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Unformatted text preview: E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 1 of 19 Lecture 8: Local Volatility Models: Implications Copyright Emanuel Derman 2008 • 4/14/08 Practical calibration of local volatility models • Implied trinomial trees • Implications: The deltas of standard options. The values of exotic options: barriers, lookbacks, etc. smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 2 of 19 Copyright Emanuel Derman 2008 8.1 Practical Calibration of Local Vol Models In practice we are given implied volatilities Σ ( K i, T i ) 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 theoretical relation between smooth differentiable implieds and their derivatives. Here is an example. Let { x i, y i } n i=1 represent the discrete implied volatility data for a given expi- ration, where x i is the moneyness, i.e. strike/spot for each option, and y i is the corresponding implied volatility. The aim is to find a smoothed regression yi = m ( xi ) + εi Eq.8.1 where m ( x ) is a smooth function with second derivatives. and ε i is the error. We then estimate m ( x ) by writing n m(x) = ∑ w i, n ( x ) y i Eq.8.2 i=1 where w i, n are n weight functions that sum to 1, and each w i, n peaks around the corresponding moneyness x i so as to give higher weight to volatilities y i 4/14/08 smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 3 of 19 closer to the moneyness x i that corresponds to a particular y i . Any argument x Copyright Emanuel Derman 2008 in the function m ( x ) gets a contribution from all { x i, y i } 4/14/08 n i = 1, with the great- est contribution coming from those x i closest to x . As an example, we can choose Kh ( x – xi ) w i, n ( x ) = --------------------------------n Eq.8.3 ∑ Kh ( x – xi ) i=1 where K h ( 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 ⁄ 2 h2 K h ( u ) = -- ---------- e h 2π Eq.8.4 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 m ( x ) 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 K h ( 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. smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 4 of 19 Copyright Emanuel Derman 2008 8.2 Trinomial Trees with Constant Volatility Trinomial trees provide another discrete representation of stock price movement, 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 reference on trinomial trees is the paper by Derman, Kani and Chriss, Implied Trinomial Trees of the Volatility Smile, Journal of Derivatives, 3(4) (1996) pp 722; 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 material 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 ----- = rdt + σ dZ S 2 or σ d ln S = ⎛ r – -----⎞ dt + σ dZ . ⎝ 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 F = Se ( r – δ )Δ t , where δ is the dividend yield. Therefore: 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 convenient 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 Copyright Emanuel Derman 2008 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. S 1-p-q o Su o Sm o p o Sd q p S u + qS d + ( 1 – p – q ) S m = 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 p ( S u – F ) 2 + q ( S d – F ) 2 + ( 1 – p – q ) ( S m – F ) 2 = S σ 2 Δ t + O ( Δ t ) Eq.8.6 2 where O ( Δ t ) denotes terms of higher order than Δt which vanish more rapidly 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 ln S that’s chosen to be more symmetric. Because of the symmetry, we have to solve only for ε and q in order to match the mean and variance of ln S ⁄ S 0 over time Δt. To make the tree even 2 σ simpler, we choose m = ⎛ r – -----⎞ Δ t 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 Copyright Emanuel Derman 2008 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. 