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Math. Meth. Oper. Res. (2006) 64: 227236DOI 10.1007/s00186-006-0086-0

O R I G I NA L A RT I C L E

Miklav Mastinek

Discretetime delta hedgingand the BlackScholes modelwith transaction costs

Received: 6 April 2005 / Accepted: 3 March 2006 / Published online: 5 August 2006 Springer-Verlag 2006

Abstract The paper deals with the problem of discretetime delta hedging anddiscrete-time option valuation by the BlackScholes model. Since in the

BlackScholes model the hedging is continuous, hedging errors appear whenapplied to discrete trading. The hedging error is considered and a discrete-timeadjusted BlackScholesMerton equation is derived. By anticipating the time sen-sitivity of delta in many cases the discrete-time delta hedging can be improved andmore accurate delta values dependent on the length of the rebalancing intervals canbe obtained. As an application the discrete-time trading with transaction costs isconsidered. Explicit solution of the option valuation problem is given and a closedform delta value for a European call option with transaction costs is obtained.

Keywords Delta hedging

Transaction costs

1 Introduction

The option valuation problem with transaction costs has been considered exten-sively in the literature. In many papers on option valuation with transaction coststhe discrete-time trading is considered by the continuous-time framework of theBlackScholesMerton partial differential equation (BSM-pde); see e.g. [Leland1985; Boyle and Vorst 1992; Hoggard et al. 1994; Avellaneda and Paras 1994;

Toft 1996]. Since in continuous-time models the hedging is instantenuous, hedg-ing errors appear when applied to discrete trading. In that case perfect replication

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228 M. Mastinek

interval, depends on the length t of the interval. When rebalancing intervals arerelatively small and thus trading takes place very frequently, then the expectedhedging error may be relatively small. For the analysis of the hedging error in

the case where option values are described by the BSM-pde; see e.g. [Boyle andEmanuel 1980; Toft 1996; Mello and Neuhaus 1998]. However in presence of trans-action costs more frequent trading may increase the accumulated costs considerably[Korn 2004; Kocinski 2004].

In continuous-time models the value of delta representing the number of sharesheld in the portfolio varies continuously with time. If V(t, S) is the option valueat time t and S the price of the underlying asset, then the delta value at time t isgiven by the partial derivative VS (t, S) of the current value of V; see e.g. [Blackand Scholes 1973; Merton 1973].

In practice where the trading takes place discretely in time, the number ofshares given by the delta is held constant over the rebalancing interval and thusthe hedging error cannot be eliminated. This means that reducing the risk and thehedging error is of main practical interest. In the case when t is sufficiently small,the current value of delta gives an acceptable value such that the hedging error issmall. However, when the rebalancing is not that frequent, due to transaction costsfor instance, then in general the hedging error will not be small. Moreover, if thedelta is more sensitive with respect to time, the error will further increase.

In the papers of Leland and others, which consider the option pricing prob-

lem with transaction costs by the BlackScholes model, the number of shares keptconstant is given by the BlackScholes delta at time tregardless of the time betweensuccessive rehedgings and the time sensitivity of delta..

In this paper the time between rehedgings and the time sensitivity of deltais considered. The objective is to incorporate the time change of delta over therebalancing interval into the model implicitly so that the resulting equation canbe directly transformed to the BSM pde and solved as the diffusion equation. Inthis way the number of shares kept constant between successive rehedgings can bederived explicitly by the BSM pde adjusted to discrete-time trading. A closed form

discrete-time delta dependent on the interval length t can be obtained. Short noteson further research and improvements of discrete hedging are given. As an appli-cation the case of option valuation with transaction costs is treated and explicitlysolved.

The paper outline is as follows: in Sect. 2 the portfolio over the rebalanc-ing interval is studied. The discrete-time adjusted BlackScholesMerton partialdifferential equation is derived and appropriate values for discrete-time hedg-ing are given. In Sect. 3 the option valuation problem with transaction costs isconsidered and an appropriate equation describing the process is obtained. A

specific example of a European call option is studied and a closed formrepresentation of the discrete-time delta is given.

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Discretetime delta hedging and the BlackScholes model with transaction costs 229

where W(t) is the Wiener process (Brownian motion), the annualized volatilityand = +(1/2)2 the expected annual rate of return. Then the change ofS = S(t)over the small noninfinitesimal interval of length t is approximately equal to

S = S(t + t) S(t) = S Zt + St, (2.2)where Zis normally distributed variable with mean zero and variance one; in shortZ N(0, 1) (Leland 1985; Hull 1997)

Let be the value of a portfolio at time t consisting of a long position in theoption and a short position in N(t) units of shares with the price S

= V N(t)S. (2.3)

In the continuous-time model where the hedging is instantanuous and the repli-cation is perfect, the number of shares N(t) is given by the current value of thederivative VS (t, S) (the delta), where V(t, S) is the solution of the BlackScholesMerton equation.

