A note on hedging: restricted but optimal delta hedging, mean, variance, jumps, stochastic...

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1 IntroductionThe relationship between continuous dynamic delta hedging

and risk-neutral pricing is a fundamental one. Take away the

possibility of such continuous portfolio rebalancing and the

edifice that is the Black–Scholes model (and its extensions

and “improvements”) starts to look very shaky. This can be

clearly seen on the wilmott.com forums where discussion of

the fundamentals of derivatives theory is in vogue like never

before.

In this note we show how a simple mean-variance frame-

work can be used to price in incomplete markets; in particu-

lar, we show that prices close to those from Black–Scholes

(Black and Scholes, 1973) are obtained even when delta hedg-

ing is rather infrequent. We suggest ways in which rebalanc-

ing can be done “optimally” and we see major reduction in

risk.

Other important work on the subject of discrete hedging

and risk minimization are Föllmer and Schweizer (1989), Schäl

(1994), Schweizer (1995), Martini and Patry (1999), Heath,

Platen, and Schweizer (2001a, 2001b), and Mastinsek (2006).

The Martini and Patry paper is most relevant to the current

work.

In Section 2 we remind the reader about the mean-vari-

ance model when volatility is stochastic. In Section 3 we sim-

ilarly outline the jump-diffusion case. For completeness, in

Section 4 we put the two together. In Section 5 we explain

how the calculated means and variances can be used to

achieve different optimized “targets,” thus allowing the

model to be used by speculators and by hedgers.1 Sections 6 to

8 consider different hedging strategies and present numerical

results. In Section 9 we briefly discuss the benefits of nonlin-

ear models and in Section 10 we explain some of the more

obvious advantages of the model under discussion. We con-

clude in Section 11.

2 The Model for the Underlying Asset:Stochastic VolatilityFirst let us consider the classical stochastic volatility model

dS = μ(S, σ, t) S dt + σ S dX1

A Note on Hedging: Restrictedbut Optimal Delta Hedging,Mean, Variance, Jumps,Stochastic Volatility, and CostsHyungsok Ahn

Nomura UK, London

Paul Wilmott

Wilmott Associates, London, e-mail: [email protected]

AbstractWe consider the pricing of options when delta hedging only takes place at discrete intervals. We show how to include transaction

costs, jumps and stochastic volatility while optimally, but discretely, dynamically hedging.

Keywordsjumps, Poisson process, stochastic volatility, option pricing, mean-variance pricing, risk minimization, discrete hedging, static

hedging, transaction costs

1. Calibrated models are, of course, useless for the speculator.

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122 | Wilmott Journal

WILMOTT Journal | Volume 1 Number 3 Published online in Wiley InterScience | DOI: 10.1002/wilj (www.interscience.wiley.com)

and

dσ = p(S, σ, t) dt + q(S, σ, t) dX2

with a correlation of ρ(S, σ, t).

As derived in Ahn and Wilmott (2003) the equation for the

mean, m, of a portfolio with an arbitrary stock position (delta), �, is

mt + 1

2σ 2S2mSS + ρσ Sqmσ σ + 1

2q2mσ σ

+ μSmS + pmσ − rm = (μ − r)S�.

The equation for the variance, v, is, for an arbitrary delta,

vt + 1

2σ 2S2vSS + ρσ SqvSσ

+ 1

2q2vσ σ + μSvS + pvσ − 2rv

+σ 2S2m2S + q2m2

σ + 2ρσ SqmSmσ

+σ 2S2�2 − 2�(σ 2S2mS + ρσ Sqmσ

) = 0.

In Ahn and Wilmott (2003) we chose delta to minimize the

variance, the natural thing to do, that is

� = mS + ρq

σ Smσ .

From this followed the two equations for m and v. However,

this assumed that delta hedging could be done continuously. In

what follows we will not make that assumption, but instead

only hedge at discrete times.

With � = 0 on the grounds that we cannot dynamically

delta hedge, and so our two equations are

mt + 1

2σ 2S2mSS + ρσ SqmSσ + 1

2q2mσ σ

+ μSmS + pmσ − rm = 0.