4/14/08 (b) Combining two steps of a Jarrow-Rudd binomial tree (a) Combining two steps of a Cox-Ross-Rubinstein binomial tree Su = Se σ Sm = S Sd = Se – σ Su = Se ( r – σ 2 ⁄ 2 )Δ t + σ 2Δt Sm = Se ( r – σ 2 ⁄ 2 )Δ t 2Δt ⎛ e r Δ t ⁄ 2 – e –σ Δ t ⁄ 2 ⎞ p = ⎜ ---------------------------------------------- ⎟ ⎝ e σ Δ t ⁄ 2 – e – σ Δ t ⁄ 2⎠ Sd = S e ( r – σ 2 ⎛ eσ Δ t ⁄ 2 – erΔt ⁄ 2 ⎞ q = ⎜ ---------------------------------------------- ⎟ ⎝ e σ Δ t ⁄ 2 – e – σ Δ t ⁄ 2⎠ p q 2 ⁄ 2 )Δ t – σ 2 Δ t p = 1/4 2 q = 1/4 Su Su S 2Δt Sm S p Sm q Sd Sd FIGURE 8.3. 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. We draw the log of the stock price here. ln(S/S0) o (1-q)/2 0 q o m+ε o m o m-ε (1-q)/2 Δt smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 7 of 19 Copyright Emanuel Derman 2008 with the expected value of ln S ⁄ S 0 and we also choose the probabilities to be symmetric about the center. 4/14/08 The expected value of the log term is then exactly m, since the probabilities are symmetric. To get the variance of returns right we must have 2 2 ( 1 – q )ε ≈ σ Δ t or Δt ε = σ ----------1–q Eq.8.7 It’s often convenient to choose q = 2/3. Then the multiplicative factors for the stock become 2 M=e σ ⎛ r – -----⎞ Δ t ⎝ 2⎠ U = Me D = Me σ 3Δt Eq.8.8 –σ 3 Δ t This is accurate only to O(Δt), but in the limit as the spacing goes to zero, higher order terms become negligible. Figure 8.4 illustrates a risk-neutral trinomial tree with constant volatility. smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 8 of 19 FIGURE 8.4. Example of a risk-neutral trinomial tree with constant volatility Copyright Emanuel Derman 2008 Risk-neutral trinomial tree with constant volatility r continuous 0.1 0.1 0.1 f 1.0101 1.0101 1.0101 dt 0.1000 0.1000 0.1000 sig 0.2000 0.2000 0.2000 m 1.0080 1.0080 1.0080 u 1.1247 1.1247 1.1247 d 0.9034 0.9034 0.9034 stock 100.0000 112.4732 100.8032 90.3441 126.5021 113.3766 101.6129 91.0697 81.6206 142.2810 127.5182 114.2872 102.4290 91.8012 82.2761 73.7394 112.4732 100.8032 90.3441 126.5021 113.3766 101.6129 91.0697 81.6206 28.4823 15.5599 5.6828 1.1223 0.0885 0.1 1.0101 0.1000 0.2000 1.0080 1.1247 0.9034 0.1 1.0101 0.1000 0.2000 1.0080 exp(r-sig^2/2)dt with prob 2/3 1.1247 m*exp(sig*sqrt( 3 dt)) with prob 1/6 0.9034 m*exp(-sig*sqrt( 3 dt)) with prob 1/6 160.0279 143.4238 128.5425 115.2052 103.2518 92.5386 82.9370 74.3316 66.6192 142.2810 127.5182 114.2872 102.4290 91.8012 82.2761 73.7394 43.2760 28.5132 15.2822 4.6552 0.5366 0.0000 0.0000 0.1 1.0101 0.1000 0.2000 1.0080 1.1247 0.9034 pv of stock 100.0000 strike 100.0000 call option 7.1968 4/14/08 15.9075 6.5036 1.6931 60.0279 43.4238 28.5425 C = [ 1 / 6 ( u p ) + 2 / 3 ( m i d d l e ) + 1 / 6 ( d n ) ] / f 15.2052 3.2518 0.0000 0.0000 0.0000 0.0000 smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 9 of 19 Copyright Emanuel Derman 2008 8.3 Trinomial Trees with Local Volatility σ(S,t) 4/14/08 In dealing with binomial local volatility trees, we discovered that for finite Δ t calibrating a binomial tree to a variable local volatility sometimes lead to negative probabilities or violations of the no-arbitrage principle. For trinomial trees, we will show that the calibration to local volatilities can be done by adjusting the probabilities after the stock price nodes are chosen independently, thereby more easily avoiding negative probabilities. o S 1-p-q o o Sm o p Su Sd q Δt In the figure at right, the conditions to satisfy are pS u + ( 1 – p – q ) S m + qS d = F 2 2 2 22 p ( Su – F ) + ( 1 – p – q ) ( Sm – F ) + q ( Sd – F ) ≈ S σ Δ t Eq.8.9 To make life easier, we choose S m ≡ F , so the middle node always coincides with the forward. Then the equations above simplifies to pS u + qS d = ( p + q ) F 2 2 22 p ( Su – F ) + q ( Sd – F ) ≈ S σ Δ t Eq.8.10 Given the nodes Su and Sd, we can solve