. Ukull and A. White, “The Pricing of Options on Asscts with Stochastic Volatilities,” Journal of Finaiice,42 (June 1987), 281-300. This result is independent of the process followed by the variance rate.T See J. C. Hull and A. White, “An Analysis of the Bias in Option Pricing Caused by a Stochastic Volatility,”Advances in Futures and Options Research, 3 (1988), 27-61; 5. L. Heston, “A Closed Form Solution for Options withStochastic Volatility with Applications to Bonds and Currency Options,” Review of Financial Studies, 6, no. 2 (1993),327-43.
5 The reason is given in footnote 2.
> See J.-C. Duan, “The GARCH Option Pricing Model,” Mathematical Finance, 5 (1995), 13-32; and J.-C. Duan,“Cracking the Smile,” R/SK, December 1996, pp. 55-59.460 CHAPTER 20
assuming first a constant volatility model and then a stochastic volatility model. Delta hedgingworks less well when the volatility is stochastic. This emphasizes the point made in Chapter 14 thattraders should monitor vega as well as delta and gamma.
20.4 THE IVF MODEL
 
A financial institution likes to use models for pricing exotic options that have the property thatthey -price all plain vanilla options correctly.'° The models we have looked at so far can becalibrated on any day so that they give a reasonably good fit to the prices of vanilla options,but they do not provide an exact fit. In 1994, Derman and Kani, Dupire, and Rubinstein developedwhat has become known as the implied volatility function (IVF) model or the implied tree model."This is a model that is designed to fit exactly the prices observed today for al! European optionprices, regardless of the shape of the volatility surface.The risk-neutral process for the asset price in the model has the form
dS = [r(t) — q(t)|S dt + o(S, tS dz
where r(t) is the instantaneous forward interest rate for a contract maturing at time t and q(t) is
the dividend yield as a function of time. The volatility a(S, t) is a function of both $ and ¢ and is
chosen so that the model prices all European options consistently with the market. It is shown
both by Dupire and by Andersen and Brotherton-Ratcliffe that o(S,£) can be calculated
analytically:'
Cmke/ OF + GT) Cmir + KET) — GF) 8e mir /AKK?(8 Cnt /OK*)
 
