Optimal Hedging com

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Optimal Hedging of Option Portfolios

with Transaction Costs

Valeri I. Zakamouline

This revision: January 17, 2006

Abstract

One of the most successful approaches to option hedging with trans-action costs is the utility-based approach, pioneered by Hodges andNeuberger (1989). Judging against the best possible tradeoff betweenthe risk and the costs of a hedging strategy, this approach seems toachieve excellent empirical performance. However, this approach hasone major drawback that prevents the broad application of this ap-proach in practice: the lack of a closed-form solution. The numericalcomputations are very cumbersome in implementation. Despite somerecent advances in finding an explicit description of the utility-basedhedging strategy by using either asymptotic, approximation, or othermethods, so far they were concerned primarily with hedging a single

plain-vanilla option. Yet in practice one faces the problem of hedginga portfolio of options on the same underlying asset. Since the knowl-edge of the optimal hedging strategy for a portfolio of options is ofgreat practical significance, in this paper we suggest a simplified pa-rameterized functional form of the utility-based hedging strategy for aportfolio of options and a method for finding the optimal parameters.The method is based on simulations and is simple in implementation.Despite the simplicity of the suggested functional form, the perfor-mance of our optimized simplified hedging strategy is close to that ofthe exact utility-based strategy. Moreover, we provide an empiricaltesting of our optimized hedging strategies against some alternativestrategies and show that our strategies outperform all the others.

Key words: option hedging, transaction costs, simulations.

JEL classification: G11, G13.

This is not a complete draft, but rater a paper in preparation. The author would liketo thank the participants of the Quantitative Methods in Finance 2005 Conference fortheir insightful comments.

Address for correspondence: Agder Regional University, Faculty of Economics, ServiceBox 422, 4604 Kristiansand, Norway. E-mail: [email protected].

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1 Introduction

One of the most successful approaches to option hedging with transac-tion costs is the utility based approach, pioneered by Hodges and Neu-berger (1989). Judging against the best possible tradeoff between the riskand the costs of a hedging strategy, the utility based approach seems toachieve excellent empirical performance (see Mohamed (1994), Clewlow andHodges (1997), Martellini and Priaulet (2002), Zakamouline (2004), and Za-kamouline (2006a)). However, this approach has one major drawback thatprevents the broad application of this approach in practice: the lack of aclosed-form solution. Therefore, the solution must be computed numerically.The numerical algorithm is cumbersome to implement and the calculationof the optimal hedging strategy is time consuming.

In particular, the implementation of the numerical algorithm presents asort of a challenge. In the case of a general utility function one needs to solvenumerically an optimal stochastic control problem in four dimensions. Theuse of the negative exponential utility function reduces the dimensionalityof the problem by one, but in this case one faces the problem of overflow orunderflow in the values of the exponential utility (see Clewlow and Hodges(1997)). When it comes to the problem of having a large computationaltime, considering the exploding development within the computer industrythis problem becomes less important when one needs to find the optimalhedging strategy for a non-exotic option. In this case the optimal transactionpolicy is not path-dependent and, hence, the evolution of the underlying

price process can be modelled as a recombining binomial tree. However,when an option payoff is path-dependent, one needs to construct a non-recombining binomial tree for the underlying price process. In this case thecomputation of the optimal hedging policy for a sufficiently large number oftrading dates is out of reach.

Since there are no explicit solutions for the utility based hedging withtransaction costs and the numerical methods are computationally hard, forpractical applications it is of major importance to use other alternatives.One of such alternatives is to obtain an asymptotic solution. In asymptoticanalysis one studies the solution to a problem when some parameters inthe problem assume large or small values. The asymptotic analysis of the

model of Hodges and Neuberger (1989) for the case of a short Europeancall option was performed by Whalley and Wilmott (1997) and Barles andSoner (1998).

Zakamouline (2004) compared the performances of asymptotic strategiesagainst the performance of the exact strategy and found that under realisticmodel parameters an asymptotic strategy performs noticeably worse thanthat obtained from the exact numerical solution. He suggested to use analternative to the asymptotic analysis, namely, the approximation method.Under approximation it is meant the following: one has a rather slow and

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cumbersome way to compute the optimal hedging policy with transaction

costs and wants to replace it with simple and efficient approximating func-tion(s). To do this, one first specifies a flexible functional form of the optimalhedging policy. Then, given a functional form, the parameters are chosento provide the best fit to the exact numerical solution. This second stage isknown as model calibration. Zakamouline performed the approximationof the utility-based hedging strategy for a short European call option andtested empirically his approximation strategy against both the asymptoticstrategies and some other well-known strategies. He evaluated the perfor-mance of different hedging strategies within the unified mean-variance andthe mean-VaR (Value-at-Risk) frameworks and found that his approxima-tion strategy outperforms all the others.

The approximation methodology presented in Zakamouline (2004) pro-duces a clearly superior result as compared to other alternative methods offinding closed-form expressions for the optimal hedging strategy for a plain-vanilla European option. However, this methodology does retain the maindisadvantage one would like to get rid off: to approximate the optimal strat-egy for a specific option one needs to start with the numerical calculationsof the optimal hedging strategy for a large set of the model parameters.Since the numerical algorithm is cumbersome to implement and the calcula-tion of the optimal hedging strategy is time consuming, the approximationmethodology is unlikely to be commonly used by the practitioners.

In practice one faces the problem of hedging a large portfolio of optionson the same underlying asset. Consequently, the knowledge of the optimal

hedging strategy for a portfolio of options is of great practical significance.The first contribution of this paper is to present a detailed description ofthe nature of the optimal hedging strategy for a portfolio of options. Weillustrate that the optimal hedging strategy for a portfolio of option is, infact, rather simple and has three essential features: (i) existence of the no-transaction region, (ii) optimal form of the no-transaction region, and (iii)the volatility adjustment.

The second contribution of this paper is to suggest a simplified param-eterized functional form of the optimal hedging strategy for a portfolio ofoptions and a method for finding the optimal parameters. That is, in theessence our idea is based on a pure financial engineering way of thinking:

We try to mimic the essential properties of the optimal hedging strategyby specifying a simple but flexible functional form of the optimal hedgingpolicy. Then, we optimize the performance of the simplified hedging strategyby changing a few parameters in order to find the best possible risk-returntradeoff. Even though our optimization method takes a longer time thanthe exact numerical computations in the case of a portfolio of non-exoticoptions, the implementation of our optimization method is simple as themethod is based on Monte-Carlos simulations instead of on a straightfor-ward numerical solution of an optimal stochastic control problem. A great

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advantage of our method is that it works equally well also for finding the op-

timal hedging strategy for a portfolio of exotic options. That is, this methodmakes feasible the finding of the optimal dynamic hedging strategy in caseswhen the direct numerical computation is out of reach.

Despite the simplicity of the suggested functional form, an optimizedstrategy showed to be very effective with the performance close to that ofthe exact utility-based strategy. We provide an empirical testing of our opti-mized strategy against the alternative strategies and show that our strategyoutperforms all the others. We tested our optimization method by simula-tions of the hedging such popular combinations of standard options as Bulland Bear spreads, long and short positions in Butterfly spread, Call Ratiospread, Put Ratio spread, Straddle, Strip, Strap, Strangle, and Wrangle.

The paper is organized as follow. In Section 2 we introduce the problemof hedging an option portfolio and review some known methods. In Section3 we review the utility-based option pricing and hedging approach as well asthe results of various asymptotic and approximation methods. The purposeof this section is to present a detailed description of the nature of the opti-mal hedging strategy in the presence of transaction costs. In Section 4 weintroduce the rational under our simplified hedging strategy and describeour optimization methodology. In Section 5 we provide an empirical testingof our optimized strategies. Section 6 concludes the paper.

2 The Hedging Problem and Some Solutions

We consider a continuous time economy with one risk-free and one riskyasset, which pays no dividends. We will refer to the risky asset as the stock,and assume that the price of the stock, St, evolves according to a diffusionprocess given by

dSt = Stdt + StdWt,

where and are, respectively, the mean and volatility of the stock returnsper unit of time, and Wt is a standard Brownian motion. The risk-free asset,commonly referred to as the bond or bank account, pays a constant interestrate of r 0. We assume that a purchase or sale of shares of the stockincurs transaction costs ||S proportional to the transaction ( 0).