The equation for the variance, v, is

vt + 1

2σ 2S2vSS + ρσ SqvSσ + 1

2q2vσ σ + μSvS + pvσ − 2rv

+σ 2S2m2S + 2ρσ SqmSmσ + q2m2

σ = 0.

We can still accommodate discrete delta hedging using the

above equations by having a pure stock component in the final

condition for m or by having jump conditions.

We leave this model at this point, moving on to jump-diffusion.

3 The Model for the Underlying Asset:Jump DiffusionWe now look at the classical jump-diffusive random walk for an

asset, S, given by

dS = μS dt + σ S dX + (J − 1)S dq,

where μ is the drift rate in the absence of jumps, σ is the

volatility in the absence of jumps, dX is a Wiener process, J is

the, possibly random, factor representing the size of the jump

(so that a stock that has value S before a jump becomes JS after

a jump), dq is a Poisson process with intensity λ. We assume

that all of the parameters in the above and the distribution of J

are known.

We can take results from Ahn and Wilmott (2007, 2008).

With an arbitrary delta the mean and variance equations

are

mt + 1

2σ 2S2mSS + μS mS + λE [m(JS, t) − m]

− (μ − r + λE[J − 1])S� − rm = 0

(in this m is simply m(S, t) and

vt + 1

2σ 2S2vSS + μS vS + λE [v(JS, t) − v] − 2rv

+ σ 2S2(mS − �)2 + λE[(

m(JS, t) − m − (J − 1)S�)2

]= 0.

Previously (Ahn and Wilmott, 2007), we chose delta in two

ways:

1. The Merton way to eliminate the diffusive risk: � = mS

2. The optimal way to reduce the variance as much as possible:

� = mS + λE

[(J − 1)(m(JS, t) − m) − (J − 1)2S mS

]S(σ 2 + λE [(J − 1)2]

)Now we propose to choose � = 0 (also discussed in Ahn and

Wilmott, 2008), again because hedging can only be done dis-

cretely. And again a discrete delta hedging can be accommo-

dated by including a stock position in the final condition for m

or by having jump conditions. This results in the two equa-

tions

mt + 1

2σ 2S2mSS + μS mS + λE [m(JS, t) − m] − rm = 0,

and

vt + 1

2σ 2S2vSS + μS vS + λE [v(JS, t) − v] − 2rv

+ σ 2S2m2S + λE

[(m(JS, t) − m

)2]

= 0.

4 Stochastic Volatility and JumpsFor completeness we shall note that incorporating both stochas-

tic volatility and stock jumps will lead to

mt + 1

2σ 2S2mSS + ρσ SqmSσ + 1

2q2mσ σ

+ μS mS + pmσ + λE [m(JS, σ, t) − m] − rm = 0

(1)

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WILMOTT Journal | Volume 1 Number 3Published online in Wiley InterScience | DOI: 10.1002/wilj (www.interscience.wiley.com)

and

vt + 1

2σ 2S2vSS + ρσ SqvSσ + 1

2q2vσ σ

+ μS vS + pvσ + σ 2S2m2S + 2ρσ SqmSmσ + q2m2

σ

+ λE [v(JS, σ, t) − v] + λE[(

m(JS, σ, t) − m)2

]− 2rv = 0.

(2)

5 Meaning of m and vIn the mean-variance framework we are using here we can

interpret and use m and v in several ways. For example, if we

are selling exotic contracts then we might interpret an

option’s value as being m + ξ√

v, with our “personal” param-

eter ξ , since such a value will result in us making an expect-

ed profit but with a certain limited degree of risk. Or we

might be content to set up a portfolio in such a way as to

keep v as small as possible, since that minimizes risk. Or if we

are seeking arbitrage opportunities then choosing a portfolio

that maximizes m while limiting v might be a sensible deci-

sion. The crucial point being that different people can use

the same m and v in different ways. (This allows this type of

model to be used by both buy side and sell side, since they

have different goals. For further discussion, see Ahn and

Wilmott, 2003.)

In what follows we shall consider the case where we are only

concerned with minimizing variance v. This goal will be repre-

sented mathematically via rehedging conditions.

