Pricing efficiency on the New Zealand Futures and Options Exchange

Journal of

Journal of Multinational Financial Management 8 (1998) 49-62

ELSEVIER

MULTINATIONAL FINANCIAL MANAGEMENT

Pricing efficiency on the New Zealand Futures and Options Exchange Paul Smith a,Paul Gronewoller b**, Lawrence C. Rose ’ a Credit Suisse Securities, b Department of’ Finance and Property Studies, Massey c Department of’ Commerce. Massey University

London. University, -Auckland

UK Palmerston Auckland,

North,

New Zealand

lVe)v Zealand

Received 30 August 1997; accepted 30 January 1998

Abstract Volatility estimates implied from option pricing models have frequently been used in conjunction with trading strategies to examine whether arbitrage profits were possible. Previous examinations focusing on markets where trading in options and underlying assets can be considered as near continuous and synchronous as markets get, indicate few arbitrage opportunities exist. We posit that such profits may be more evident in markets that rarely exhibit continuous and/or synchronous trading. Two trading rules are used to identify mis-priced options which were traded via a delta neutral spread to attempt to profit from expected corrections. Results indicate that options are efficiently priced on the NZFOE when trading costs are considered. Our results imply that continuous and synchronous trading did not appear to be necessary conditions for efficient option pricing on the NZFOE. 0 1998 Elsevier Science B.V. All rights reserved. JEL clussificutinn:

G13; G14; G15

Keywords:

market pricing

Option

efficiency; Implied

volatility;

New Zealand

1. Introduction Volatility implied by standard option pricing formulas are often utilised when developing trading strategies to test pricing efficiency in option markets.’ The premise underlying these tests is that the market’s collective expectation of the underlying asset return volatility should be consistent across all options with the same exercise price written on that asset. If not, assuming one has the ability to adjust positions * Corresponding author. E-mail: [email protected] ’ See for example Black and Scholes (1972). Brenner and Galai (1984), Chiras and Manaster ( 1978). Figlewski ( 1989). Gemmill and Dickins ( 1986). Latane and Rendleman (1976), and Nisbet ( 1992). 1042-444X/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved PII

s 1042-444x

( 98 )OOO 17-b

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instantaneously and without cost, arbitrage opportunities could exist. In general, results indicate that arbitrage profits are possible. But, these arbitrage profits are typically less than the costs incurred by the general investor entering into the necessary transactions. Most of the past analysis is based on data generated from option and underlying asset markets that exhibit nearly continuous and synchronous trading. One question that has yet to be resolved completely is whether similar results can be observed in markets where one is much less likely to be able to trade two securities nearly instantaneously. Markets differ dramatically in regard to the extent shares and options are traded. For example, in the US most shares are traded at least daily while outside the US that is not the case. In markets where thin and/or non-synchronous trading is prevalent, opportunities to construct or close a potential arbitrage position are likely to be less prevalent. In addition, Jeffrey et al. (1997) find that, on the London Stock Exchange, price changes of infrequently traded shares tend to lag the movement of frequently traded shares. As the market prices of thinly traded securities lag their fundamental values, opportunities for larger arbitrage gains are likely to be more prevalent, provided the arbitrage can be executed. During our sample period, daily trading on the London Traded Options Market averaged less than 8000 contracts, which can be considered relatively thin. Daily trading on the New Zealand Futures and Options Exchange averaged less than 5000 contracts per day. The objective of this paper is to examine the efficiency of option trading on the New Zealand Futures and Options Exchange (NZFOE), where trading activity in either the underlying asset or options is considered to be rarely continuous or synchronous. The NZFOE is owned by the Syndey Futures Exchange (SFE). The NZFOE’s trading hours fit between the close of the US markets and the open of the Australian and Japanese markets. Six of the seven New Zealand shares in our sample are also listed on the Australian Stock Exchange, four of which are also listed on either the New York Stock Exchange, the London Stock Exchange and/or the SEAQ International system. As over 50% of New Zealand securities are held by non-residents, the daytime trading hours of the NZFOE provide an extended risk management market for offshore holders of New Zealand shares. Growing interest in the Asia-Pacific region and the resultant exposures are significantly increasing the strategic importance of the NZFOE. In this paper the Black and Scholes (1973) option pricing model is used to identify mis-priced options from the standard deviations implied from their observed prices. Mis-priced options are then traded via a delta neutral spread to attempt to profit from expected corrections. The results show efficient pricing on the NZFOE when transactions costs are considered. Our results indicate that instantaneous and synchronous trading in the equity and option markets does not appear to be necessary conditions for efficient option pricing.