[o(K, T)? =2 (20.4)
where Cut(K, 7) is the market price of a European call option with strike price K and maturity T.If a sufficiently large number of European call prices are available in the market, this equation canbe used to estimate the o(S, f) function.'?
Andersen and Brotherton-Ratcliffe implement the model by using equation (20.4) in conjunc-tion with the implicit finite difference method. An alternative approach, known as the implied treemethodology, is suggested by Derman and Kani and by Rubinstein. This involves constructing atree for the asset price that is consistent with option prices in the market. We will now describe theDerman-Kani version of the implied tree approach.
An implied tree is a binomial! tree where a forward induction procedure is used to determine thepositions of nodes at the end of each time step and the probabilities on branches. As in the case ofthe regular binomial tree, we branch from the jth node at time (n — 1) dt to either the (+ 1)th
© There is a practical reason for this. If the bank does not use a model with this property, there is a danger thattraders working for the bank will spend their time arbitraging the bank’s internal models.
| See B. Dupire, “Pricing with a Smile,” RISK, February 1994, pp. 18-20; E. Derman and I. Kani, “Riding onaSmile,” RISK, February 1994, pp. 32-39; M. Rubinstein, “Implied Binomial Trees,” Journal of Finance, 49, no. 3(uly 1994), 771-818.
"2 See B. Dupire, “Pricing with a Smile,” RISK, February 1994, pp. 18-20; L. B. G Andersen and R. Brotherton-Ratcliffe, “The Equity Option Volatility Smile: An Implicit Finite Difference Approach,” Journal of ComputationalFinance, 1, no. 2 (Winter 1997/98), 5-37. Dupire considers the case where r and q are zero; Andersen and Brotherton-Ratcliffe consider the more general situation.
3 Some smoothing of the observed volatility surface is typically necessary.More on Models and Numerical Procedures 461
node or the jth node at time 7 ét. It is constructed using forward induction. To understand theapproach, note that there are n + | nodes at time nét.'4 Assume that the tree has already beenconstructed up to time (7 — 1)ét. The next step involves:
{. Choosing the positions of the m+ 1 nodes at time n dt
2. Choosing the n ‘“‘up”’ probabilities on the branches between times (m — 1)éd¢ and nét. (The“down” probabilities are |] minus the “up” probabilities.)
These choices provide 2n + 1 degrees of freedom. The interest rate for the pcriod between (n — 1) étand n 4t is set equal to the forward rate. The expected return from the asset at each of the nodes attime (n — 1) 6t must equal this interest rate. This uses up n degrees of freedom. The tree is alsoconstructed to ensure that » European-style options maturing at time nét are priced correctly.These options have strike prices equal to the stock prices at the nodes at time (n — 1) t.'° This usesup an additional n degrees of freedom. ‘he tinal degree of freedom is used up in ensuring that thecenter of the tree equals today’s stock price.
The requirements just mentioned lead to 2n + 1 equations in 2n + 1 unknowns. By solving theequations, the construction of the tree is advanced by one time step. One problem with theapproach is that negative probabilities do sometimes arise. When a particular probability turnsout to be negative, it is necessary to introduce a rule to override the option price responsible for thenegative probability.
When it is used in practice the LVF model is recalibrated daily to the prices of plain vanillaoptions. It is a tool to price exotic options consistently with plain vanilla options, As discussed inChapter 15, plain vanilla options define the risk-neutral probability distribution of the asset priceat all future times. It follows that the IVF model gets the risk-neutral! probability distribution of theasset price at all future times correct. This means that options providing payoffs at just one time(eg., all-or-nothing and asset-or-nothing options) are priced correctly by the IVF model. However,the model does not necessarily get the joint distribution of the asset price at two or more timescorrect. This means that exotic options such as compound options and barrier options may bepriced incorrectly. Hull and Suo (2001) tested the [VF model by assuming that all derivative pricesare determined by a stochastic volatility model.'® They found that the model works reasonably wellfor compound options, but sometimes gives serious errors for barrier options.
20.5 PATH-DEPENDENT DERIVATIVES
 
 
A path-dependent derivative (or history-dependent derivative) is a derivative where the payoffdepends on the path followed by the price of the underlying asset, and not just on its final value.Asian options and lookback options are examples of path-dependent derivatives. As explained inChapter 19, the payoff from an Asian option depends on the average price of the underlying asset;
™ The brief description of the implied tree methodology here is based on the work of E. Derman and I. Kanipublished in RISK in 1994.
'S Th practice, it is necessary to interpolate between the implied volatilities of actively traded options to determineuuplied vyolaiilities for the options used in ihe tree consiruciion. These implied volatifities are then couveried iniooption prices using Black-Scholes.
'6 See J.C. Hull and W. Suo, “A Methodology for the Assessment of Model Risk and Its Application to the ImpliedVolatility Function Model,” Journal of Financial and Quantitative Analysis, 37, no. 2 (June 2002).462 CHAPTER 20
 