6 Discrete Hedging: Case 0, Control Case,Rehedging at Fixed IntervalsMany authors have considered the case of rehedging at fixed

intervals (see Boyle and Vorst, 1992; Wilmott, 1994; and

Kamal and Derman, 1999, for example). We shall take this as

our control case before considering “optimal” rehedging. In

the present framework all we have to do to build this into the

model is to introduce a new variable, to represent the

amount of stock held for hedging purposes. So we now have

m(S, σ, t; �) and v(S, σ, t; �). These are to satisfy Equations (1)

and (2).

6.1 Final condition

The final conditions at expiration, time T, are then for a call

simply

m(S, σ, T; �) = max(S − E, 0) + �S and v(S, σ, T; �) = 0.

The rehedging conditions are slightly less trivial.

6.2 Rehedging conditions

Given the goal, mentioned above, of simply minimizing vari-

ance, we can make several observations concerning rehedg-

ing:

• When we rehedge the variance is continuous.

• We rehedge to a new delta that minimizes the variance

going forward (i.e. backwards in time!).

• The mean will change by any costs incurred on rehedg-

ing.

Supposing that we rehedge n times during the life of the

option, at equal time intervals, then we will have the following

continuity conditions.

First, continuity of variance while simultaneously minimiz-

ing by choosing a new delta:

v(S, σ, ti−; �) = min�

v(S, σ, ti+; � ). (3)

ti is the time at which the ith rehedging takes place,

ti = t0 + (i − 1)(T − t0)/n, where t0 is the time at which we can

start hedging the option.

The condition on m is then

m(S, σ, ti−; �) = m(S, σ, ti+; � Best ) + (� − � Best )

S − Costs(S, �, � Best )(4)

where crucially the � Best is the same � that comes from the

minimization in (3) (which will be a function of S, σ and t, of

course) and the Costs(S, �, � Best ) is the loss due to hedging

costs.

6.3 Results: A single vanilla call, case 0, rehedging at fixed

intervals

This short note is simply setting out the problem of optimal dis-

crete hedging, etc., in various models. Our focus is on the

dependence of the “solution” on the strategies. Therefore, in

what follows our examples will be on the easily understood, sim-

plest case, when volatility is constant and there are no jumps,

and no transaction costs.

The problem is then

mt + 1

2σ 2S2mSS + μS mS − rm = 0

and

vt + 1

2σ 2S2vSS + μS vS + σ 2S2m2

S − 2rv = 0

with

m(S, T; �) = max(S − E, 0) + �S and v(S, T; �) = 0

and

v(S, ti−; �) = min�

v(S, ti+; � )

and

m(S, ti−; �) = m(S, ti+; � Best ) + (� − � Best )S.




The final results that we want will be

v(S, t0; � Best ) and m(S, t0; � Best ) − � Best S.

The following results were found by finite-difference solution

of the governing equations. The code was relatively straightfor-

ward, fewer than 100 lines. Solution by Monte Carlo would be

rather complicated because of the nonlinear nature of the model.

Parameters were: σ = 0.2, r = 0.05, T − t0 = 1, E = 100.

Example 1: μ = 0.05. Consider first the case when the

drift is the same as the interest rate. In this simple case the

mean function m gives the same values as a function of S and

t as the classical Black–Scholes model for all n. This is

because it doesn’t matter how you delta hedge, the expecta-

tion is unaffected. The interesting results are therefore for

the variance. And these are shown in Figures 1 and 2. These

show how variance varies with asset S and number of hedges

n (not including the very first hedge at initiation). Once you

are up to hedging more than 75 times in the year then the

risk becomes negligible. The value of m at the money is

10.45. To put the variance into context, with 25 hedges the

variance is 1.095, so the standard deviation in the price is

about 1.046. The variance decreases approximately as the

inverse of the number of rehedges, as in theory it should

(Boyle and Vorst, 1992).

Figure 3 shows how the “value” converges to Black–Scholes,

in the sense of how the mean plus/minus the standard deviation

for the at-the-money call varies with the number of optimal

hedges. Figure 4 shows how similar the hedge ratio is to simple

Black–Scholes.

Example 2: μ = 0.25. We now consider a large μ so that the

mean is not the same as Black–Scholes. Results are shown in the

following four figures, Figures 5–8.

Figure 1: Variance versus S, and n, three-dimensional plot.

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Figure 3: m ±√v.

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Number of optimal hedges

Mea

n +

/- S

D

Figure 4: Delta versus S, different n, and Black–Scholes.

