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An Endogenous Volatility Approach

to Pricing and Hedging Call Options with

Transaction Costs

Leonard C. MacLean Yonggan Zhao William T. Ziemba

September 28, 2009

JEL Classication: B23 C15 C61 G13

Leonard C. MacLean: School of Business Administration, Dalhousie University, Halifax, NS,

Canada B3H 3J5. Email: lmaclean@mgmt.dal.ca Yonggan Zhao: RBC Center for Risk Management and School of Business Administration, Dal-

housie University, Halifax, NS, Canada B3H 3J5. E-mail: Yonggan.Zhao@Dal.Ca. William T. Ziemba: Sauder School of Business, University of British Columbia, Vancouver, BC,

Canada, V6T 1Z2 (Emeritus), and Visiting Professor, Mathematical Institute, Oxford University, 24-

29, St. Giles Street, Oxford, OX1 3LB, UK. E-mail: wtzimi@mac.com.

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An Endogenous Volatility Approach

to Pricing and Hedging Call Options with TransactionCosts

Abstract

Standard delta hedging fails to exactly replicate a European call option in the

presence of transaction costs. We study a pricing and hedging model similar to the

delta hedging strategy with an endogenous volatility parameter for the calcula-

tion of delta over time. The endogenous volatility depends on both the transaction

costs and the option strike prices. The optimal hedging volatility is calculated

using the criterion of minimizing the weighted upside and downside replication

errors. The endogenous volatility model with equal weights on the up and down

replication errors yields an option premium close to the Leland (1985) heuristic

approach. The model with weights being the probabilities of the options money-

ness provides option prices closest to the actual prices. Option prices from the

model are identical to the Black-Scholes option prices when transaction costs are

zero. Data on S&P 500 index cash options from January to June 2008 illustrate

the model.

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1. Introduction

In the theory of option pricing, the goal is to create a replicating portfolio whose pay-

off equals the value of the option at some exercise date. In the Black-Scholes-Merton

(BSM) framework, delta hedging achieves that goal. The celebrated BSM formula

for pricing a European call option is based on two assumptions: (i) the underlying

stock price follows geometric Brownian motion; and (ii) trading has neither restric-

tions nor costs. Assumption (ii) does not hold and the theory fails, so the replicating

strategy has hedging error. One symptom of this failure is implied volatility skew-

ness. It is observed that equity option prices are consistently higher (lower) than the

Black-Scholes-Merton model prices for in- (out-of-) the-money call options, so implied

volatilities of in- (out-of-) the-money options are higher (lower) than at-the-money

ones.

Several researchers have considered the shortcomings of the BSM pricing formula.

An alternative to the geometric Brownian motion approach is the binomial pricing

model (Cox, Ross, and Rubinstein 1979.) Boyle and Vorst (1992) designed a perfect

hedging strategy in the binomial model with transaction costs. The perfect hedge is

possible due to the assumption of a binomial process for the underlying stock price.

They also developed a similar risk neutral valuation approach that is a two state

Markov process. Edirisinghe, et al. (1993) developed a general replicating strategy,

in the framework of optimization, by minimizing the initial cost subject to the hedging

portfolio payoff being at least as large as the options payoff. They indicated that it is

not necessarily optimal to revise the portfolio at each rebalance time.

Accepting the geometric Brownian motion assumption, but introducing transac-

tion costs on trading , Leland (1985) proposed a strategy in which the volatility for

the calculation of delta is increased by a term which is related to the transaction cost

rate and the length of the rebalance interval. As the option premium is positively

related to the volatility, the intention in augmenting the volatility was to increase

the option premium with trading frequency for a given proportional transaction cost

rate. However, as pointed out by Kabanov and Safarian (1997), this strategy does not

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guarantee a perfect replication of a call option in the limit (continuous trading). The

mathematical error in the Leland claim of perfect replication is discussed by Zhao

and Ziemba (2007a and 2007b) and Leland (2007).

A neglected aspect of the Leland approach is that the strike price of an option

does not appear in the expression for the augmented volatility, even though the im-plied volatilities are not constant across all levels of strike prices. This suggests that

relating the augmented volatility to the options strike price can adjust for skewness.

In this paper, we develop an option pricing and hedging model based on portfolio

replication techniques. The objective is to modify the Leland approach by linking the

hedging volatility to both transaction costs and the strike prices. Instead of using

an exogenous augmented volatility, we endogenize the volatility in an optimization

problem. The hedging volatility is selected to minimize the weighted mean absolutereplication error. The total replication error is divided into two quantities, the down-

side shortfall and the upside gain, which are weighted by the probabilities that the

option is in or out of the money, respectively. As the probability of the moneyness of

the option increases, the downside shortfall is considered to be more signicant. Simi-

larly, as the probability of the moneyness increases, the upside gain is considered less

signicant. With the endogenized volatility, it is possible to examine whether and

how transaction costs explain the skewness of equity options across a range of strikeprices. Since the probability that the call option is in the money decreases with the

level of the strike price, the weighing of the downside shortfall and the upside gain

generates skewness of the hedging volatilities across different strike prices.

The endogenous volatility approach is compared with the Leland and the Black-

Scholes approaches in numerical experiments. Using S&P 500 cash options, simula-

tion results show that option prices for the endogenous volatility model are closer to

the actual prices than prices from other models, especially for deep in-the-money and

deep out-of-the-money options. This result is consistent with the volatility skewness.

Option prices from the endogenous volatility model are identical to the Black-Scholes

option prices if transaction costs are ignored.

