There are two common approaches to pricing options in the presence of variable volatilities or interest rates: the first consists of calibrating a trinomial tree using observed market prices of options and then using the same tree to consistently price other, unquoted options (Derman et al., 1996). The second approach uses the Fokker Planck equation (9.3) to extract a continuous expression for the volatility as a function of strike and time to maturity. The results can then be used in a variety of types of computation (trees, Monte Carlo, finite differences). We now examine the first approach.
(i) The procedure is best explained with a concrete example, and we will try to make this as simple as possible. Assume we have a stock with priceS0=100, interest rater =8% and continuous dividendq =3%. Prices of call options quoted in the market are as given in Table 9.1.
Table 9.1 Quoted European call option prices ($)
Strike: 1 month 4 months 7 months
80 20.28 21.30 22.43
90 10.48 12.56 14.31
100 2.62 5.69 7.86
110 0.12 1.83 3.64
120 0 0.36 1.32
130 0 0.04 0.37
We set ourselves the objective of building a tree to price 6-month options. For the reasons given in the last section, this is best achieved by means of a trinomial tree. We choose a three-step tree, so that the length of each step is 2 months. We further choose the spacing as e where =0.25×√
3δt =0.1786 or e =1.1934. This is in line with the spacings suggested for a tree with variable volatility in the last section. The tree is set out in Figure 9.8;
to make the tree fully functional, we need to find out the transition probabilities at all the nodes.
(ii)State Prices in a Trinomial Tree:We now apply the analysis of Section 9.3 to a trinomial tree.
The state priceλni is the value at node zero of an Arrow Debreu security which pays out $1 if (and only if ) the node at timenand stock price leveliis reached. It was shown that the node zero value of a European option maturing at time stepncan be written f0 =
iλni fin, i.e. the payoff multiplied by the state price, summed over all final nodes.
Let us now imagine that we know the market prices of a put or a call option for any strikeX, maturing in 6 months. We can write the market prices of these options asC6 mandP6 m; their
9.5 DERMAN KANI IMPLIED TREES
6 m
S = 100
04m
?+1 = .2555
4m
?+2 = .0175
S
+1= 119.34
S
+2= 142.41
S
+3= 169.95
S = 70.22
-2S = 58.84
-3.0455
−
4 m
2 .0175
?? ?
4 m
1 .2555
?? ?
0 2 m 4m
4m
?+1 = .2555
4m
?+2 = .0175
=
S = 83.80
-16 m
l−2 = .1965
6 m
l−1 = .2736 l+6 m1 =
.0434 l+26 m=
.0008 l+36 m=
.3958
6 m
l0 =
−6 m3 .0051 l
4 m
2 .0175 l+ =
4 m
1 .2555 l+ =
Figure 9.8 Trinomial tree
payoffs in 6 months are of course, max[S6 m−X,0] and max[X−S6 m,0]. Referring back to the trinomial tree of the last subsection, we can write
C142.416 m =
i
λ6 mi max[S6 m−142.41,0]
The payoffs at all the nodes below the top right-hand one are zero, so this last equation reduces to
C6 m142.41 =λ6 m+3(169.95−142.41)
The next state price down in the final column can be obtained from the analogous equation C1196 m.34 =λ6 m+3(169.95−119.34)+λ6 m+2(142.41−119.34)
Since we already know the top state priceλ6 m+3 from the previous equation, we can obtainλ6 m+2; and so on down the final column of nodes. We will need one additional relationship to calculate the final node down. This is provided by the normalizing relationshipiλ6 mi =e−8%×6 months: remember that the state prices are really probabilities of reaching final nodes multiplied by discount factors, and the probabilities sum to unity.
119
9 Variable Volatility
We have very simply derived the final state prices from observed market prices, and can therefore price any 6-month option whose payoff depends only on the price in 6 months; all this without reference to volatilities or transition probabilities!
(iii)Interpolations:The last subsection begs a huge question: where do we get market prices of call options for the precise strikes needed in the trinomial tree? The only practical way of obtaining something useful is by a process of interpolation, starting with the observed market prices quoted in the table.
Table 9.2 Implied vols of quoted calls (%) Strike: 1 month 4 months 7 months
80 25.10 23.71 23.06
90 23.19 22.89 22.46
100 21.02 21.34 21.55
110 19.23 20.42 20.85
120 17.87 19.24 19.87
130 17.04 18.38 19.07
As was previously explained, interpolation between option prices is best carried out via an interpolation between implied volatilities, since these move slowly with changes in strike price or maturity. The call option prices of Table 9.1 translate into the implied volatilities of Table 9.2 if we apply the Black Scholes formula. In the real world, we would at this point be finessing the data to make sure that there are no obvious anomalies, outliers or mistakes.
Interpolations and extrapolations have to be made in two directions: with respect to time and with respect to stock price. The technique most commonly used in practice is the cubic spline which is described in Appendix A.11, but here we use simple linear interpolation.
Our objective is first to obtain implied volatilities and hence option prices, for maturities and stock prices corresponding to the nodes of the trinomial tree. This is done in two steps, first interpolating the rows for maturities 2, 4, and 6 months; then the new columns are interpolated for the specific strikes equal to stock values at the nodes. Finally the interpolated implied volatilities are turned back into “market prices”, with the desired strikes and maturities. See Table 9.3.