2. Background Extensions of the Black and Scholes (1972, 1973) papers have become the basis for many tests of option market pricing efficiency. It was evident from their work

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asset did not yield the same implied

volatility. This noise in implied volatility estimates was attributed to heterogeneous volatility forecasts across option traders, transactions costs, non-simultaneity of prices and model mis-specification. Noting the inconsistencies in the implied volatility estimates of the BS model, various schemes have been proposed to estimate true volatility. For example, Schmalensee and Trippi (1978) weighted implied volatility equally across different strike prices to estimate the market’s collective expectations. Latane and Rendleman (1976) created a WISD (weighted implied standard deviation) by weighting the implied standard deviations of individual stock options by the first derivative of the Black Scholes (BS) model with respect to the standard deviation. This WISD was found to be useful in identifying relatively over- and underpriced options and was shown to be highly correlated with the observed volatility of the underlying asset. Chiras and Manaster (1978) point out that Latane and Rendleman’s WISD was not a true weighted average as the resulting weights do not sum to one, biasing the ISD toward zero. Chiras and Manaster extended Latane and Rendleman’s ISD weighting by the elasticity of the call price with respect to standard deviation, and included dividends. Day and Lewis (1988) weighted the ISDs of different exercise prices by relative trading volume. Alternatively, Beckers ( 198 1) and Whaley ( 1982) assume errors in option prices with different exercise prices exist and develop implied volatility measures that minimize the pricing errors. Whaley employs non-linear regression and Beckers employs a quadratic loss function. Beckers (1981) also tested the predictive performance of three ISDs using closing CBOE option and share prices: Latane and Rendleman’s WISD, the ISD of the nearest at-the-money option (ATISD), and an extension of Whaley’s derived from a quadratic loss function (BISD). He concluded that the nearest to actual results were produced by the ATISD, followed by Beckers’ BISD and ultimately Latane and Rendleman’s WISD. Gemmill and Dickins ( 1986) employed Chiras and Manaster’s WISD to examine call pricing efficiency on the London Traded Options Market (LTOM) between May 1978 and July 1983. Before transactions costs, the BS model was able to identify 800 profitable arbitrage positions. The profitable positions disappeared when the implications of transactions costs and trading at the bid-ask spread instead of closing prices were analysed. Gemmill and Dickins also investigated the impact of dividends by employing the Merton (1973) constant dividend model where appropriate to derive a new set of ISDs and reset the arbitrage positions. No dividend impact was observed. Gemmill (1986) compared the ability of six ISD schemes to predict observed variance during the option’s approximately 20 week time to maturity. The six estimation schemes were: Chiras and Manaster’s WISD, Beckers’ BISD, an arithmetic mean ISD (MISD), the ISD of the nearest at-the-money option (ATISD), the ISD of the furthest in-the-money option (INISD), the ISD of the furthest out-ofthe-money option (OUTISD). Only the implied volatility of the closest in-the-money option performed better than a historical standard deviation. However, no significant economic difference was found between the predictive ability of the ISDs. Gemmill

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then examined the predictive ability of the various volatility estimates by pooling the cross-section and time-series data and forecasting the actual standard deviation. The ISDs made a significant contribution, but the historical standard deviations made no significant contribution when forecasting actual standard deviations. The estimated ISDs seem to contain different information than historical standard deviations. Corrado and Miller ( 1996) note that there is no generally accepted ISD estimation method, and show that virtually unbiased estimates of the underlying assets return standard deviation are obtained using the BS model and the at-the-money options.