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2. A Pricing and Hedging Model

The Black-Scholes-Merton theory for options pricing has important restrictions in

terms of the dynamics of the underlying stock price and the frictionless trading mar-

ket. Lelands modication of the theory to incorporate transactions costs also has

limitations. In this section a new approach to option pricing with transactions costs

is developed. As in Lelands model, transactions costs affect the hedging volatility,

and the geometric Brownian motion and trading frequency assumptions are the same

as those in the BSM model. An optimization problem linking volatility to strike price

is dened and the corresponding option prices and hedging errors are related to the

BSM and Leland approaches .

2.1. Volatilities and the Hedging Strategies

Standard option pricing theory suggests that the implied volatilities for all option con-

tracts with different strikes should be the same in an orderly market. However, evi-

dence from the equity option market has shown that implied volatilities are heavily

skewed. Market prices of options are usually higher (lower) than the Black-Scholes-

Merton prices for in (out of) the money call options. This market imperfection has

been documented as volatility skewness in Rubinstein (1994).One reason for this imperfection is the presence of transaction costs. Leland (1985)

considered the idea that the increment of the option premium induced by an aug-

mented volatility can offset the necessary transaction costs. However, the hedging

portfolio does not replicate the option payoff. A possible explanation for the price

deviation with Lelands approach is that the augmented volatility does not depend

on the options strike prices. In this section we develop a method which relates the

hedging volatilities to both transaction costs and strike prices. We use the following notation:

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q round trip transaction cost.

K option strike price.

T option exercise date.

r risk free rate.

N () cumulative normal distribution function. instantaneous stock volatility.

hedging volatility for the delta hedging strategy.

To set up an alternative adjusted volatility model within the BSM setting, we

assume the dynamics of the underlying security follow geometric Brownian motion.

Let S t be the security price at time t. Then the price dynamics are

dS S

= r dt + dZ, (1)

where Z is a standard Brownian motion. This setting indicates that we are work-

ing under a risk neutral probability measure. We only consider hedging that takes

place at a nite number of time points. Two popular strategies are: (i) the delta

hedging strategy with xed time rebalancing, and (ii) the delta hedging strategy with

times determined by the price movement in the underlying security. We consider

xed times, so that comparisons with the BSM and the Leland models are based onthe alternative approaches to the hedging volatility.

Let i , i = 1 , ...n , be the time epochs when hedging trades take place for a given

horizon with 0 = 0 and n +1 = T . While the option writer holds N ( d i ) shares at

trading epoch i , where d i = ln

S iK +( r +

12

2 )( T i ) T i

, the amount in the trading account

is, for i = 1 , . . . ,n ,

B 0 = C 0

(1 + q/ 2) S 0

B i +1 = B i er i (1 + sgn (N ( d i +1 ) N ( d i )) q/ 2) S i +1 (N ( d i +1 ) N ( d i )) ,where C 0 is the initial price of the call option given by the BSM formula

C 0 = S 0 N ( d 0 ) Ke rT N ( d 0 T ) .

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Hence, the terminal payoff of the hedging portfolio at the options expiration date is

P T = B n e r ( T n ) + (1 q/ 2) S T N ( d n ) . (2)The focus is on the volatility estimate . In BSM the estimate is simply the stan-

dard deviation of stock returns and is independent of transaction costs and strike

prices. The hedging volatility proposed by Leland (1985) is

= 1 + 2 q twhich depends on the transaction cost q but not on the strike prices. In our endoge-

nous approach, the hedging volatility will depend on both the transactions cost and

strike price.

2.2. Hedging Error and Endogenous Volatility

The payoff on the call option with strike price K is C T = ( S T K )+ at the expira-tion date T . For a trajectory of stock prices, the delta hedging strategy has replication

error, with absolute value |P T C T |. The mean absolute error is E [|P T C T |]. Theupside and downside replication errors are E [(P T C T )+ ] and E [(P T C T ) ], re-spectively. The rationale for separating the error into scenarios with gains and lossesis the different perceptions or utility that an investor would assign to such scenarios.

Investors may view the upside replication gains differently from the downside repli-

cation losses. Empirical evidence shows that the implied volatilities for equity options

are heavily skewed, which suggests that risk perception is not symmetric. It is pro-

posed that risk aversion depends on the moneyness of the option. If the probability

that the option expires in the money is large, investors may be concerned about the

downside replication losses, considering the issued option as a liability. Similarly, if the probability that the option expires out of the money is small, the replication gains

will not be signicant to the writer of the option. An investor would like to weigh the

upside gains and downside losses according to the options moneyness.

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Let K be the probability that the option with strike price K ends valueless at the

expiry date, i.e.,

K = Pr[ S T K ],which implies that

K = N (ln( S 0 /K )+( r 12

2

) T T ).

With the xed parameters = {S 0 ,K,r,T,K,,q, 1 , ... n }, consider the weightedmean errors criterion for determining volatility

( , ) = K ( ) E (P T C T )+ | , + (1 K ) E (P T C T ) | , . (3)

Equation ( 3) can be viewed as the expected performance, in terms of hedging error,

using the probability measure induced by the moneyness of the option. Since the

probability that the option is out of the money at expiration increases with the level

of the strike prices, the interest in hedging the downside error should decrease as the

option strike price increases. Correspondingly, the investor prefers to charge a higher

premium for deep in the money options and lower premium for deep out-of-the-money

options than the BSM options prices.

The performance measure depends on several parameters. The main attention

is on the hedging volatility parameter . So, holding other parameters constant, the

objective is to determine the adjusted volatility with the best performance. The writer

of the call option will solve the nonlinear optimization model

min> 0

( , ) . (4)

The solution to ( 4) depends on the parameter vector and particularly the transaction

costs and the strike prices. There are some important properties of the endogenous

volatility problem as an alternative to the BSM or Leland hedging approaches.