We consider hedging a portfolio of European options on the same under-lying stock with the same maturity T. We denote the value of the optionportfolio at time t as V(t, St). The terminal payoff of the option portfolioone wishes to hedge is given by V(T, ST). The general set-up of the hedgingproblem is as follows: When a hedger sells/buys a portfolio of options, hereceives/pays the price V(t, St) and sets up a hedging portfolio by buying shares of the stock and putting V(t, St) (1 + )St in the bank account.As time goes, the hedger rebalances the hedging portfolio according to someprescribed rule/strategy.

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2.1 The Black-Scholes Model

When a market is friction-free ( = 0), Black and Scholes (1973) showedthat it is possible to replicate the payoff of an option by constructing a self-financing dynamic trading strategy consisting of the risk-free asset and thestock. As a consequence, the absence of arbitrage dictates that the optionprice be equal to the cost of setting up the replicating portfolio.

The Black-Scholes model can be easily extended for the case of a portfolioof options. That is, in a friction-free market it is possible to replicate thepayoff of a portfolio of options. As for a single option, the value of thehedging portfolio that replicates the payoff of a portfolio of options shouldsatisfy the following PDE

Vt + rSVS + 12 2S2

2

VS2 rV = 0, (1)

with a boundary condition given by V(T, ST). The general solution of thePDE is given by

V(t, St) = er(Tt)EQ[V(T, ST)].

That is, the (Black-Scholes) price of the portfolio of options equals thepresent value of the expected portfolio payoff on maturity in the so-calledrisk-neutralworld where the stock price process is given by

dSt = rStdt + StdWt.

The Black-Scholes hedging strategy consists in holding (delta) shares ofthe stock and some amount in the bank account, where

(t) =V(t, St)

S. (2)

It should be emphasized that the Black-Scholes hedging is a dynamic repli-cation policy where the trading in the underlying stock has to be donecontinuously. In the presence of transaction costs in capital markets theabsence of arbitrage argument is no longer valid, since perfect hedging isimpossible. Due to the infinite variation of the geometric Brownian motion,the continuous replication policy mandated by the Black-Scholes model in-

curs an infinite amount of transaction costs over any trading interval nomatter how small it might be. How should one hedge a portfolio of optionsin the market with transaction costs?

2.2 The Black-Scholes Hedging at Fixed Regular Intervals

One of the simplest and most straightforward hedging strategies in the pres-ence of transaction costs is to rehedge in the underlying stock at fixed regularintervals. One would simply implement the delta hedging according to the

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Black-Scholes strategy, but in discrete time. More formally, the time inter-

val [t, T] is subdivided into n fixed regular intervals t, such that t =

Tt

n .The hedging proceeds as follows: at time t the hedger receives/pays V(t, St)and constructs a replicating portfolio by purchasing (t) shares of the stockand putting V(t, St) (t)(1 + )St into the bank account. At time t + t,an additional number of shares of the stock is bought or sold in order tohave the target hedge ratio

(t + t) =V(t + t,St+t)

S. (3)

At the same time, the bank account is adjusted by

(t + t) (t) (t + t) (t)St+t .Then the hedging is repeated in the same manner at all subsequent timest + it, i = 2, 3, . . . , n 1.

The choice of the number of hedging intervals, n, is somewhat unclear.Obviously, when n is small, the volume of transaction costs is also small,but the variance of the replication error is large. An increase in n reducesthe variance of the replication error at the expense of increasing the volumeof transaction costs. Moreover, as t 0, the volume of transaction costsapproach infinity.

2.3 The Method of Hoggard, Whalley and Wilmott

A variety of methods have been suggested to deal with the problem of optionpricing and hedging with transaction costs. Leland (1985) was the first toinitiate this stream of research. He adopted the rehedging at fixed regularintervals and proposed a modified Black-Scholes strategy that permits thereplication of a single option with finite volume of transaction costs no mat-ter how small the rehedging interval is. The hedging strategy is adjusted byusing a modified volatility.

In particular, the central idea of Leland was to include the expectedtransaction costs in the cost of a replicating portfolio. That is, accord-ing to Leland, the price of an option must equal the expected costs of the

replicating portfolio including the transaction costs. As a result, the mar-ket maker, who writes, for example, a European call option and constructsthe replicating portfolio, should sell it with a premium (as compared tothe Black-Scholes price) which offsets the expected transaction costs. Onthe contrary, the market maker, who buys a European call option and con-structs the replicating portfolio, should buy the option with a discount tooffset, again, the expected transaction costs. The Lelands pricing and hedg-ing method is an adjusted Black-Scholes method where one uses a modifiedvolatility in the Black-Scholes formulas for the option price and delta. In

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hedging a short European call option the modified volatility is higher than

the original volatility which gives a higher option price. On the contrary,in hedging a long European call option the modified volatility is lower thanthe original volatility which gives a lower option price. Using the Lelandsmethod one hedges an option with a delta calculated similarly as the Black-Scholes delta, but with adjusted volatility.

Hoggard, Whalley, and Wilmott (1994) extended the method of Lelandfor the case of a portfolio of options. They showed that to find the op-tion portfolio price and hedging strategy one needs to solve the followingnonlinear PDE

V

t+ rS

V

S+

1

22S2

1 K sign

2V

S2

2V

S2 rV = 0, (4)

where

K =

8

t, (5)

and where sign() is the sign function. The comparison of the PDEs (1) and(4) allows us to introduce a new parameter

2m = 2

1 K sign

2V

S2

, (6)

and interpret it as the modified volatility. This volatility adjustment de-pends on the sign of the second derivative of the option portfolio price with

respect to the underlying asset price. Recall that this second derivative isknown as gamma (the sensitivity of the option portfolio delta to the under-lying asset price)

=

S=

2V

S2. (7)

It is widely known that the modified volatility increases/decreases an optionprice to account for the amount of hedging transaction cost. However, only afew know that the Lelands hedging very often outperforms the Black-Scholeshedging at fixed regular intervals even in the case when the hedger startswith the same initial value of the replicating portfolio (see, for example, Mo-hamed (1994), Clewlow and Hodges (1997) or Zakamouline (2006b)). The

only difference in between the Black-Scholes hedging strategy and the Le-lands hedging strategy is in the value of hedging volatility. This implies thatthe Lelands modification of volatility optimizes somehow the Black-Scholeshedging strategy in the presence of transaction costs. Zakamouline (2006b)explains in details how the Lelands modified hedging volatility works. Nowwe present shortly the explanation.

To understand how the Lelands modified hedging volatility improvesthe risk-return tradeoff of the Black-Scholes hedging strategy, we need first

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to make the following two observations: Observe that the (total) replica-

tion error of a hedging strategy can be subdivided into a hedging error andtransaction costs. Note that both the hedging error and the amount oftransaction costs of a discretely adjusted delta-neutral replicating strategyare path-dependent. In particular, the amount of transaction costs dependson the absolute value of the option gamma (see Leland (1985) or Hoggardet al. (1994)). The hedging error depends on the values and the signs of theoption theta and gamma (see Boyle and Emanuel (1980)). In short, the Le-lands modified hedging volatility makes the hedging error be negativelycorrelated with transaction costs: the hedging error becomes positive whentransaction costs are large, and the hedging error becomes negative whentransaction costs are small. Thus, the Lelands modified volatility equal-izes the replication error across different stock paths. This reduces the riskof the hedging strategy as measured by the variance of the replication error.To illustrate the aforesaid, Figure 1 presents the simulation results of hedg-ing a plain vanilla European call option according to the Lelands strategyas compared to hedging according to the Black-Scholes strategy.

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50 60 70 80 90 100 110 120130 140 150

MeanofReplicationError

Stock Price at Maturity

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50 60 70 80 90 100 110 120130 140 150


Stock Price at Maturity

(a) Short vanilla call (b) Long vanilla call

Figure 1: Comparison of the expected replication errors of the Lelands strategy(dashed line) against the Black-Scholes strategy (solid line) across different stockprices at maturity. The option is a vanilla call with strike K = 100 that is rehedgedevery t = 1/100.

