|
:''For other meanings, see lattice model (disambiguation)'' 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.〔Hull, J. C. (2006). Options, futures, and other derivatives. Pearson Education India.〕 The method is also used for valuing certain exotic options, where because of path dependence in the payoff, Monte Carlo methods 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.〕 ==Equity and commodity derivatives== = S \cdot d; *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, , p 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. 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.() The next step is to value the option recursively, stepping backwards from the final time-step, and applying risk neutral valuation at each node, where option value is the probability-weighted present value of the up- and down-nodes in the later time-step. See Binomial options pricing model#Method for more detail, as well as Rational pricing#Risk neutral valuation for logic and formulae derivation. As 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 true also 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; note though that for other Path-Dependent Options, simulation would be preferred. 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,() 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, where valuation is based on the value of the option at the up-, down- and middle-nodes in the later time-step. As for the binomial, a similar (although smaller) range of methods exist. Note that 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. 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. 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." () 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 Derman-Kani 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. 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. 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. Note that this approach is limited as to the set of skewness and kurtosis pairs for which valid distributions are available. 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 ()() 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lattice model (finance)」の詳細全文を読む スポンサード リンク
|