In practice where the trading takes place discretely in time, the number ofshares given by the delta is held constant over the rebalancing interval (t, t + t).Thus when continuous-time models are applied to discrete-time trading, hedgingerrors appear. The hedging error, defined as the difference between the return tothe portfolio value and the return to the riskless asset over the rehedging inter-val, depends on the length t of the interval. This means when t is sufficientlysmall, the current value of the derivative gives an acceptable approximation forthe discrete-time hedging with small error. However, when the rebalancing is notfrequent, especially in the case where the delta is sensitive with respect to time,higher errors may occur (cf. Remark 2.2). For the mean-variance analysis of theerror, where Vis given by the BSM equation (Boyle and Emanuel 1980; Toft 1996).

Remark 2.1 Suppose that after discrete hedging, for instance from a time pointt+ t to expiry, continuous hedging is assumed. Then the delta value at time t+ tis determined by the partial derivative VS (t

+t, S

+ S) where V(t,S) is the solution

of the BlackScholesMerton equation. In that case for example the constant valueof delta over the rehedging interval (t, t+t) can be approximated by VS (t+t, S).

In the continuation we will consider the case, where hedging is in discrete timeuntil expiry. We assume that hedging takes place at equidistant time points withrebalancing intervals of (equal) length t. Throughout we assume that partial deriv-atives of V(t, S) are continuous functions on their respective domains. Moreoverwe let the trading be relatively frequent so that the interval length t is relativelysmall. Then we have

Proposition 2.1 Let V = V(t, S) be the option value at time t and price S. Let be the annualized volatility and r the annual interest rate of a riskless asset

i l d d If h i b f h N( ) h ld h

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230 M. Mastinek

with parameters

0=

1 + rt and r 0 = r1 + rt, (2.6)then the mean and the variance of the hedging error is of order O(t2).

Proof Let us consider the return to the portfolio over the period [t, t + t],t [0, T t]. By assumption over the period of length t the value of theportfolio changes by

= V N(t)S (2.7)as the number of shares N(t) is held fixed during the time step t.

First we consider the change V of the option value V(t, S) over the timeinterval [t, t + t] of length t. We give the Taylor series expansion of thedifference in the following way:

V = V(t + t, S + S) V(t, S)= (V(t + t, S + S) V(t + t, S))

+(V(t + t, S) V(t, S)) = I + I I (2.8)where:

I = (V(t + t, S + S) V(t + t, S))= VS (t + t, S)( S) +

1

2VS S (t + t, S)( S)2

+ 16

VS S S (t + t, S)( S)3 + O(t2), (2.9)

and

I I

=(V(t

+ t, S)

V(t, S))

=Vt(t

+t, S)( t)

+O(t2). (2.10)

By (2.2) we get the approximation

V = Vt(t + t, S)( t) + VS (t + t, S)( S)+1

2VS S (t + t, S)( S)2 +

1

6VS S S (t + t, S)( S)3 + O(t2)

= Vt(t + t, S)( t) + VS (t + t, S)( S)+1

2VS S (t + t, S)(S Z

t + S t)2

+16

VSS S (t + t, S)(S Zt + S t)3 + O(t2) (2.11)

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Thus by (2.7) it follows:

=

V

N(t) S

= Vt(t + t, S)( t) + [VS (t + t, S) N(t)] ( S)+ 1

2VS S (t + t, S)(2 S2Z2t + 2 S2Z t

32 )

+ 16

VS S S (t + t, S)3 S3Z3t32 + O(t2). (2.13)

By choosing the number of shares in the following way

N(t) = VS (t + t, S) (2.14)

the S term can be eliminated completely and the risk in (2.13) can be reduced.Hence we get

= Vt(t + t, S)( t) +1

2VS S (t + t, S)(2 S2Z2t + 2 S2Z t

32 )

+1

6

VS S S (t

+t, S)3 S3Z3t

32

+O(t2). (2.15)

By assumption the amount can be invested in a riskless asset with an interestrate r compounded continuously. Then over the rehedging interval of length t thereturn to the riskless investment is equal to

B = exp(rt) = rt + O(t2). (2.16)