Note that the Lelands volatility adjustment reduces the risk of a repli-cating strategy. However, the reduction of risk can happen at the expense

of reducing the returns of a replicating strategy. When a hedger is shortgamma, the option is hedged with an increased hedging volatility. An in-creased hedging volatility decreases the absolute value of the option gammain the region where the option gamma is high. This reduces the amountof hedging transaction costs (see Zakamouline (2006b)). On the contrary,when a hedger is long gamma, the option is hedged with a decreased hedgingvolatility. A decreased hedging volatility increases the absolute value of theoption gamma in the region where the option gamma is high. This increases

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the amount of hedging transaction costs. This is illustrated in Table 1.

Strategy Black-Scholes LelandMean of replication error -3.632 -3.276

Std. deviation of replication error 1.443 0.938

(a) Short vanilla call

Strategy Black-Scholes LelandMean of replication error -3.632 -4.361

Std. deviation of replication error 1.329 0.778

(b) Long vanilla call

Table 1: Comparison of the risk-return tradeoffs of the Lelands strategyagainst the Black-Scholes strategy. The option is a vanilla call that is re-

hedged every t = 1/100.

2.4 The Delta Tolerance Strategy

This commonly used strategy prescribes rehedging to the Black-Scholes deltawhen the hedge ratio1 moves outside of the prescribed tolerance from theperfect hedge position. More formally, the series of stopping times is recur-sively given by

1 = t, i+1 = inf

i < < T :

V

S

> H

, i = 1, 2, . . . ,

where VS is the Black-Scholes hedge, and H is a given constant tolerance.The intuition behind this strategy is pretty obvious: The parameter H is aproxy for the measure of risk of the replicating portfolio. More risk aversehedgers would choose a low H, while more risk tolerant hedgers will accepta higher value for H. This strategy for hedging a single option seems tobe first suggested by Whalley and Wilmott (1993). It is straightforward toapply this strategy to the hedging an option portfolio.

The hedging proceeds as follows: at time t the hedger constructs a repli-cating portfolio by purchasing/selling (t) = VS shares of the stock. Thenthe hedger monitors continuously until T the discrepancy between the hedgeratio and the perfect hedge position. When this discrepancy exceeds H, the

rebalancing occurs so as to bring the hedge ratio to the perfect hedge posi-tion.

2.5 The Asset Tolerance Strategy

This strategy is based on monitoring the moves in the underlying asset priceand was suggested first by Henrotte (1993) for hedging a single option. The

1This is defined as the relative quantity of the underlying asset held in the hedgingportfolio.

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strategy prescribes rehedging to the Black-Scholes delta when the percentage

change in the value of the underlying asset exceeds the prescribed tolerance.More formally, the series of stopping times is recursively given by

1 = t, i+1 = inf

i < < T :

S() S(i)S(i)

> h , i = 1, 2, . . . ,where h is a given constant percentage. The intuition behind this strategyis similar to that of the delta tolerance strategy. It is also straightforwardto apply this strategy to the hedging an option portfolio.

The hedging proceeds as follows: at time t the hedger constructs a repli-cating portfolio by purchasing/selling (t) = VS shares of the stock. Thenthe hedger monitors continuously until T the percentage change in the value

of the underlying asset. When this percentage change exceeds h, the rebal-ancing occurs so as to bring the hedge ratio to the perfect hedge position.

3 The Utility Based Hedging

3.1 The Method

Hodges and Neuberger (1989) pioneered the utility-based option pricingand hedging approach based that explicitly takes into account the hedgersrisk preferences. The key idea behind the utility based approach is the indif-ference argument: The so-called reservation price of an option is definedas the amount of money that makes the hedger indifferent, in terms of ex-pected utility, between trading in the market with and without the option.In many respects such an option price is determined in a similar mannerto a certainty equivalent within the expected utility framework, which isa well grounded pricing principle in economics. The difference in the twotrading strategies, with and without the option, is interpreted as hedgingthe option.

Initially, the utility-based approach was applied to the pricing and hedg-ing of single options. However, this method can be easily applied for pricingand hedging a portfolio of options. The starting point for the utility basedoption pricing and hedging approach is to consider the optimal portfolioselection problem of the hedger who faces transaction costs and maximizes

expected utility of his terminal wealth. The hedger has a finite horizon [t, T].For the simplicity of the exposition we assume that there are no transac-tion costs at terminal time T. The hedger has the amount xt in the bankaccount, and yt shares of the stock at time t. We define the value functionof the hedger with no options as

J0(t, xt, yt, St) = max Et[U(xT + yTST)], (8)

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where U(z) is the hedgers utility function. Similarly, the value function of

the hedger who, for example, sells the option portfolio is defined by

J1(t, xt, yt, St) = max Et[U(xT + yTST V(T, ST))]. (9)

Finally, the price of the option portfolio is defined as the compensation psuch that

J1(t, xt + p, yt, St) = J0(t, xt, yt, St). (10)

The solutions to problems (8), (9), and (10) provide the unique price and,above all, the optimal hedging strategy. Unfortunately, there are no closed-form solutions to all these problems. As a result, the solutions have to beobtained by numerical methods. The existence and uniqueness of the so-lutions were rigorously proved by Davis, Panas, and Zariphopoulou (1993).For implementations of numerical algorithms, the interested reader can con-sult Davis and Panas (1994) and Clewlow and Hodges (1997).

It is usually assumed that the hedger has the negative exponential utilityfunction

U(z) = exp(z); > 0,where is a measure of the hedgers (constant) absolute risk aversion. Thischoice of the utility function satisfies two very desirable properties: (i) thehedgers strategy does not depend on his holdings in the bank account, (ii)the computational effort needed to solve the problem is much lower thanthat in case of a utility function that exhibits a non-constant absolute risk

aversion. This particular choice of utility function might seem restrictive.However, as it was conjectured by Davis et al. (1993) and showed in An-dersen and Damgaard (1999), an option price is approximately invariant tothe specific form of the hedgers utility function, and mainly only the levelof absolute risk aversion plays an important role.

The numerical calculations show that when the hedgers risk aversion israther low, the hedger implements mainly a so-called static hedge, whichconsists in buying shares of the stock at time t and holding them until theoption maturity T. When the hedgers risk aversion increases, he starts torebalance the hedging portfolio in between (t, T). Recall that in the frame-work of the utility based hedging approach the hedging strategy is definedas the difference, () = y1()

y0(),

[t, T], between the hedgers

optimal trading strategies with and without options. When the hedgersrisk aversion is moderate, it is impossible to give a concise description ofthe optimal hedging strategy. However, when the hedgers risk aversion israther high, then we can assume that y0() 0 (that is, the hedger does notinvest in the underlying stock in the absence of option contracts) and theoptimal hedging strategy can be conveniently described as () = y1().In the latter case the numerical calculations show that the optimal hedgeratio is constrained to evolve between two boundaries, l and u, such

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-0.1

0

0.1

0.2

0.3

0.4

0.5

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Stock Price

Hedging BoundariesBlack-Scholes Delta

Figure 2: Optimal hedging strategy versus the Black-Scholes delta for a shortButterfly Spread and the following model parameters: = 2.0, = 0.01, St = 100,

= 0.25, = r = 0.05, T t = 0.5, and exercise prices 80, 100, and 120.

that l < u. As long as the hedge lies within these two boundaries,l u, no rebalancing of the hedging portfolio takes place. That iswhy the region between the two boundaries is commonly denoted as the notransaction region. As soon as the hedge ratio goes out of the no transac-tion region, a rebalancing occurs in order to bring the hedge to the nearest

boundary of the no transaction region. In other words, if moves belowl, one should immediately transact to bring it back to l. Similarly, if moves above u, a rebalancing trade occurs to bring it back to u. Figures2 and 3 illustrate the optimal hedging strategy.

