method. For similar reasons, real options and employee stock options are often modeled using a lattice framework, though with modified assumptions. Some exotic options, such as barrier options, are also easily modeled here; note though that for other Path-Dependent Options, simulation would be preferred.

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; see local volatility. 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. Both Implied binomial trees (often Rubinstein IBTs [5]) and Implied trinomial trees (often Derman-Kani-Chriss [6]) exist. 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.

As an alternative, Edgeworth binomial trees [7] 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" [8] 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 a Rubinstein IBT. Note that this approach is limited as to the set of skewness and kurtosis pairs for which valid distributions are possible. One recent proposal, Johnson binomial trees, is to use Johnson's system of distributions, as this is capable of accommodating all possible pairs; see Johnson SU distribution.

For multiple underlyers multinomial lattices [9][10] 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" [11] via an Edgeworth (or Johnson) tree.

Interest rate derivatives

Tree-based bond option valuation:

0. Construct a short-rate tree, which, as described in the text, will be consistent with the current term structure of interest rates.

1. Construct a corresponding tree of bond-prices, where the underlying bond is valued at each node by "backwards induction":

  • at its final nodes, bond value is simply face value (or $1), plus coupon (in cents) if relevant, discounted to the start of the time-step using the corresponding short-rate;
  • at each earlier node, it is the discounted expected value of nodes in the later time step, plus coupon payments during the current time step, similarly discounted to the start of the time-step.

2. Construct a corresponding bond-option tree, where the option on the bond is valued similarly:

  • at option maturity, value is based on moneyness for all nodes in that time-step;
  • at earlier nodes, value is a function of the short-rate-discounted expected value of the option at the nodes in the later time step; where non-european value is the greater of this and the exercise value given the corresponding bond value.

Interest rate lattices are commonly used in valuing Bond options, Swaptions, and other interest rate derivatives [12][13] In these cases the 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;[14] this is usually the case for Hull-White trees.

The short-rate lattices are, in turn, further categorized: these will be either equilibrium-based (Vasicek and CIR) or arbitrage-free (Ho–Lee and subsequent). This distinction means that for equilibrium-based models the yield curve is an output from the model, while for arbitrage-free models the yield curve is an input to the model.[15]

In the latter case, one "calibrates" the model parameters to fit both the current term structure of interest rates (i.e. the yield curve), and the corresponding volatility structure. Here, calibration means that the interest-rate-tree reproduces the prices of the zero-coupon bonds — and any other interest-rate sensitive securities — used in constructing the yield curve; note the parallel to implied trees above, and compare Bootstrapping (finance). For models assuming a normal distribution (such as Ho-Lee), calibration may be performed analytically, while for log-normal models the calibration is via a root-finding algorithm; see boxed-description under Black–Derman–Toy model.

The volatility structure, i.e. vertical node-spacing, here reflects the volatility of rates during the quarter — or other period — corresponding to the lattice time-step. (Some analysts use "realized volatility", i.e. of the rates applicable historically during the time-step; others prefer to use current interest rate cap prices, and the implied volatility for the Black-76-prices of each component caplet; see Interest rate cap#Implied Volatilities.) Given this functional link to volatility, note the resultant difference in the construction relative to implied trees: here, the volatility is known for each time-step, and the node-values must be solved for specified risk neutral probabilities; for implied trees, on the other hand, a single volatility cannot be specified per time-step, i.e. we have a "smile", and the tree is built by solving for the probabilities corresponding to specified values of the underlying at each node.

Once calibrated, the lattice can be used in the valuation of various of the fixed income instruments and derivatives. The approach for bond options is described aside — note that this approach addresses the problem of pull to par experienced under closed form approaches; see Black–Scholes model#Valuing bond options. For swaptions the logic is almost identical, substituting swaps for bonds in step 1, and swaptions for bond options in step 2. For caps (and floors) step 1 and 2 are combined: at each node the value is based on the relevant nodes at the later step, plus, for any caplet (floorlet) maturing in the time-step, the difference between its reference-rate and the short-rate at the node (and reflecting the corresponding day-count fraction and notional-value exchanged). For bonds with "embedded options" a third step would be required: at each node in the time-step incorporate the effect of the embedded option on the bond price and / or the option price there before stepping-backwards one time-step. (And noting that these options are not mutually exclusive, and so a bond may have several options embedded. [16]) For other, more exotic interest rate derivatives, similar adjustments are made to steps 1 and onward.

For the forward rate-based models, dependent on volatility assumptions, the lattice might not recombine.[17] 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.

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