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8 Hedging to a bandwidth: The model of Whalley & Wilmott (1993) and Henrotte (1993)

8 Hedging to a bandwidth: The model of Whalley & Wilmott (1993) and Henrotte (1993)

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risk, as measured by the variance over a time step δt of this imperfectly hedged position is, to

leading order,

σ 2S2 D −



∂V

∂S



2



δt.



I can make two observations about this expression. The first is simply to confirm that when

D = ∂V /∂S this variance is zero. The second observation is that a natural hedging strategy is

to bound the variance within a given tolerance and that this strategy is equivalent to restricting

D so that

∂V

(48.9)

σS D −

≤ H0 .

∂S

The parameter H0 is now a measure of the maximum expected risk in the portfolio. When

the perfect hedge (∂V /∂S) and the current hedge (D) move out of line so that (48.9) is violated,

then the position should be rebalanced. Equation (48.9) defines the bandwidth of the hedging

position.

The model of Whalley & Wilmott (1993) and Henrotte (1993) takes this as the hedging

strategy: The investor prescribes H0 and on rehedging rebalances to D = ∂V /∂S.

We find that the option value satisfies the non-linear diffusion equation





1

2

2

2 4 2

H

∂V

∂V

∂V

σ S

k1 + (k2 + k3 S) 0  ,

(48.10)

+ 12 σ 2 S 2 2 + rS

− rV =

∂t

∂S

∂S

H0

S

where is the option’s gamma and the parameters k1 , k2 and k3 are the cost parameters for

the cost structure introduced in Section 48.7. Note that again there is a non-linear correction to

the Black–Scholes equation that depends on the gamma.



48.9 UTILITY-BASED MODELS

48.9.1



The Model of Hodges & Neuberger (1989)



All of the above models for transaction costs take the hedging strategy as exogenously given.

That is, the investor chooses his strategy and then prices his option afterwards. Strategies like

this have been called local-in-time because they only worry about the state of an option at the

present moment in time. An alternative, first examined by Hodges & Neuberger (1989), is to

find a strategy that is in some sense optimal. These have been called global-in-time models

because they are concerned with what may happen over the rest of the life of the option.

The seminal work in this area, combining both utility theory and transaction costs, was by

Hodges & Neuberger (HN), with Davis, Panas & Zariphopoulou (DPZ) making improvements

to the underlying philosophy. HN explain that they assume that a financial agent holds a

portfolio that is already optimal in some sense but then has the opportunity to issue an option

and hedge the risk using the underlying. However, since rehedging is costly, they must define

their strategy in terms of a ‘loss function.’ They thus aim to maximize expected utility. This

entails the investor specifying a ‘utility function.’ The case considered in most detail by HN and

DPZ is of the exponential utility function. This has the nice property of constant risk aversion.

Mathematically, such a problem is one of stochastic control and the differential equations

involved are very similar to the Black–Scholes equation.



transaction costs Chapter 48



1.2



1

Black-Scholes

delta



Delta



0.8

Edges of

hedging

bandwidth



0.6



0.4



0.2



0



S



Figure 48.6 The optimal hedging strategy with proportional costs.



48.9.2



The Model of Davis, Panas & Zariphopoulou (1993)



The ideas of HN were modified by DPZ. Instead of valuing an option on its own, they embed

the option valuation problem within a more general portfolio management approach. They then

consider the effect on a portfolio of adding the constraint that at a certain date, expiry, the

portfolio has an element of obligation due to the option contract. They introduce the investor’s

utility function, in particular, they assume it to be exponential. They only consider costs proportional to the value of the transaction (κνS), in which case they find that the optimal hedging

strategy is not to rehedge until the position moves out of line by a certain amount. Then, the

position is rehedged as little as possible to keep the delta at the edge of this hedging bandwidth. This result is shown schematically in Figure 48.6. Here we see the Black–Scholes delta

position and the hedging bandwidth.

In HN and DPZ the value of the option and, most importantly, the hedging strategy are given

in terms of the solution of a three-dimensional free boundary problem. The variables in the

problem are asset price S, time t, as always, and also D, the number of shares held in the

hedged portfolio.

48.9.3



The Asymptotic Analysis of Whalley & Wilmott (1993)



The models of HN and DPZ are unwieldy because they are time consuming to compute. As

such it is difficult to gain any insight into the optimal hedging strategy. Whalley & Wilmott

did an asymptotic analysis of the DPZ model assuming that transaction costs are small, which

is, of course, the case in practice. This analysis shows that the option price is given by the

solution of an inhomogeneous diffusion equation, similar to the Black–Scholes equation.



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This asymptotic analysis also shows that the HN optimal hedging bandwidth is symmetric

about the Black–Scholes delta so that

∂V

D−



∂S



1/3



3k3 Se−r(T −t) F (S, t, )2





,



where

F (S, t, ) =







e−r(T −t) (µ − r)

.