3.2 Hedging to a Fixed Bandwidth Around Delta

Despite a sound economical appeal of the utility based option hedging ap-proach, it does have a number of disadvantages: the model is cumbersometo implement and the numerical computations are time consuming. Onecommonly used simplification of the utility-based hedging strategy (see, for

example, Martellini and Priaulet (2002)) is known as hedging to a fixedbandwidth around delta. This strategy prescribes to rehedge when thehedge ratio moves outside of the prescribed tolerance from the correspond-ing Black-Scholes delta. More formally, the boundaries of the no transactionregion are defined by

=V

S H, (11)

where VS is the middle of the hedging bandwidth that equals the Black-Scholes hedge, and H is some constant which gives a constant size of the

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0

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40 60 80 100 120 140 160 180

Stock Price

Hedging BoundariesBlack-Scholes Delta

Figure 3: Optimal hedging strategy versus the Black-Scholes delta for a long BullSpread with exercise prices 80 and 120, and the rest of the model parameters as in

Figure 2.

hedging bandwidth. This strategy is closely related to the delta tolerancestrategy. The value of H is determined by the hedgers risk tolerance. Theessential difference between these two strategies is that in the hedging to afixed bandwidth around delta a rebalancing brings the hedge ratio to thenearest boundary of the bandwidth, while in the delta tolerance strategy

a rebalancing brings the hedge ratio to the perfect hedge position in theabsence of transaction costs.

However, this strategy is, in fact, an over-simplified utility-based hedgingstrategy. As it is clearly seen from Figures 2 and 3, the middle of the hedgingbandwidth does not coincide with the Black-Scholes hedge. Moreover, thesize of the hedging bandwidth in the utility-based hedging is not constant.We will illustrate this below.

3.3 The Asymptotic Analysis of Whalley and Wilmott

Since there are no explicit solutions for the utility based hedging strategy

with transaction costs and the numerical methods are computationally hard,for practical applications it is of major importance to use other alternatives.One of such alternatives is to obtain an asymptotic solution. In asymptoticanalysis one studies the solution to a problem when some parameters in theproblem assume large or small values.

Whalley and Wilmott (1997) were the first to provide an asymptoticanalysis of the model of Hodges and Neuberger (1989) assuming that trans-action costs are small. Using formal matched asymptotics, they showed that

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the boundaries of the no transaction region are given by

=V

S Hww = V

S

3

2

er(Tt)S2

13

, (12)

where, again, VS is the Black-Scholes hedge. It is important to note thatthe asymptotic method of Whalley and Wilmott (1997) is of general appli-cability. Even though the authors considered the pricing and hedging of ashort European call option, the final expressions for the option price andhedging strategy are valid for any option, including an option portfolio as aspecial case of a complex option.

It is easy to see that the optimal hedging bandwidth is not constant,

but depends in a natural way on a number of parameters. As 0,the optimal hedge approaches the Black-Scholes hedge. As increases, thehedging bandwidth decreases in order to decrease the risk of the hedgingportfolio. The dependence of the hedging bandwidth on the option gammais also natural, as we expect to rehedge more often in regions with highgamma. Moreover, it agrees quite well with the results of exact numericalcomputations, see Figures 4 and 5. The importance of the dependence of thesize of the hedging bandwidth on the option gamma was emphasized by Za-kamouline (2005) by comparing the empirical performances of the Whalleyand Wilmott asymptotic strategy against the hedging to a fixed bandwidthstrategy: without this dependence there are lots of unnecessary rebalanc-ing when the price of the underlying changes insignificantly. That is why

the Whalley and Wilmott asymptotic strategy outperforms the hedging to afixed bandwidth strategy. Moreover, Zakamouline (2005) showed that whenthe option gamma is huge and the size of a constant hedging bandwidthis small relative to the range of the option delta, the performance of thehedging to a fixed bandwidth strategy might be worse than that of theBlack-Scholes hedging at fixed intervals strategy.

3.4 The Asymptotic Analysis of Barles and Soner

Barles and Soner (1998) performed an alternative asymptotic analysis ofthe same model assuming that both the transaction costs and the hedgers

risk tolerance are small. They found that to find the option value one needsto solve the following nonlinear PDE

V

t+ rS

V

S+

1

22S2

1 f

er(Tt)2S2

2VS2

rV = 0. (13)

Again, formally the option value is equal to the Black-Scholes value with anadjusted variable volatility given by

2m = 2 (1 Kbs) = 2

1 f

er(Tt)2S2

. (14)

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40 60 80 100 120 140 160 180 200Stock Price

Total Hedging BandwidthAbsolute Value of Gamma

Figure 4: The form of the optimal hedging bandwidth (u l)S, obtainedusing the exact numerics, versus the absolute value of the option portfolio gamma

times the stock price, ||S, for a short Butterfly Spread and the following modelparameters: = 0.1, = 0.01, St = 100, = 0.25, = r = 0.05, T t = 0.5, andexercise prices 80, 100, and 120.

40 60 80 100 120 140 160 180 200

Stock Price

Total Hedging BandwidthAbsolute Value of Gamma

Figure 5: The form of the optimal hedging bandwidth (ul)S, obtained usingthe exact numerics, versus the absolute value of the option portfolio gamma times

the stock price, ||S, for a long Bull Spread with exercise prices 80 and 120, andthe rest of the model parameters as in Figure 4.

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-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

40 60 80 100 120 140 160 180 200

Stock Price

Middle of Hedging BandwidthHoggard, Whalley and Wilmott Delta

Black-Scholes Delta

Figure 6: The middle of the optimal hedging bandwidth, obtained using the exactnumerics, versus the Black-Scholes delta and the Hoggard, Whalley and Wilmott

delta with K = 0.54 for a short Butterfly Spread and the following model parame-

ters: = 3.0, = 0.01, St = 100, = 0.25, = r = 0.05, T t = 0.5, and exerciseprices 80, 100, and 120.

In contrast to (6), this volatility adjustment depends not only on the sign,but also on the value of the option gamma. The optimal hedging strategyconsists in keeping the hedge ratio inside the no transaction region given by

=V(m)

S Hbs = V(m)

S 1

Sg

2S2

, (15)

whereV(m)

S is the Black-Scholes hedge with an adjusted volatility.The comparative statics for the Barles and Soner optimal hedging band-

width is similar to that of the Whalley and Wilmott one: the hedging band-width increases when either the level of the transaction costs, the hedgersrisk tolerance, or the option gamma increases.

Unfortunately, the functions f() and g() depend on the option payoffand Barles and Soner gave their forms only for the case of a plain vanilla call.Consequently, one cannot use the asymptotic strategy of Barles and Sonerfor hedging a portfolio of options. However, it is important to emphasize thatthe Barles and Soner volatility adjustment works similarly2 to the Lelandsvolatility adjustment. For a short call option the modified volatility is higherthan the original volatility. On the contrary, for a long call option themodified volatility is lower than the original volatility.

Even though Barles and Soner performed the asymptotic analysis of theutility based pricing and hedging for a particular type of option, the follow-

2This is also clearly seen from the comparison of equations (13) and (4).

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-0.8

-0.6

-0.4

-0.2

0

40 60 80 100 120 140 160 180 200

Stock Price

Middle of Hedging BandwidthHoggard, Whalley and Wilmott Delta

Black-Scholes Delta

Figure 7: The middle of the optimal hedging bandwidth, obtained using the exactnumerics, versus the Black-Scholes delta and the Hoggard, Whalley and Wilmott

delta with K = 0.54 for a long Bull Spread with exercise prices 80 and 120, and

the rest of the model parameters as in Figure 6.

ing conclusion becomes obvious: the middle of the hedging bandwidth inthe utility-based hedging does not coincides with the Black-Scholes delta.This agrees quite well with the results of the exact numerical computations,see Figures 6 and 7. This essential feature of the utility-based hedging

strategy was already observed by Hodges and Neuberger (1989) and em-phasized by Clewlow and Hodges (1997) using the results of Monte Carlosimulations. Indeed, as the hedgers risk aversion increases, the hedgingbandwidth decreases in order to decrease the risk of the hedging portfolio.Without volatility adjustment, the Whalley and Wilmott asymptotic, deltatolerance, asset tolerance, and hedging to a fixed bandwidth around deltastrategies converge to the continuous time Black-Scholes strategy. That is,in the limit as we increase the hedgers risk aversion, the amount of trans-action costs tends to infinity. On the contrary, in the correct utility-basedstrategy, as the hedgers risk aversion increases, the decrease in the size of thehedging bandwidth is largely compensated by the increase in the volatility

adjustment. Figures 6 and 7 also show that the optimal volatility adjust-ment in the utility-based hedging is very similar to the Hoggard, Whalley,and Wilmott volatility adjustment when we appropriately choose the valueof parameter K in the PDE (4). Note that K determines the degree ofvolatility adjustment.

