4 The model of Hoggard, Whalley & Wilmott (1992)
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where this has been evaluated at time t and asset value S. I have not given the details, but this
choice minimizes the risk of the portfolio, as measured by the variance, to leading order. After
a time step δt and rehedging, the number of assets we hold is
∂V
(S + δS, t + δt).
∂S
Note that this is evaluated at the new time and asset price. We can subtract the former from the
latter to ﬁnd the number of assets we have traded to maintain a ‘hedged’ position. The number
of asset traded is therefore
ν=
∂V
∂V
(S + δS, t + δt) −
(S, t).
∂S
∂S
Since the time step and the asset move are both small we can apply Taylor’s theorem to expand
the ﬁrst term on the right-hand side:
∂ 2V
∂V
∂V
∂ 2V
(S + δS, t + δt) =
(S, t) + δS 2 (S, t) + δt
(S, t) + · · · .
∂S
∂S
∂S
∂S∂t
Since δS = σ Sφ δt 1/2 + O(δt), the dominant term is that which is proportional to δS; this term
is O(δt 1/2 ) and the other terms are O(δt). To leading order the number of assets bought or
sold is
ν≈
∂ 2V
∂ 2V
(S,
t)
δS
≈
σ Sφ δt 1/2 .
∂S 2
∂S 2
We don’t know beforehand how many shares will be traded, but we can calculate the expected
number, and hence the expected transaction costs. The expected transaction cost over a time
step is
E [κS|ν|] =
2
∂ 2V
κσ S 2
δt 1/2 ,
π
∂S 2
(48.2)
√
where the factor 2/π is the expected value of |φ|. We can now calculate the expected change
in the value of our portfolio from (48.1), including the usual Black–Scholes terms and also the
new cost term:
E[δ ] =
∂V
∂ 2V
2 ∂ 2V
+ 12 σ 2 S 2 2 − κσ S 2
∂t
∂S
π δt ∂S 2
δt.
(48.3)
Except for the modulus sign, the new, non-Black–Scholes, term in the above is of the same
form as the second S derivative that has appeared before; it is a gamma term, multiplied by
the square of the asset price, multiplied by a constant.
Now assuming that the holder of the option expects to make as much from his portfolio as if he
had put the money in the bank, then we can replace the E[δ ] in (48.3) with r(V − S(∂V /∂S)) δt
transaction costs Chapter 48
as before to yield an equation for the value of the option:
∂V
∂ 2V
2 ∂ 2V
+ 12 σ 2 S 2 2 − κσ S 2
∂t
∂S
π δt ∂S 2
∂V
+ rS
− rV = 0.
∂S
(48.4)
There is a nice ﬁnancial interpretation of the term that is not
present in the usual Black–Scholes equation. The second derivative of the option price with
respect to the asset price, the gamma, = ∂ 2V /∂S 2 , is a measure of the degree of mishedging
of the hedged portfolio. The leading-order component of randomness is proportional to δS
and this has been eliminated by delta-hedging. But this delta hedging leaves behind a small
component of risk proportional to the gamma. The gamma of an option or portfolio of options
is related to the amount of rehedging that is expected to take place at the next rehedge and
hence to the expected transaction costs.
The equation is a non-linear parabolic partial differential equation, one of the ﬁrst such in
ﬁnance. It is obviously also valid for a portfolio of derivative products. This is one of the ﬁrst
times in this book that we distinguish between single options and a portfolio of options. But for
much of the rest of the book this distinction will be important. In the presence of transaction
costs, the value of a portfolio is not the same as the sum of the values of the individual
components. We can best see this by taking a very extreme example.
We have positions in two European call options with the same strike price and the same
expiry date and on the same underlying asset. One of these options is held long and the other
short. Our net position is therefore exactly zero because the two positions exactly cancel each
other out. But suppose that we do not notice the cancelation effect of the two opposite positions
and decide to hedge each of the options separately. Because of transaction costs we lose money
at each rehedge on both options. At expiry the two payoffs will still cancel, but we have a
negative net balance due to the accumulated costs of all the rehedges in the meantime. This
contrasts greatly with our net balance at expiry if we realize beforehand that our positions
cancel. In the latter case we never bother to rehedge, leaving us with no transaction costs and
a net balance of zero at expiry.
Now consider the effect of costs on a single vanilla option held long. We know that
∂ 2V
>0
∂S 2
for a single call or put held long in the absence of transaction costs. Postulate that this is true
for a single call or put when transaction costs are included. If this is the case then we can drop
the modulus sign from (48.4). Using the notation
σˇ 2 = σ 2 − 2κσ
2
.
π δt
(48.5)
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the equation for the value of the option is identical to the Black–Scholes value with the exception that the actual variance σ 2 is replaced by the modiﬁed variance σˇ 2 . Thus our assumption
that ∂ 2V /∂S 2 > 0 is true for a single vanilla option even in the presence of transaction costs.