for p and q: 22 S σ Δt p = ----------------------------------------( Su – F ) ( Su – Sd ) 22 S σ Δt q = ----------------------------------------( F – Sd ) ( Su – Sd ) Eq.8.11 We can therefore choose a grid of stock prices in the future that allows us to determine p’s and q’s that lie strictly between 0 and 1 and still match the correct forward and variance. Below are two examples of trees built with different grids and that lead to different probabilities p and q on the tree, but will nevertheless produce the same options prices in the limit as Δ t → 0 . We can first choose the grid and then determine the probabilities. In the example below we choose stock prices that lie on an initial grid formed simply by using a CRR stock price generators. smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 10 of 19 Copyright Emanuel Derman 2008 We choose U = exp ( σ g 2 Δ t ) and D = exp ( – σ g 2 Δ t ) , Note that volatility 4/14/08 σ g (the generator volatility) used to generate the grid is not the true local volatility, but just some constant (convenient, fake, approximately representative local) volatility used to generate the lattice of prices. Here below is a risk-neutral trinomial tree with local volatility σ ( S ) = 0.1 + ( S ⁄ 100 – 1 ) built on a lattice generated with a 15% CRR volatility of prices. Risk-neutral trinomial tree with constant volatility r continuous 0 f 1.0000 dt 0.0100 local sig a+b(S/100 - a 0.1000 b For generation of initial lattice vol generator 0.1500 U 1.0214 UD=M^2 to close tree D 0.9790 M 1.0000 stock state space 100.0000 p_up 102.1440 100.0000 97.9010 104.3339 102.1440 100.0000 97.9010 95.8461 106.5708 104.3339 102.1440 100.0000 97.9010 95.8461 93.8343 1.0000 σ ( S ) = 0.1 + ( S ⁄ 100 – 1 ) U = exp ( σ 2 Δ t ) = exp ( 0.15 0.02 ) = 1.0214 108.8557 106.5708 104.3339 102.1440 100.0000 97.9010 95.8461 93.8343 91.8647 S^2*sig^2*dt/((S_u - F)(S_u - S_d)) 0.1099 q_dn 0.1621 0.1099 0.0686 0.2259 0.1621 0.1099 0.0686 0.0376 Jarrow-Rudd generated lattice with 15% volatility 1-p-q 0.3019 0.2259 0.1621 0.1099 0.0686 0.0376 0.0162 0.7778 0.6723 0.7778 0.8613 0.5434 0.6723 0.7778 0.8613 0.9241 0.3898 0.5434 0.6723 0.7778 0.8613 0.9241 0.9673 S^2*sig^2*dt/((F-S_d)(S_u - S_d)) 0.1123 0.1656 0.1123 0.0701 0.2307 0.1656 0.1123 0.0701 0.0384 0.3083 0.2307 0.1656 0.1123 0.0701 0.0384 0.0165 2.4103 0.7004 0.0645 0.0011 0.0000 4.5708 2.3339 0.4752 0.0158 0.0000 0.0000 0.0000 strike 102.0000 call option 0.1955 0.8723 0.1273 0.0054 6.8557 4.5708 2.3339 0.1440 0.0000 0.0000 0.0000 0.0000 0.0000 discounted call value for strike 102 smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 11 of 19 Copyright Emanuel Derman 2008 Below is another risk-neutral trinomial tree built on 20% vol-generating lattice with local volatility = 0.15 - 0.1(S/100 - 1) and time steps of one year, just as an example. Of course, for accurate convergence to the continuous time limit, one needs much smaller time steps. 4/14/08 Risk-neutral trinomial tree r continuous 0.04879 f 1.0500 dt 1.0000 local sig a+b(S/100 - a 0.1500 b For generation of initial lattice vol generator 0.2000 U 1.3932 UD=M^2 to close tree D 0.7913 M 1.0500 stock state space 100.0000 p_up 139.3241 105.0000 79.1320 194.1121 146.2903 110.2500 83.0886 62.6188 270.4449 203.8176 153.6048 115.7624 87.2430 65.7497 49.5515 -0.1000 σ ( S ) = 0.15 – 0.1 ( S ⁄ 100 – 1 ) 376.7949 283.9671 214.0085 161.2850 121.5505 91.6051 69.0371 52.0290 39.2111 S^2*sig^2*dt/((S_u - F)(S_u - S_d)) 0.1089 q_dn 0.0593 0.1018 0.1413 0.0151 0.0521 0.0945 0.1348 0.1699 0.1445 0.0787 0.1350 0.1875 0.0201 0.0691 0.1254 0.1789 0.2255 101.5950 53.8486 20.0519 4.0986 0.3864 strike 102.0000 call option 20.5266 51.4482 20.3209 5.3875 0.7466 0.8620 0.7632 0.6712 0.9648 0.8789 0.7800 0.6862 0.6046 0.9953 0.9760 0.8953 