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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. smilelecture8.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 illposed
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 smilelecture8.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 NadarayaWatson 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. smilelecture8.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 riskneutral 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 riskneutral 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 smilelecture8.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 1pq 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 riskneutral 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 riskneutral 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 smilelecture8.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 JarrowRudd binomial tree (a) Combining two steps of
a CoxRossRubinstein 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 (1q)/2
0 q o m+ε o m o mε (1q)/2
Δt smilelecture8.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 riskneutral trinomial tree with constant volatility. smilelecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 8 of 19 FIGURE 8.4. Example of a riskneutral trinomial tree with constant volatility Copyright Emanuel Derman 2008 Riskneutral 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(rsig^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 smilelecture8.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 noarbitrage 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 1pq 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. smilelecture8.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 riskneutral trinomial tree with local volatility
σ ( S ) = 0.1 + ( S ⁄ 100 – 1 ) built on a lattice generated with a 15% CRR volatility of prices. Riskneutral 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 JarrowRudd generated lattice
with 15% volatility 1pq 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/((FS_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 smilelecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 11 of 19 Copyright Emanuel Derman 2008 Below is another riskneutral trinomial tree built on 20% volgenerating 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 Riskneutral 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 JarrowRudd generated lattice
with 20% volatility 1pq 0.0020
0.0103
0.0450
0.0872
0.1282
0.1643
0.1945 S^2*sig^2*dt/((FS_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 smilelecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 12 of 19 Copyright Emanuel Derman 2008 Here is one more riskneutral trinomial tree built on a 13% volgenerating 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 Riskneutral 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 JarrowRudd generated lattice
with 13% volatility 1pq 0.0292
0.0833
0.1494
0.2190
0.2870
0.3506
0.4084 S^2*sig^2*dt/((FS_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 smilelecture8.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% volatilitygenerating 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 noarbitrage principle.
Riskneutral 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 JarrowRudd generated
with 5% volatility 1pq 0.9991
1.1901
1.3831
1.5760
1.7671
1.9549
2.1384 S^2*sig^2*dt/((FS_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 smilelecture8.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 noarbitrage conditions. 8.4 Deltas and Exotics in Local Volatility Models
8.4.1 Four rules of thumb for local volatilities in the small
slope atthemoney 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 smilelecture8.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 BlackScholes 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 oneyear S&P option with a BS 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 BlackScholes 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. smilelecture8.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 riskneutral 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 UpandOut 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 flatvolatility world, the value of both of these calls is determined by the constant BlackScholes 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 BlackScholes implied volatility for the upandout 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 BlackScholes implied volatility of about 0.09, so this
intuition about averaging works reasonably. smilelecture8.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 UPANDOUT 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
"' )
" smilelecture8.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 BlackScholes 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 upandout option under consideration.
25.00 implied volatility Copyright Emanuel Derman 2008 Example 1. An Upand Out Call that has no BlackScholes 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 UpandOut 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 BlackScholes volatility does this call price correspond to?
No skew: UpandOut call value as a function of BlackScholes
Implied Volatility smilelecture8.fm E4718 Spring 2008: Derman: Lecture 8:Local Volatility Models: Implications Page 19 of 19 Copyright Emanuel Derman 2008 The maximum BlackScholes value in a noskew 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 BlackScholes implied volatility which
gives the localvolatility “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 upandout
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 BlackScholes 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 BlackScholes.1 1. Derman, Kamal, Zou: The Local Volatility Surface. 4/14/08 smilelecture8.fm ...
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