The BSM option price is a special instance of the solution to the optimization

model ( 4) if transaction costs are zero. Moreover, as the trading interval ap-

proaches zero, which implies a continuous trading in the limit, both the upside

replication gains and the downside replication losses must tend to zero, since

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an exact hedge can be achieved in the framework of continuous trading without

transaction costs. Thus, the optimization model ( 4) is consistent with the BSM

model.

If the weights reecting risk aversion are selected as K ( ) = 0 .5 (indifference

between gains and losses), so that the average absolute replication error is min-

imized, the optimal volatility is close to Lelands formula, which approximately

minimizes mean squared errors.

The weights from the moneyness probability will introduce a correction for the

bias of hedging prices. The weights have an effect of tilting the prices, aligning

them closer to true prices.

Since there is only one variable in the nonlinear optimization model ( 4), an efcientalgorithm can be applied to the solution for the optimal endogenous volatility. A

Matlab code was written to solve this problem.

3. Empirical Tests

The objective in proposing a new method for delta hedging is to address some of

the problems with the BSM and Leland approaches. The outcome measures are: (i)

volatility skewness; (ii) pricing bias; and (iii) hedging error. The endogenous volatility

approach is expected to improve performance on those dimensions over the standard

methods. Evidence of improved performance is provided in this section.

The comparator option pricing and hedging models have the same assumptions

for the stock dynamics - geometric Brownian motion, and option type - European call

option. Data on S&P 500 index options are used for the model input to compare the

simulation results with the actual option prices. The instantaneous stock volatilityis estimated as the sample standard deviation, based on the maximum likelihood

method. The risk free rate is set to be the average of the US six month term deposit

rates for that period. The index level was S 0 = 892 .5 on December 18, 2008. Esti-

mated as the sample standard deviation, the annualized volatility of the index return

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is 34.61%. The range of the strike prices used in this study is from 750 to 1100 in

25 point increments. The round-trip proportional transaction costs used to study how

the option prices are affected by the level of transaction cost rate are 0 (no cost), 0.3% ,

0.6% , 0.9% , 1.2% , and 1.5% . The options expired on June 18, 2009, which indicates

a total of 130 days for the hedging to take place. For the signicance of this study,the number of scenarios is set to 10000. We simulate 10000 paths of the underlying

asset prices using the estimated parameters from the data. We then apply a process

control strategy to implementing portfolio hedging over time, taking transaction costs

into consideration. The key innovation in the approach is the calculation of the ad-

justed volatility. Instead of exogenously specifying an adjusted volatility for the BSM

formula, we endogenize the hedging volatility for the calculation of delta hedging.

The optimal hedging volatility is determined by minimizing the weighted upside anddownside hedging errors at the expiry date of the call option.

For the rest of this section, we study model performance based on the different

criteria. For comparison with the BSM and the Leland models, we consider the en-

dogenized volatility model with two weighting schemes for the upside and downside

hedging errors, (i) EV1: equal weights, (ii) EV2: moneyness probability weights.

3.1. Volatility SkewnessIt is observed that equity option prices are consistently higher (lower) than the BSM

model prices for in (out-of) -the-money options, so implied volatilities of in (out-of) -

the-money call options are higher (lower) than at-the-money ones. A motivating prin-

ciple for endogenizing volatility is to link the hedging volatility not only to transaction

costs but also to strike prices. The hedging volatilities for the various models are given

in Table I and plotted in Figure 1. The BSM volatility, which was estimated as 34.61%

per annum, does not vary by transaction cost or strike price, the Leland volatilities

vary by transaction cost, and the endogenous volatilities vary by both transaction cost

and strike price.

From Table I, the hedging volatility increases with transaction costs for all models.

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Table I: Hedging Volatilities for Various Models

Model StrikeCost

0% 0.3% 0.6% 0.9% 1.2% 1.5%

Leland 0.3461 0.3649 0.3828 0.3999 0.4163 0.4321

EV1

725 0.3451 0.3761 0.4046 0.4307 0.4559 0.4799

750 0.3455 0.3746 0.4018 0.4271 0.4515 0.4748

775 0.3454 0.3732 0.3993 0.4241 0.4475 0.4701

800 0.3452 0.3724 0.3977 0.4214 0.4442 0.4660

825 0.3452 0.3715 0.3962 0.4193 0.4415 0.4629

850 0.3453 0.3709 0.3947 0.4175 0.4395 0.4604

875 0.3453 0.3701 0.3935 0.4159 0.4372 0.4578

900 0.3453 0.3696 0.3925 0.4142 0.4352 0.4555

925 0.3454 0.3689 0.3914 0.4131 0.4337 0.4537

950 0.3450 0.3682 0.3907 0.4120 0.4325 0.4522975 0.3447 0.3679 0.3901 0.4110 0.4313 0.4506

1000 0.3450 0.3676 0.3892 0.4101 0.4303 0.4497

1025 0.3448 0.3672 0.3888 0.4094 0.4293 0.4484

1050 0.3445 0.3667 0.3881 0.4084 0.4281 0.4472

1075 0.3440 0.3663 0.3876 0.4078 0.4273 0.4461

1100 0.3441 0.3660 0.3871 0.4070 0.4263 0.4449

EV2

725 0.3629 0.3949 0.4257 0.4552 0.4834 0.5107

750 0.3595 0.3894 0.4180 0.4452 0.4712 0.4966

775 0.3559 0.3844 0.4112 0.4369 0.4614 0.4852

800 0.3526 0.3795 0.4053 0.4298 0.4533 0.4760

825 0.3495 0.3756 0.4002 0.4237 0.4463 0.4683

850 0.3465 0.3721 0.3960 0.4188 0.4406 0.4616

875 0.3436 0.3685 0.3917 0.4139 0.4349 0.4554

900 0.3408 0.3648 0.3876 0.4097 0.4303 0.4502

925 0.3379 0.3617 0.3841 0.4057 0.4258 0.4454

950 0.3350 0.3583 0.3805 0.4015 0.4213 0.4404

975 0.3319 0.3549 0.3767 0.3972 0.4168 0.43591000 0.3293 0.3520 0.3734 0.3938 0.4133 0.4320

