8 Hedging to a bandwidth: The model of Whalley & Wilmott (1993) and Henrotte (1993)
Tải bản đầy đủ - 0trang
794
Part Five advanced topics
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.
795
Part Five advanced topics
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
2γ
,
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.
797
798
Part Five advanced topics
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
=β
∂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.
799
800
Part Five advanced topics
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.
801
Part Five advanced topics
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.
803
Part Five advanced topics
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