the payoff from a lookback option depends on its maximum or minimum price. One approach tovaluing path-dependent options when analytic results are not available is Monte Carlo simulationas discussed in Chapter 18. A sample value of the derivative can be calculated by sampling arandom path for the underlying asset in a risk-neutral world, calculating the payoff, anddiscounting the payoff at the risk-free interest rate. An estimate of the value of the derivative isfound by obtaining many sample values of the derivative in this way and calculating their mean.
The main problem with using Monte Carlo simulation to value path-dependent derivatives isthat the computation time necessary to achieve the required level of accuracy can be unacceptablyhigh. Also, American-style path-dependent derivatives (i.c., path-dependent derivatives where oneside has exercise opportunities or other decisions to make) cannot easily be handled. In thissection, we show how the binomial tree methods presented in Chapter 18 can be extended to copewith some path-dependent derivatives.'? The procedure can handle American-style path-dependentderivatives and is computationally more efficient than Monte Carlo simulation for European-stylepath-dependent derivatives.
For the procedure to work, two conditions must be satisfied:
1. The payoff from the derivative must depend on a single function, F, of the path followed bythe underlying asset.
2. It must be possible to calculate the value of F at time t + é¢ from the value of F at time t andthe value of the underlying asset at time t+ 6t.
ilustration Using Lookback Options
As a first illustration of the procedure, we consider an American lookback put option on a non-dividend-paying stock.'® If exercised at time t, this pays off the amount by which the maximumstock price between time zero and time t exceeds the current stock price. We suppose that theinitial stock price is $50, the stock price volatility is 40% per annum, the risk-free interest rate is10% per annum, the total life of the option is three months, and that stock price movements arerepresented by a three-step binomial tree. With our usual notation, this means that Sy = 50,o = 0.4, r= 0.10, dt = 0.08333, uv = 1.1224, d = 0.8909, a = 1.0084, and p = 0.5073.
The tree is shown in Figure 20.L. The top number at each node is the stock price. The next levélof numbers at each node shows the possible maximum stock prices achievable on paths leading tothe node. The final level of numbers shows the values of the derivative corresponding to each of thepossible maximum stock prices. The values of the derivative at the final nodes of the tree arecalculated as the maximum stock price minus the actual stock price.
To illustrate the rollback procedure, suppose that we are at node A, where the stock price is $50.The maximum stock price achieved thus far is either 56.12 or 50. Consider first the situation whereit is equal to 50. If there is an up movement, the maximum stock price becomes 56.12 and the valueof the derivative is zero. If there is a down movement, the maximum stock price stays at 50 and thevalue of the derivative is 5.45. Assuming no early exercise, the value of the derivative at A when themaximum achieved so far is 50 is therefore
(0 x 0.5073 + 5.45 x 0,4927)e7! *9-08353 — 9 66
"7 This approach was suggested by J. Hull and A. White. “Efficient Procedures for Vatuing European and AmericanPath-Dependent Options,” Journal of Derivatives, 1, no. | (Fall 1993), 21-31.
18 This example is used as a first illustration of the general procedure for handling path dependence. We present amore etfticient approach to valuing American-style lookback options in the next section.More on Models and Numerical Procedures 463
70.0070.000.00
56.1262.99 56.126.87 0.00
     
  
56.12 50.006.12 2.6644,5556.12 50,0011.57 5.45
35.3650.0014.64
Figure 20.1 Tree for valuing an American lookback option
Clearly, it is not worth exercising at node A in these circumstances because the payoff from doingso is zero. A similar calculation for the situation where the maximum value at node A is 56.12 givesthe value of the derivative at node A, without early exercise, to be
In this case, early exercise gives a value of 6.12 and is the optimal strategy. Rolling back throughthe tree in the way we have indicated gives the value of the American lookback as $5.47.
Generalization
The approach just described is computationally feasible when the number of alternative values of thepath function, 7, at cach node does not grow too fast as the number of time steps is increased. Theexample we used, a lookback option, presents no problems because the number of alternative valuesfor the maximum asset price at a node in a binomial tree with n time steps is never greater than n.
Luckily, the approach can be extended to cope with situations where there are a very largenumber of different possible values of the path function at each node. The basic idea is as follows.At a node, we carry out calculations for a small number of representative values of F. When thevalue of the derivative is required for other values of the path function, we calculate it from theknown values using interpolation.
The first stage is to work forward through the tree establishing the maximum and minimumvalues of the path function at each node. Assuming the value of the path function at time t+ dtdepends only on the value of the path function at time t and the value of the underlying variable at464 CHAPTER 20
 