1025 0.3265 0.3491 0.3701 0.3900 0.4092 0.4276

1050 0.3236 0.3460 0.3668 0.3867 0.4056 0.4238

1075 0.3214 0.3435 0.3643 0.3838 0.4027 0.4206

1100 0.3182 0.3404 0.3610 0.3806 0.3992 0.4170

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00.005

0.010.015

500

1000

1500

1

0

1

2

H e

d g

i n g

V o

l a t i l i t y

BSM Model

Transaction CostsStrike Prices 00.005

0.010.015

500

1000

1500

0.35

0.4

0.45

0.5

Transaction Costs

Leland Model

Strike Prices

H e

d g

i n g

V o

l a t i l i t y

00.005

0.010.015

500

1000

1500

0.35

0.4

0.45

0.5

Transaction Costs

EV1 Model

Strike Prices

H e

d g

i n g

V o

l a t i l i t y

00.005

0.010.015

500

1000

15000.2

0.4

0.6

Transaction Costs

EV2 Model

Strike Prices

H e

d g

i n g

V o

l a t i l i t y

Figure 1. Hedging Volatilities for Alternative Models

The Leland hedging volatility does not depend on the strike price. As the strike price

increases, the volatility decreases for the endogenous models. The changing volatility

with changing strikes is greater for the EV2 model than the EV1 model. Also, the

volatility skewness is more pronounced for all levels of transaction cost with the EV2

model than with the EV1. This suggests that the option prices with the endogenous

method EV2 should be closer to actual prices.

3.2. Predicted Option Prices

In Table II , the actual option prices at various strike levels are compared to the model

prices at strike levels and transaction cost levels. For each model the prices which areclosest to the actual price are in bold.

There are several observations we can make based on the presented prices in Table

II and Figure 2. Within each model, the best price for in-the-money options is at the

high transaction cost level. In fact, as the strike price increases, the transaction cost

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Table II: Predicted Option Prices by Model

Model Strike Cost Actual 0% 0.3% 0.6% 0.9% 1.2% 1.5%

Leland

725 212.10 183.28 186.34 189.34 192.28 195.15 197.96750 193.95 164.92 168.36 171.69 174.93 178.08 181.15775 176.55 147.72 151.50 155.14 158.65 162.05 165.34

800 159.90 131.72 135.79 139.69 143.44 147.05 150.54825 144.00 116.94 121.26 125.37 129.31 133.10 136.75850 128.75 103.38 107.88 112.16 116.25 120.17 123.94875 114.65 91.01 95.64 100.03 104.22 108.24 112.10900 100.90 79.81 84.50 88.95 93.19 97.26 101.18925 88.55 69.72 74.41 78.87 83.13 87.22 91.15950 76.85 60.69 65.33 69.75 73.98 78.05 81.97975 66.20 52.64 57.18 61.52 65.69 69.71 73.58

1000 56.50 45.51 49.91 54.14 58.21 62.14 65.951025 47.75 39.22 43.45 47.53 51.48 55.31 59.021050 40.00 33.70 37.73 41.64 45.44 49.14 52.741075 33.20 28.88 32.68 36.40 40.04 43.60 47.071100 27.20 24.68 28.25 31.77 35.23 38.62 41.96

EV1

725 212.10 183.12 188.22 193.10 197.72 202.29 206.72750 193.95 164.82 170.16 175.29 180.19 184.96 189.60775 176.55 147.584 153.188 158.529 163.669 168.58 173.35800 159.90 131.53 137.42 142.97 148.18 153.23 158.08825 144.00 116.74 122.77 128.46 133.80 138.94 143.89850 128.75 103.21 109.31 115.01 120.45 125.72 130.71875 114.65 90.82 96.92 102.65 108.14 113.35 118.38900 100.90 79.63 85.66 91.36 96.74 101.94 106.99925 88.55 69.56 75.42 81.02 86.42 91.56 96.54950 76.85 60.42 66.15 71.70 76.99 82.06 86.96975 66.20 52.31 57.90 63.30 68.40 73.38 78.16

1000 56.50 45.25 50.56 55.67 60.65 65.52 70.241025 47.75 38.93 43.98 48.91 53.69 58.37 62.891050 40.00 33.36 38.12 42.81 47.36 51.83 56.231075 33.20 28.47 32.98 37.42 41.75 46.01 50.201100 27.20 24.31 28.47 32.63 36.70 40.73 44.72

EV2

725 212.10 186.00 191.41 196.82 202.16 207.37 212.54750 193.95 167.35 172.93 178.40 183.73 188.87 193.96775 176.55 149.69 155.46 160.99 166.36 171.51 176.57800 159.90 133.12 138.97 144.63 150.04 155.26 160.30825 144.00 117.73 123.71 129.37 134.80 140.04 145.14850 128.75 103.49 109.59 115.31 120.77 125.99 131.00875 114.65 90.39 96.52 102.20 107.64 112.79 117.81900 100.90 78.49 84.47 90.13 95.61 100.73 105.68925 88.55 67.70 73.61 79.19 84.57 89.59 94.47950 76.85 57.96 63.71 69.17 74.36 79.28 84.04975 66.20 49.24 54.76 60.04 65.03 69.84 74.52

