• 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.

• 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.

• [39] An alternate approach, originally published by Goldman Sachs (1994),[40] does not decouple the components, rather, discounting is at a conversion-probability-weighted
  risk-free and risky interest rate within a single tree.

• 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.

• In the latter case, the calibration is directly on the lattice: the fit is to both the current term structure of interest rates (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.

• The second step is to then incorporate any term structure of volatility by building a corresponding DKC tree (based on every second time-step in the CRR tree: as DKC is trinomial
  whereas CRR is binomial) and then using this for option valuation.

• 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.

• [31] In the former case, the approach is to “calibrate” the model parameters, such that bond prices produced by the model, in its continuous form, best fit observed market
  prices.

• [4] 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.

• price, are approximated given differences between option prices – with their related spot – in the same time step.

• 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).

• 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.

• 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.

• For a bond with an embedded option, the standard yield to maturity based calculations of duration and convexity do not consider how changes in interest rates will alter the
  cash flows due to option exercise.

• Interest rate derivatives Lattices are commonly used in valuing bond options, swaptions, and other interest rate derivatives[23][24] 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.

• 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.

• [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 similar reasons, real options and employee stock options are often modeled using a lattice framework, though with modified assumptions.

• Since the 2007–2012 global financial crisis, swap pricing is (generally) under a “multi-curve framework”, whereas previously it was off a single, “self discounting”, curve;
  see Interest rate swap § Valuation and pricing.

• 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.

• [26] 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.

• The trinomial model is considered[12] 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.

• 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.

• For callable- and putable bonds 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.

• [14] 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 vanilla options, as the number of steps increases, the results rapidly converge, and the binomial model is then preferred due to its simpler implementation.

• 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.

• [35] With the zeroeth step thus accomplished, the valuation will proceed largely as previously, using steps 1 and onwards, but here with cashflows based on the LIBOR “dimension”,
  and discounting using the corresponding nodes from the OIS “dimension”.

• 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.

• 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.

• The next step also differs: the underlying price here is built via “backward induction” i.e.

• 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.

• 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”.

• [2] 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,[3] though methods now exist for solving this problem.

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

• 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).

• of the rates applicable historically for the time-step; to be market-consistent, analysts generally 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.)

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

 

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