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Figure 2: Variance versus S, different n.




7 Discrete Hedging: Case 1, RestrictedNumber of Hedges, Hedging Time OptimalThe above assumes delta hedging at prescribed times, the ti. If

we want to model optimal discrete delta hedging at arbitrary

times then all we have to do to build this into the model is to

introduce another new variable, this one to represent the num-

ber of hedges remaining before expiration.

The problem is now to find m(S, σ, t; n, �) and v(S, σ, t; n, �).

The governing equations are still Equations (1) and (2). To

these we must add final conditions and conditions representing

the rehedging.

7.1 Final conditions

The final conditions at expiration, time T, are for a call again

m(S, σ, T; n, �) = max(S − E, 0) + �S and v(S, σ, T; n, �) = 0.

7.2 Rehedging conditions

But now continuity of variance while simultaneously minimiz-

ing by choosing a new delta becomes

v(S, σ, t; n + 1, �) < min�

v(S, σ, t; n, � ). (5)

The above obviously has similarities with the early exercise of an

American option. Note that there are no ti in these expressions. The

times at which to rehedge come out of the solution of the problem

in the same way as the optimal time to exercise an American

option. In that problem the early-exercise (free-boundary) con-

straint maximizes the option’s value by choosing the optimal time

to exercise. In our problem the free-boundary constraint optimizes

the times at which to rehedge so as to minimize the variance.

Note how the number of hedges left goes from n + 1 to n as

we rehedge.

Also note that this condition would change if our goal was

not to minimize v but was another goal such as one of those

mentioned in Section 5 above.


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Figure 7: m ±√v.

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12

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0 5 10 15 20 25


Mea

n +

/- S

D


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The condition on m is then

m(S, σ, t; n + 1, �) = m(S, σ, t; n, � Best ) + (� − � Best )S

− Costs(S, �, � Best )(6)

where the � Best is again the same � that comes from the min-

imization in (5) (which will be a function of S, t and n, of

course) and the Costs(S, �, � Best ) is the loss due to hedging

costs.

7.3 Results: A single vanilla call, case 1, restricted number of

hedges, hedging time optimal

As above, for the numerical results we concentrate on the dis-

crete hedging problem in the absence of jumps and with con-

stant volatility. The problem is therefore

mt + 1

2σ 2S2mSS + μS mS − rm = 0

and

vt + 1

2σ 2S2vSS + μS vS + σ 2S2m2

S − 2rv = 0

with

m(S, T; n, �) = max(S − E, 0) + �S and v(S, T; n, �) = 0

and

v(S, t; n + 1, �) < min�

v(S, t; n, � )

and

m(S, t; n + 1, �) = m(S, t; n, � Best ) + (� − � Best )S.

The final results that we want will be

v(S, t0; n, � Best and m(S, t0; n, � Best ) − � Best S.

Parameters were as above.

Example 1, again: μ = 0.05. Consider first the case when the

drift is the same as the interest rate. Again, the mean function

m gives the same values as a function of S and t as the classical

Black–Scholes model for all n. The variances are shown in

Figures 9 and 10. Once you are up to hedging more than 25

times in the year then the risk becomes negligible. With 25

hedges the variance is 0.018, so the standard deviation in the

price is about 0.134. This is a factor of eight less than when

rehedging was done at fixed time intervals.

In Figure 11 is shown how the “value” converges to

Black–Scholes, in the sense of how the mean plus/minus the

standard deviation for the at-the-money call varies with the

number of optimal hedges. Figure 12 similarly shows conver-

gence to the Black–Scholes delta.

It appears that the variance is much smaller when rehedg-

ing is done optimally rather than at fixed intervals, Case 1 is

much better, in the sense of reducing risk, than Case 0. That it

is better is obvious, since there is more freedom, but it is not so

obvious that it should be vastly better as it appears to be.

This is particularly clear from Figure 13 in which are plotted

the logarithms of the variance against the logarithm of the

number of rehedges. The dashed line is Case 0, which shows the

inverse relationship between variance and number of hedges.

The solid line is Case 1, optimally rehedging. Clearly not only is

optimal rehedging everywhere better at reducing risk the vari-

ance reduction also increases more dramatically for a relatively

small number of rehedges.