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3.5 The Approximation Method of Zakamouline

Recall that in asymptotic analysis one studies the limiting behavior of theoptimal hedging policy as one or several parameters of the problem approachzero. Even though asymptotic analysis can reveal the underlying structureof the solution, under realistic parameters this method provide not quiteaccurate results. Zakamouline (2004) compared the performance of asymp-totic strategies against the exact strategy and found out that under realisticmodel parameters an asymptotic strategy performs noticeably worse thanthat obtained from the exact numerical solution. The explanation lies inthe fact that, when some of the model parameters are neither very smallnor very large, an asymptotic solution provides not quite accurate results.In particular, as compared to the exact numerical solution, under realistic

parameters the size of the hedging bandwidth and the volatility adjustmentobtained from asymptotic analysis are overvalued. What is more impor-tant, an asymptotic solution showed to be unable to sustain a correct inter-relationship between the size of the hedging bandwidth and the degree ofvolatility adjustment. The significance of the correct interrelationship couldhardly be overemphasized: The empirical testing of the hedging strategiesrevealed that either undervaluation or overvaluation of the volatility adjust-ment (with respect to the size of the hedging bandwidth) results in a drasticdeterioration of the performance of a hedging strategy.

Zakamouline (2004) and Zakamouline (2006a) summarized the stylizedfacts about the nature of the utility-based hedging strategy and suggested a

general specification of the optimal hedging policy for a single plain vanillaEuropean option. The careful visual inspection of the numerically calculatedoptimal hedging policy together with the insights from asymptotic analy-sis advocate for the following general specification of the optimal hedgingstrategy

=V(m)

S (H1 + H0), (16)

where m is the adjusted volatility given by

2m = 2 (1 K) . (17)

The term H1 is closely related to Hww in the Whalley and Wilmott hedging

strategy (see equation (12)) and to Hbs in the Barles and Soner hedgingstrategy (see equation (15)). The main feature of H1 is that this termdepends on the option gamma. Note that the option gamma approaches zeroas the option goes farther either out-of-the-money (S 0) or in-the-money(S ). This means, in particular, that the size of the no transactionregion in an asymptotic strategy also approaches zero. On the contrary, theexact numerics show that, when the option gamma goes to zero, the size ofthe no transaction region times the stock price approaches a constant value,see, for example, Figures 4 and 5. It turns out that this constant value

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is actually the size of the no-transaction region in the optimal portfolio

selection problem without options. To reflect this feature of the optimalhedging policy Zakamouline introduced the term H0. The reason for theabsence ofH0 in an asymptotic strategy is the fact that when either or 0 then the size of H0 becomes much less than the size of H1 (see,for example, equations (20) and (21) below). However, when the hedgersrisk aversion is not very high, the presence of H0 is important. This factwas emphasized in Zakamouline (2004). Finally, K is closely related to Kbsin the Barles and Soner volatility adjustment (see equation (14)) and to Kin the Hoggard, Whalley, and Wilmott volatility adjustment (see equation(6)).

Zakamouline (2004) suggested to use an alternative to the asymptoticanalysis, namely, the approximation method. The general description of theapproximation technique he employed can be found in, for example, Judd(1998) Chapter 6. Under approximation it is meant the following: one hasa rather slow and cumbersome way to compute the optimal hedging policywith transaction costs and wants to replace it with simple and efficientapproximating function(s). To do this, one first specifies a flexible functionalform of the optimal hedging policy. Then, given a functional form, theparameters are chosen to provide the best fit to the exact numerical solution.This second stage is known as model calibration.

In short, Zakamouline assumed the following functional form of the ap-proximating function for H0, H1, and K:

H0 =

1

2

3

S

4

(T t)5

, (18)H1 = K =

123S45e6r(Tt)(T t)7 , (19)where , 1, . . . , k are parameters to be chosen in order to achieve the bestfit. The results of his estimations of the best-fit parameters for hedging ashort European call option, after some rounding off the values of parameters and i, give the following approximating functions

H0 =

S 2(T t) , (20)

H1 = 1.12 0.31(T t)0.05

er(Tt)

0.25

||

0.5

, (21)

K = 5.76 0.78

(T t)0.02

er(Tt)

0.25 S2||0.15 . (22)

Then, Zakamouline tested empirically his approximation strategy againstboth the asymptotic strategies and some other well-known strategies. Heevaluated the performance of different hedging strategies within the unifiedmean-variance and the mean-VaR frameworks and found that his approxi-mation strategy outperforms all the others.

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4 A Simplified Utility-Based Hedging Strategy and

the Method of Finding the Optimal Parameters

4.1 A Simplified Parameterized Description of the Utility-

Based Hedging Strategy

The approximation methodology presented in Zakamouline (2004) producesa clearly superior result as applied to the problem of finding a closed-formexpressions for the optimal hedging strategy for a specific option or an optionportfolio. However, this methodology does retain the main disadvantage wewould like to get rid off: to approximate the optimal strategy for a specificoption one needs to start with the numerical calculations of the optimalhedging strategy for a large set of the model parameters. Since the nu-

merical algorithm is cumbersome to implement and the calculation of theoptimal hedging strategy is time consuming, the approximation methodol-ogy is unlikely to be commonly used by the practitioners.

In the preceding section we presented the description of the nature ofthe optimal hedging strategy for a portfolio of options. This presentationsuggests that the optimal hedging strategy is rather simple and has threeessential features:

The presence of the no transaction region such that if the hedge ratiolies outside of the no transaction region a rebalancing occurs in order tobring the hedge to the nearest boundary of the no transaction region.

The form of the no transaction region is mainly derived from the formof the absolute value of the option portfolio gamma. In addition weknow that when the option portfolio gamma tends to zero, the size ofthe no transaction region tends to some constant.

The middle of the no transaction region does not coincide with theBlack-Scholes delta. This feature of the optimal hedging strategy canbe conveniently described as a modified hedging volatility.

The hedging to a fixed bandwidth around delta strategy given by

=V

S H,

is, in fact, an over-simplified utility-based hedging strategy. That is, thisstrategy lacks two essential features of the utility-based hedging strategy:the optimal form of the hedging bandwidth and the volatility adjustment.The absence of these features leads to a bad empirical performance of thishedging strategy when the size of the no transaction region, H, is rathersmall (i.e., when the hedgers risk aversion is high) and when the optionpayoff is either discontinuous or a rather complicated function of the under-lying, see Zakamouline (2005). However, in hedging an option position that

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has more or less smooth payoff and when the hedgers risk aversion is not

very high, the hedging to a fixed bandwidth strategy shows a good empiricalperformance that is close to that of the exact utility-based hedging strategy,see Martellini and Priaulet (2002) and Zakamouline (2005).

To reflect all the three essential features of the utility-based hedgingstrategy, we propose to describe it similarly to as in Zakamouline (2004)and Zakamouline (2006a)

=V(m)

S (H1 + H0),

where the delta of an option portfolio, V(m)S , is obtained by solving thefollowing PDE with a proper boundary condition

V

t+ rS

V

S+

1

22mS

2 2V

S2 rV = 0,

and where m is the adjusted volatility given by

2m = 2 (1 K) .

We know from Zakamouline (2004) that H0 is the same for all option posi-tions. The challenge now is to find simple functional forms for H1 and Kthat are suitable for any option portfolio.

As a motivation for the choice of the functional form for H1, let us con-sider the Whalley and Wilmott asymptotic strategy given by (12). Notethat some parameters in (12) are constant during the life of an option posi-tion, but the others are variable. For practical applications it makes senseto present the Whalley and Wilmott asymptotic strategy in the followingform

=V

S Hww = V

S h

er(Tt)S2

13

, (23)

where h =32

13

is some constant parameter reciprocal to the hedgers

risk aversion. Similarly to the asymptotic result of Whalley and Wilmott,we assume that the form of the hedging bandwidth H1 is given by

H1(t, S) = h1er(Tt)

S

||

, (24)

where , and are some parameters to be estimated. To estimate theseparameters we need to compute the bandwidth H1(t, S) and estimate thebest-fit parameters3 for , and . To do this, we define an option portfo-lio, fix the set of parameters r,,,,, and calculate numerically the up-per yu0 (t, S) and the lower y

l0(t, S) boundaries of the no transaction region

3See Zakamouline (2004) for a detailed description of the estimation procedure.

