An empirical test of the pricing of VPO contracts.

Abstract:

A variable purchase option (VPO) is a call option issued by a company on a stochastic rather than on a fixed number of its ordinary shares. This paper tests the arbitrage-free pricing model of Handley (2000) using a transactions dataset of actual market prices covering the five VPOs traded on the Australian Stock Exchange during the six-year period from May 1992 to May 1998. It is initially found that the model systematically overprices VPOs. A subsequent case-based explanatory analysis of the pricing errors, however, shows that this mispricing substantially disappears under different estimates of two key parameters. The results are consistent with investors using risk-adjusted discount rates rather than the risk-free rate in valuing the bond component of the VPO and, when material using a narrow range of volatility estimates, rather than historic volatility estimates, in valuing the option component of the VPO.

Keywords:

VPO; OPTION; EQUITY CAPITAL; DILUTION.

1. Introduction

A variable purchase option (VPO) is a call option issued by a company on a stochastic rather than on a fixed number of its ordinary shares. Handley (2000) examines the design of the security and derives a theoretical arbitrage-free pricing model within a continuous-time framework. A VPO is structured so that rational exercise at maturity is ex ante virtually certain. As a result, the security represents an innovative capital-raising tool available to corporations which have a certain requirement to raise a fixed amount of new equity capital on some known date in the future. From an investor's view, a VPO offers a right to participate in a future capital raising at a fixed discount to the underlying share price at that time and may also be used by investors who wish to hedge a fixed-dollar exposure to the underlying stock. The purpose of this paper is to test a number of empirical implications which result from the analysis in Handley (2000) and in particular:

(i) to test a simple theoretical bound on VPO prices;

(ii) to examine the performance of the pricing model; and

(iii) to examine the theoretical probability of rational exercise.

The empirical tests in this paper are based on a transactions dataset of actual market prices covering the five VPOs traded on the Australian Stock Exchange ('ASX') during the six-year period from May 1992 to May 1998. The paper is organised as follows. In section 2, the security is briefly reviewed and the pricing model is specified in section 3. Section 4 provides details of the dataset and the approach to estimating key parameters. The empirical results are presented in section 5 and a summary and conclusion is set out in section 6.

2. The Security

In its most basic form, a VPO is a European call option on a variable number of ordinary shares where the actual number of shares underlying each option is calculated at maturity in accordance with a formula which is specified at the time of issue. In the specific case of a bounded VPO, the number of shares underlying each VPO at maturity is:

(1) c ;[S.sub.T] < [x.sub.c]

[N.sub.T] = k / [S.sub.T](1 - d) ;[x.sub.c] [less than or equal to] [S.sub.T] < [x.sub.f]

f ;[x.sub.f] [less than or equal to] [S.sub.T]

where: k = the (fixed) exercise price per option;

[S.sub.T] = the market price of the underlying shares at maturity;

d = the exercise discount (expressed as a proportion);

c and f = the cap and floor;

[x.sub.c] = k / c(1 - d) = is the corresponding cap price; and

[x.sub.f] = k / f(1 - d) = the corresponding floor price. In order for the formula to be well defined, it is necessary to impose the constraint that [S.sub.T] is strictly greater than zero. The number of shares received is a function of a stochastic stock price so it too is stochastic. Since the VPO is a type of call option then the payoff at maturity on a long VPO is:

(2) [F.sub.T] = max[0, [N.sub.T][S.sub.T] - k]

which after substituting from (1) becomes:

(3) 0 ;[S.sub.T] < [x.sub.c](1 - d)

c[S.sub.T] - k ;[x.sub.c](1 - d) [less than or equal to] [S.sub.T] < [x.sub.c]

FT = kd / 1 - d ;[x.sub.c] [less than or equal to] [S.sub.T] < [x.sub.f]

f[S.sub.T] - k ;[x.sub.f] [less than or equal to] [S.sub.T]

By way of example, consider the issue of 26.9 million VPOs by West Australian Newspapers Limited ('WAN Limited') as part of a capital reconstruction undertaken in August 1994. Each VPO had an exercise price of $5.00 and was exercisable at a 10% discount to the underlying share price on 30 June 1999 subject to a minimum of one share and a maximum of three shares. From (1) and (3), if the WAN Limited stock price at maturity was between the cap price of $1.85 and the floor price of $5.56, then each VPO was exercisable into $5.56 worth of shares, for a fixed outlay of $5.00 per VPO, resulting in a fixed payoff to the investor of $0.56 per VPO. If the share price at maturity exceeded $5.56, then each VPO was exercisable into one share (for the $5.00 outlay) resulting in a payoff increasing linearly in the stock price. If the share price at maturity was between $1.00 and $1.85, then each VPO was exercisable into three shares (for the $5.00 outlay) resulting in a payoff decreasing linearly in the stock price. Rational investors would not exercise the VPO if the share price at maturity was below $1.00

Since the number of shares underlying each VPO is bounded, then for a given stock price at maturity, the payoff at maturity on the VPO is also bounded and may be expressed as:

(4) f max [??][0, [S.sub.T] - [x.sub.f] (1 - d)][??] [less than or equal to] [F.sub.T] [less than or equal to] c max[0, [S.sub.T] - [x.sub.c] (1 - d)]

The lower bound in (4) is equivalent to the payoff at maturity on f call options each on one share with a strike price of [x.sub.f](1 - d) while the upper bound is equivalent to the payoff at maturity on c call options each on one share with a strike price of [x.sub.c] (1 - d). Accordingly, one interpretation of a bounded VPO is that it is at worst f call options (each on one share with a strike price of [x.sub.f](1 - d)) and at best c call options (each on one share with a strike price of [x.sub.c](1 - d)). This provides a simple arbitrage-free bound on the value of a VPO at any time prior to its maturity.