3. Methodology

and data

The BS model is used to estimate implied volatility from observed option prices. Although algebraic solutions exist for obtaining implied volatility, an iterative procedure is used in this paper to produce more accurate results. In a second pass and where dividends are likely to impact the price of the underlying asset, implied volatility estimates are derived from the Merton ( 1973) constant dividend model. Two market based ISDs are then used to identify mis-priced options. The WISD derived by Chirds and Manaster and later used in Gemmill and Dickins was initially applied. Gemmill and Dickins (1986) argue that their weighting scheme is appropriate for relatively new and illiquid markets. The WISD weights ISDs across all exercise prices by call price elasticity with respect to volatility. This ISD measure is used in all procedures with the exception of two. In practice, WISD may misrepresent the markets expected future variance of returns and this could distort any efficiency tests. Beckers ( 1981), Gemmill ( 1986), and Corrado and Miller (1996) have noted the implied volatility of the nearest at-the-money option to be as good as, if not better than, WISDs in estimating the markets expected future volatility of returns to an underlying stock. Thus a parallel procedure, employing the ISD of the nearest at-the-money option (ATISD) as a measure of the markets expected future variance of returns, is also employed. Potential arbitrage positions or spreads are discovered by substituting estimated ISDs (WISD and ATISD) into the BS formula in place of actual standard deviations, and repricing each contract. The methodology identifies mis-specified volatility parameters in observed option prices and trades those options in anticipation of profits from volatility induced price corrections. Estimates of the markets expected future volatility of returns were implied and used to approximate equilibrium volatility parameters and prices for each option contract. Mis-priced contracts are then traded in delta neutral spreads on the premise that prices will revert to equilibrium over time. But, arbitrage positions set using ATISDs in place of market ISDs do not always result in simultaneous long and short positions. As the ATISD was the market ISD of at least one contract, correctly priced option contracts were sometimes used in constructing spreads. For each spread to be delta neutral, a given change in the stock price was required

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to affect the value of the first and second option positions (ni) equally:’

(1) where ?C,/?S is the change in the value of option i for a unit change in the value of the underlying stock (the BS delta). From Eq. (l), the correct hedge ratio in over- and underpriced options for that position to be instantaneously immunised to changes in the underlying stock was derived as:

To set the spread, the ‘expensive’ option was sold and the ‘cheap’ option bought in the ratio described in Eq. (2). This yielded a spread position instantaneously immunised from movements in the underlying stock and, given the relative inelasticity of the BS option pricing model with respect to the risk free rate, positioned to re-price based substantially on changes in the volatility parameter. The assumption of the spread is that the volatility variable in ‘expensive’ and/or ‘cheap’ options was temporarily mis-priced. Spreads are closed out on the first occasion that both contracts are traded on the same day.3 The aim is to reduce the downward bias on profits which Gemmill and Dickins suggested resulted from a longer holding period. A truer representation of spread profits should strengthen the test of market efficiency. One additional benefit of a shorter holding period was a reduction in the risk of significant movements in the price of the underlying stock. There is also an implicit assumption that early exercise would not occur, which is consistent with Merton (1973). Two trading rules are tested in this paper. The first is consistent with Gemmill and Dickins’ most valuable trading rule (MVTR).4 Their MVTR maximises joint percentage profits from both legs of a potential spread: Maximise [(c - Ci) -I-(Cj - q)]/[C,

+ Cj]

(3)

where * denotes a model value. The second trading rule used in this paper selects options for the short and long legs that carry the highest price elasticity with respect to volatility. As the spread is set to re-price based largely on changes in volatility, this trading rule selects those options (in the ratio of delta) whose prices were most sensitive to a unit change in volatility. The combinations of implied volatility estimates, trading rules and dividend adjustments tested are summarised in Table 1 as Procedures l-5. ’ Each spread contained one long and one short position only. 3 Spreads could not always be opened and closed on the same day. was one day. 4 An illustration of this trading rule is found in Appendix A.