1000 56.50 41.64 46.89 51.90 56.74 61.42 65.931025 47.75 34.91 39.90 44.63 49.18 53.65 57.971050 40.00 29.01 33.68 38.14 42.49 46.72 50.841075 33.20 24.08 28.36 32.55 36.61 40.64 44.541100 27.20 19.66 23.62 27.49 31.32 35.09 38.77

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00.005

0.010.015

500

1000

1500

0

100

200

Transaction Costs

BSM Model

Strike Prices

P r e

d i c t e d P r i c e s

00.005

0.010.015

500

1000

1500

0

100

200

Transaction Costs

Leland Model

Strike Prices

P r e

d i c t e d P r i c e s

00.005

0.010.015

500

1000

15000

100

200

300

Transaction Costs

EV1 Model

Strike Prices

P r e

d i c t e d P r i c e s

00.005

0.010.015

500

1000

15000

100

200

300

Transaction Costs

EV2 Model

Strike Prices

P r e

d i c t e d P r i c e s

Figure 2. Predicted Prices for Various Models

for the best matched price decreases. For almost all strike prices, the closest model

price is produced by the EV2 model for some transaction cost level.The options prices

from the EV2 model are very accurate for in-the-money options. For each of the deeply

in- or out-of-the-money options there exists a transaction cost level such that the EV2

model price is closest to the actual price, as bolded prices in the table indicate.

3.3. Hedging Error

Realized Error . The estimation error with the option price is a factor in the perfor-

mance of the hedging portfolio. The portfolio performance for the period of the data

based on option prices from the various models is provided in Table III , in which theerror is the difference between the portfolio value and the option payoff.

As expected the superiority of the EV2 model occurs for deeply in- or out-of-the-

money options for some transaction cost level, where the predicted option prices were

closer to actual option prices. In the presence of transaction costs, we propose that

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Table III: Realized Errors for the Various ModelsModel Strike Cost0% 0.3% 0.6% 0.9% 1.2% 1.5%

BSM

725 22.5076 16.6535 10.7993 4.9451 -0.9091 -6.76322750 23.4656 17.1585 10.8515 4.5444 -1.7626 -8.06976775 24.5768 17.8350 11.0932 4.3514 -2.3904 -9.13213800 25.8243 18.6799 11.5356 4.3912 -2.7532 -9.89758825 27.2804 19.7815 12.2826 4.7837 -2.7152 -10.21415850 29.2287 21.4312 13.6337 5.8362 -1.9614 -9.75894875 33.1912 25.1650 17.1387 9.1125 1.0862 -6.94007900 39.0564 31.2463 23.4363 15.6263 7.8163 0.00638925 42.5350 35.3584 28.1818 21.0053 13.8287 6.65216950 33.2905 26.9553 20.6202 14.2850 7.9498 1.61473975 26.9386 21.5411 16.1435 10.7460 5.3485 -0.04916

1000 20.9479 16.4414 11.9349 7.4284 2.9219 -1.584601025 15.3613 11.6266 7.8920 4.1574 0.4228 -3.311941050 10.7064 7.6393 4.5721 1.5050 -1.5621 -4.629271075 7.1018 4.5916 2.0813 -0.4289 -2.9392 -5.449481100 4.3057 2.2485 0.1913 -1.8659 -3.9231 -5.98030

Leland

725 22.5076 17.3185 12.1842 7.1064 2.0849 -2.8814750 23.4656 18.0064 12.6324 7.3402 2.1257 -3.0154775 24.5768 18.8555 13.2548 7.7647 2.3763 -2.9184800 25.8243 19.8748 14.0846 8.4359 2.9140 -2.4936825 27.2804 21.1794 15.2741 9.5382 3.9507 -1.5058850 29.2287 23.1092 17.2065 11.4878 5.9282 0.5081875 33.1912 27.1816 21.4014 15.8197 10.4125 5.1600900 39.0564 33.4957 28.1752 23.0584 18.1166 13.3272925 42.5350 37.5091 32.6753 28.0054 23.4773 19.0731950 33.2905 28.4193 23.7313 19.2008 14.8073 10.5340975 26.9386 22.2825 17.7660 13.3758 9.0997 4.9268

1000 20.9479 16.6096 12.3621 8.2014 4.1229 0.12141025 15.3613 11.2853 7.2711 3.3183 -0.5744 -4.40921050 10.7064 6.8227 2.9879 -0.7976 -4.5346 -8.22421075 7.1018 3.3705 -0.3231 -3.9764 -7.5888 -11.16071100 4.3057 0.7310 -2.8245 -6.3539 -9.8531 -13.3205

EV1

725 22.4737 17.6824 12.8881 8.1075 3.3890 -1.2812750 23.4427 18.4089 13.4228 8.4886 3.6329 -1.1493775 24.5422 19.2740 14.1002 9.0297 4.0470 -0.8364800 25.7722 20.3186 15.0034 9.8002 4.7411 -0.2089825 27.2218 21.6439 16.2623 11.0324 5.9642 1.0276850 29.1688 23.6207 18.2814 13.1426 8.1903 3.3498875 33.1130 27.7227 22.5619 17.6275 12.8521 8.2443900 38.9719 34.0349 29.3454 24.8440 20.5395 16.4045925 42.4621 37.9574 33.6717 29.5824 25.6122 21.7803950 33.2117 28.6778 24.3989 20.2977 16.3449 12.5262975 26.8911 22.4082 18.1300 14.0016 10.0332 6.1798