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without the option portfolio, and the upper yu1 (t, S) and the lower yl1(t, S)

boundaries of the no transaction region with the option portfolio. Then

H0(t, S) =yu0 (t, S) yl0(t, S)

2,

H1(t, S) =yu1 (t, S) yl1(t, S)

2 H0(t, S).

We measure the goodness of fit using the L2 norm. This largely amounts tousing the techniques of ordinary linear regression after the log-log transfor-mation of (24). That is, we find the parameters , and by solving theproblem

minh1,,,m

log(H1(t, S)) log(h1) + r(T t) log(S)log(|(t, S)|)2,

where m is the number of different data points in t and S.Our estimations of the best fit parameters for show that this parameter

is almost insignificant. That is, in the most cases we cannot reject thehypothesis that the value of is significantly different from zero. This meansthat we can safely assume the following form for the hedging bandwidth H1

H1(t, S) = h1S||, (25)

Unfortunately, our estimations of the best fit parameters for and show

that the values of and depend not only on a particular option portfolio,but also on the hedgers risk aversion. How do we proceed further? Wereformulate the question as follows: How important is the variable S in(24)? It is easy to find out by comparing the goodness of fit, given by R2,(equal to the regression sum of squares divided by the total sum of squares),from the estimation of (25) and the goodness of fit, R2, from the estimationof

H1(t, S) = h1||. (26)The goodness of fits for some popular combinations of standard options isgiven in Table 2.

By studying the table it becomes clear that the gamma of an option

portfolio alone explains at least 84% of the variation of the bandwidth H1.Even though is significant in the linear regression and the variable S helpsto improve the goodness of fit, we can disregard S because the marginalimprovement of the goodness of fit provided by this variable is small, andoften even insignificant.

Now we are left with only and estimate (26) for different option port-folios and different values of the hedgers risk aversion. Our study showsthat (0.3, 2.5) and depends on the composition of an option portfolioand the hedgers risk aversion. That is, there is no single value of that

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Combination R2,, % R2, % R

2=1, % R

2=2/3, %

Bull Spread 95 92 96 95Bear Spread 95 87 87 90

Long Butterfly Spread 94 93 81 93

Short Butterfly Spread 92 90 95 92

Long Call Ratio Spread 94 92 84 94

Short Call Ratio Spread 92 84 96 88

Long Put Ratio Spread 94 92 63 92

Short Put Ratio Spread 98 91 95 72

Long Straddle 94 92 80 47

Short Straddle 98 91 63 92

Long Strap 99 97 76 42

Short Strap 97 96 54 88Long Strip 99 96 76 39

Short Strip 97 96 54 88

Long Strangle 90 87 87 69

Short Strangle 90 89 80 97

Long Wrangle 99 94 85 73

Short Wrangle 96 95 81 98

Table 2: The goodness of fits for some popular combinations of standard options.The chosen model parameters are: = 1.0, = 0.01, = 0.25, = r = 0.05,

T t = 0.5, and exercise prices 80, 100, and 120. The goodness of fit was measuredin the stock price interval 50

St

150. Note that in the estimation of (25) and

(26) we use a log-log transformation, but in the estimation of (27) we do not use

one. As a result, R2, and R2

are not quite directly comparable with R2

=1 and

R2=2/3.

suits all possible option portfolios and the hedgers risk aversions. What wecan do is the following: we can either define a single value for that belongsto the interval (0.3, 2.5), for example = 1, or we can left it as a parameterto be determined. To illustrate how much we explain in the variation of

H1(t, S) = h1||=value (27)

if we hold fixed at some value4, Table 2 reports also the goodness offits for = 1 and = 2/3 (recall that the latter value is obtained in theasymptotic analysis of Whalley and Wilmott). We see that even if is fixed,the goodness of fit remains quite high and when = 1 in the worst case weexplain at least 54% of the variation of the bandwidth H1. Consequently,for the sake of simplicity we propose to use = 1.

4That is, in this case we estimate only the value of h1.

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Now we turn on to the determination of the functional form for K. As

in Zakamouline (2004), we assume that the form of the volatility adjustmentis given byK = ksign()S

|| (28)such that the sign of the volatility adjustment depends on the sign of theoption portfolio gamma and the volatility is adjusted more in regions wherethe absolute value of gamma is high5. k in (28) is some parameter that de-termines the degree of the volatility adjustment. This parameter is constantduring the life of the option position and is proportional to the hedgers riskaversion and the level of transaction costs.

When it comes to the estimation of and in (28), our study showsthat the values of and also depend not only on a particular option

portfolio, but also on the hedgers risk aversion. Again, the option gammais the major factor that explains the variation of the form of K. Whenwe try to estimate the best-fit parameter for different option positionsand different values of the hedgers risk aversion, we find that (0, 0.3).Again, the optimal value of is not unique. We suggest the choice =0 which might seem a bit surprising, but is completely equivalent to theHoggard, Whalley, and Wilmott volatility adjustment and could probablybe justified by the following: the presence of the volatility adjustment ismuch more important than its slight dependence on the option gamma. Asan illustration of the fact that the Hoggard, Whalley, and Wilmott deltawith appropriately chosen volatility adjustment is very close to the middle

of the hedging bandwidth in the utility-based hedging strategy, see Figures6 and 7.Consequently, after all the experiments, we propose the following simpli-

fied description of the utility-based hedging strategy for any option portfolio

=V(m)

S

h1|| + h0S(T t)

, (29)

where the delta of an option portfolio is obtained by solving the followingPDE with a proper boundary condition

V

t+ rS

V

S+

1

22S2 1 ksign

2V

S22V

S2 rV = 0. (30)

In contrast to the hedging to the fixed bandwidth strategy given by (11), oursimplified utility-based hedging strategy is not over-simplified, but preservesall the essential features of the utility-based hedging strategy: (i) existence ofthe no-transaction region, (ii) optimal form of the no-transaction region, and(iii) the volatility adjustment. However, despite a seemingly very drastic

5This is clearly seen for a single option, see, for example, Zakamouline (2004) or Za-kamouline (2006a).

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plified utility-based hedging strategy that constitute the efficient frontier of

the strategy. The essential feature of a point belonging to the efficient fron-tier is that the risk-return tradeoff given by this point cannot be improved.That is, any other combinations of (h1, h0, k) which produce a lower riskhave a lower return, or, similarly, any other combinations of (h1, h0, k)which produce a higher return have a higher risk. Thus, the main ideabehind the our method of finding the optimal parameters of the simplifiedhedging strategy is to use the simulation analysis in order to find the efficientfrontier of the hedging strategy in some risk-return space.

Before proceeding to the formal presentation of our method, we need toagree on a suitable risk-return framework. The expected replication erroris the only sensible candidate for the return measure of a hedging strategy.As to the risk measure, there are many metrics of risk. In the context ofoption hedging, the most popular risk metric is the variance of the replica-tion error of a hedging strategy. Therefore, we propose to find the optimalparameters of the simplified utility-base hedging strategy in the followingrisk-return space: mean replication error - variance of replication error. Asa justification for the chosen risk-return space consider the following: Wesuppose that a hedger has a relatively high risk aversion so that he does notinvest in the risky asset if he does not have to. The hedger faces the optionhedging problem where he uses the amount of p = xt + ytSt to construct aself-financing replicating portfolio to hedge the option position. It is widelyknown that in case of the negative exponential utility the hedgers choicein y, t

< T, is independent of the hedgers wealth. Consequently, we

can assume that the hedger has zero initial wealth. Recall that the hedgersproblem is

max E[ exp((xT + yTST V(T, ST)))] = max E[ exp(RE)], (31)

where xT + yTST V(T, ST) is now the replication error, RE, of a hedgingstrategy. Using a Taylor series expansion it is easy to show that the problem(31) roughly amounts to

max E[RE] 12

Var[RE]. (32)

That is, the hedgers problem can be interpreted as finding the optimal risk-return tradeoff between the risk and return of a hedging strategy where thereturn is given by the mean replication error and the risk is measured bythe variance of the replication error.