Let us define the hedging error H as the difference between the return to

the portfolio value and the return to the riskless value B over the rehedginginterval of length t. By (2.13) and (2.14) we obtain

H = B= Vt(t + t, S)( t) +

1

2VS S (t + t, S)

2 S2Z2t + 2 S2Z t32

+ 1

6VS S S (t + t, S)3 S3Z3t

32 rt + O(t2)

= Vt(t + t, S)( t) + 12

VS S (t + t, S)2 S2Z2t + 2 S2Z t32 1

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232 M. Mastinek

hence it follows:

H

=

B

=(1

+rt)Vt(t

+t, S)(t)

+12

VS S (t + t, S)

2 S2Z2t + 2 S2Zt32

+ 16

VS SS (t + t, S)3 S3Z3t32 (V(t + t, S)

SVS (t + t, S))rt + O(t2). (2.19)Let us assume for a moment that there exists a solution V(t, S) of the followingpartial differential equation

(1 + rt)Vt(t + t, S) + 12

2 S2 VS S (t + t, S)(V(t + t, S) SVS (t + t, S))r = 0 (2.20)

for t [0, T t]. Then from (2.20) we obtain

H = 12

VSS (t + t, S)

2 S2(Z2 1)t + 2 S2Z t32

+

1

6

VS S S (t

+t, S)3 S3Z3t

32

+O(t2). (2.21)

Taking expectations and noting that Z N(0,1) we get the meanE(H) = O(t2) (2.22)

and the variance

V(H) =

1

2VS S (t + t, S)2 S2

22t2 + O(t3) (2.23)

Therefore when the option value V(t, S) satisfies the following partial differentialequation:

Vt(t + t, S) +1

220 S

2 VS S (t + t, S)+ r0 SVS (t + t, S)) r0 V(t + t, S) = 0, (2.24)

where 0 and r0are given by (2.6), then the hedging error has mean zero of orderO(t2) and variance of order O(t2). By a shift V1(t) = V(t+ t) from (2.24) theBlackScholesMerton equation for V1(t) with expiry at t

=T

t is obtained.

Hence the BSM Eq. (2.5) with adjusted volatility 0 and interest rate r0 readilyfollows.

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Remark 2.3 By Expansion (2.11) it readily follows that higher order derivativesVS S and VS S S can be included into N(t) and hedging further improved. Since inProposition 2.1 the form of the BlackScholesMerton pde is preserved, the model

can be generalized to models with dividends and models with time dependentvolatility and interest rates.

By Proposition 2.1 it follows directly that in the discrete-time case the numberof shares held fixed over the rebalancing interval depends on the length t of theinterval. In addition by Remark 2.2, ift is not small then the values of delta at timetand t+ t may considerably differ, especially when the delta is more sensitive tothe change in time. If in (2.21) |Z| is near one, the gamma term is near zero andso other terms of the hedging error prevail. In that case by (2.4) the hedging error

can be substantially reduced. Moreover, when options vega and rho are high, thenby (2.6) the adjustment of parameters with respect to t can be considered.

3 Transaction costs

It is known that transaction costs can be included into the BlackScholesMertonequation by considering appropriately adjusted volatilility (Leland 1985;Avellaneda and Paras 1994; Toft 1996).

We denote here transaction costs by ctr. These costs depend on the volume oftransactions at rehedging and a constant percentage k.

Let us consider again the portfolio at time t consisting of one option long anda short position in N(t) units of stockS with the value = V N(t)S.

Let k represent the round trip transaction costs, measured as a fraction of thevolume of transactions. Then the transactions costs of rehedging over the rehedginginterval of length t are equal to

ctr

=k

2

S

|N(t

+t)

N(t)

|, (3.1)

for the details (Leland 1985; Avellaneda and Paras 1994). If the discrete time trad-ing is considered, then by (2.4) we have N(t) = VS (t+ t, S)and the volume canbe approximated in the following way:

N(t + t) N(t) = VS (t + 2t, S + S) VS (t + t, S)= VS S (t + t, S) S + O(t)

Let us assume

k <

t

(3.2)

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234 M. Mastinek

In some cases it is possible to assume that S2 VSS is of order O(t1/2); for instance

see Leland (1985) for a European call option.In the same way as in Sect. 2 [relations (2.13)(2.15)] the change in value of

the portfolio can be studied. By (2.14) then it follows:

= V N(t) S ctr = Vt(t + t, S)( t)+ 1

2VS S (t+t, S)(2 S2Z2t

k

2S2 |VS S (t+t, S)| |Z|

t+O

t

32

(3.4)