1000 20.9411 16.6487 12.5023 8.4926 4.6081 0.83231025 15.3874 11.2596 7.2508 3.3534 -0.4413 -4.14681050 10.7787 6.7591 2.8425 -0.9707 -4.6931 -8.33191075 7.2348 3.2901 -0.5613 -4.3119 -7.9801 -11.57061100 4.4629 0.6481 -3.1148 -6.7915 -10.4032 -13.9520

EV2

725 23.0339 18.2398 13.4942 8.7993 4.1593 -0.4118750 23.9992 18.9788 14.0360 9.1699 4.3763 -0.3236775 25.0496 19.8070 14.6680 9.6470 4.7232 -0.0901800 26.1889 20.7286 15.4459 10.3054 5.2987 0.4110825 27.5078 21.9278 16.5486 11.3554 6.3243 1.4482850 29.2652 23.7231 18.3945 13.2655 8.2966 3.4688875 32.9434 27.5547 22.3688 17.4011 12.5957 7.9738900 38.4696 33.4842 28.7553 24.2852 19.9256 15.7290925 41.6664 37.1496 32.8233 28.7018 24.6590 20.7609950 32.5135 27.9060 23.5343 19.3407 15.2839 11.3684975 26.4853 21.8731 17.4686 13.2248 9.1324 5.1825

1000 20.9214 16.4740 12.1872 8.0430 4.0253 0.11861025 15.8749 11.5550 7.3830 3.3314 -0.6139 -4.46191050 11.8874 7.6444 3.5492 -0.4359 -4.3187 -8.11231075 8.9112 4.7819 0.7577 -3.1533 -6.9899 -10.73381100 6.7489 2.7567 -1.1475 -4.9883 -8.7516 -12.4411

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investors view gains and losses from a hedging portfolio differently depending on the

moneyness of the option.

The performance of the hedging portfolio depends on the hedging volatility and

corresponding option price. The hedging volatility depends on the criteria for perfor-

mance of the hedging portfolio. The results by the alternative performance criteriaare considered separately.

Hedging Error by Mean Absolute Deviation . The values for the mean ab-

solute deviation criterion are illustrated in Figure 3 and Table IV. Those results by

strike price and transaction cost show that the EV1 is the best model for this criterion,

and also how far the other models are from the optimum.

00.005

0.010.015

500

1000

15000

10

20

Transaction Costs

BSM Model

Strike Prices

H e

d g

i n g

E r r o r

00.005

0.010.015

500

1000

15000

2

4

6

Transaction Costs

Leland Model

Strike Prices

H e

d g

i n g

E r r o r

00.005

0.010.015

500

1000

15001

2

3

4

Transaction Costs

EV1 Model

Strike Prices

H e

d g

i n g

E r r o r

00.005

0.010.015

500

1000

15002

3

4

5

Transaction Costs

EV2 Model

Strike Prices

H e

d g

i n g

E r r o r

Figure 3. Hedging Error by Mean-Absolute Deviation for Alternative Models

For the mean absolute deviation criterion the values for the Leland, EV1 and EV2

models are close, and very close in the mid-range of strikes. The errors increase in

transaction cost for each model. However, the pattern with strikes differs for the

models. Leland and EV1 have a hill pattern, whereas the EV2 has a valley shape. So

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Table IV: Hedging Errors by Mean-Absolute DeviationModel Strike Cost0% 0.3% 0.6% 0.9% 1.2% 1.5%

BSM

725 1.6407 2.8783 5.2932 7.9109 10.5519 13.1938750 1.8288 3.0946 5.6181 8.3779 11.1687 13.9605775 1.9962 3.2653 5.8775 8.7624 11.6789 14.5989800 2.1342 3.3941 6.0694 9.0506 12.0672 15.0854825 2.2536 3.5033 6.2089 9.2459 12.3277 15.4115850 2.3513 3.5721 6.2853 9.3471 12.4600 15.5777875 2.4198 3.6061 6.2789 9.3332 12.4482 15.5667900 2.4438 3.5846 6.2010 9.2172 12.2993 15.3873925 2.4463 3.5480 6.0840 9.0369 12.0576 15.0861950 2.4216 3.4604 5.9030 8.7616 11.6950 14.6363975 2.4000 3.3612 5.6764 8.4209 11.2447 14.0739

1000 2.3416 3.2474 5.4382 8.0507 10.7424 13.44021025 2.2814 3.1142 5.1566 7.6231 10.1681 12.72121050 2.1930 2.9503 4.8513 7.1602 9.5483 11.94541075 2.0688 2.7713 4.5351 6.6742 8.8971 11.13071100 1.9534 2.5938 4.2062 6.1805 8.2363 10.3026

Leland

725 1.6407 2.0093 2.7585 3.6462 4.5909 5.5668750 1.8288 2.1467 2.8142 3.6286 4.5098 5.4269775 1.9962 2.2600 2.8429 3.5828 4.3947 5.2487800 2.1342 2.3564 2.8672 3.5286 4.2714 5.0598825 2.2536 2.4435 2.8844 3.4728 4.1492 4.8748850 2.3513 2.5172 2.9008 3.4241 4.0320 4.6940875 2.4198 2.5676 2.8984 3.3609 3.9090 4.5103900 2.4438 2.5787 2.8702 3.2813 3.7692 4.3114925 2.4463 2.5693 2.8300 3.1995 3.6392 4.1294950 2.4216 2.5323 2.7636 3.0891 3.4809 3.9214975 2.4000 2.5028 2.7077 2.9935 3.3412 3.7377