Thus,

the minimum

holding

period

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procedures

Procedure

Volatility

estimate

I 2 3 4 5

C&M (1978) C&M ( 1978) C&M ( 1978) ATISD ATISD

WISD WISD WISD

Trading

rule

G&D MVTR Elasticity G&D MVTR G&D MVTR G&D MVTR

Dividend

adjustment

No No Yes No Yes

3.1. Data

The NZFOE provided intraday prices for call options written on seven stocks for the period 5 October 1990 to 2 April 1996. The call options were on the following stocks: Brierley Investments Ltd, Fletcher Challenge Ltd - Forestry, Fletcher Challenge Ltd - Ordinary, Lion Nathan Ltd, Telecom Corporation NZ Ltd, Goodman Fielder and Carter Holt Harvey Ltd. No bid or offer identifier was available for each trade. Unadjusted close of trade New Zealand Stock Exchange stock prices and dividend yields were provided by Datastream Incorporated. Reserve Bank 90-day bill successful tender yields were provided by the Reserve Bank of New Zealand.5 Transaction costs were estimated at $1.25 per lot traded, with four lots necessary per spread.6

4. Results Analysis of intraday prices for call options on stock for the period 5 October 1990 to 2 April 1996 yielded 162 mis-priced options. However, in 119 of the 162 mis-priced opportunities identified, no offsetting over-/underpriced option could be identified. Setting spreads for these observations did not require the use of a trading rule as the counter position was presumed to be a fairly priced option. The high number of spreads not requiring a trading rule was attributed to non-synchronous trading in the various options on a stock. This is a characteristic of the NZFOE’s illiquidity. Results were analysed and reported over the total sample ( 162 spreads), and for the subset of spreads (43) that required the use of a trading rule. Implied standard deviations for some options were not possible to calculate as their price fell outside the pricing boundaries specified in the BS (1973) model.’ These options were omitted from that set used in setting spreads. All spreads are closed out at the first opportunity. The average holding period for spreads was 1.4 days with a minimum holding period of 1 day and maximum 5 The risk free rate was approximated using [successful tender] yields for Reserve Bank Bills whose maturity was closest to that of the option. This was consistent with the risk free rate approximation used by Schmalensee and Trippi (1978). 6 Transaction cost information is supplied by the NZFOE and verified by several option traders. ’ The number of options that could not be solved for their ISD numbered 446 of 3074, or 14.5% of the sample. Comparative statistics from earlier studies were not available.

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of 12 days. This is a more realistic holding period than the one month holding period in Gemmill and Dickins. Non-parametric sign tests are used to analyse the results due to the high number of opening spread positions that required no capital outlay.8 With a positive opening spread position, a rate of return measure was inappropriate as there was no capital outlay required. Sign tests are used to examine the null hypothesis that the median return to the sample was less than or equal to zero (market efficiency), against the alternative hypothesis that the median return to the sample is greater than zero.9 The null hypothesis of market efficiency is rejected if the binomial probability of seeing a value less than or equal to the number of unprofitable outcomes (the p value) is less than or equal to the level of significance.” These tests are conducted before and after trading costs (transaction costs and the bid ask spread) are included. Procedure 1 weights ISDs using WISD, and sets spreads using MVTR. Before trading costs, this produced 102 profitable spreads (dollar value >O) and 57 unprofitable spreads (dollar value CO). The average per trade dollar return to the entire sample was $1.81, and the null hypothesis of market efficiency is rejected at the 1% level of significance. After including trading costs, 38 spreads remained profitable while 124 became unprofitable, with an average dollar return to the sample of -$4.03. The null hypothesis of market efficiency cannot be rejected at the 1% level of significance. Of the 43 spreads whose construction required the use of the trading rule, 30 are profitable before trading costs, with an average per trade dollar return of $2.12. The null hypothesis of market efficiency is rejected at the 1% level of significance. After trading costs, nine spreads remained profitable and 34 became unprofitable. The average per trade dollar return for the trading rule subsample is -$3.39, and the null hypothesis of market efficiency cannot be rejected at the 1% level of significance. The results of Procedure 1 are summarised in Table 2. Table 2 Hypothesis

tests for the Chiras

and Manaster

All spreads

Before trading costs After trading costs

( 1978) WISD PC 1’; n. n)