0.7971 0.7017 0.6177 0.5475 adjust probabilities 0.0027 0.0137 0.0597 0.1157 0.1701 0.2180 0.2581 173.3020 106.6748 56.4619 19.7653 2.3873 0.0000 0.0000 Jarrow-Rudd generated lattice with 20% volatility 1-p-q 0.0020 0.0103 0.0450 0.0872 0.1282 0.1643 0.1945 S^2*sig^2*dt/((F-S_d)(S_u - S_d)) 104.999983 274.7949 181.9671 112.0085 59.2850 19.5505 0.0000 0.0000 0.0000 0.0000 discounted call value for strike 102 is 20.5266 smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 12 of 19 Copyright Emanuel Derman 2008 Here is one more risk-neutral trinomial tree built on a 13% vol-generating lattice: stock prices are different, probabilities are different, but options prices are about the same, and will be identical as Δ t → 0 4/14/08 Risk-neutral trinomial tree r continuous 0.04879 f 1.0500 dt 1.0000 local sig a+b(S/100 - a 0.1500 b For generation of initial lattice vol generator 0.1300 U 1.2619 UD=M^2 to close tree D 0.8737 M 1.0500 stock state space 100.0000 p_up 126.1924 105.0000 87.3665 159.2453 132.5020 110.2500 91.7349 76.3291 200.9555 167.2075 139.1271 115.7624 96.3216 80.1456 66.6861 -0.1000 σ ( S ) = 0.15 – 0.1 ( S ⁄ 100 – 1 ) 253.5906 211.0032 175.5679 146.0834 121.5505 101.1376 84.1528 70.0204 58.2614 S^2*sig^2*dt/((S_u - F)(S_u - S_d)) 0.2735 q_dn 0.1863 0.2555 0.3215 0.1001 0.1678 0.2374 0.3044 0.3666 0.3286 0.2239 0.3071 0.3863 0.1203 0.2017 0.2853 0.3659 0.4406 66.7283 40.0265 19.5063 7.1391 1.8658 strike 102.0000 call option 20.2896 38.4824 19.9541 8.6451 0.3979 0.5898 0.4374 0.2922 0.7796 0.6306 0.4774 0.3297 0.1929 0.9356 0.8166 0.6710 0.5178 0.3680 0.2280 0.1008 adjust probabilities 0.0351 0.1001 0.1796 0.2632 0.3450 0.4214 0.4908 103.8126 70.0647 41.9842 18.8357 5.3443 0.0000 0.0000 Jarrow-Rudd generated lattice with 13% volatility 1-p-q 0.0292 0.0833 0.1494 0.2190 0.2870 0.3506 0.4084 S^2*sig^2*dt/((F-S_d)(S_u - S_d)) 104.999983 151.5906 109.0032 73.5679 44.0834 19.5505 0.0000 0.0000 0.0000 0.0000 discounted call value for strike is 20.2896 smile-lecture8.fm Copyright Emanuel Derman 2008 E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications 4/14/08 Page 13 of 19 Finally, here is a trinomial tree built on a 5% volatility-generating lattice This generating volatility is too small to properly match or represent the much larger local volatilities generated by the formula, and so the nodes are not far enough apart to give probabilities that lie between 0 and 1.This is an illustration of a lattice that doesn’t work. But, because of the greater number of degrees of freedom in building trinomial trees, one can always find a lattice that doesn’t violate the no-arbitrage principle. Risk-neutral trinomial tree r continuous 0.04879 f 1.0500 dt 1.0000 local sig a+b(S/100 - a 0.1500 b For generation of initial lattice vol generator 0.0500 U 1.1269 UD=M^2 to close tree D 0.9783 M 1.0500 stock state space 100.0000 p_up 112.6934 105.0000 97.8318 126.9980 118.3281 110.2500 102.7234 95.7106 143.1184 133.3479 124.2444 115.7624 107.8595 100.4961 93.6354 -0.1000 σ ( S ) = 0.15 – 0.1 ( S ⁄ 100 – 161.2850 150.2743 140.0153 130.4566 121.5505 113.2525 105.5209 98.3171 91.6051 S^2*sig^2*dt/((S_u - F)(S_u - S_d)) 1.9679 q_dn 1.6489 1.8389 2.0252 1.3232 1.5164 1.7081 1.8971 2.0820 2.1121 1.7697 1.9736 2.1736 1.4202 1.6275 1.8333 2.0361 2.2346 1.0723 1.2773 1.4845 1.6915 1.8965 2.0981 2.2951 34.4810 25.8110 17.7329 24.4765 2.6732 45.9755 36.2050 27.1016 18.6196 10.7166 10.7124 7.1706 strike 102.0000 call option -152.2057 24.5819 43.7111 -34.8299 Jarrow-Rudd generated with 5% volatility 1-p-q 0.9991 1.1901 1.3831 1.5760 1.7671 1.9549 2.1384 S^2*sig^2*dt/((F-S_d)(S_u - S_d)) 104.999983 -3.0799 -2.4186 -2.8125 -3.1987 -1.7 43 -2.1 43 -2.5 41 -2.9 33 -3.3 16 adjust probabilities: ARBITRAGE VIOLATIONS 59.2850 48.2743 38.0153 28.4566 19.5505 11.2525 3.5209 0.0000 0.0000 discounted call valu is -152.2057 NONSENSE smile-lecture8.fm Copyright Emanuel Derman 2008 E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 14 of 19 Thus, we have more flexibility in building trinomial trees; we can choose a lattices of stock prices that don’t violate arbitrage, and then adjust the probabilities to match the stochastic process, provided the lattice was reasonable. In contrast, with binomial trees, we were forced to a definite lattice which sometimes violated the no-arbitrage conditions. 