Now we turn on to the formal presentation of our optimization methodin the chosen risk-return space where we search for the combinations ofthe triple of parameters (h1, h0, k) which belong to the efficient frontier.We choose some triple of parameters of the hedging strategy, (h1, h0, k),perform path simulations, and estimate the return, = (h1, h0, k), and

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the risk, = (h1, h0, k), associated with this particular hedging strategy.

Observe that for any risk averse hedger the hedging strategy with parameters(h1, h0, k

) is considered to be better than the strategy with parameters

(h1, h0, k) if

(h1, h0, k

) (h1, h0, k) and (h1, h0, k) (h1, h0, k). (33)

That is, if the strategy with (h1, h0, k

) provides either higher return with

less or equal risk, or less risk with higher or equal return than the strategywith (h1, h0, k).

The optimization method we propose is based on a sequential improve-ment of the risk-return tradeoff of a hedging strategy. That is, starting withsome (h1, h0, k) we search for a new (h

1, h

0, k

) = (h1+h1, h0+h0, k+

k) such that the risk-return tradeoff of the strategy with (h1, h

0, k

) is

better than that of the strategy with (h1, h0, k). The most general proce-dure for finding the risk-return improvement step could be interpreted asfinding a unit vector u = a,b,c such that

Du(h1, h0, k) 0, (34)

andDu(h1, h0, k) 0, (35)

where Du(h1, h0, k) and Du(h1, h0, k) are the directional derivatives of(h1, h0, k) and (h1, h0, k), respectively, in the direction of the unit vectoru

. The reader is reminded that the directional derivative of some functionf(x,y ,z) in the direction of the unit vector u = a,b,c is defined as

Duf(x,y ,z) = limh0

f(x + ha,y + hb,z + hc) f(x,y ,z)h

= fx(x,y ,z)a + fy(x,y ,z)b + fz(x,y,z)c = f(x,y,z)u.Consequently, to implement an improvement step from the point with(h1, h0, k) we need first to find the partial derivatives of and at thispoint and then to find a vector u such that both (34) and (35) are satisfied.Note that the vector u is not unique. However, since our goal is to findthe points on the efficient frontier, the actual path from some (h1, h0, k) to

(h

1, h

0, k

) belonging to the efficient frontier is not important.A simple practical realization of the optimization method is based on

the implementation of an improvement step with respect to (h1, h0) and aconsequent improvement step with respect to (h1, k). Note that the start-ing point for our optimization method is the risk-return tradeoff given bythe benchmark strategy (h1, 0, 0). The introduction of h0 > 0 and k > 0helps to improve the risk-return tradeoff of the benchmark strategy. Sup-pose that some (h1, h0, k) is an improvement of the benchmark strategyand (h1, h0, k) and (h1, h0, k) are the corresponding return and the risk

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of this improvement. An improvement step with respect to (h1, h0) is carried

out in the following manner. We define the step sizes h1 and h0, imple-ment the simulations and find the risk-returns tradeoffs in the points with(h1 + h1, h0, k) and (h1, h0 + h0, k). Then we estimate the derivativesof with respect to h1 and h0

h1=

(h1 + h1, h0, k) (h1, h0, k)h1

,

h0=

(h1, h0 + h0, k) (h1, h0, k)h0

.

Similarly, we estimate the derivatives of with respect to h1 and h0. Nowgiven a step size h0 we want to find a step size h1 such that

= (h1 + h1, h0 + h0, k) (h1, h0, k) > 0,

= (h1 + h1, h0 + h0, k) (h1, h0, k) < 0.The step size h1 is not unique. That is, there is some degree of freedom indetermining h1. Therefore, we need to impose an additional condition inorder to determine the step size h1. For example, this condition could be

= . (36)

That is, we want to move in the north-west direction in the chosen risk-return space. It is easy to check that this condition gives us

h1 = h0

+ h0h1

+ h1

h0.

In addition we need to check that

=

h1h1 +

h0h0 < 0. (37)

Alternatively, we can check that > 0. If we are able to find the step sizeh1 such that the condition (37) is satisfied, we conclude that an improve-ment step with respect to (h1, h0) is feasible and the risk-return tradeoff of

the point with (h1 + h1, h0 + h0, k) becomes a new improvement of thebenchmark strategy.

Similarly we can implement an improvement step with respect to (h1, k).In this case given the step size k we also search for h1 such that

= (h1 + h1, h0, k + k) (h1, h0, k) > 0,

= (h1 + h1, h0, k + k) (h1, h0, k) < 0.

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Despite the simplicity of the stepwise risk-return improvement algorithm

described above, in a practical realization the algorithm is very time con-suming since in order to estimate the risk and return of a hedging strategywith a sufficiently high precision we need to simulate a great number ofstock paths. For the purpose of comparison, a numerical computation ofthe exact hedging strategy for a portfolio of non-exotic options in a 250-periods model takes 2-10 minutes6 on a computer with a 2.6 GHz processor,given that the program is implemented on C++. To implement the abovedescribed algorithm and get robust optimization results, in order to estimatethe risk and return of a hedging strategy we need to simulate approximately500000 paths, where each path consists of 250 equally spaced trading datesover the life of an option portfolio. In this case given that the algorithmis also implemented on C++, the computational time for finding the opti-mal parameters (h1, h0, k) of the risk-return optimized hedging strategy bystarting from some benchmark (h1, 0, 0) amounts, on average, to more thanan hour.

To summarize, the optimization algorithm described above is very sim-ple, but computationally time intensive. We want to compute an answermore quickly. After having carried out many experiments, we arrived toa faster modification of the algorithm to determine the optimal triple ofparameters (h1, h0, k). The algorithm is given in the Appendix in ordernot to overload the paper. Using this algorithm, the computational timeto find the optimal parameters of the simplified utility-based hedging strat-egy is close to that of the exact numerical calculations of the utility-based

hedging strategy for an option position with a non path-dependent payoff.In addition, we want to emphasize one more time that this algorithm alsoworks similarly for an option position with a path-dependent payoff, whilethe exact numerical computations in this case are not feasible.

5 The Simulation Analysis

In this section we provide an empirical testing of our optimized strategyagainst the Black and Scholes, Hoggard, Whalley and Wilmott modifiedvolatility, delta tolerance, asset tolerance, hedging to a fixed bandwidtharound delta, and the Whalley and Wilmott asymptotic strategy. We pro-

vide the simulation results for a long Bull Spread (one long call with strikeK1 and one short call with strike K2 > K1, both call options with assetsettlements) and a short Butterfly Spread (one long call with strike K1,one long call with strike K3, and 2 short calls with strike K2 such thatK1 < K2 < K3, all call options with asset settlements). The model param-eters are T t = 1 year to maturity, St = 100, the volatility = 0.25, andthe drift = r = 0.05. The proportional transaction costs were = 0.01.

6The computational time depends on the specified precision of computation.

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The exercise prices for the long Bull Spread are K1 = 80 and K2 = 120.

The exercise prices for the short Butterfly Spread are K1 = 80, K2 = 100,and K3 = 120.The simulation proceeds as follows: At the beginning, the writer of a

complex option position receives the Black-Scholes value of the complexoption and sets up a replicating portfolio. The underlying path of the stockis simulated according to

S(t + t) = S(t)exp

( 0.52)t +

t

,

where is a normally distributed variable with mean 0 and variance 1. Ateach t a check is made to see if the option needs to be rehedged. If so,the rebalancing trade is performed and transaction costs are drawn from

the bank account. Finally, at expiration, we compute the replication error,that is, the cash value of the replicating portfolio minus the due exercisepayment.

For the Black and Scholes and Hoggard, Whalley and Wilmott strate-gies, we vary the parameter t (rehedging interval). For the other strategies,each path consists of 250 equally spaced trading dates over the life of theoption. In the delta tolerance, asset tolerance, and hedging to a fixed band-width strategies, H and h take values in [0.01, 0.35]. For each value of theparameter of these hedging strategies we generate 100,000 paths and com-pute the mean and variance of the replication error. By varying the value ofthe parameter of a hedging strategy, we span all the possible combinations

of risk and return.For the benchmark strategy we varied the parameter h1 in [0.025, 2.5]. To

find the optimal strategy we used fixed steps for h0 and k and adaptivesteps for h1. To implement the optimization and compute the mean andvariance of the replication error with higher precision we generated 200,000paths for each triple of parameters (h1, h0, k).