In both the general bounded VPO contract and (for simplicity) the WAN Limited example described above, the stock price used to calculate the number of shares underlying each VPO in (1) and hence the stock price which determines the effective exercise price per share is the stock price at maturity [S.sub.T]. However, due to market imperfections, the VPO contract may specify that [N.sub.T] is in fact to be based on a weighted average of stock prices over some specified trading period, referred to here as the price setting period rather than on the single stock price at maturity. (For example, in the WAN Limited case, the number of shares per VPO was to be based on a weighted average stock price over a 10-business-day period in June 1999). Further, Australian securities law requires that the price setting period must finish no earlier than 20 business days prior to the maturity date. (1) If a price setting period applies, then a VPO may be interpreted as a type of Asian call option since the payoff at maturity is based on an average of stock prices rather than on a single stock price.

Since a VPO is a call option written by a company on its own ordinary shares, then it is in fact a type of warrant. When a warrant is exercised, new shares are issued and the exercise proceeds, if any, are paid into the firm. As a result, the wealth of the firm's existing (non-participating) shareholders may be diluted. Dilution complicates the analysis and valuation of warrants compared to otherwise equivalent call options. The need to take account of dilution in examining a VPO is further complicated by the fact that a VPO is a warrant with a stochastic rather than a fixed exchange ratio. However, since, in an efficient market, the stock price of a firm conditionally reflects dilution at all times following the time that a warrant issue is first announced, then taking the stock price of the actual firm with VPOs as the underlying state variable for pricing purposes means that no explicit adjustment for dilution is necessary. (2) Whether dilution is material in any particular case is obviously determined by the terms of the particular issue and the prevailing stock price. By way of example, in the WAN Limited case, the expected dilution factor, expressed as the ratio of the number of ordinary shares underlying each VPO to the existing number of ordinary shares on issue averaged 17% over the period from October 1994 to May 1998.

3. The Pricing Model

Handley (2000) derives the theoretical arbitrage-free pricing model for a bounded VPO on a non-dividend-paying stock within a Black-Scholes setup. Here the model is extended to take account of the payment of discrete deterministic dividends on the underlying stock. (3) Consider a long position in a bounded VPO on an underlying stock with price process S = {[S.sub.t]; t [member of] [0, T]} where [S.sub.t] represents the ex-dividend stock price at time t. Let [S.sup.&] = {[S.sup.&.sub.t], t [member of] [0, T]} where [S.sup.&.sub.t] = [S.sub.t] - [DELTA] is the process for the stock price net of the present value of the escrowed future dividends to be paid on the underlying stock over the life of the VPO. (4) Assume the price process of the VPO is a function of the underlying stock price net of the escrowed future dividends and time i.e. [F.sub.t] = F([S.sup.&.sub.t], t). Then the arbitrage-free price of the VPO at time t [member of] [0, T] is:

(5) [F.sub.t] = kd / 1 - d [e.sup.-r(T - t)] + f x [C.sub.BS]([S.sup.&.sub.t], [x.sub.f], T) - c x [P.sub.BS]([S.sup.&.sub.t], [x.sub.c], T)+ c x [P.sub.BS] ([S.sup.&.sub.t], [x.sub.c] (1-d), T)

where: k, d, c, f, [x.sub.c] and [x.sub.f] = are as previously defined;

r = the instantaneous risk-free interest rate;

T = the maturity date of the VPO;

[C.sub.BS]([S.sup.&.sub.t], [x.sub.f], T) = the Black-Scholes value of a call option on one share with current price [S.sup.&.sub.t], strike of [x.sub.f] and maturity of T years [P.sub.BS](*) is the corrresponding Black-Scholes put value and kd / 1 - d is referred to as the VPO discount.

The key assumptions underlying (5) are that the VPO is treated as a contingent claim on the underlying stock price of the firm net of the escrowed future dividends, [S.sup.&] follows geometric Brownian motion with constant volatility [[sigma].sup.&.sub.S] [[sigma].sup.&.sub.S], the term structure of interest rates is flat and known with certainty, frictionless markets and security trading takes place continuously over time. The proof and intuition for the pricing formula follows along similar lines to that in Handley (2000) in the case of VPOs on non-dividend-paying stocks. The pricing model may also be extended to allow for a price-setting period. If, however, the price-setting period is small relative to the maturity of the VPO then it can reasonably be ignored.

A key design feature of a VPO is that rational exercise is ex ante virtually certain. It is straightforward to obtain the theoretical risk-neutral probability of rational exercise of a VPO conditional on the stock price at any time. A sufficient condition for the rational exercise of a bounded VPO at maturity is [S.sup.&.sub.T] > [x.sub.c] (1 - d). Then the risk-neutral probability of rational exercise of a bounded VPO conditional on the stock price at time t [member of] [0, T) is:

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where:

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [P.sup.*]([S.sup.&.sub.T]|[S.sup.&.sub.t]) is the conditional probability density function of [S.sup.&.sub.T] under the equivalent martingale measure Q.

4. Data

The first issue of VPOs was undertaken by General Property Trust in May 1992. Another five issues occurred prior to 30 June 1998; one issue by each of West Australian Newspapers Limited, Colonial Limited, Mirvac Limited, Westfield America Trust and a second issue by General Property Trust. In each case the security has been traded on the ASX. The key terms of each issue are set out in the appendix. This paper is based on a transactions dataset provided by SIRCA which, subject to one exception, includes all transactions in the five VPOs, and corresponding stocks, that were traded on the ASX from 7 May 1992 to 22 May 1998. (5) The information provided includes the price, volume, date and time of each trade recorded to the nearest second. Such a rich dataset allows problems associated with nonsynchronous trading between the stock and the VPO to be minimised.