1000 2.3416 2.4411 2.6266 2.8849 3.1949 3.55161025 2.2814 2.3788 2.5471 2.7790 3.0594 3.37821050 2.1930 2.2869 2.4430 2.6515 2.9062 3.19641075 2.0688 2.1616 2.3080 2.5000 2.7327 3.00021100 1.9534 2.0449 2.1834 2.3591 2.5706 2.8133

EV1

725 1.6394 1.8520 2.2112 2.6188 3.0467 3.4804750 1.8283 2.0033 2.3118 2.6736 3.0598 3.4582775 1.9953 2.1415 2.4051 2.7240 3.0723 3.4353800 2.1329 2.2585 2.4843 2.7629 3.0747 3.4063825 2.2522 2.3612 2.5574 2.8069 3.0880 3.3923850 2.3502 2.4485 2.6228 2.8454 3.0993 3.3742875 2.4186 2.5110 2.6727 2.8756 3.1097 3.3622900 2.4423 2.5351 2.6838 2.8716 3.0892 3.3254925 2.4453 2.5335 2.6694 2.8429 3.0417 3.2605950 2.4189 2.5084 2.6384 2.7993 2.9826 3.1847975 2.3971 2.4862 2.6100 2.7595 2.9308 3.1190

1000 2.3387 2.4269 2.5460 2.6872 2.8486 3.02541025 2.2783 2.3684 2.4842 2.6195 2.7711 2.93651050 2.1883 2.2817 2.3960 2.5268 2.6715 2.82711075 2.0627 2.1589 2.2736 2.4029 2.5442 2.69431100 1.9480 2.0432 2.1558 2.2828 2.4193 2.5634

EV2

725 2.0216 2.2873 2.7415 3.2578 3.7860 4.3291750 2.1075 2.3128 2.6716 3.0884 3.5213 3.9802775 2.1739 2.3407 2.6232 2.9690 3.3399 3.7325800 2.2249 2.3515 2.5901 2.8817 3.2057 3.5494825 2.2858 2.3929 2.5902 2.8419 3.1283 3.4393850 2.3529 2.4511 2.6257 2.8487 3.1021 3.3772875 2.4248 2.5168 2.6789 2.8839 3.1182 3.3711900 2.4854 2.5805 2.7323 2.9156 3.1372 3.3762925 2.5573 2.6456 2.7881 2.9573 3.1701 3.3953950 2.6184 2.7117 2.8471 3.0209 3.2269 3.4451975 2.7077 2.8076 2.9422 3.1186 3.3132 3.5133

1000 2.7635 2.8694 3.0132 3.1771 3.3591 3.55611025 2.8203 2.9219 3.0752 3.2584 3.4432 3.64471050 2.8468 2.9566 3.1151 3.2900 3.4834 3.68761075 2.7970 2.9290 3.0835 3.2730 3.4580 3.66681100 2.7927 2.9210 3.0869 3.2646 3.4609 3.6731

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the errors for Leland and EV1 decrease for the extreme strikes, whereas the errors

increase for EV2 in the extremes.

Hedging Error by Option Moneyness . With the mean absolute deviation,

gains and losses are perceived the same - equally weighted. The more realistic propo-

sition is that hedging error is perceived differently depending on the moneyness of theoption. Table V presents the weights by strike (moneyness) for the gains and losses.

When the option is deep in the money, the greater weight is on losses, while for deep

out of the money the greater weight is on gains.

Table V: Weighting probabilitiesStrike Pr : Gains 1 Pr : Losses

725 0.2426 0.7574750 0.2880 0.7120

775 0.3355 0.6645800 0.3840 0.6160825 0.4329 0.5671850 0.4814 0.5186875 0.5287 0.4713900 0.5743 0.4257925 0.6177 0.3823950 0.6586 0.3414975 0.6967 0.3033

1000 0.7319 0.26811025 0.7642 0.23581050 0.7935 0.20651075 0.8198 0.1802

1100 0.8435 0.1565

The values for the weighted mean absolute deviation criterion are illustrated in

Figure 4 and Table VI. Those results by strike price and transaction cost show that

the EV2 is the best model for the new criterion.

In Figure 4 the pattern with strikes is somewhat changed for the Leland and EV1

models, with increasing error for deep out of the money options. The pattern for

the EV2 model is ipped, so that the errors for deep in/out of the money options in

decreased. This is exactly the effect that underlies the hedging strategy.

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Table VI: Hedging Errors By Option MoneynessModel Strike Cost0% 0.3% 0.6% 0.9% 1.2% 1.5%

BSM

725 1.6326 4.2301 8.0050 11.9826 15.9835 19.9853750 1.8294 4.2786 7.9856 11.9288 15.9031 19.8784775 1.9959 4.2257 7.7988 11.6445 15.5219 19.4027800 2.1329 4.0928 7.4680 11.1491 14.8656 18.5837825 2.2526 3.9159 7.0351 10.4856 13.9810 17.4783850 2.3508 3.6877 6.5170 9.6948 12.9237 16.1575875 2.4214 3.4287 5.9224 8.7976 11.7335 14.6731900 2.4519 3.1337 5.2912 7.8484 10.4715 13.1006925 2.4602 2.8486 4.6715 6.9112 9.2186 11.5340950 2.4443 2.5499 4.0593 5.9848 7.9849 9.9930975 2.4300 2.2776 3.4793 5.1102 6.8205 8.5361

1000 2.3671 2.0209 2.9596 4.3201 5.7597 7.20541025 2.3072 1.7906 2.4836 3.6007 4.7963 6.00001050 2.2194 1.5692 2.0629 2.9643 3.9450 4.93461075 2.0970 1.3698 1.7039 2.4132 3.2065 4.01041100 1.9818 1.2009 1.3921 1.9453 2.5798 3.2250