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Example 2, again: μ = 0.25. We now consider a large μ so

that the mean is not the same as Black–Scholes. Results are

shown in the four figures, 14–17.

8 Discrete Hedging: Case 2, AcceptableLocal VarianceThe model can be further adapted to accommodate a hedging

strategy in which one only rehedges when variance can be

reduced by some prescribed threshold level. In this case we do

not limit the number of rehedges. The governing equations and

final condition remain the same as earlier. The rehedging con-

ditions are now as follows.

Figure 11: m ±√v.

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Mea

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/- S

D

Figure 12: Delta versus S different n, and Black–Scholes.

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Figure 13: ln(v) versus ln(n) for Cases 0 and 1.

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ln(v

aria

nce

)

Optimal dynamic hedging

Fixed-interval hedging

1 3


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Figure 19: Variance versus S and ε, three-dimensional plot.

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Figure 20: Variance versus S, different ε.

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4210.50.250.1250.06250.031250.0156250.00781250.003906250.0019531250.0009765630.0004882810.0002441410.000122076.10352E-053.05176E-051.52588E-057.62939E-063.8147E-06

If min�

v(S, σ, t; � ) < v(S, σ, t; �) − ε then

v(S, σ, t; �) = min�

v(S, σ, t; � )

at which “point”

m(S, σ, t; �) = m(S, σ, t; � Best ) + (� − � Best )S.

Here ε is the prescribed variance threshold, meaning that

rehedging will only take place if there is a gain of at least this

amount upon rehedging. Mathematically this is similar to the

early-exercise free-boundary problem, but now with a simulta-

neous jump condition.

Figure 18 shows this rehedging schematically. This “hedging

bandwidth” has been seen before in Whalley and Wilmott (1996,

1997) in the context of transaction costs for another non-linear model.

Example 1, again: μ = 0.05. Figures 19–22 show results for

the same first example as before.

Figure 18: Schematic diagram of rehedging according to a threshold.

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1 1.2

Delta

Variance, v

Threshold, εOutside of these points you will rehedge to here





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Local variance threshold

Mea

n +

/- S

D

Figure 22: Delta versus S, different ε, and Black–Scholes.

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Figure 23: Results for optimal delta and hedging band.

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Figure 24: Variance versus S, and ε, three-dimensional plot.

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Figure 23 shows results for the optimal delta and hedging

band as functions of the asset.

Example 2, again: μ = 0.25. Figures 24–27 show results for

the same second example as before.

9 Nonlinearity and Static HedgingThe models presented here can be used for speculation, if one

believes one has a better model for the underlying than the

“market,” or for valuing and hedging exotics. We can make sev-

eral observations about this model, and the “advanced” versions

with jumps and/or stochastic volatility, concerning the nonlin-

earity:

• Since the above model is nonlinear there will be significant

advantages to be gained by statically hedging the exotic

with vanillas. (See Ahn and Wilmott, 2003, 2007 and 2008,

and Wilmott, 2006 for plenty of details and examples using

similar models.) Figure 25: Variance versus S, different ε.

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ian

ce

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number of liquid contracts. Despite the complexity of these mod-

els and their solutions this approach is naive. Such models mere-

ly match a snapshot of the market and rarely have any ability to

match the dynamics of the market going forward. Indeed, the per-

fect static matching of a model’s output with market prices leads

many otherwise clever people to think they are working with a

good model. But the minute that model has to be recalibrated, as

it always does, is the moment at which the model’s very founda-

tions are destroyed. Let us repeat, the current crop of popular cal-

ibrated models are almost invariably exceedingly poor models

since they capture little of future dynamics. Many people will

respond to this by saying that it is not possible to predict the

future and therefore they are doing the best that they can. Yes and

no. The future may be impossible to predict but there are now

many models with the flexibility to accommodate many future

market dynamics, the present model is one of them.

Here is a partial list of the advantages of our proposed

model.

• PPeerrffeecctt ““ccaalliibbrraattiioonn””:: Nonlinearity ensures that reasonable

market prices of liquid instruments are matched by default.