Starting with the SIRCA dataset, each VPO trade is matched to the closest trade in the corresponding stock resulting in 12,799 matched VPO-stock pairs. Of these, 20 VPO-stock pairs are excluded as the time of trade was absent and a further 2,004 VPO-stock pairs are excluded as the trade occurred after the start of the averaging period. This leaves 10,775 VPO-stock pairs available for testing. Since the pricing model in this paper is arbitrage-free, then apparent pricing errors may simply arise as a result of nonsynchronous trading between the stock and the VPO. To minimise this potential problem, the final sample of VPO-stock pairs to be tested here are those where the trades occur within one minute of each other. Descriptive statistics for the VPO-stock pairs matched to within one minute, and also for VPO-stock pairs matched to within five minutes and all VPO-stock pairs are set out in table 1.

Referring to panel A, the final sample used in this paper consists of 4,256 matched VPO-stock pairs representing approximately 40% of all matched trades. Of these, approximately 58% relate to one security, WANO while the most frequently traded VPO is the CGHO security. It is likely that specific contractual features account for the greatest trading volume occurring in the WANO security and the greatest trading frequency (measured by the ratio of the number of trades to the number of days in the period covered) occurring in the CGHO security. Based on the issue terms, these VPOs are the 'cleanest' for pricing purposes whereas each of the other three VPOs has one or more characteristics which complicates their pricing. For example, the GPTO security has a Bermudan feature (i.e. is exercisable at one of five possible exercise dates) while both the GPTOA and MRVO securities have an unknown maturity date. The median time between each VPO trade and the matched stock trade is 19 seconds. A simple comparison of median trading volumes and values in the VPOs and underlying stocks in each of panels A, B and C indicate that, except for the MRVO security, the final sample of trades for pairs matched to within one minute are of similar size and value to the set of trades for pairs matched to within five minutes and the set of all matched trades.

Other required data broadly falls into two categories: contract specification and parameter estimation. The key issue terms of each VPO including number issued, exercise price, cap and floor on the number of shares underlying each VPO, exercise discount, maturity and the start and end dates of the price setting period are sourced from the prospectus relating to each issue. In relation to two issues (GPTOA and MRVO) the maturity date and price setting period of the VPO are initially uncertain. The prospectus states that the issuer has the right to nominate the maturity date, subject to a specified earliest possible maturity date and a specified latest possible maturity date. It is assumed here that the latest possible maturity date will be nominated by the issuer.

Parameters required to be estimated are interest rates, stock price volatility and expected future dividends. Daily interest rate data is sourced from the Reserve Bank of Australia's Occasional Paper No. 10 and subsequent updates provided direct by the RBA. The proxy for the riskfree interest rate for each day t is based on a linear interpolation of the 90-day Bank Accepted Bill rate and the 5-year Treasury Bond rate on that day, where the interpolation weights are based on the maturity of the VPO. The proxy for stock price volatility for each day t is the sample standard deviation of the continuously compounded return on the share over the previous 60 trading days. These rolling historic volatility estimates are calculated using daily stock price data, adjusted for dividend and capitalisation changes, sourced from IRESS. There were no exchange traded options on any of the underlying stocks during the sample period. The final parameter to be estimated is, for each day t, the present value of the escrowed future dividends to be paid on the underlying stock prior to the start of the price setting. There are two sources of potential measurement error which makes this task particularly problematic. First, a reliable time series of the market's expectations of future dividends is not available. Second the Australian tax system, and in particular the valuation of imputation tax credits and the taxation of trusts introduces further complications in estimating the expected drop off in the stock price on an ex dividend date. Following Galai (1977), Whaley (1982), Rubinstein (1985) and Bakshi, Cao and Chen (1997) and consistent with the assumptions of the Black (1975) model, it is assumed that investors have perfect foresight in setting their dividend expectations such that for each day t the present value of the escrowed future dividends is proxied by the present value of the actual realised dividends paid on the underlying stock and fully grossed up for any attached imputation credits, discounted from the corresponding ex dividend dates at the riskfree rate. Dividend amounts and ex-dates are sourced from the ASX Annual Stockmarket Summaries.

5. Empirical Results

5.1 Arbitrage Bounds

Table 2 reports the results from a test of the theoretical arbitrage bounds on VPO prices discussed in section 2 above.

Descriptive statistics of violations of these bounds for each VPO using the sample of VPO-Stock pairs matched to within one minute are shown. Recall, the lower arbitrage bound is equal to the value of f call options on the stock at a strike of [x.sub.f](1 - d) per share and the upper arbitrage bound is equal to the value of c call options on the stock at a strike of [x.sub.c] (1 - d) per share where d is the fixed exercise discount expressed as a proportion, c and f are the cap and floor on the number of shares per VPO, and [x.sub.c] and [x.sub.f] are the corresponding cap and floor prices. Theoretical call option prices are calculated using Black's (1975) extension to the Black-Scholes model for the case of discrete, deterministic dividends paid on the underlying stock. In relation to two securities (GPTO and WANO), the number of shares underlying each VPO was subject to a cap and floor and therefore the theoretical VPO price is subject to both a lower and upper bound. For the remaining three securities (GPTOA, MRVO and CGHO), the number of shams underlying each VPO was subject to a cap only and therefore only an upper bound on the theoretical VPO price is determined.