Table 9.3 Interpolations
Interpolated Implied Vols (%) Interpolated “Market Prices” ($) Strike: 2 months 4 months 6 months 2 months 4 months 6 months
58.84 — — 24.72 — — 41.98
70.22 26.15 24.51 23.95 30.21 30.65 31.11
83.80 24.05 23.40 23.02 16.91 17.82 18.75
100.00 21.13 21.34 21.48 3.84 5.69 7.19
119.34 18.41 19.32 19.73 0.04 0.40 1.04
142.41 16.44 17.31 17.82 0 0 0.02
169.95 — — — — — —
9.5 DERMAN KANI IMPLIED TREES
(iv) Returning to the theme of subsection (ii) above, we have derived the state prices for each final node of the tree and can therefore price any European option. Obviously, if we were trying to price a European call or put, it would be simpler just to interpolate as we did to find the
“market prices”. However, there are European options other than puts and calls which may need to be priced. And then there is the matter of American options, which cannot be priced without a knowledge of all the intermediate probabilities in the tree. These are obtained from a knowledge of all the state prices in the tree. The first step therefore is to repeat the calculations of subsection (ii) for each column of nodes in the tree, using the interpolated “market prices”
of the call options at each node. The results are given in Table 9.4.
Table 9.4 State prices in trinomial tree Node Level λ00 λ2 monthsi λi4 months λ6 monthsi
58.84 0.0008
70.22 0.0175 0.0434
83.80 0.1984 0.2555 0.2736
100.00 1 0.6088 0.4763 0.3958
119.34 0.1796 0.1950 0.1965
142.41 0.0293 0.0455
169.95 0.0051
6m
0
S+2= 142.41
4m
?+2 = .0175
S+1= 119.34
4m
?+1 = .2555
4m
(pu)2 4m
(pm)2 4m
(pd)2 4m
(pu)1
4m
(pd)1 4m
(pu)0 4m
(pm)1
S
+3= 169.95S+2= 142.41
4m
?+2 = .0175
S+1= 119.34
4m
(pu)2 4m
(pm)2 4m
(pd)2 4m
(pu)1
4m
(pd)1 4m
(pu)0 4m
(pm)1
06 m .3958 l =
6 m +1 .2736 l =
+26 m .0434 l =
6 m +3 .0008 l =
4 m 2 .0175
?? ?
l4 m=
4m
?+2 = .0175
4m
(pu)2 4m
(pm)2 4m
(pd)2 4m
(pu)1
4m
(pd)1 4m
(pu)0 4m1
(pm)
4 m .0175 l+2 =
.2555
+1
Figure 9.9 Detail from trinomial tree (v)Transition Probabilities:Just as we found a
simple iterative process for calculating state prices from call option prices, so we can derive the transition probabilities from the state prices. This is illustrated in Figure 9.9, which is a snapshot of the top right-hand corner of the trinomial tree of Figure 9.8.
The calculation proceeds in a recursive, two- step process which alternately uses the for- ward and backward induction described in Section 9.3(v):
(A1) Forward induction connects state prices at successive time steps, and is given explicitly for the binomial tree by equation (9.5). A precisely analogous relationship holds for the trinomial tree: taking the very top branch in the diagram, the formula for λ6 m3 has the simple form
λ6 m3 =e−rδt(pu)4 m2 λ4 m2
where (pu)4 m2 , (pm)4 m2 and (pd)4 m2 are the transition probabilities in 4 months; e−rδt =0.9868 is the one-period discount factor. This leads immediately to (pu)4 m2 =0.0439 for the top 2- month probability.
(B1) Backward induction (risk neutrality) gives
S2e(r−q)δt=(pu)4 m2 S3+(pm)4 m2 S2+(pd)4 m2 S1
121
9 Variable Volatility
e−(r−q)δt =0.9917 and probabilities sum to unity: (pu)4 m2 +(pm)4 m2 +(pd)4 m2 =1. Using the result we found for (pu)4 m2 , these last two equations may be solved to give (pm)4 m2 =0.9555 and (pd)4 m2 =0.0007.
(A2) Forward induction for thesecondstate price down in the last column is λ6 m2 =e−rδt
(pm)4 m2 λ4 m2 +(pu)41mλ4 m1 We already know (pm)4 m2 so we can calculate (pu)4 m1 =0.1066.
(B2) Backward induction applied to the second cell down gives S1e(r−q)δt=(pu)4 m1 S2+(pm)4 m1 S1+(pd)4 m1 S0 And so on . . . .
The process is continued for the entire column of cells and the same method is used for all columns in the tree, finally yielding the values of all probabilities given in Figure 9.10.
.0439
.1466 .0553 .6106
.9555
.8179 .1434
.7372
.7059
.7074 .1357 .0007 .1066 .0755
.1570 .1883
.1195 .1576
.1365 .0896
.8551
.7300
.7181 .1298 .1233 .1521 .2011
Figure 9.10 Transition probabilities
(vi)Use of Put Options:For most practical purposes, call options are easier to work with than puts. For one thing, American and European calls are usually the same price so that we can use market data on American traded options to build our trees; this is not true of put options.
However, from the iterative methods of calculating state prices (and hence probabilities) in subsections (ii) and (v), it is clear that any errors or anomalies in the market price of a call option are transmitted to the calculated probabilities at all the lower nodes. If European put option