Figures 8 and 9 summarize the results of simulations. For both thecomplex options our optimized strategies outperform all the others. In ad-dition, we have carried on similar simulations for different values of themodel parameters r, , , T t, and , and different types of complexoptions. Due to the space limitations, the results of these simulations are

not presented. Qualitatively, the relative performance of different hedgingstrategies remains the same as that in Figures 8 and 9.

PROVIDE SIMULATIONS RESULTS OF HEDGING A PORTFOLIOOF ASIAN OPTIONS. NOTE THAT IN THIS CASE THE DIRECT NU-MERICAL COMPUTATIONS OF THE OPTIMAL HEDGING STRAT-EGY IS OUT OF REACH.

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-3.5

-3

-2.5

-2

-1.5

-1

-0.5

1 1.5 2 2.5 3 3.5 4


Std Deviation of Replication Error

Optimized SimplifiedWhalley and Wilmott Asymptotic

Fixed BandwidthHoggard, Whalley and Wilmott

Delta ToleranceAsset Tolerance

Black and Scholes

Figure 8: Comparison of hedging strategies for a short Butterfly Spread.

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0.5 1 1.5 2 2.5 3 3.5 4



Optimized SimplifiedWhalley and Wilmott Asymptotic

Fixed BandwidthHoggard, Whalley and Wilmott

Delta Tolerance

Asset ToleranceBlack and Scholes

Figure 9: Comparison of hedging strategies for a long Bull Spread.

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6 Conclusion

This section concludes the paper.

Appendix

A Faster Algorithm to Find the Optimal Param-

eters of the Simplified Hedging Strategy

The optimization algorithm described in Section 4.2 is very simple, but com-putationally time intensive. We want to compute an answer more quickly.One possibility is to decrease the number of steps in the stock path. How-ever, this might yield non-optimal computed results. In addition, note thatwhen we simulate a hedging strategy we always estimate the risk and returnwith some error. In order to compute the risk and return without any er-ror, we can use an n-period binary tree for the stock path. Unfortunately,since the transaction policy is path dependent, a binary tree for the stockpath becomes non-recombining. Since in this case the number of tree nodesincreases exponentially as the number of trading dates increases, this alsoleads to very time intensive computations. As a matter of fact, using a non-recombining binary tree it is practically feasible to compute the risk andreturn of a hedging strategy with no more than approximately 32-35 trad-ing dates over the life of an option position. After having carried out many

experiments, we found out that we can use a non-recombining binary treewith a few periods of trading for implementing a risk-return improvementstep with respect to (h1, h0), but for implementing a risk-return improve-ment step with respect to (h1, k) we need many trading dates over the lifeon an option position.

Consequently, we arrived to the following fast two-step procedure todetermine the optimal triple of parameters (h1, h0, k): First using a non-recombining binary tree with a few numbers of periods we implement a fullimprovement of the risk-return tradeoff of a hedging strategy with respect to(h1, h0) by starting from some (h1, 0) (that is, at this step k = 0). Then, fora given pair of (h1, h0) using the simulations of a hedging strategy with many

periods we search for k 0 which minimizes the risk of a hedging strategy.That is, instead of searching for a small improvement step with respect to(h1, h0), and then searching for a small improvement step with respect to(h1, k), we implement a full risk-return improvement by introducing thehedging bandwidth H0 while keeping K = 0, and then we implement afull risk-return improvement by introducing the volatility adjustment K.This separation of the risk-return improvements by means of H0 and Kis possible because the bandwidth H0 is important when the hedger has

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relatively small risk aversion, while for a highly risk averse hedger H0 0.To get the idea, see, for example, formulas (20) and (21) and observe thatas increases, H0 decreases much faster than Hw. On the other hand, thevolatility adjustment is more important for a highly risk averse hedger thanfor a more risk tolerant hedger. It is clearly seen from, for example, formula(22). Note that as increases, the volatility adjustment also increases.

Now we turn on to the detailed presentation of our fast risk-return op-timization algorithm and the illustration of the improvements by meansof Figure 10. In the beginning, we define an initial benchmark set of thetriples of parameters (h1, 0, 0). For example, in our experiments we variedh1 in (0.005, 50). The empirical performance of this benchmark strategy forhedging a short Butterfly Spread is illustrated in Figure 10.

In the first stage, we use a 20-period non-recombining binary tree forthe stock path in order to find the hedging strategy and compute its riskand return. Using this tree we search for a full risk-return improvement ofa hedging strategy by introducing a hedging bandwidth H0. In particular,in accordance with the algorithm described in Section 4.2, starting withsome (h1, 0, 0) we perform a number of improvements steps with respect to(h1, h0) until the improvement is possible. The outcome of this stage is a setof new triples of parameters (h1, h

0, 0). This set of parameters become a new

benchmark for the second improvement stage. The empirical performanceof the new benchmark is illustrated in Figure 10.

In the second stage we use simulations of a hedging strategy with 250trading dates. In particular, using the algorithm proposed in Zakamouline

(2005), we search for k that minimizes the risk of a hedging strategy. Inshort, Zakamouline (2005) proposed to use the risk reduction properties ofthe modified hedging volatility to improve the performances of some well-known hedging strategies in the presence of transaction costs. As applied tothis particular case, the algorithm starts with the benchmark triple of pa-rameters (h1, h

0, k = 0) obtained in the first stage. We simulate the hedging

strategy N times and estimate the risk. Then we change the value of k bysome step k, simulate the new strategy and estimate the risk again. Weproceed in such a manner and find the optimal k = ik, i {0, 1, 2, . . .},which gives the strategy with the lowest possible risk. For the purpose of il-lustration, Figure 10 shows the performances of two hedging strategies with

the optimal volatility adjustment. One strategy has no hedging bandwidthH0 (that is, the starting point for this strategy is the risk-return tradeoffgiven by (h1, 0, 0)), and the other starts with the optimal triple (h

1, h

0, 0)

obtained in the first stage.The computational time to find the optimal combination of (h1, h0) de-

pends on the initial value of h1 and the step size h0. The higher the valueof h1 the longer the computational time (that is, the smaller the hedgersrisk aversion, the larger the hedging bandwidths H1 and H0). If h1 = 50

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-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5 1 1.5 2 2.5 3 3.5 4



Fixed BandwidthBenchmark

Inproved by H0 onlyImrpoved by K

only

Imrpoved first by H0 and then by KExact Numerical

Figure 10: This figure illustrate the performances of the simplified utility-basedhedging strategy for the different stages of optimization. For the purpose of com-

parison, the figure shows the performance of the exact utility-based strategy and

the performance of the hedging to a fixed bandwidth strategy.

then the computational time amounts7 to approximately 5-15 minutes de-pending on the step size h0. The computational time to find the optimalvolatility adjustment depends on the number of simulations N, the step sizek, and the initial value ofh1. Note that in this case the smaller the valueofh1 the longer the computational time (that is, the higher the hedgers riskaversion, the smaller the bandwidth H1 and the higher the optimal volatilityadjustment K). If N = 200000 and h1 = 0.005, the computational timeamounts to approximately 10-20 minutes depending on the step size k.Consequently, the computational time to find the optimal parameters of the

simplified hedging strategy is close to that of the exact numerical calcula-tions of the utility-based hedging strategy for a non path-dependent optionposition.

For the purpose of comparison, Figure 10 shows also the performanceof the exact (numerically computed) utility based hedging strategy and theperformance of the hedging to a fixed bandwidth strategy. Observe that the

7As before, all the times are given under conditions that the program is written onC++ and is run on a computer with 2.6 GHz processor.

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performance of the optimized simplified hedging strategy is very close to

that of the exact hedging strategy when the hedgers risk aversion is ratherlow. However, as the hedgers risk aversion increases, the performance ofthe optimized simplified hedging strategy becomes worse than that of theexact strategy. We guess that the exact utility based hedging strategy hasprobably some other feature besides the volatility adjustment that improvesthe performance when the hedgers risk aversion is very large. However,the performance of the optimized simplified hedging strategy is much betteras compared to that of the hedging to a fixed bandwidth strategy. Forthis particular option position, the optimized simplified hedging strategy iseither 50 % less expensive at the same level of risk exposure, or 50 % lessrisky at the same level of the mean hedging error than the hedging to a fixedbandwidth strategy.

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