Smile-Lecture8.Fm 4/14/08 Lecture 8:Local Derman: 2008: Spring E4718

E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 1 of 19 Lecture 8: Local Volatility Models: Implications • Practical calibration of local volatility models • Implied trinomial trees • Implications: The deltas of standard options. The values of exotic options: barriers, lookbacks, etc. Copyright Emanuel Derman 2008 4/14/08 smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 2 of 19 8.1 Practical Calibration of Local Vol Models In practice we are given implied volatilities Σ()Ki, Ti over a range of discrete strikes and expirations, and must calibrate a smooth local volatility function to these discretely specified values. Earlier we mentioned that this is an ill-posed problem, and finding methods to solve it are critically important to the practical use of local volatility models. To use Dupire’s equation, we need a smooth implied volatility surface that is at least twice differentiable. We must therefore create a smooth implied volatility surface. The most straightforward way to do this is to write down a smooth parametric form for the implied volatilities, and then compute the parameters that minimize the distance between computed and observed standard options prices. One can then calculate the local volatilities by taking the appropriate derivatives of the implieds. One difficulty with this method is how to determine a realistic form of the parametrization, particularly on the wings where prices are hard to obtain. Wilmott’s book has one parametrization.There are a variety of other papers on this topic, some of them mentioned in Chapter 4 of Fengler’s book. The method illustrated here will be semi parametric. The idea is to smooth the variations in market implied volatilities by averaging the data in a series of small contiguous and overlapping regions using a parametric smoothing function. One can again determine the resultant local volatilities from the theoreti- cal relation between smooth differentiable implieds and their derivatives. Here is an example. n Let {}xi, yi i = 1 represent the discrete implied volatility data for a given expi- ration, where xi is the moneyness, i.e. strike/spot for each option, and yi is the corresponding implied volatility. The aim is to find a smoothed regression yi = mx()i + εi Eq.8.1 where mx() is a smooth function with second derivatives. and εi is the error. We then estimate mx() by writing n mx()= ∑ win, ()x yi Eq.8.2 Copyright Emanuel Derman 2008 i = 1 where win, are n weight functions that sum to 1, and each win, peaks around the corresponding moneyness xi so as to give higher weight to volatilities yi 4/14/08 smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 3 of 19 closer to the moneyness xi that corresponds to a particular yi . Any argument x n in the function mx() gets a contribution from all {}xi, yi i = 1 , with the great- est contribution coming from those xi closest to x . As an example, we can choose Kh()xx– i win, ()x = ------------------n ---------------- Eq.8.3 ∑ Kh()xx– i i = 1 where Kh()u is a function that peaks around zero with a degree of peaking determined by h . One example is the Gaussian with standard deviation h, 1 1 –u2 ⁄ 2h2 Kh()u = ------------- e Eq.8.4 h 2π a function which integrates to 1. Small h produces greater localization of the smoothing, As h → 0 , all smoothing vanishes and the function is defined only at the observed moneyness values. The greater the number of observed implied volatilities n , the greater the density of information, and so the more information there is in a small region around the moneyness x , and so one can choose a smaller h and still obtain smoothing. One can show that this Nadaraya-Watson estimator for mx() converges to the true regression function as h → 0 and n → ∞ with their product kept finite. One can also show that minimizing the weighted squares of the differences between the observed volatilities and the estimated volatilities, where the weights are given by Equation 8.4, leads to the solution Equation 8.3. Fengler discuss how to choose the Kh()u so as to minimize the bias between the true regression and the smoothed estimator while avoiding the oversmoothing that makes the estimator function follow every wiggle in the data. Fengler’s Chapter 4 provides much more information on this method. Copyright Emanuel Derman 2008 4/14/08 smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 4 of 19 8.2 Trinomial Trees with Constant Volatility Trinomial trees provide another discrete representation of stock price move- ment, analogous to binomial trees1. Their advantage is a greater flexibility in the description of the implied stochastic process for the stock price in discrete steps, so that one can avoid