No more tying yourself in numerical knots to calibrate your

unstable stochastic volatility model. Nonlinearity means

that there are genuine model-based reasons for static hedg-

ing, a benefit of which is perfect “calibration.”2

• SSppeeeedd:: The models above not having n dependence will be

almost as fast to solve numerically as their equivalent linear

models. (There will be a factor of 2, because we must solve

for m and v, and finding the optimal delta will slow things

down slightly as well.) On the other hand, calibration, a

major reason for some models’ computational sluggishness,

is not necessary.

• EEaassyy ttoo aadddd ccoommpplleexxiittyy ttoo tthhee mmooddeell:: The modular form of

the equations means that it is easy to add jumps, stochastic

volatility, etc.

• RReeaalliissttiicc rriisskk ccoonnttrrooll:: There is no reliance on either the

impossibility of continuous dynamic hedging nor on arbi-

trary rehedging.

• OOppttiimmaall ddyynnaammiicc hheeddggiinngg:: Rehedging with the underlying

is done optimally.

• OOppttiimmaall ssttaattiicc hheeddggiinngg:: Hedging exotics with traded vanil-

las will increase “value” because of the nonlinearity. This

can be optimized. See Wilmott (2006).

• FFrriiccttiioonnss ccaann bbee mmooddeelllleedd:: For no extra computational

time, market frictions such as transactions costs can be

incorporated.

• CCaann bbee uusseedd bbyy bbuuyy aanndd sseellll ssiiddeess:: By changing the “target

function” (in our examples we only considered minimizing

the variance) the model can be used by hedge funds looking

for statistical arbitrage and by investment banks selling

exotics. See Ahn and Wilmott (2003).


0

2

4

6

8

10

12

14

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Local variance threshold

Mea

n +

/- S

D

Figure 27: Delta versus S, different ε, and Black–Scholes.

-1.2

-1

-0.8

-0.6

-0.4

-0.2

050 60 70 80 90 100 110 120 130 140 150

Asset

Op

tim

al d

elta

BS4210.50.250.1250.06250.031250.0156250.00781250.00390625

• Incorporating transaction costs requires no extra effort.

• In the above we have chosen to minimize variance v; one

could instead maximize m, or minimize/maximize a combi-

nation, as discussed in Ahn and Wilmott (2003).

We could say a lot more about the pros and cons of nonlin-

earity in derivatives models, but we assume that the reader is

familiar with the models mentioned above. If not, then we sug-

gest that Wilmott (2006) provides the best starting point.

10 Advantages of the ModelMany researchers are going down the route of finding increas-

ingly complicated models with an increasing number of parame-

ters. These parameters are then chosen to calibrate an increasing

2. Here calibration is within inverted commas to emphasize that calibration can mean several things. And in the context of nonlinear models its meaning makes more

practical sense than in the linear models in which it is really just a form of curve fitting. Again the reader unfamiliar with the advantages of nonlinear models should read

Wilmott (2006).




• TThhee mmooddeell llooookkss ffaammiilliiaarr:: The two governing equations

are similar to classical Black–Scholes, albeit with a non-

linear forcing term in one of them. The constraints are

similar to those seen with early exercise. As such, the

model should not “frighten the horses” or risk manage-

ment.

11 ConclusionsAs we have seen elsewhere (Ahn and Wilmott, 2008) dynamic

delta hedging need not be performed continuously for its

effects to be felt. Static hedging of options with other options

and occasional delta hedging can eliminate enough risk. We

say “enough” risk since there are inevitably going to be other

risks that we cannot eliminate and so why bother being

obsessive about one risk source? In this note we have focused

on making the occasional delta hedging “optimal.” Here we

have chosen optimal to mean variance minimizing, although

other choices are possible. But optimality is also about choos-

ing when to rehedge. We have discussed various possibilities

about timing of rehedging, and it seems that one can reduce

far more risk than seen before in the discrete-hedging litera-

ture.

HHyyuunnggssookk AAhhnn is a leading mathematician modelling financial market and

derivative products. He had taught mathematical finance and probability

theory in universities and private institutions, including University of Oxford

and University of California, and published many articles on the subject.

After leaving academia, he has been working for financial institutions,

developing front-office pricing models and risk analysis tools for structured

products in equity, energy, commodity, credit, interest rate, long-dated FX,

and various hybrid markets. His expertise includes volatility modelling, exotic

derivative structuring, and cutting-edge numerical computation such as

high-dimensional finite-difference and Monte Carlo methods. Currently he is

the global head of equity derivative quantitative research at Nomura

International plc.