time r+ ét, the maximum and minimum values of the path function for the nodes at time t+ dtcan be calculated in a straightforward way from those for the nodes at time t. The second stage isto choose representative values of the path function at each node. There are a number ofapproaches. A simple rule is to choose the representative values as the maximum value, theminimum value, and a number of other values that are equally spaced between the maximumand the minimum. As we roll back through the tree, we value the derivative for each ofrepresentative values of the path function.
We illustrate the nature of the calculation by considering the problem of valuing the average pricecall option in Example 19.2. We examine the case where the payoff depends on the arithmeticaverage stock price. The initial stock price is 50, the strike price is 50, the risk-free interest rate is10%, the stock price volatility is 40%, and the time to maturity is one year. We use a tree with 20 timesteps. The binomial tree parameters are 6¢ = 0.05, u = 1.0936, d= 0.9144, p= 0.5056, and1 — p = 0.4944. The path function is the arithmetic average of the stock price.
Figure 20.2 shows the calculations that are carried out in one small part of the tree. Node X isthe central node at time 0.2 year (at the end of the fourth time step). Nodes Y and Z are the twonodes at time 0.25 year that are reachable from node X. The stock price at node X is 50. Forwardinduction shows that the maximum average stock price that is achievable in reaching node X is53.83. The minimum is 46.65. (We include both the initial and final stock prices when calculatingthe average.) From node X, we branch to one of the two nodes Y and Z. At node Y, the stock priceis 54.68 and the bounds for the average are 47.99 and 57.39. At node Z, the stock price is 45.72 andthe bounds for the average stock price are 43.88 and 52.48.
Suppose that we have chosen the representative values of the average to be four equally spacedvalues at each node. This means thal, at node X, we consider the averages 46.65, 49.04, 51.44, and53.83. At node Y, we consider the averages 47.99, 51.12, 54.26, and 57.39. At node Z, we considerthe averages 43.88, 46.75, 49.61, and 52.48. We assume that backward induction has already beenused to calculate the value of the option for each of the alternative values of the average at nodes Yand Z. The values are shown in Figure 20.2. For example, at node Y when the average is 51.12, thevalue of the option is 8.101.
S= 54.68Average S Option price47.99 7.575$1.12 8.10154.26 8.63557.39 9.178S=50.00Average S. Option price46.65 5.64249.04 5.923 x54.44 6.20653.83 6.492 S=45,72Average S_ Option price43,88 3.430Z 46.75 3.75049.61 4.07952.48 4.416
Figure 20.2 Part of tree for valuing option on the arithmetic averageMore on Models and Numerical Procedures 465
Consider the calculations at node X for the case where the average is 51.44. If the stock pricemoves up to node Y, the new average will be
5 «x 51.44 + 54.68
S = 51.98
The value of the derivative at node Y for this average can be found by interpolating between thevalues when the average is 51.12 and when it is 54.26. It is
(51.98 — 51.12) x 8.635 + (54.26 — 51.98) x 8.10154.26 — 51.12
 
= 8.247
Similarly, if the stock price moves down to node Z, the new average will be
5 x 51.444 45.72
= 50.4!6 9
and by interpolation the value of the derivative is 4.182.The value of the derivative at node X when the average is 51.44 is therefore
(0.5056 x 8.247 + 0.4944 x 4.182)e7°1%0"5 — 6,206
The other values at node X are calculated similarly. Once the values at all nodes at time 0.2 yearshave been calculated, we can move on to the nodes at time 0.15 years.
The value given by the full tree for the option at time zero is 7.17. As the number of time steps andthe number of averages considered at each node are increased, the value of the option converges tothe correct answer. With 60 time steps and 100 averages at each node, the value of the option is 5.58.The analytic approximation for the value of the option calculated in Example 19.2 is 5.62.
A key advantage of the method described here is that it can handle American options. Thecalculations are as we have described them cxcept that we test for early cxcrcise at each node foreach of the alternative values of the path function at the node. (In practice, the early exercisedecision is liable to depend on both the value of the path function and the value of the underlyingasset.) Consider the American version of the average price call considered here. The valuecalculated using the 20-step tree and four averages at each node is 7.77; with 60 time steps and100 averages, the value is 6.17.
. The approach just described can be used in a wide range of different situations. The twoconditions that must be satisfied were listed at the beginning of this section. Efficiency is improvedsomewhat if quadratic rather than linear interpolation is used at each node.
20.6 LOOKBACK OPTIONS
 