The results show that there were no violations of the upper bound in respect of all five VPOs and no violations of the lower bound in respect of one of the two VPOs for which a lower bound was calculated. However, in relation to the WANO security, 379 violations of the lower bound were recorded representing 15.3% of the sample of matched VPO-stock pairs. The mean (median) value of these violations, calculated as the difference between the market price and the lower bound and expressed as a proportion of the market price is 18.04% (16.12%). Although not shown, a time series plot indicates these violations are clustered around the period from September 1997 to January 1998. Investors wishing to arbitrage these violations would have been required to either short sell call options (with a strike of [x.sub.f](1 - d)) or replicate the short position by trading in the underlying stock and risk-free bonds.

Since the lower bound is represented by the value of f call options, then it is possible to impose a lower bound on this lower bound using the well known formula for the lower bound on a European call option on a dividend paying stock. (6) In the WANO case, only 48 of the above 379 lower bound violations also breach this lower-lower bound representing less than 2% of the sample of matched VPO-stock pairs. The mean (median) value of these violations is 2.97% (1.87%).

5.2 Pricing Errors

The pricing model is tested using the final sample of 4,256 matched VPO-stock pairs by comparing observed VPO prices with the theoretical price generated by the model. Pricing errors are calculated as follows:

(8) [e.sub.t] = T[P.sub.t] - O[P.sub.t]

where: [e.sub.t] = the pricing error at time t;

O[P.sub.t] = the observed market price of the VPO at time t; and

T[P.sub.t] = the theoretical VPO price at time t given by equation (5). The mean absolute pricing error is defined by:

(9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where: n = the number of errors and the mean absolute relative pricing error, expressed as a proportion of the observed market price, is:

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Absolute rather than raw errors are used in order to avoid the potential problem of positive errors cancelling out negative errors. Since pricing errors reflect a model's static performance then the least misspecified and hence preferred model is the one which generates the smallest mean pricing errors.

Table 3 reports descriptive statistics of the pricing errors for each VPO and for the total sample. Two main conclusions may be drawn from the results. First, the pricing model overprices VPOs on average. The mean pricing error is positive for four of the five VPOs and for the sample in total. In respect of the fifth VPO (CGHO), the resultant pricing errors have been influenced by the short maturity and low strike price of the VPO. (7) Second, overpricing appears to be systematic as positive pricing errors persist over most of the sample period. The proportion of negative pricing errors is low with only 679 out of the total sample of 4,256. The mean absolute relative pricing error for the entire sample is 24.49% and the median absolute relative pricing error for the entire sample is 21.81%, indicating negative skewness in the pricing errors. The best-priced security is WANO with a mean (median) absolute relative pricing error of 20.34% (15.41%), consistent with the fact that it is one of the least complicated securities for pricing purposes. (8) The worst priced security is MRVO with a mean (median) absolute relative pricing error of 187.3% (178.9%). This however is largely due to the fact that the security contains a valuable cash repurchase right which is not taken into account by the theoretical model. A cash repurchase right provides the issuer with the right but not the obligation to buy back the VPO for a fixed cash payment at some later date. Being a type of issuer call option, a cash repurchase right will have a non positive effect on the theoretical value of the VPO and lead to an overstatement of the resultant pricing errors. Finally, the results presented in table 3 (and subsequently referred to as the base-case) are robust to theoretical prices alternatively being determined using 180-day historic volatility estimates (rather than 60-day estimates) and using first, the 90-day Bank Accepted Bill rate and second, the 5-year Treasury Bond rate as alternative proxies for the risk-free interest rate.

5.3 Explanatory Factors

The above section documents evidence of substantial pricing errors. A possible next step would be to follow MacBeth and Merville (1979), Whaley (1982) and Bakshi, Cao and Chen (1997), amongst others and analyse the relationship between these errors and a number of explanatory factors including volatility, moneyness, maturity and interest rates, within a regression framework. However, given: (i) the highly non-linear nature of the theoretical pricing model; (ii) the sample of matched trades consists of a time series of irregularly spaced data (9); and (iii) joint test considerations in relation to market efficiency, model specification and parameter estimation, then the structure of the remaining pricing errors is examined instead using a 'simulation like' approach. In relation to (iii) in particular, the hypothesis examined in this section is whether the base-case pricing errors can principally be explained by differences in the parameter estimates on which theoretical verses market prices have been based.

From (5), the theoretical value of a bounded VPO can be partitioned into two components: a bond component equal to the present value of the VPO discount and an option component equal to the Black-Scholes value of a portfolio of long call options, short put options and long put options. The theoretical value of the VPO is therefore dependent on two key parameters: the discount rate used in calculating the bond component and the volatility of the underlying stock price used in calculating the option component. In the base-case, the bond component discount rate is a risk-free rate, estimated on a daily basis, and the stock price volatility is a rolling, daily historic volatility estimate. The approach in this section is to simulate theoretical VPO prices and resultant pricing errors using some simple alternatives to these base-case parameters estimates. In particular, there is anecdotal evidence to suggest that practitioners use a risk-adjusted discount rate, rather than the risk-free rate in calculating the value of the bond component in these types of securities, (10) whilst it is possible that market prices are determined using a narrow range of 'reasonable' volatility estimates, rather than historic volatility estimates. Having regard to the slightly different contractual features of each security, this analysis is necessarily conducted on a case-by-case basis and only in relation to three of the five VPOs (WANO, GPTO and GPTOA). The MRVO and CGHO securities are excluded for the reasons outlined earlier. In addition, the assumption of frictionless markets is maintained. (11) Each VPO is now considered in turn.