K>O

n
IT=0

102 38

57 124

3 0

0.0002** 0.99

and the Gemmill Trading

n

1.81 -4

and Dickins

rule spreads

x>o

n
n=o

30 9

13 34

0 0

(1986)

MVTR

P( y; n, TC) II

0.0069** 0.99

2.12 -3.4

A Represents dollar profits. II Represents average dollar profits. P( J: n, n) Represents the cumulative binomial probability of the count rt
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Procedure 2 employs WISD to estimate the market’s expected future volatility, and set spreads according to the relative elasticity of each contract. Before trading costs 101 spreads were profitable and 58 unprofitable, with an average per trade dollar return for the entire sample of $1.75. The null hypothesis of market efficiency is rejected at the 1% level of significance. After trading costs, 36 spreads remained profitable and 126 became unprofitable. The average per trade dollar return to the entire sample is -$4.07, and the null hypothesis of market efficiency cannot be rejected at the 1% level. Of the 43 spreads that required the use of the trading rule, 16 (37%) changed their structure when relative elasticity replaced MVTR. There are 29 profitable spreads before trading costs, with an average per trade dollar return of $1.88. Interestingly, the null hypothesis of market efficiency cannot be rejected at the 1% level of significance, but is rejected at the 5% level of significance. After trading costs, the number of profitable spreads dropped to seven, and the null hypothesis of market efficiency cannot be rejected at the 1% level of significance. The average per trade dollar return is - $3.54. Acceptance of the null hypothesis at the 1% level after trading costs seems to reflect the power of the respective trading rules as opposed to conflicting findings of efficiency. This is evidenced in the difference in average per trade dollar returns to the respective samples when trading costs are excluded. The subsample spreads based on MVYR produced average dollar returns over 12% higher than the sample spreads constructed using relative elasticity, and average dollar losses were 4% lower with trading costs. The results of Procedure 2 are given in Table 3. Procedure 3 is the first of two procedures to test the impact of dividends on market efficiency. Including dividends will produce changes in implied volatility and changes in contracts indicating profitable spread possibilities. Structural changes in spreads could potentially impact trading profits. The composition of 51% (22/43) of trading rule spreads changed when WISDs were recalculated by the Merton ( 1973) dividend adjusted model. Alternatively, the composition of 60% (26/43) of ATISD spreads requiring the use of a trading rule changed when ATISDs were dividend adjusted.

Table 3 Hypothesis

tests for the Chiras

and Manaster

All spreads

Before trading costs After trading costs

(lY78) P( )‘; 12, n)

n>O

n
x:0

101 36

58 126

3 0

0.0004* 0.99

WISD n

1.75 -4.1

and the elasticity Trading

trading

rule spreads

K>O

n-co

n=O

29 7

14 36

0 0

K Represents dollar profits. I2 Represents average dollar profits. P( y: n, n) Represents the cumulative binomial probability of the count n ~0. ** Indicates rejection of the null hypothesis at the 1% level of significance. * Indicates rejection of the null hypothesis at the 5% level of significance.

rule P( ~1: n, n)

