Lattice model (finance)

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 outcomes and probabilities flow backwards through the tree until a fair value of the option today is calculated.

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.

[4] 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.

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.

Beginning with the 1987 crash, and especially since the 2007–2008 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 Financial economics § Derivative pricing and Valuation of options § Post crisis.

Here, the tree is solved such that it successfully reproduces selected (all) market prices, across various strikes and expirations.

As regards the construction, for an R-IBT the first step is to recover the "Implied Ending Risk-Neutral Probabilities" of spot prices.

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.

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

Construct a corresponding tree of bond-prices, where the underlying bond is valued at each node by "backwards induction": 2.

Under HJM,[27] 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.

[31] 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).

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 the implied trees for equity above, and compare Bootstrapping (finance).

Given this functional link to volatility, note now the resultant difference in the construction relative to equity implied trees: for interest rates, the volatility is known for each time-step, and the node-values (i.e. interest rates) must be solved for specified risk neutral probabilities; for equity, 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 interest rate lattice is then used in the valuation of various of the fixed income instruments and derivatives.

An alternative approach to modeling (American) bond options, particularly those struck on yield to maturity (YTM), employs modified equity-lattice methods.

A related development is that banks will make a credit valuation adjustment, CVA – as well as various of the other XVA – when assessing the value of derivative contracts that they have entered into.

The purpose of these is twofold: primarily to hedge for possible losses due to the other parties' failures to pay amounts due on the derivative contracts; but also to determine (and hedge) the amount of capital required under the bank capital adequacy rules.

[37] [38] [39] In the case of a swap, for example,[37] the potential future exposure, PFE, facing the bank on each date is the probability-weighted average of the positive settlement payments and swap values over the lattice-nodes at the date; each node's probability is in turn a function of the tree's cumulative up- and down-probabilities.

This PFE is combined with the counterparty's (tree-exogenous) probability of default and recovery rate to derive the expected loss for the date.

[40] For convertible bonds (CBs) the approach of Tsiveriotis and Fernandes (1998)[41] is to divide the value of the bond at each node into an "equity" component, arising from situations where the CB will be converted, and a "debt" component, arising from situations where CB is redeemed.

[43] An alternate approach, originally published by Goldman Sachs (1994),[44] does not decouple the components, rather, discounting is at a conversion-probability-weighted risk-free and risky interest rate within a single tree.

[48] Relatedly, as regards corporate debt pricing, the relationship between equity holders' limited liability and potential Chapter 11 proceedings has also been modelled via lattice.

There is however an additional requirement, particularly for hybrid securities: that is, to estimate sensitivities related to overall changes in interest rates.

Here, similar to rho and vega above, the interest rate tree is rebuilt for an upward and then downward parallel shift in the yield curve and these measures are calculated numerically given the corresponding changes in bond value.