5.3.1 WANO As detailed in the Appendix, WANO is a single, bounded VPO with a cap and floor, a European exercise feature, a known maturity date and no cash repurchase right. Figure 1 shows a plot of the base-case pricing errors over time.

[FIGURE 1 OMITTED]

The substantial mispricing under the base-case is clear with a mean (median) market price over the sample period of $0.7376 ($0.7400) compared to a mean (median) theoretical price of $0.8614 ($0.7992) of which 53% is represented by the bond component. As an alternative, theoretical prices and resultant pricing errors are recalculated over the entire sample period assuming: (i) a bond component discount rate equal to the base-case risk-free rate plus an 'arbitrary' risk premium; and (ii) an 'arbitrary' volatility estimate in calculating the value of the option component. These simulations are then repeated over an assumed grid of possible premiums and volatilities. Specifically, pricing errors are calculated for all pair-wise combinations of premium, ranging from 0% to 5% in increments of 0.5% and volatility, ranging from 10% to 30% in increments of 0.5%. The objective is to find that unique premium-volatility combination (assumed constant over the entire sample period) which produces the minimum median absolute relative pricing error from the combinations considered. The resultant pricing errors are referred to as interim pricing errors. It is, however, unlikely that investors would use one and only one premium-volatility estimate over such an extended period of time. To examine whether investors instead use a narrow range of possible premiums (in this case, 2.0%, 2.5% and 3.0%) and a narrow range of possible volatilities (16% to 19% in increments of 0.5%) in setting market prices, the theoretical price at each point in time is based on that premium-volatility combination, from this subset of combinations, which results in the minimum absolute relative pricing error at that time. The resultant pricing errors are referred to as minimum pricing errors. (12) As shown in table 4, whilst this selection is done on an ex-post basis, it nonetheless produces some quite remarkable results given the simplicity of the assumed parameter estimates.

Panel A reports descriptive statistics of base-case, interim and minimum pricing errors for the WANO VPO. Recall, the base-case pricing errors, shown here for comparison, are based on a zero premium and rolling, daily historic volatility estimates. The interim pricing errors are based on a 2.5% premium and a stock price volatility of 18.0%, both of which are assumed constant over the entire sample period. These simple alternative parameter estimates lead to a substantial improvement in the performance of the theoretical pricing model with the mean (median) absolute relative pricing error falling from 20.34% (15.41%) to 7.72% (5.45%) which, using the Wilcoxon Paired-Sample Signed-Rank test (in relation to the median) is statistically significant at the 1% level. The minimum pricing errors are based on a premium of 2.0 to 3.0% and a volatility of 16.0 to 19.0%. Again there is a statistically significant improvement in pricing with the (mean) median absolute relative pricing error falling further to (4.14%) 0.95% which corresponds to a mean (median) absolute pricing error of 2.5 cents (0.7 cents) per VPO. This compares to a mean (median) bid-ask spread, based on monthly observations over the entire sample period, of 2.3 cents (1.0 cents) per VPO. Figure 2 shows a plot of these pricing errors over time. Except for some initial underpricing and some unexplained overpricing during the period from September to November 1997, the theoretical model, based on these simple parameter estimates, performs exceptionally well in explaining market prices.

[FIGURE 2 OMITTED]

In summary, these results are consistent with the hypothesis that investors use a risk-adjusted discount rate, rather than the risk-free rate in valuing the bond component of the WANO VPO and use a narrow range of 'reasonable' volatility estimates (however determined), rather than historic volatility estimates, in valuing the option component of the VPO. Although the reasons for this behaviour are not clear, it is noted that a risk-adjusted discount rate may be motivated by default risk and or liquidity considerations, with the latter possibly taking on particular importance given the low level of trading in the security.

5.3.2 GPTO GPTO is a single, bounded VPO with a cap and floor, a Bermudan exercise feature (the VPO is exercisable, at the investor's option, at one of five possible known exercise dates up to and including the maturity date), a known maturity date and no cash repurchase right. In the base-case, the Bermudan feature is (approximately) taken into account by treating the VPO as a package of five mutually exclusive European options with maturity dates corresponding to the five possible exercise dates. The value of the VPO is then equal to the maximum of these European values, with the corresponding maturity date taken as the optimal exercise date. This approach is also consistent with the ex post exercise decision of investors.

Panel B reports descriptive statistics of the base-case, interim and minimum pricing errors for the GPTO VPO. Again there is substantial mispricing under the base-case with a mean (median) market price over the sample period of $15.59 ($15.80) compared to a mean (median) theoretical price of $20.26 ($20.73). Of this amount, 98% is attributable to the bond component, which suggests that the estimate of the bond-component discount rate rather than the option component volatility estimate is the most important parameter. This is subsequently reflected in the fact that the Interim pricing errors are based on a 12.5% premium and assuming zero value is attributed to the option component, that is, the VPO is effectively being priced as a risky zero-coupon bond. The minimum pricing errors are based on a premium of 10.0 to 27.5% (in increments of 2.5%) and similarly assuming zero value is attributed to the option component. Although this is a wide range of risk premia, pricing is consistent with investors using an increasing risk premium over time i.e. the minimum pricing errors tend to be based on a 10% premium in the beginning of the sample period increasing reasonably steadily over the sample period to a 27.5% premium at the end of the sample period. Like in the WANO case, these simple changes lead to a substantial and statistically significant improvement in the performance of the theoretical pricing model with the mean (median) absolute relative pricing error ultimately falling to (3.41%) 1.62%, which corresponds to a mean (median) absolute pricing error of 54.7 cents (24.4 cents) per VPO. This compares to a mean (median) bid-ask spread, based on monthly observations over the entire sample period, of 60.6 cents (30.0 cents) per VPO. Whilst it appears difficult to theoretically justify the magnitude of the risk premium over the latter part of the sample period, the results are nonetheless consistent with the hypothesis that investors priced the GPTO VPO as a risky zero-coupon bond using an increasing discount rate akin to that used in the pricing of equity securities.