I7

0.0158** 0.99

1.88 -3.5

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Procedure 3 is a second pass where the Procedure 1 spreads are reset after adjusting ISDs for dividends. When including dividend effects and before trading costs, 105 spreads are profitable and 54 unprofitable with an average per trade dollar return of $2.04. The null hypothesis of market efficiency was rejected at the 1% level of significance. After trading costs, 40 spreads remained profitable and 122 became unprofitable, with an average dollar return of -$3.80. The null hypothesis of market efficiency cannot be rejected at the 1% level of significance. Of the 43 spreads whose construction required the use of a trading rule, 32 were profitable before trading costs with an average per trade dollar return of $2.93. The null hypothesis of market efficiency is rejected at the 1% level of significance. After trading costs, the number of profitable spreads dropped to 10 and the average per trade dollar return is -$2.56. The null hypothesis of market efficiency cannot be rejected at the 1% level of significance. The results of Procedure 3 are summarised in Table 4. The structure of spreads with and without the inclusion of dividends differed significantly. Of the 43 spreads in place using WISD and MVTR in Procedure 1, 22 ( 5 1%) change their structure when dividends are included in Procedure 3. Although the effect of this is difficult to quantify, the summaries in Tables 2 and 4 can be compared with the knowledge that the only difference between the two procedures is the inclusion of dividends leading to the results in Table 4. Average per trade dollar profits from all spreads in Table 4 before trading costs were higher than the equivalent sample return in Table 5 ($2.04 versus $1.81), and average dollar losses including trading costs lower (- $3.80 versus - $4.03). As WISD may misrepresent the market’s expected future variance of returns, Procedure 4 substitutes ATISD for WISD and runs a parallel test to Procedure 1. The market’s expected future return variance is estimated using ATISD, and spreads were set by MVTR. Initially, spread setting under this set of assumptions required the ISD of the nearest at-the-money option to fall within the ISDs of at least two other options of the same maturity in order for there to be an over- and underpriced option to spread. However, this condition excluded all existing spreads selected from only two contracts ( 119/162), and eliminated 119 contracts mis-priced in terms of

Table 4 Hypothesis

tests for WISD,

MVTR,

with

All spreads

Before trading costs After trading costs

dividend PC .K

n>O

n
n=o

105 40

54 I22

3 0

adjustment Il.

0.0001* 0.99

n)

Il

2.04 -3.8

Trading

rule spreads

n>o

n
ll=O

32 10

II 33

0 0

ri Represents dollar profits. I7 Represents average dollar profits. P( J: n, n) Represents the cumulative binomial probability of the count n < 0. ** Indicates rejection of the null hypothesis at the 1% level of significance. * Indicates rejection of the null hypothesis at the 5% level of significance.

P( y; 12, a)

I7

0.0009** 0.99

2.93 -2.6

58 Table 5 Hypothesis

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All spreads

Before trading costs After trading costs

and Dickins P(y:n,rc)

rr>o

iT
II=0

108 36

51 126

3 0

Financial

0.0001** 0.99

Management

( 1986)

IT

2.3 -3.6

8 c 199X) 4%62

MVTR

Trading

rule spreads

rr>o

n
n=O

36 7

7 36

0 0

P(y;

N, T)

0.0001*+ 0.99

17

3.98 -1.6

n Represents dollar profits. II Represents average dollar profits. P( J; n, rr) Represents the cumulative binomial probability of the count rt < 0. ** Indicates rejection of the null hypothesis at the 1% level of significance. * Indicates rejection of the null hypothesis at the 5% level of significance.

the new volatility estimate, ATISD.” To ensure all pricing inefficiencies are addressed, correctly priced option contracts are bought or sold as counter positions for the mis-priced legs of each spread. Before trading costs, 108 spreads under Procedure 4 are profitable and 51 unprofitable with an average per trade dollar return of $2.30. The null hypothesis of market efficiency is rejected at the 1% level of significance. After trading costs, 36 spreads remained profitable and 126 became unprofitable, with an average per trade dollar return of -$3.56. The null hypothesis of market efficiency cannot be rejected at the 1% level of significance. Of the 43 spreads whose construction required the use of a trading rule, 36 are profitable before trading costs with an average per trade dollar return to the sample of $3.98. The null hypothesis of market efficiency is rejected at the 1% level of significance. After trading costs, the number of profitable spreads dropped to seven and the average per trade dollar return is -$1.63. The null hypothesis of market efficiency cannot be rejected at the 1% level of significance. These results are consistent with earlier results and provide evidence of pricing efficiency on the NZFOE. The results of Procedure 4 are summarised in Table 5. Procedure 5 is a second pass where the Procedure 4 spreads are reset after adjusting ISDs for dividends. When including dividend effects, and before trading costs, 112 spreads are profitable and 47 unprofitable with an average per trade dollar return of $2.07. The null hypothesis of market efficiency was rejected at the 1% level of significance. After including trading costs, 48 spreads remain profitable and 114 became unprofitable, with an average dollar return of -$3.81. The null hypothesis of market efficiency cannot be rejected at the 1% level of significance. Of the 43 spreads whose construction require the use of a trading rule, 32 are profitable excluding trading costs with an average per trade dollar return of $4.29. The null hypothesis of market efficiency is rejected at the 1% level of significance. After trading costs, the number of profitable spreads dropped to 14 and the average I’ As the ISD of the nearest at-the-money future variance of returns to the underlying priced and one ‘incorrectly’ priced contract, equilibrium, and no corrections in its price