γ S2σ 2



The parameter γ is the index of risk aversion in the utility function.

These results are important in that they bring together all the local-in-time models mentioned above and the global-in-time models of HN and DPZ into the same diffusion equation

framework.

This hedging bandwidth has been tested using Monte Carlo simulations by Mohamed (1994)

and found to be the most successful strategy that he tested. The model has been extended by

Whalley & Wilmott (1994) to an arbitrary cost structure, which is described below.

48.9.4



Arbitrary Cost Structure



The above description concentrates on the proportional cost case. If there is a fixed cost component then shares are traded to position the number of shares to be at some optimal rebalance

point. This is illustrated schematically in Figure 48.7.

1.2



1

Black–Scholes

delta

0.8

Optimal rebalance points

Delta



796



Edges of hedging

bandwidth



0.6



0.4



0.2



0



S



Figure 48.7 The optimal hedging strategy with arbitrary cost structure.



transaction costs Chapter 48



I do not give any of the details but note that the algorithm for finding the optimal rebalance

point and the hedging bandwidth is as follows.

Assume that costs take the form K(S, ν), and that this is symmetric for buying and selling.

The bandwidth is given by

− A(S, t) ≤ D ≤

where



+ A(S, t)



is the Black–Scholes delta. The optimal rebalance points are given by

D=



± B(S, t).



A and B come from solving

γ AB(A + B) = 3e−r(T −t)



2 ∂K



∂ν



ν=A−B



and

γ (A + B)3 (A − B) = 12e−r(T −t)

where



2



K(S, A − B),



is the Black–Scholes gamma and γ is the index of risk aversion.



48.10 INTERPRETATION OF THE MODELS

Non-linear and inhomogeneous diffusion equations appear throughout the physical science literature. Thus there is a ready-made source of theoretical results and insights. I describe some

of these in this section.

48.10.1



Nonlinearity



The effect of the nonlinearity on the valuation equations is that

the sum of two or more solutions is not necessarily a solution

itself. As I have said, a portfolio consisting of an equal number

of the same options but held long and short (which has value

identically zero), is not equal to the sum of the values of the

two sub-portfolios of all the long and short options. This makes

sense because in valuing each sub-portfolio separately we are

assuming that each would be hedged separately, with attendant transaction costs to be taken

into account. Upon recombining the two, the intrinsic values cancel, but the two sets of costs

remain, giving a negative net value. The importance of nonlinearity extends far beyond this

however.

Consider the following. Transaction cost models are non linear. The value of a portfolio

of options is generally not the same as the sum of the values of the individual components.

We can add contracts to our portfolio by paying the market prices, but the marginal value of

these contracts may be greater or less than the amount that we pay for them. Is it possible to

optimize the value of our portfolio by adding traded contracts until we give our portfolio its best

value? This question is answered (in the affirmative) in Chapter 60. In a sense, the optimization

amounts to finding the cheapest way to reduce the gamma of the portfolio globally, since the

costs of hedging are directly related to the gamma.



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48.10.2



Negative Option Prices



The transaction cost models above can result in negative option prices for some asset values depending on the hedging strategy implied by the model. So for example in the Hoggard–Whalley–Wilmott model with fixed transaction costs, k1 > 0, option prices can become

negative if they are sufficiently far out of the money. This model assumes that we rehedge at

the end of every time step, irrespective of the level of risk associated with our position and

also irrespective of the option value. Thus there is some element of obligation in our position,

and the strategy should be amended so that we do not rehedge if this would make the option

value go negative. In the case of a call therefore, there may be an asset price below which we

would cease to rehedge and in this case we would regard the option as worthless.

Note that this is not equivalent to discarding the option; if the asset price were subsequently

to rise above the appropriate level (which will change over time), we would begin to hedge

again and the option would once more have a positive value. So we introduce the additional

conditions for a moving boundary:

V (Ss (t), t) = 0

and

∂V

(Ss (t), t) = 0.

∂S

The value Ss (t) is to be found as part of the solution. This problem is now a ‘free boundary

problem,’ similar mathematically to the American option valuation problem. In our transaction

cost problem for a call option, we must find the boundary, Ss (t), below which we stop hedging.

This is illustrated in Figure 48.8.

In this figure we are valuing a long vanilla call with fixed costs at each rehedge. The top

curve is the Black–Scholes option value as a function of S at some time before expiry. The

bottom curve allows for the cost of rehedging but with the obligation to hedge at each time

step. The option value is thus negative far out of the money. The middle curve also incorporates

costs but without the obligation to rehedge. It thus always has a positive value and is, of course,

also below the Black–Scholes option value.

48.10.3



Existence of Solutions



Linear diffusion equations have many nice properties, as we discussed in Chapter 6. The solution

to a ‘sensible’ problem exists and is unique. This need not be the case for non-linear equations.