 
A number of researchers have suggested an interesting and instructive approach to valuing lookbackoptions.!? To illustrate it, we again consider the American-style lookback put in Figure 20.1. Whenexercised, this provides a payoff equal to the excess of the maximum stock price over the current
'° The approach was proposed by Eric Reiner in a lecture at Berkeley. It is also suggested by S. Babbs, “BinomialValuation of Lookback Options,” Working Paper, Midland Global Markets, 1992; and T. H. F. Cheuk andT. C. F. Vorst, “Lookback Options and the Observation Frequency: A Binomial Approach,’ Working Paper,Erasmus University, Rotterdam.466 CHAPTER 20
stock price. We define G(t) as the maximum stock price achieved up to time t and set
Gi)
Y= sp .
We next use the Cox, Ross, and Rubinstein tree for the stock price to produce a tree for Y. Initially,
Y — 1 because G = S at time zero. If there is an up movement in S during the first time step. both G
= I because G = S at time zero. If there is an up movement in S$ during the first time step, both Gand § increase by a proportional amount w and Y = 1. If there isa down movement in S during thefirst time step, G stays the same, so that Y = 1/d =u. Continuing with these types of arguments,we produce the tree shown in Figure 20.3 for Y. (Note that in this example u = 1.1224, d = 0.8909,a = 1.0084, and p = 0.5073.) The rules defining the geometry of the tree are as follows:
1, When Y = | at time f, it is either u or I at time ¢ + dt.2. When Y =u” at time t for m > 1, it is either u”*! or wu! at time t+ 6t.
An up movement in Y corresponds to a down movement in the stock price and vice versa.Therefore, the probability of an up movement in Y is always 1 — p, and the probability of a downmovement in Y is always p.
We use the tree to value the American lookback option in units of the stock price rather than indollars. In dollars, the payoff from the option is
 
 
SY-S
1.41400.4140
1.2598ry. 2598Ao. 2598
0.2598
11004 1.12241.12240.12241.0000 1.0000
0.1094 0.0834 0.0533 0
Figure 20.3. Efficient procedure for valuing an American-style lookback optionMore on Models and Numerical Procedures 467
 
In stock price units, the payoff from the option, therefore, isY¥-1
We roll back through the tree in the usual way, valuing a derivative that provides this payoff exceptthat we adjust for the differences in the stock price (i.e., the unit of measurement) at the nodes.If f;,; 18 the value of the lookback at the jth node at time i 6¢ and Y; ; is the value of Y at this node,the rollback procedure gives
fi, = max {¥,;—1, e UL = p) fie jnid + Pfisij-14D
when j > I. Note that, in this equation, fj,),;,; is multiplied by d, and f,,; ;-) is multiplied by uw.This takes into account that the stock price at node (i, j) is the unit of measurement. The stockprice at node (i + 1, 7 + 1), which is the unit of measurement for f,4) j,1, 18 d times the stock priceat node (i, j); and the stock price at node (i+ 1, j— 1), which is the unit of measurement forfisi,j-1> iS u times the stock price at node (i, j). Similarly, when j = 0, the rollback procedure gives
fig = max {¥,;-1, eo (1 - PY Fi jid + Pfigi jul}
The calculations for our example are shown in Figure 20.3. The tree estimates the value of theoption at time zero (in stock price units) as 0.1094. This means that the dollar value of the option is0.1094 x 50 = 5.47. This is the same as the value calculated from the tree in Figure 20.1. For agiven number of time steps, the two procedures are equivalent. The advantage of the proceduredescribed here is that it considerably reduces the number of computations.
The value of the option given by the tree in Figure 20.3, when it is European, is 5.26. The exactvalue of the European option, as shown in Example 19.1, is 7.79. The value given by the treeconverges very slowly to this as the number of time steps is increased. For example, with 100, 500,1,000, and 5,000 time steps the values given by the tree for the European option in our exampleare 7.24, 7.54, 7.61, and 7.71.
20.7_ BARRIER OPTIONS
 
 
 
 
In Chapter 19 we presented analytic results for standard barrier options. Here we consider thenumerical procedures that can he used for harrier options when there are no analytic results.
In principle, a barrier option can be valued using the binomial and trinomia! trees discussed inChapter 18. Consider an up-and-out option. We can value this in the same way as a regularoption except that, when we encounter a node above the barrier, we set the value of the optionequal to zero.
Unfortunately, convergence is very slow when this approach is used. A large number of timesteps are required to obtain a reasonably accurate result. The reason for this is that the barrierbeing assumed by the tree is different from the true barrier.”° Define the inner barrier as the barrierformed by nodes just on the inside of the true barrier (i.e., closer to the center of the tree) and theouter barrier as the barrier formed by nodes just outside the true barrier (e., farther away from thecenter of the tree). Figure 20.4 shows the inner and outer barrier for a trinomial tree on the
0 See P. P. Boyle and S. H. Lau, “Bumping Up Against the Barrier with the Binomial Method,” Journal ofDerivatives, 1, no, 4 (Summer 1994), 6-14, for a discussion of this.468 CHAPTER 20
Outer barrier
 