The modiﬁed volatility will be recognized as the Leland volatility correction mentioned at the
start of this chapter.
For a short call or put option position we simply change all the signs in the above analysis
with the exception of the transaction cost term, which must always be a drain on the portfolio.
We then ﬁnd that the call or put is valued using the new variance
σˆ 2 = σ 2 + 2κσ
2
.
π δt
(48.6)
Again this is the Leland volatility correction.
The results (48.5) and (48.6) show that a long position in a single call or put with costs
incorporated has an apparent volatility that is less than the actual volatility. When the asset
price rises the owner of the option must sell some assets to remain delta hedged. But then the
effect of the bid-offer spread on the underlying is to reduce the price at which the asset is sold.
The effective increase in the asset price is therefore less than the actual increase, being seen as
a reduced volatility. The converse is true for a short option position.
The above volatility adjustments are applicable when you have an option or a portfolio of
options having a gamma of one sign. If the gamma is always and everywhere positive use the
lower volatility value for a long position, and the higher value for a short position. If gamma
is always and everywhere negative, swap these values around.
For a single vanilla call or put, we can get some idea of the total transaction costs associated
with the above strategy by examining the difference between the value of an option with the
cost-modiﬁed volatility and that with the usual volatility; that is, the difference between the
Black–Scholes value and the value of the option taking into account the costs. Consider
V (S, t) − Vˆ (S, t),
where the hatted function is Black–Scholes with the modiﬁed volatility. Expanding this expression for small κ we ﬁnd that it becomes
(σ − σˆ )
∂V
+ ···.
∂σ
This is proportional to the vega of the option. We know the formula for a European call option
and therefore we ﬁnd the expected spread to be
√
2κSN (d1 ) (T − t)
,
√
2π δt
where N (d1 ) has its usual meaning.
The most important quantity appearing in the model is
K=
κ
.
σ δt 1/2
(48.7)
transaction costs Chapter 48
K is a non-dimensional quantity, meaning that it takes the same value whatever units are
used for measuring the parameters. If this parameter is very large, we write K
1, then the
transaction costs term is much greater than the underlying volatility. This means that costs are
high and that the chosen δt is too small. The portfolio is being rehedged too frequently. If the
transaction costs
√ are very large or the portfolio is rehedged very often then it is possible to
have κ > σ / 8 π δt. In this case the equation becomes forward parabolic for a long option
position. Since we are still prescribing ﬁnal data, the equation is ill-posed. Although the asset
price may have risen, its effective value due to the addition of the costs will have actually
dropped. I discuss such ill posedness later.
If the parameter K is very small, we write K
1, then the costs have only a small effect on
the option value. Hence δt could be decreased to minimize risk. The portfolio is being rehedged
too infrequently.
We can see how to use this result in practice if we have data for the bid-offer spread, volatility
and time between rehedges for a variety of stocks. Plot the parameter κ/σ against the quantity
1/δt 1/2 for each stock. An example of this for a real portfolio is shown in Figure 48.1. In this
ﬁgure are also shown lines on which K is constant. To be consistent in our attitude towards
transaction costs across all stocks we might decide that a value of K = K is ideal. If this
0.025
A
C
0.02
B
0.015
G
H
κ/σ
I
D
E
F
J
K
L
0.01
N
O
Q
M
P
R
0.005
1/(dt)1/2
0
0
2
4
6
8
10
12
14
Figure 48.1 The parameter κ/σ against 1/δt1/2 for a selection of stocks. On each curve the
transaction cost parameter K is the same.
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Part Five advanced topics
is the bold line in the ﬁgure then options on those stocks above the line, such as A, are too
infrequently hedged, while those below, such as R, are hedged too often. Of course, this is a
very simple approach to optimizing a hedging strategy. A more sophisticated approach would
also take into account the advantage of increased hedging: The reduction of risk.
48.5 NON-SINGLE-SIGNED GAMMA
For an arbitrary portfolio of options, the gamma, ∂ 2V /∂S 2 , is not of one sign. If this is the case
then we cannot drop the modulus sign. Since the Hoggard–Whalley–Wilmott equation is non
linear we must in general solve it numerically.
In Figures 48.2 and 48.3 is shown the value of a long bull spread consisting of one long call
with E = 45 and one short call with E = 55 and the delta at six months before expiry for the
two cases, with and without transaction costs. The volatility is 20% and the interest rate 10%.
In this example K = 0.25. The bold curve shows the values in the presence of transaction costs
and the other curve in the absence of transaction costs. The latter is simply the Black–Scholes
value for the combination of the two options. The bold line approaches the other line as the
transaction costs decrease.
In Figures 48.4 and 48.5 is shown the value of a long butterﬂy spread and its delta, before
and at expiry. In this example the portfolio contains one long call with E = 45, two short calls
with E = 55 and another long call with E = 65. The results are with one month until expiry
for the two cases, with and without transaction costs. The volatility, the interest rate and K are
as in the previous example.