8.4 Deltas and Exotics in Local Volatility Models 8.4.1 Four rules of thumb for local volatilities in the small slope at-the-money approximation: Rule of Thumb 1: The Rule of 2: Local volatility varies with market level about twice as rapidly as implied volatility varies with strike. implied volatility local volatility vol strike Comment: In equity markets with negative skew, the implied volatility for all strikes and maturities decrease as the market level increases. Rule of Thumb 2: Relation between sensitivity of implied volatility to spot and strike. The change in implied volatility of a given option for a change in market level is about the same as the change in implied volatility for a change in strike level. negative skew index level up lower volatility subtree down current later higher volatility subtree time Σ ( S, K ) ≈ σ 0 – β ( S + K ) + 2 β S 0 4/14/08 smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 15 of 19 Copyright Emanuel Derman 2008 Hedge ratios of standard options in the presence of (negative) skew are therefore smaller than Black-Scholes hedge ratios. 4/14/08 Rule of Thumb 3: The correct exposure Δ of an option is approximately given by the chain rule formula Δ = Δ BS + V BS × β Eq.8.12 For example, a one-year S&P option with a B-S hedge ratio of 60% probably has a true hedge ratio of 50%, because volatility moves down as the market moves up. Suppose S = 1000. VBS = 400 dollars; β = -0.0002 vol point per strike pt.: V BS β ∼ 0.1 Black-Scholes tree Cu C Cd implied tree C'u constant volatility subtrees C subtree C'd subtree Rule 4. For short times to expiration, the inverse of the implied volatility for a given strike is the harmonic average of the local volatilities across ln(S) from spot to strike. smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 16 of 19 Copyright Emanuel Derman 2008 8.4.2 Theoretical Value of Barrier Options in Local Vol Models 4/14/08 In this section we illustrate the effect of local volatility models on exotic options, taking barrier options as an example. Barrier options values are especially sensitive to the risk-neutral probability of index remaining in the region between the strike and the barrier, and hence to the local volatility in this region. strike and barrier, which depends on the skew:. Here we are going to calculate their value in local volatility models and try to gain some intuition about them. Example 1: An Up-and-Out Call. with Strike 100 and Barrier 110 In the lecture on static hedging, we showed that you can approximately replicate a down and out call by means of a long position in the call itself combined with a short position in a put whose strike is (logarithmically) reflected through the barrier. In a flat-volatility world, the value of both of these calls is determined by the constant Black-Scholes volatility. In a skewed world, however, each call has an implied volatility which is approximately the average of the local volatilities between spot and strike. For an option with strike at 100 and barrier at 110, the reflected strike is approximately at 120. Thus, in a local volatility model, the approximate value of the Black-Scholes implied volatility for the up-and-out call is the average of the local volatilities between 100 and 120.In the figure below, the local volatility varies between 0.1 and 0.07 in this range, with an average of a about 0.085. The value of the down and out option in the local volatility model is about 1.1, which corresponds to a Black-Scholes implied volatility of about 0.09, so this intuition about averaging works reasonably. smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 17 of 19 Copyright Emanuel Derman 2008 . LOCAL VOLATILITY AS A FUNCTION OF SPOT            " reflected strike strike ! *+ ,!