Leland

725 1.6326 2.5594 3.8621 5.2996 6.7912 8.3115750 1.8294 2.5855 3.6881 4.9351 6.2468 7.5930775 1.9959 2.5823 3.4859 4.5449 5.6749 6.8462800 2.1329 2.5692 3.2933 4.1673 5.1222 6.1225825 2.2526 2.5584 3.1148 3.8188 4.6107 5.4520850 2.3508 2.5467 2.9603 3.5136 4.1517 4.8439875 2.4214 2.5265 2.8142 3.2335 3.7380 4.2956900 2.4519 2.4848 2.6733 2.9804 3.3635 3.8001925 2.4602 2.4351 2.5454 2.7626 3.0483 3.3829950 2.4443 2.3732 2.4190 2.5557 2.7558 3.0021975 2.4300 2.3290 2.3245 2.3962 2.5255 2.6999

1000 2.3671 2.2507 2.2129 2.2412 2.3157 2.43171025 2.3072 2.1840 2.1223 2.1159 2.1510 2.21811050 2.2194 2.0942 2.0199 1.9883 1.9943 2.02831075 2.0970 1.9774 1.8985 1.8538 1.8401 1.85221100 1.9818 1.8709 1.7928 1.7391 1.7102 1.7025

EV1

725 1.6736 1.9122 2.3184 2.8114 3.3067 3.8198750 1.8500 2.0516 2.3935 2.8069 3.2405 3.6882775 2.0178 2.1790 2.4655 2.8096 3.1947 3.5939800 2.1544 2.2781 2.5127 2.8166 3.1496 3.5085825 2.2647 2.3718 2.5711 2.8316 3.1237 3.4414850 2.3531 2.4508 2.6267 2.8513 3.1065 3.3849875 2.4146 2.5075 2.6672 2.8683 3.0984 3.3466900 2.4361 2.5295 2.6741 2.8518 3.0607 3.2895925 2.4390 2.5201 2.6483 2.8177 3.0030 3.2099950 2.3980 2.4813 2.6150 2.7709 2.9418 3.1290975 2.3617 2.4551 2.5872 2.7220 2.8873 3.0553

1000 2.3029 2.3875 2.4970 2.6344 2.7968 2.97071025 2.2264 2.3129 2.4317 2.5592 2.7118 2.86031050 2.1136 2.2030 2.3195 2.4415 2.5770 2.73071075 1.9612 2.0670 2.1901 2.3099 2.4407 2.57951100 1.8511 1.9435 2.0596 2.1683 2.2869 2.4172

EV2

725 1.3100 1.5115 1.8609 2.2589 2.6677 3.0744750 1.5897 1.7603 2.0685 2.4319 2.8189 3.2134775 1.8482 1.9936 2.2607 2.5856 2.9429 3.3143800 2.0611 2.1857 2.4168 2.7047 3.0261 3.3695825 2.2320 2.3385 2.5390 2.7942 3.0842 3.3958850 2.3505 2.4479 2.6236 2.8482 3.1039 3.3818875 2.4085 2.5015 2.6602 2.8613 3.0896 3.3375900 2.3950 2.4848 2.6275 2.8067 3.0113 3.2334925 2.3325 2.4153 2.5423 2.7006 2.8820 3.0784950 2.2085 2.2902 2.4078 2.5516 2.7127 2.8850975 2.0688 2.1484 2.2560 2.3835 2.5250 2.6760

1000 1.8934 1.9685 2.0651 2.1767 2.3006 2.43171025 1.7101 1.7788 1.8656 1.9658 2.0747 2.19001050 1.5034 1.5692 1.6490 1.7382 1.8334 1.93441075 1.3007 1.3625 1.4336 1.5125 1.5969 1.68461100 1.1130 1.1687 1.2326 1.3025 1.3761 1.4523

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00.005

0.010.015

500

1000

1500

0

10

20

Transaction Costs

BSM Model

Strike Prices

H e

d g

i n g

E r r o r

00.005

0.010.015

500

1000

1500

0

5

10

Transaction Costs

Leland Model

Strike Prices

H e

d g

i n g

E r r o r

00.005

0.010.015

500

1000

15001

2

3

4

Transaction Costs

EV1 Model

Strike Prices

H e

d g

i n g

E r r o r

00.005

0.010.015

500

1000

15001

2

3

4

Transaction Costs

EV2 Model

Strike Prices

H e

d g

i n g

E r r o r

Figure 4. Hedging Error by Option Moneyness for Alternative Models

4. Conclusion

In the last thirty years, nancial derivatives have grown from a marginal activity

to occupy center-stage position in nancial economic theory and nancial practice.

At the same time, mathematical nance has grown to be one of the main branches

of applied mathematics. The single largest credit for these remarkable developments

are due to Fisher Black, Myron Scholes, and Robert Merton, whos classic 1973 papers

gave a theory of how to price options. Without this prescription, option pricing would

have remained more of an art than a science, and trading in options would have been

less liquid and less important, as traders would have had a less rm idea on how

to fairly value and hedge the options. However, this great achievement rests on theassumptions of no arbitrage, lognormality for spot price dynamics, and frictionless

trading. In reality, even though the condition of arbitrage free and the assumption of

lognormality are arguably to be satisfactory most of the time, transaction costs always

exist. As well the evidence suggests that the option price depends on the moneyness

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of the option, that is the strike price.

In this paper a pricing method is introduced which links the volatility to both

the transaction cost and the strike price. The Black-Scholes-Merton formula is used,

with the endogenous volatility, to price options. The results are clear. The endoge-

nous volatility approach produces more accurate prices. The optimization problemfor calculating the endogenous volatilities is simple to implement so that using the

approach is practical.

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