The form of the equation and the final data V (S, T ) for the equation, (the payoff at expiry), may

result in the solution ‘blowing up,’ that is, becoming infinite and thus financially unrealistic.

This can occur in some models even if transaction costs are small because of the effect of the

option’s gamma, which in those models is raised to some power greater than one in the extra

transaction cost term. So wherever the gamma is large this term can dominate. For example,

near the exercise price for a vanilla call or put option, ∂ 2V /∂S 2 (S, 0) is infinite. We consider

the case for the model of Equation (48.10).

The governing equation in this case has a transaction cost term proportional to 2 . Close to

expiry and near the exercise price, E, we write t = T − τ and S = E + s where |s|/E

1



transaction costs Chapter 48



10



8



V



6



4



2



0

0



5



10



15



20



25



30



35



−2



S



Figure 48.8 When to stop hedging if there are fixed costs.



and then the equation can be approximated by

∂V



∂τ



∂ 2V

∂s 2



2



,



(48.11)



where

β=



σ 2E4

H0



1



k1 + H02



k2

+ k3

E



.



(48.12)



Taking H0 to be a constant, which is equivalent to a fixed bandwidth for the delta, it can be

shown that equation (48.11) is ill-posed, that it is has no solution, if (S, 0) > 0.2

So a long vanilla call or put hedged under this strategy has no finite value. Note that for

short vanilla options, (S, 0) < 0 and a solution does exist, so they can be valued under such

a strategy, and can be hedged with a constant level of risk throughout the life of the option.

However, returning to the case of payoffs with positive gamma, as the option approaches expiry,

the number of hedging transactions required, and hence the cost of maintaining the hedging

strategy, increases unboundedly unless the level of risk allowed (H0 ) is itself allowed to become

unbounded.

2 We can see this intuitively as follows. If (S, 0) > 0 then the right-hand side of (48.11) is positive. Thus V increases

in time, increasing fastest where the gamma is largest. This in turn further increases the gamma, making the growth in

V even faster. The result is a blow up. The linear diffusion equation also behaves in this way but the increase in V does

not get out of control, since the gamma is raised only to the power one.



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48.11 NON-NORMAL RETURNS

We know that returns are not Normally distributed but is this important?

Suppose that returns are given by

δS

= µ δt + σ ψ δt 1/2

S

for some random variable ψ of empirically determined distribution. What matters as far as

expected transaction costs are concerned is not the mean of ψ, nor its standard deviation. What

is most important is the mean of the absolute value of ψ i.e. the average value of |ψ|. We can

examine the data to see if this number is greater or less than the theory says, the ratio to the

theoretical value giving us a transaction cost factor.

In Table 48.1 are given the transaction cost factors for a selection of stocks, scaled with time

step and volatility so that all numbers would be one if the underlying distribution were Normal.

You can see that they are all less than one, but not by an enormous amount. So costs are going

to be slightly less important than you might think. As we saw in Chapter 47 electricity prices

stand out as being the furthest from the theory.

Table 48.1 Transaction

cost factors.

Asset

Asahi Breweries

Asda Group

Cable & Wireless

Std Chartered

Equifax

Fleetwood Ents

Ford

Nepool

Sumitomo Bank

Toshiba



Factor

0.93

0.93

0.91

0.68

0.88

0.92

0.96

0.50

0.88

0.88



48.12 EMPIRICAL TESTING

In this section we look at empirical results for transaction costs and hedging error using various

hedging strategies described above. We will look at the following four strategies:

• Basic Black–Scholes strategy, delta hedging at fixed intervals

• Leland volatility-modified delta, hedging at fixed intervals

• Delta tolerance, hedging to the Black–Scholes delta when the difference between quantity

of the underlying held and ideal delta move too far out of line

• The asymptotic version of the utility model

We will use stock price data that is generated randomly, with known and constant volatility,

and we will also use real data. Many stock path realizations will be used so that we can examine

the statistical properties of the total costs and hedging errors.



transaction costs Chapter 48



Finally, we examine











Average total transaction costs

Average price (i.e. Black–Scholes value plus average costs)

Standard deviation of price

Price of 95th percentile



The third of these includes the hedging error that would be present even if there were no

costs at all. The last simply means the price at which the contract must be sold to ensure that

95% of the time we do not lose money.

When we come to look at real data we also examine for each stock price time series which

of the four strategies is the winner. Here ‘winner’ means the strategy that gives the lowest total

cost plus hedging error for that particular realized asset path.

To understand how the random simulations were done, see Chapter 80.

To start with we value and hedge an at-the-money call with a volatility of 20%, risk-free rate

of 5%. The underlying is currently at 100 and there is one year to expiry. The values for k1 ,

k2 and k3 are all 0.01.