5.3.3 GPTOA GPTOA is a package of three, separate, bounded VPOs, each with a cap but no floor, a European exercise feature, an unknown maturity date and subject to a cash repurchase right. (13) Under the base-case, the mean (median) market price over the sample period is $1.26 ($1.35) compared to a mean (median) theoretical price of $1.69 ($1.68). The pricing of the GPTOA VPO is consistent with that of the GPTO VPO. As shown in panel C, there is significant improvement in pricing performance in moving from the base-case, through the interim pricing errors (which are based on a 22.5% premium and assuming zero value is attributed to the option component) to the minimum pricing errors (which are based on a premium of 17.5 to 27.5% in increments of 2.5% and similarly assuming zero value is attributed to the option component). In this case, the mean (median) absolute relative pricing error is (1.67%) 0.96% and the mean (median) absolute pricing error is 1.9 cents (1.3 cents) per VPO. This compares to a mean (median) bid-ask spread, based on monthly observations over the entire sample period, of 4.5 cents (3.5 cents) per VPO. Again, the results are consistent with the hypothesis that investors priced the GPTOA VPO as a risky zero-coupon bond using increasing discount rates akin to those in pricing equity securities.

5.4 The Probability of Rational Exercise

Table 5 sets out descriptive statistics of the theoretical risk-neutral probability of rational exercise of the WANO, GPTO and GPTOA VPOs each conditional on the observed stock price at each time t and based on the sample of VPO-Stock pairs matched to within one minute. The results indicate that there is a very high ex ante probability of rational exercise of a VPO consistent with the key design feature of the security. Ex post exercise of the options is consistent with these high ex ante probabilities.

6. Conclusion

In this paper, three empirical tests of the pricing of VPO contracts are conducted, being a test of a simple theoretical bound, a test of the performance of an arbitrage-free pricing model, and an examination of the theoretical risk-neutral probability of rational exercise. While there was some evidence of lower bound violations, this was found to be isolated to one security over a five-month period at the end of 1997. In relation to the performance of the pricing model, there initially was evidence that the model systematically overprices VPOs. A subsequent case-based explanatory analysis of the pricing errors, however, shows that this mispricing substantially disappears under different estimates of two key parameters. In one case, the results are consistent with investors using risk-adjusted discount rates rather than the risk-free rate in valuing the bond component of the VPO and using a narrow range of 'reasonable' volatility estimates, rather than historic volatility estimates, in valuing the option component of the VPO. In two other cases where the option component represents a negligible proportion of the total value of the VPO, the results are consistent with investors pricing the VPO as a risky zero-coupon bond using an increasing discount rate akin to that used in the pricing of equity securities. Finally, it is also shown that there is a very high ex ante probability of rational exercise of a VPO consistent with the key design feature of the security.

Appendix

Details of VPO Issues

This table sets out a summary of the key terms of VPO issues to 30 June
1998

      Issuer              General Property          West Australian
                               Trust               Newspapers Limited

ASX Code               GPTO                      WANO
Issue date             May 1992                  Aug 1994
Issue Price            $100.00 (a)               nil
Number Issued          1.2 million               26.9 million
Amount to be Raised    $120 million              $135 million
on Exercise
Type of Issue          1 for 450 non             1 for 7 entitlements
                       renounceable rights       issue as part of a
                       offer of debenture +      capital restructuring
                       detachable VPO
Exercise Price         $100.00                   $5.00
Exercise Discount      14% if exercised on       10%
                       19 Jun 95 else 18%
Floor/Cap Prices       both                      both
Cash Repurchase        no                        no
Right
Averaging Period       10 business days          10 business days
Exercise Type          Bermudan                  European
                       (qtrly between 19 Jun
                       95 and maturity)
Maturity               17 Jun 96                 30 Jun 99
Exercise Date(s)       no                        no
Nominated by Issuer
Underwritten           issue only                no
Dividend Protected     no                        no
Special Early          no                        issuer subject to
Exercise Events                                  takeover
Protected Against      reconstructions only      reconstructions only
Capital Changes

      Issuer              General Property          Colonial Limited
                               Trust

ASX Code               GPTOA                     CGHO
Issue date             Oct 1994                  May 1997 (b)
Issue Price            $1.35                     nil
Number Issued          30.7 million              576.8 million
Amount to be Raised    $460 million              $230 million
on Exercise
Type of Issue          1 for 20 renounceable     1 for 1 issue as part
                       rights offer of a         of a demutualisation
                       package of 3 separate     and share issue
                       VPOs
Exercise Price         $5.00 each                $0.40
Exercise Discount      15%                       15%
Floor/Cap Prices       cap only                  cap only
Cash Repurchase        issuer has right to       no
Right                  buy back each VPO for
                       $0.8824
Averaging Period       10 business days          15 business days
Exercise Type          European                  American
                                                 (not less than 15
                                                 business days)
Maturity               no earlier than 15 Mar    between 30 Sep 97
                       96, 15 Sep 96 15 Sep      and 15 Jun 98
                       97 and no later than
                       30 Sep 96, 30 Jun 97
                       and 30 Jun 98 for
                       1st, 2nd and 3rd VPO
                       respectively
Exercise Date(s)       yes                       yes
Nominated by Issuer
Underwritten           both                      exercise only
Dividend Protected     no                        no
Special Early          no                        no
Exercise Events
Protected Against      reconstructions and       reconstructions and
Capital Changes        winding up                winding up