12
10
Value without costs
Value with costs
8
V
790
6
4
2
0
0
20
40
60
80
100
S
Figure 48.2 The value of a bull spread with (bold) and without transaction costs. The payoff is also
shown.
transaction costs Chapter 48
1.2
1
Delta without costs
Delta with costs
Delta
0.8
0.6
0.4
0.2
0
0
20
40
60
80
100
S
Figure 48.3 The delta for a bull spread prior to and at expiry with (bold) and without transaction
costs.
12
10
Value without costs
Value with costs
V
8
6
4
2
0
0
20
40
60
80
100
S
Figure 48.4 The value of a butterﬂy spread with (bold) and without transaction costs.
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Part Five advanced topics
Delta without costs
Delta with costs
0.8
0.3
Delta
792
−0.2
20
30
40
50
60
70
80
90
S
−0.7
−1.2
Figure 48.5 The delta for a butterﬂy spread with (bold) and without transaction costs.
48.6 THE MARGINAL EFFECT OF TRANSACTION COSTS
Suppose we hold a portfolio of options, let’s call its value P (S, t), and we want to add another
option to this portfolio. What will be the effect of transaction costs? Call the value of the new,
larger, portfolio P + V : What equation is satisﬁed by the marginal value V ? This is not just
the cost equation applied to V because of the nonlinearity of the problem. We can write
∂P
∂ 2P
2 ∂ 2P
∂P
+ rS
+ 12 σ 2 S 2 2 − κσ S 2
− rP = 0
∂t
∂S
π δt ∂S 2
∂S
since the original portfolio must satisfy the costs equation. But now the new portfolio also
satisﬁes this equation:
∂(P + V ) 1 2 2 ∂ 2 (P + V )
+ 2σ S
− κσ S 2
∂t
∂S 2
+ rS
2 ∂ 2 (P + V )
π δt
∂S 2
∂(P + V )
− r(P + V ) = 0.
∂S
Both of these equations are non linear. But if the size of the new option is much less than the
original portfolio we can linearize the latter equation to examine the equation satisﬁed by the
marginal value V . We ﬁnd that
∂V
∂ 2P
∂ 2V
+ 12 σ 2 S 2 2 − sgn
∂t
∂S
∂S 2
κσ S 2
∂V
2 ∂ 2V
+ rS
− rV = 0.
2
π δt ∂S
∂S
transaction costs Chapter 48
This equation is now linear. The important point to note is that the volatility correction only
depends on the sign of the gamma of the original portfolio, P . If the gamma of the original
portfolio is positive then the addition of another contract with positive gamma only makes the
cost situation worse. However, the addition of a new contract with negative gamma will reduce
the level of transaction costs. The beneﬁts of reducing gamma may even make it worthwhile
to buy/sell a contract for more/less than the Black–Scholes value, theoretically.
48.7
OTHER COST STRUCTURES
The above model can be extended to accommodate other transaction cost structures. Suppose
that the cost of buying or selling ν assets is κ(ν, S). We follow the analysis of Section 48.4
up to the point where we take expectations of the transaction cost of hedging. The number of
assets traded is still proportional to the gamma, and the expected cost of trading is
E κ σ S δt 1/2 φ, S
.
The option pricing equation then becomes
1
∂V
∂ 2V
∂V
+ 12 σ 2 S 2 2 + rS
− rV = E κ σ S δt 1/2 φ, S
∂t
∂S
∂S
δt
.
For example, suppose that trading in shares costs k1 + k2 ν + k3 νS, where k1 , k2 and k3 are
constants. This cost structure contains ﬁxed costs (k1 ), a cost proportional to volume traded
(k2 ν), and a cost proportional to the value traded (k3 νS). The option value satisﬁes the non-linear
diffusion equation
k1
∂V
∂V
∂ 2V
+ 12 σ 2 S 2 2 + rS
− rV =
+
∂t
∂S
∂S
δt
48.8
2
∂ 2V
.
σ S(k2 + k3 S)
π δt
∂S 2
(48.8)
HEDGING TO A BANDWIDTH: THE MODEL
OF WHALLEY & WILMOTT (1993) AND
HENROTTE (1993)
We have so far seen how to model option prices when hedging takes place at ﬁxed intervals
of time. Another commonly used strategy is to rehedge whenever the position becomes too
far out of line with the perfect hedge position. Prices are therefore monitored continuously but
hedging still has to take place discretely.
Due to the complexity of this problem and those that follow, I only give a brief sketch of
the ideas and results.
With V (S, t) as the option value, the perfect Black–Scholes hedge is given by
=
∂V
.
∂S
Suppose, however, that we are not perfectly hedged, that we hold −D of the underlying
asset but do not want to accept the extra cost of buying or selling to reposition our hedge. The
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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.