48.12.1



Black–Scholes and Leland Hedging



This is a straight fight between using the Black–Scholes delta or the Leland volatility-adjusted

delta.

Figure 48.9 shows the average amount of costs incurred and the standard deviation of the

option price when the time between hedging varies. For each hedging period 5000 simulations

were used. The number of days between rehedging varied from 1 up to 25.

Not surprisingly, as the time gap increases the amount of transaction costs paid decreases.

The figure also shows the standard deviation of the price. This includes the hedging error

that would still be present in the absence of costs. It is very clear that the level of risk increases

5

Average transaction costs (BS)

Standard deviation of price (BS)

Average transaction costs (Leland)

Standard deviation of price (Leland)



4.5



Average costs



4

3.5

3

2.5

2

1.5

1

0.5

0

0



5



10

15

Days between rehedging



20



25



Figure 48.9 Average costs and standard deviation of price under the Black–Scholes and Leland

strategies.



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20

18

16

14

12

Price



802



10

8



Average price (BS)

95th percentile price (BS)

Average price (Leland)

95th percentile price (Leland)



6

4

2

0

0



5



10



15



20



25



Days between rehedging



Figure 48.10

strategies.



Average price and 95th percentile of price under the Black–Scholes and Leland



when the number of trades decreases. The kinks in the graph between 13 and 20 days reflect

the fact that as the number of days increases, the gap between the last trade and the penultimate

trade sometimes decreases, or sometimes increases, which affects the level of risk taken on.

Figure 48.10 shows the average total price of hedging the option. This is a straight average

of the 5000 prices accumulated from the simulations. This graph suggests that the fewer trades

the better. The Leland strategy produces a lower price than the Black–Scholes.

Figure 48.10 also shows the 95th percentile price. This is the price we must charge for the

option for us to make a profit 95% of the time. This is the most informative picture from a risk

management point of view.

Frequent hedging is good for risk control but bad for pricing, infrequent hedging is the

opposite. The 95th percentile price is a compromise between taking on risk and incurring

transaction costs. With this as our option ‘value’ the figure shows that the Leland model

outperforms the Black–Scholes method.

The optimal number of days between rehedging for the Leland method was 4, giving a 95th

percentile of 14.90.

Now we repeat all of this analysis and plotting for the delta-tolerance strategy.

48.12.2



Market Movement or Delta-tolerance Strategy



In this model the risk in the hedged position is restricted by the parameter H0 , see

Equation (48.9).

Figure 48.11 shows the number of trades required on average as H0 varies.

The next figure, 48.12, shows how much the strategy affects the average total transaction

costs. Observe that instead of plotting the costs against the bandwidth I have plotted costs against



transaction costs Chapter 48



60



Average number of trades



50



40



30



20



10



0

0



5



10

Bandwidth



15



20



Figure 48.11 Average number of trades for the delta-tolerance strategy.



6

Average transaction costs (DT)

Standard deviation of price (DT)



5



Costs and SD



4



3



2



1



0

0



25



50



75



100



125



150



175



200



'Days between rehedging'



Figure 48.12 Average costs and standard deviation of price for the delta-tolerance strategy.



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25



20



15

Price



804



10



Average price (DT)

95th percentile price (DT)



5



0

0



Figure 48.13



25



50



75

100

125

'Days between rehedging'



150



175



200



Average price and 95th percentile of price for the delta-tolerance strategy.



the inverse of the average number of trades. This is not quite the same as average number of

days between rehedges, hence the inverted commas. As before restricting the number of trades

restricts the amount of costs, but increases the standard deviation of price, the risk.

The average contract price is shown in Figure 48.13 along with the 95th percentile of price. As

in the Leland strategy, there is a compromise point. With the 95th percentile ‘value’ determining

this point we find that we get an option ‘value’ of 15.15. This is slightly worse than the Leland

strategy but better than normal Black–Scholes hedging.

48.12.3



The Utility Strategy



In this strategy the parameter to be varied is γ , the level of risk aversion.

Figure 48.14 shows the average number of trades versus the risk aversion parameter. Again,

this is used to convert from risk aversion to a measure of the number of days between rehedges,

so that all plots can be better compared across strategies.

From the 95th percentile plot the best option ‘value’ is 15.03. Thus this method turned out

to be better than the delta-tolerance method, but still not as good as the Leland fixed-time

step hedging strategy. However over the whole range of values considered for γ this utility

method produced a far lower 95th percentile price than the ranges produced from the other

strategies; the 95th percentile seems to be quite insensitive to the risk aversion parameter. See

Figures 48.15 and 48.16.

There are a couple of points to note about the use of the utility strategy. First, we have only

used the asymptotic version since the computational time necessary for the solution of the full

partial differential equation would be prohibitively large. Second, we looked at a fairly general

cost model, not just proportional costs. The addition of the extra transaction costs, fixed and



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