      Issuer               Mirvac Limited          Westfield America
                                                         Trust

ASX Code               MRVO                      WFACA (d)
Issue date             Oct 1997                  June 1998
Issue Price            nil                       $0.20 each
Number Issued          8.3 million               9.3 million
Amount to be Raised    $100 million              $465 million
on Exercise
Type of Issue          1 for 20 entitlements     Public offer of a
                       issue of a package of     package of 3 separate
                       2 separate VPOs as        VPOs
                       part of a 1 for 4
                       rights offer
Exercise Price         $6.00 each                $50 each
Exercise Discount      15%                       5%
Floor/Cap Prices       cap equivalent (c)        neither
Cash Repurchase        issuer has right to       issuer has right to
Right                  buy back each VPO for     buy back each VPO for
                       $0.18                     $2.64
Averaging Period       10 business days          20 business days
Exercise Type          European                  European
Maturity               no earlier than 1 Jul     29 Jun 01, 28 Jun 02
                       98, 1 Jul 99 and no       and 30 Jun 03 for 1st,
                       later than 30 Jun 00,     2nd and 3rd VPO
                       30 Jun01 for 1st and      respectively
                       2nd VPO respectively
Exercise Date(s)       yes                       no
Nominated by Issuer
Underwritten           exercise only             both
Dividend Protected     no                        no
Special Early          no                        issuer subject to
Exercise Events                                  takeover
Protected Against      reconstructions and       no
Capital Changes        winding up

SOURCE: Prospectus for each issue.

(a) Each debenture and VPO was issued as a package for a combined price
of $100 and therefore no amount was separately identified as being
attributable to the VPO at the time of issue.

(b) Initially each VPO is stapled to one underlying share of common
stock which trades as a combined security. On a date to be nominated by
the issuer, the VPO and share will be destapled and each will then
trade separately.

(c) The issuer has the ability to reset the averaging period and
exercise date if the stock trades below $1.50 during the averaging
period.

(d) The issued security is referred to as Stapled Partly Paid Units.

Table 1

Descriptive Statistics of Sample of Matched VPO-Stock Trades

This table sets out descriptive statistics for VPO trades matched to
the closest trade in the underlying stock in relation to the five VPOs
that were traded on the ASX from 7 May 1992 to 22 May 1998. Number
represents the number of matched VPO-stock pairs, VPO Volume (Value)
represents the median number (dollar value) of VPOs traded per matched
pair, Stock Volume (Value) represents the median number (dollar value)
of shares of stock traded per matched pair and Time represents the
median time difference, in seconds, between the VPO trade and the
matched stock trade.

Security      Period Covered       Number     VPO       VPO
                                             Volume    Value

            Panel A: VPO-Stock Pairs: Matched to Within One Minute

GPTO        12/5/92 to 8/5/96         387        21     319
GPTOA       20/10/94 to 15/4/98       778       175     191
WANO        21/10/94 to 22/5/98     2,480     1,000     760
MRVO        21/10/97 to 22/5/98        72       431     162
CGHO        4/12/97 to 29/12/97       539     4,000     280

            Panel B: VPO-Stock Pairs: Matched to Within Five Minutes

GPTO        12/5/92 to 8/5/96         845        25     403
GPTOA       20/10/94 to 17/4/98     1,356       200     211
WANO        21/10/94 to 22/5/98     4,870     1,300     959
MRVO        10/10/97 to 22/5/98       141       711     260
CGHO        4/12/97 to 29/12/97       769     4,900     350

            Panel C: VPO-Stock Pairs: All Trades

GPTO        7/5/92 to 8/5/96        1,290        29     445
GPTOA       20/10/94 to 17/4/98     1,691       200     243
WANO        21/10/94 to 22/5/98     6,650     1,354     939
MRVO        10/10/97 to 22/5/98       344     1,013     420
CGHO        4/12/97 to 29/12/97       800     4,491     350

Security    Stock     Stock    Time
            Volume    Value

            Panel A: VPO-Stock Pairs:
            Matched to Within One Minute

GPTO        1,800     4,144     23
GPTOA       2,847     7,003     18
WANO        1,300     6,400     20
MRVO        3,181     7,070     27
CGHO        1,188     5,148     16

            Panel B: VPO-Stock Pairs:
            Matched to Within Five Minutes

GPTO        2,000     4,683     68
GPTOA       3,300     8,162     44
WANO        1,400     6,493     59
MRVO        2,993     6,584     60
CGHO        1,045     4,490     29

            Panel C: VPO-Stock Pairs:
            All Trades

GPTO        2,000     4,810    157
GPTOA       3,200     7,843     74
WANO        1,400     6,380    110
MRVO        2,580     5,814    444
CGHO        1,032     4,479     32

Table 2

Violations of Theoretical Arbitrage Bounds

This table sets out descriptive statistics of arbitrage bounds
violations for each VPO using the total sample of VPO-Stock pairs
matched to within one minute. Number represents the number of matched
VPO-stock pairs, Market represents the mean (median) market price of
the VPO in dollars per contract, Lower Bound represents the mean
(median) lower bound on the value of the VPO in dollars per contract,
Lower Violations represents the number (relative frequency) of
violations of the lower bound and the mean (median) value of these
violations expressed as a proportion of the market price of the VPO.
Upper Bound, and Upper Violations are defined similarly.