By assumption that the amount can be invested in a riskless asset, it follows thatthe return to the riskless investment is equal to

B = rt + O(t2) = (V(t, S) VS (t + t, S)S)rt + O(t2)= (V(t + t, S) Vt(t + t, S)t VS (t + t, S)S)rt + O(t2) (3.5)

In the same way as in (2.19) the hedging error can be obtained

H = B= (1 + rt)Vt(t + t, S)( t) +

1

2VS S (t + t, S)2 S2Z2t

(V(t + t, S) SVS (t + t, S))rt k

2S2 |VS S (t + t, S)| |Z|

t + O

t

32

(3.6)

We note that E(|Z|) = 2/ for Z N(0,1). Therefore by assuming that V(t, S)satisfies the following partial differential equation:

(1 + rt)Vt(t + t, S) +1

22 S2 VS S (t + t, S)

k2

S2

2

t|VS S (t + t, S)|

(V(t + t, S) SVS (t + t, S))r = 0 (3.7)

it follows that the hedging error can be written in the following way:

H = 12

2 S2 VSS (t + t, S)(Z2 1)t

k

2 S2 |.VS S (t + t, S)||Z| 2 t + O t32 (3.8)

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By (2.6) and a transformation t + t t from Eq. (3.7) we get

Vt(t, S) +1

2 2

0 S2

VS S (t, S) k

2 0 S2 2

t |VS S (t, S)|(V(t, S) SVS (t, S))r0 = 0 (3.10)

Eq. (3.10) is nonlinear. In the case where VS S (t, S) is of the same sign for all valuestand S the equation can be reduced to a linear equation. For instance a Europeancall or put option satisfy the requirement. Therefore the following result can beobtained.

Proposition 3.1 Let C(t, S) be the value of the European call option, be the

annualized volatility, r the annual interest rate of a riskless asset and let the trad-ing take place discretely with rebalancing intervals of lengtht. If k satisfies (3.2)and if the approximate number of shares N(t) held short over the rebalancinginterval is equal to: N(t) = CS (t+ t, S),where C(t, S) satisfies the BlackScho-lesMerton equation

Ct(t, S) +1

221 S

2 CS S (t, S) + r1 SCS (t, S)) r1C(t, S) = 0, (3.11)

With parameters

21 =2

1 + rt

1 k

2

t

and r1 =

r

1 + rt, (3.12)

then the mean and the variance of the hedging error are of order O(t3/2)andO(t2) respectively.

Proof By relations (3.4)(3.10) the proof can be given in the same way as that ofProposition 2.1. We only note that the second derivative CSS of the solution of theBSM equation is given by the Gaussian or normal density function and hence it isa positive function. Thus the Eq. (3.11) is a direct consequence of Eq. (3.10).

For a short position in a call option and a long position in the shares we have = V + N(t)S. Thus in (3.4) and in subsequent equations all the signs withthe exception of the transaction costs term have to be changed. Therefore we con-clude: Corolarry 3.1 Let the portfolio consist of a short position in the European calloption and a long position in N(t) shares. If N(t) = CS (t+ t, S), where C(t, S)satisfies the BSM equation (3.11) with parameter1 given by :

21

=(2/1

+rt)

1 + (k/ )2/ t, then the mean and the variance of the hedging error are oforder O(t3/2)and O(t2)respectively.

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236 M. Mastinek

Boyle P, Emanuel D (1980) Discretely adjusted option hedges. J Financ Econ. 8:259282Boyle P, Vorst T (1992) Option replication in discrete time with transaction costs. J Finance

47:271293

Hoggard T, Whalley AE, Wilmott P (1994) Hedging option portfolios in the presence of trans-action costs. Adv Fut Opt Res 7:2135Hull JC (1997) Option, futures & other derivatives. Prentice-Hall, New JerseyKocinski M (2004) Hedging of the European option in discrete time under proportional transac-

tion costs. Math Meth Oper Res 59:315328Korn R (2004) Realism and practicality of transaction cost approaches in continuous-time port-

folio optimisation: the scope of the Morton-Pliska approach. Math Meth Oper Res 60:165174Leland HE (1985) Option pricing and replication with transaction costs. J Finance 40:12831301Mello AS, Neuhaus HJ (1998) A portfolio approach to risk reduction on discretely rebalanced

oprion hedges. Manag Sci 44(7):921934Merton RC (1973) Theory of rational option pricing. Bell J Econ Manag Sci 4:141183Toft KB (1996) On the mean-variance tradeoff in option replication with transactions costs. J

Finance Quant Analysis 31 (2):233263

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