contract was used as the estimate of the market’s expected stock, any two party spreads involved trading one ‘correctly’ The implied price of the nearest at-the-money option was in could be expected.

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per trade dollar return was -$1.37. The null hypothesis of market efficiency cannot be rejected at the 1% level of significance. The results of Procedure 5 are summarised in Table 6. Again, the summaries in Tables 5 and 6 can be compared with the knowledge that the only difference between the two procedures is the inclusion of dividends leading to the results in Table 5. Average per trade dollar profits from spreads requiring the use of a trading rule before trading costs in Table 6 were higher than the equivalent sample return in Table 5 ($4.29 versus $3.98), and average dollar losses including trading costs lower (-$1.37 versus -$1.63).

5. Conclusions

This paper examined the pricing efficiency of call options on the NZFOE by using spreads to exploit the information contained in ISDs. Market implied standard deviations are either weighted by volatility elasticity to construct weighted implied standard deviations, or approximated by the ISD of the nearest at-the-money option. Then, Black and Scholes implied values for traded options are derived by substituting these ISDs into the Black and Scholes (1973) option pricing model. Mis-priced options are identified by comparing the model values to observed market prices, and delta neutral spreads are constructed from long and short positions in those mis-priced options. Trading rules were used to select amongst eligible contracts for the spreads. It was expected that the null hypothesis of market efficiency would not be rejected for the period under study if the average per trade dollar return to the spreads after trading costs was less than or equal to zero. Non-parametric sign tests on the significance of the results consistently supported the null hypothesis of pricing efficiency for call options on stocks traded on the NZFOE. This pricing efficiency obtained within the boundaries of transaction costs. Significant positive profits were consistently found before transaction costs were considered, yet the introduction of transaction costs saw those profits move to significant losses in every procedure. This result indicates that options are efficiently priced on the NZFOE. Table 6 Hypothesis

tests for ATISD,

the Gemmill

All spreads

Before trading costs After trading costs

and Dickins P(.Y;n,n)

?T>o

w
rr=o

112 48

47 114

3 0

0.0001** 0.99

(1986) I7

2.07 -3.8

MVTR,

and the dividend

Trading

rule spreads

n>O

n
rr=o

32 14

11 29

0 0

n Represents dollar profits. 17 Represents average dollar profits. P( y; n, rc) Represents the cumulative binomial probability of the count n < 0. ** Indicates rejection of the null hypothesis at the 1% level of significance. * Indicates rejection of the null hypothesis at the 5% level of significance.

elect P( y; n. n)