++ ++      +++  - +   ' .+ +/, 0 1 23 $ &$ barrier     #        " $ % $ " $  $ '$    $ VALUE OF UP-AND-OUT CALL AS A FUNCTION OF IMPLIED VOLATILITY (   4/14/08 The call value in the local volatility model has an implied BS volatility which is about half the local volatility at strike and reflected strike, that is 1/2(0.1 + \0.7) = 0.85     "' ) " smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 18 of 19 4/14/08 In some cases, the local volatilities can produce options values that cannot be matched by any Black-Scholes implied volatility. No amount of intuition can get you the exactly correct value. Consider the case below, with a spot and strike at 100, and the barrier at 130, and the skew as shown in Figure 8.5. FIGURE 8.5. A hypothetical volatility skew for options of any expiration. We assume a constant riskless discount rate of 5% and a zero dividend yield. The arrows show the strike (100) and barrier (130) level of the upand-out option under consideration. 25.00 implied volatility Copyright Emanuel Derman 2008 Example 1. An Up-and Out Call that has no Black-Scholes Implied Volatility 20.00 15.00 10.00 5.00 0.00 50.00 100.00 150.00 200.00 strike (% of spot) We can value the Up-and-Out Call by building an implied tree calibrated to this skew. The resultant value of the barrier option in this local volatility model is 6.46. What Black-Scholes volatility does this call price correspond to? No skew: Up-and-Out call value as a function of Black-Scholes Implied Volatility smile-lecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 19 of 19 Copyright Emanuel Derman 2008 The maximum Black-Scholes value in a no-skew world is 6.00 corresponding to a 9.5% implied volatility. This value is smaller than the “correct” value in the local volatility model. There is NO Black-Scholes implied volatility which gives the local-volatility “correct” option value. The implied volatility that comes closest to it is about 10%. We can understand this as follows.The slope of the skew is 1 vol pt. per 10 strike points. The rule of 2 then indicates that the slope of the local volatilities will be about 1 vol pt. per 5 strike points. Now, we showed in the previous lecture that you can think of an up-and-out option with strike 100 and barrier 130 as being replicated by an ordinary call with strike 100 and a reflected call with strike 160. Therefore, the local volatility that is relevant to valuation ranges between spot prices 100 and 160 with a slope of approximately 1 vol pt. per 5 strike points, that is from values of 15% to 15 – ( 60 ⁄ 5 ) = 3% . The average local volatility in this range is about 9%, which substantiates the approximate claim the implied volatility is the average of the local volatilities between spot and strike. Local volatility models have analogous effects on the values of other exotic options, moving their values away from Black-Scholes values. Lookback calls (that pay out the final value of the index less the minimum value of the index between inception and expiration), for example, have higher deltas in a local volatility model than they do in Black-Scholes.1 1. Derman, Kamal, Zou: The Local Volatility Surface. 4/14/08 smile-lecture8.fm ...
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