Paul Wilmott is a financial consultant, specializing in derivatives, risk

management and quantitative finance. He has worked with many leading

US and European financial institutions. Paul studied mathematics at St

Catherine’s College, Oxford, where he also received his D.Phil. He founded

the Diploma in Mathematical Finance at Oxford University and the journal

Applied Mathematical Finance. He is the author of Paul Wilmott

Introduces Quantitative Finance (Wiley 2007), Paul Wilmott On

Quantitative Finance (Wiley 2006), Frequently Asked Questions in

Quantitative Finance (Wiley 2006) and other financial textbooks. He has

written over 100 research articles on finance and mathematics. Paul was a

founding partner of the volatility arbitrage hedge fund Caissa Capital

which managed $170 million. His responsibilities included forecasting,

derivatives pricing, and risk management. Paul is the proprietor of

www.wilmott.com, the popular quantitative finance community website,

the quant magazine Wilmott and is the Course Director for the Certificate

REFERENCESAhn, H. and Wilmott, P. 2003. Stochastic Volatility and Mean-Variance

Analysis. Wilmott magazine NNoovveemmbbeerr: 84–90.

Ahn, H. and Wilmott, P. 2007. Jump Diffusion, Mean and Variance: How to Dynamically

Hedge, Statically Hedge and to Price. Wilmott magazine MMaayy: 96–109.

Ahn, H. and Wilmott, P. 2008. Dynamic Hedging is Dead! Long Live Static

Hedging! Wilmott magazine JJaannuuaarryy: 80–87.

Black, F. and Scholes, M. 1973. The Pricing of Options and Corporate

Liabilities. Journal of Political Economy 8811: 637–59.

Boyle, P. and Vorst, T. 1992. Option Replication in Discrete Time with

Transaction Costs. Journal of Finance 4477: 271.

Föllmer, H. and Schweizer, M. 1989. Hedging by Sequential Regression: An

Introduction to the Mathematics of Option Trading. The ASTIN Bulletin 11: 147–160.

Kamal, M. and Derman, E. 1999. Correcting Black–Scholes. Risk Magazine

1122((11)): 82–85.

Heath, D., Platen, E., and Schweizer, M. 2001a. A Comparison of Two

Quadratic Approaches to Hedging in Incomplete Markets. Mathematical

Finance 1111: 385–413.

Heath, D., Platen, E. and Schweizer, M. 2001b. Numerical Comparison of Local

Risk-Minimisation and Mean-Variance Hedging. In E. Jouini, J. Cvitanic and M.

Musiela (Eds). Option Pricing, Interest Rates and Risk Management. 509–537.

Cambridge University Press.

Martini, C. and Patry, C. 1999. Variance Optimal Hedging for a Given Number

of Transactions. INRIA.

Mastinsek, M. 2006. Discreteñtime Delta Hedging and the Black–Scholes

Model with Transaction Costs. Math. Meth. Oper. Res. 6644: 227–236.

Schäl, M. 1994. On Quadratic Cost Criteria for Option Hedging. Mathematics

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of Operation Research 2200: 1–32.

Whalley, A.E. and Wilmott, P. 1996. Key Results in Discrete Hedging and

Transaction Costs. In A. Konishi and R. Dattatreya (Eds). Frontiers in

Derivatives. 183–196. Irwin Professional Publishing.

Whalley, A.E. and Wilmott, P. 1997. An Asymptotic Analysis of an Optimal

Hedging Model for Option Pricing with Transaction Costs. Mathematical

Finance 77: 307–324.

Wilmott, P. 1994. Discrete Charms. Risk Magazine 77(3): 48–51.

Wilmott, P. 2006. Paul Wilmott On Quantitative Finance, Second Edition.

Wiley: Chichester.

in Quantitative Finance www.7city.com/cqf. The CQF teaches how to

apply mathematics to finance, focusing on implementation and pragma-

tism, robustness and transparency. Paul was a professional juggler with

the Dab Hands troupe. He also has three half blues from Oxford University

for Ballroom Dancing.


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