17

0.0009** 0.99

4.29 - I .4

60

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Consistent with Latane and Rendleman ( 1976) and Gemmill and Dickins (1986). this paper found the BS option pricing model to be useful in identifying over- and underpriced options on the NZFOE. The unilateral rejection of the null hypothesis of market efficiency prior to the introduction of trading costs confirmed the BS model could be used to exploit mis-priced volatility. Employing the constant dividend model altered the structure of a large proportion of spreads for the ISD weighting schemes, yet had little effect on the profitability of spreads. The lack of an effect on pricing efficiency corresponds with the findings of Gemmill and Dickins ( 1986). The elasticity based trading rule altered the structure of a large percentage of the spreads, yet yielded similar results to that of MVTR. The null hypothesis of pricing efficiency held here as well. Testing pricing efficiency using the ISD of the nearest at-the-money option as a measure of the market’s expected future variance of returns also confirmed earlier test results. The null hypothesis of pricing efficiency again holds. This finding was significant as ATISDs are considered by some to be an empirically superior measure of the market’s expected future variance of returns. Analysis of short term spread holding periods augments existing literature that has found pricing efficiency when examining ISD based profit opportunities for monthly and intraday holding periods. The results imply that continuous and synchronous trading were not necessary conditions for efficient option pricing on the NZFOE.

Acknowledgements The authors are grateful to Ike Mathur (Editor), an anonymous referee, and David Dubofsky whose valuable comments and suggestions substantially improved the article. Any remaining errors are naturally ours.

Appendix A Constructing spreads using the Gemmill and Dickins (1986) MVTR This appendix illustrates spread construction using the Chiras and Manaster (1978) WISD and the Gemmill and Dickins (1986) MVTR. The following data was obtained on 9 October 1990 for three call options maturing in February 199 1 written on Brierley Investments Ltd (BRY ) which was trading at NZ$l.36.

ISD $1.20 $1.40 $1.60

60.24 $0.13 $0.05

0.27 0.32 0.31

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ISDs for each option are estimated using this data. The WISD for BRY stock options maturing in February 1991 were calculated using the Chiras and Manaster (1978)approach. (a) Derive the ISD for each option contract. (b) Derive d, for each contract using the Black and Scholes (1973) model. (c) Use d, to calculate the elasticity of each contract using the following:

elasticity =( sct’zmd”2)(7)

(AlI

(d) Restate each contract’s elasticity as a percentage of the sum of all elasticities for the given stock and maturity. (e) For each contract, multiply the elasticity weights obtained in (d) by the ISD to derive the contribution of each contract to the WISD. (f) Sum all contributions to yield the WISD.

Strike

ISD

d,

Elasticity

Percent

$1.20 $1.40 $1.60

0.27 0.32 0.31

1.17 0.24 -0.46

0.75 0.88 2.32

0.19 0.22 0.59

of total

Contribution

WISD

0.05 0.07 0.18

0.30 0.30 0.3

(sum

of contributions)

The calculation of the WISD for BRY call options is summarised below: After the WISD is estimated, it is substituted back into the Black and Scholes (1973) in place of the options ISD and each contract is repriced. The resulting model prices will identify over- and underpriced options which are bought and sold in a delta neutral spread. The Gemmill and Dickins (1986) MVTR maximizes joint percentage profits from each leg of the spread. The sum of the differences between model and market prices for each leg of a spread is expressed as a percentage of the sum of the market values of each option. The combination that maximises this percentage return is selected as the optimal spread. The options with $1.40 and $1.60 strike prices are found to be overpriced. So, the ex ante value of combining contracts 1 and 2 in a spread would be calculated as follows. (a) Substitute the WISD for the ISD in the Black and Scholes (1973) formula and re-value each contract. (b) Calculate the difference between the model and observed market values. (c) Calculate the MVTR value. A spread over contracts $1.20 and the $1.60 strike price options produced a MVTR value of 0.0283, thus a spread over the $1.20 and the $1.40 strike price options was selected and the MVTR value results are summarised below.The $1.40 strike price option was sold and the $1.20 strike price option bought in the delta

62

P. Smith

et al. J Journal

Strike

WISD

BS model

$1.20 %I.40

0.30 0.30

0.25 0.12

of’kfultinational

value

Market

Financial

Management

value

0.24 0.13

8 (I 998) 49-62

Over-/underpriced

MVTR

Over Under

0.0312 0.0312

value

ratio. The results from the spread are detailed below. The spread was not profitable in this case. Open

spread

Close

spread

Gross

Overpriced

0.0746

-0.05

0.0246

Underpriced Total

- 0.097 -0.0224

0.06 0.01

-0.037 -0.0124

profit

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