 
 
 
Figure 20.4 Barriers assumed by trinomial trees
assumption that the true barrier is horizontal. Figure 20.5 does the same for a binomial tree. Theusual tree calculations implicitly assume that the outer barrier is the true barrier because thebarrier conditions are first used at nodes on this barrier. When the time step is ét, the vertical
emacing hetween the nadcs is of order ./8t. This means that crrors created by the difference betweenSpacing OCtWecn Uie HOGCS i$ Of OTACT Voi. 1 iS MCAS Uiat CiTOTs CiCatea oy tne aincrence ociwecn
the true barrier and the outer barrier also tend to be of order V/8t.We now present three alternative approaches for overcoming this problem. For all threeapproaches, it turns out to be more efficient to use a trinomial tree rather than a binomial tree.
Positioning Nodes on the Barriers
 
Suppose that there are two horizontal barriers, H; and Hy, with H, > Hy, and that thestock price follows geometric Brownian motion. In a trinomial tree, there are three possiblemovements in the asset’s price at each node: up by a proportional amount uw; stay the same;and down by a proportional amount d, where d = 1/u. We can always choose u so that nodes lie
on both barriers. The condition that must be satisfied by u is
Hy= Hyu®orIn Hy =In A, + Ninu
for some integer N.More on Models and Numerical Procedures 469
When discussing trinomial trees in Section 18.5, the value suggested for u was e’%?*, so thatInu = o-/36¢. In the situation considered here, a good rule is to choose Inu as close as possible tothis value, consistent with the condition given above. This means that we set
Iny =e =ANwhere__ fin A, — In Ay |N = int} ———— + 0.5| aJ3 dt
and int[x] ts the integer part of x. Normally, the trinomial stock price tree is constructed so that thecentral node is the initial stock price. In this case, the stock price at the first node is the initial stockprice. After that, we choose the central node of the tree to be H,u™, where M is the integer thatmakes this quantity as close as possible to the initial stock price, that is,
. in Sy — In H,M= inf Sens + 05]
 
Inu
This leads to a tree of the form shown in Figure 20.6. The probabilities on all branches of the treeare chosen, as usual, to match the first two moments of the stochastic process followed by the assetprice. The approach works well except when the initial asset price is close to a barrier.
     
Outer barrier
True barrier
Inner barrier <—
 
f
Figure 20.5 Barriers assumed by binomial trees470 CHAPTER 20
Barrier 2
 
Barrier 1
Figure 20.6 Tree with nodes lying on each of two barriers
Adjusting for Nodes Not Lying on Barriers
An alternative procedure for coping with barriers is to make no changes to the tree and adjust forthe fact that the barrier is specified incorrectly by the tree.”! The first step is to calculate an innerbarrier and an outer barrier, as described earlier. We then rollback through the tree, calculatingtwo values of the derivative on the nodes that form the inner barrier. The first of these values isobtained by assuming that the inner barrier is correct; the second is obtained by assuming that theouter barrier is correct. A final estimate for the value of the derivative on the inner barrier is thenobtained by interpolating between these two values. Suppose that at time i 5t, the true barrier is 0.2from the inner barrier and 0.6 from the outer barrier. Suppose further that the value of thederivative on the inner barrier is 0 if the inner barrier is assumed to be correct and 1.6 if the outerbarrier is assumed to be correct. The interpolated value on the inner barrier is 0.4. Once we haveobtained a value for the derivative at all nodes on all inner barriers, we can roll back through thetree to obtain the initial value of the derivative in the usual way.For a single horizontal barrier, this approach is equivalent to the following:
1. Calculate the price of the derivative on the assumption that the