arbitrage violations more easily. Both trinomial and binomial trees are simple discrete methods of solving the partial differential equation for the options valuation model. An initial refer- ence on trinomial trees is the paper by Derman, Kani and Chriss, Implied Tri- nomial Trees of the Volatility Smile, Journal of Derivatives, 3(4) (1996) pp 7- 22; a version of this is on my web site, and the appendix of that paper has describes the construction and calibration of trinomial trees. Some of the notes below are taken from there. Other references are the book by Clewlow and Strickland, and the book by Espen Haug. Rebonato’s book also has some mate- rial on this. Binomial and trinomial trees are merely special instances of more general methods of solving partial differential equations, some of which may be much more efficient. Wilmott has a thorough and more general discussion of these methods. We want to model the risk-neutral process dS σ2 ------ = rdt + σdZ or dSln = ⎛⎞r – ------ dt + σdZ. S ⎝⎠2 Figure 8.1 below illustrates a single time step in a trinomial tree. The stock price at the beginning of the time step is S. During this time step the stock price can move to one of three nodes: with probability p to the up node, value Su; with probability q to the down node, value Sd; and with probability 1 – p – q to the middle node, value Sm. At the end of the time step, there are five unknown parameters: the two probabilities p and q, and the three node prices Su, Sm and Sd. There are two conditions – on the mean and the variance of the process – that must be satisfied in order for the tree to represent geometric Brownian motion in the continuum limit. First, for a risk-neutral trinomial tree, as in the binomial case, the expected value of the stock at the end of the period must be its forward price FSe= ()Δr – δ t , where δ is the dividend yield. Therefore: Copyright Emanuel Derman 2008 1.Both trinomial and binomial trees approach the same continuous time theory as the number of periods in each is allowed to grow without limit. Nevertheless, one kind of tree may sometimes be more conve- nient than another when you are working in discrete time, before you reach the continuous limit. 4/14/08 smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 5 of 19 FIGURE 8.1. In a single time step of a trinomial tree the stock price can move to one of three possible future values, each with its respective probability. The three transition probabilities sum to one. o Su p 1-p-q S o o Sm q o Sd pSu ++qSd ()1 – p – q Sm = F Eq.8.5 Second, if the stock price volatility during this time period is σ, then the node prices and transition probabilities must produce the appropriate variance, so that 2 2 2 2 2 2 p()Su – F ++qS()d – F ()1 – p – q ()Sm – F = S σ ΔtO+ ()Δt Eq.8.6 2 where O()Δt denotes terms of higher order than Δt which vanish more rap- idly as we approach the continuum limit. Different discretizations of risk-neutral trinomial trees have different higher order terms in Equation 8.6. They all become negligible in the continuum limit. Because there are two constraints on five parameters in the tree, one has much more flexibility in building the tree. In contrast, in the binomial case, the mean and variance conditions determined the location of the nodes and the risk-neutral probability with no flexibility in avoiding arbitrage violations. Figure 8.2 below illustrates two methods of combining binomial trees to produce a trinomial tree. Because trinomial trees are more general there are more ways to build them. Figure 8.3 illustrates a trinomial tree for the lnS that’s chosen to be more sym- metric. Because of the symmetry, we have to solve only for ε and q in order to match the mean and variance of lnSS⁄ 0 over time Δt. To make the tree even Copyright Emanuel Derman 2008 σ2 simpler, we choose mr= ⎛⎞– ------ Δt so that the central node always coincides ⎝⎠2 4/14/08 smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 6 of 19 FIGURE 8.2. Two equivalent methods for building constant volatility trinomial trees with spacing Δτ. (a) Combining two steps of a CRR binomial tree with a spacing of Δτ/2. (b) Combining two steps of a JR binomial tree with spacing Δτ/2. (a) Combining two steps of (b) Combining two steps of a Cox-Ross-Rubinstein binomial tree a Jarrow-Rudd binomial tree σ 2Δt ()Δr – σ2 ⁄ 2 t + σ 2Δt Su = Se Su = Se S = m S 2 Sm = Se()Δr – σ ⁄ 2 t Sd = Se–σ 2Δt ()Δr – σ2 ⁄ 2 t–2σ Δt 2 Sd = Se ⎛⎞erΔt ⁄ 2 – e–σΔt ⁄ 2 p = ⎜⎟----------------------------------------------- ⎝⎠σΔt ⁄ 2 –σΔt ⁄ 2 e – e p = 1/4 2 ⎛⎞eσΔt ⁄ 2 – erΔt ⁄ 2 q = ⎜⎟----------------------------------------------- q = 1/4 ⎝⎠eσΔt ⁄ 2 – e–σΔt ⁄ 2 Su Su p p S Sm Sm S q q S Sd d FIGURE 8.3.

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