Security    Number     Market       Lower      Upper Bound
                                    Bound

GPTO           387     15.5881      3.4564      119.9912
                      (15.8000)    (2.6358)    (124.2098)
GPTOA          778      1.2578        --         13.2676
                       (1.3500)                 (13.6010)
WANO         2,480      0.7376      0.6015        7.9143
                       (0.7400)    (0.5117)      (8.2165)
MRVO            72      0.3760        --          8.4325
                       (0.3800)                  (8.2260)
CGHO           539      0.0754        --          1.6270
                       (0.0700)                   1.6545)

Security    Lower Violations      Upper
                                Violations

GPTO           0        --       0      --
              (0%)              (0%)
GPTOA         --        --       0      --
                                (0%)
WANO         379      0.1804     0      --
             (15%)    0.1612    (0%)
MRVO          --        --       0      --
                                (0%)
CGHO          --        --       0      --
                                (0%)

Table 3

Descriptive Statistics of Pricing Errors

This table sets out descriptive statistics of pricing errors for each
VPO based on the total sample of VPO-Stock pairs matched to within one
minute. Pricing errors are calculated as [e.sub.t] = T[P.sub.t]
- O[P.sub.t] where [e.sub.t] is the pricing error at time t, O[P.sub.t]
is the observed market price of the VPO at time t and T[P.sub.t] is the
theoretical VPO price at time t given by equation (5). Number
represents the number of matched VPO-stock pairs, Negative represents
the number of negative pricing errors with relative frequency shown in
brackets, MPE represents the mean pricing error in dollars per VPO,
MAPE represents the mean absolute pricing error in dollars per VPO,
MARPE represents the mean absolute relative pricing error as a
proportion of observed market price and Med ARPE represents the median
absolute relative pricing error as a proportion of observed market
price.

Security   Number   Negative     MPE      MAPE    MARPE    Med ARPE

GPTO         387       0        4.6724   4.6724   0.3029    0.2976
                      (0%)
GPTOA        778       0        0.4322   0.4322   0.3245    0.3293
                      (0%)
WANO       2,480     501        0.1238   0.1402   0.2034    0.1541
                     (21%)
MRVO          72       0        0.6968   0.6968   1.8730    1.7894
                      (0%)
CGHO         539     178       -0.0053   0.0055   0.0617    0.0027
                     (33%)
Total      4,256     679        0.5871   0.5981   0.2449    0.2181
                     (16%)

Table 4

Descriptive Statistics of Interim and Minimum Pricing Errors

This table sets out descriptive statistics of interim pricing errors
and minimum pricing errors for three VPOs based on the sample of
VPO-Stock pairs matched to within one minute. Number represents the
number of matched VPO-stock pairs, Negative represents the relative
frequency of the pricing errors which are negative, MPE represents the
mean pricing error in dollars per VPO, MAPE (MedAPE) represents the
mean (median) absolute pricing error in dollars per VPO, MARPE
(MedARPE) represents the mean (median) absolute relative pricing error
as a proportion of observed market price, and Rho represents the sample
correlation coefficient between theoretical and observed market prices.
The base-case is included for comparative purposes. The Wilcoxon
Paired-Sample Signed-Rank Test is used to test for equality of the
median absolute pricing errors and for equality of the median absolute
relative pricing errors. * denotes statistical significance at the 1%
level.

Errors       Number    Negative      MPE       MAPE      MedAPE

                               Panel A: WANO

Base-case    2,480        21        0.1238    0.1402      0.0853
Interim      2,480        47        0.0136    0.0511      0.0379
Minimum      2,480        49        0.0056    0.0251      0.0069
Wilcoxon Statistic (Interim-Base-case)                  -31.5 *
Wilcoxon Statistic (Minimum-Interim)                    -42.7 *

                               Panel B: GPTO

Base-case      387         0        4.6724    4.6724      4.6854
Interim        387        36        0.8784    1.4805      1.3166
Minimum        387        46        0.2781    0.5471      0.2439
Wilcoxon Statistic (Interim-Base-case)                  -17.0 *
Wilcoxon Statistic (Minimum-Interim)                    -15.5 *

                               Panel C: GPTOA

Base-case      778         0        0.4322    0.4322      0.4488
Interim        778        46       -0.0280    0.0706      0.0536
Minimum        778        43        0.0006    0.0194      0.0134
Wilcoxon Statistic (Interim-Base-case)                  -24.2 *
Wilcoxon Statistic (Minimum-Interim)                    -22.5 *

                Errors                    MARPE     MedARPE      Rho

                                                 Panel A: WANO

Base-case                                 0.2034      0.1541    0.9179
Interim                                   0.0772      0.0545    0.9818
Minimum                                   0.0414      0.0095    0.9916
Wilcoxon Statistic (Interim-Base-case)      --      -31.3 *       --
Wilcoxon Statistic (Minimum-Interim)        --      -42.7 *       --

                                                 Panel B: GPTO

Base-case                                 0.3029      0.2976    0.7358
Interim                                   0.0933      0.0851    0.9046
Minimum                                   0.0341      0.0162    0.9284
Wilcoxon Statistic (Interim-Base-case)      --      -17.0 *       --
Wilcoxon Statistic (Minimum-Interim)        --      -15.5 *       --

                                                 Panel C: GPTOA

Base-case                                 0.3245      0.3293    0.9841
Interim                                   0.0547      0.0428    0.9718
Minimum                                   0.0167      0.0096    0.9969
Wilcoxon Statistic (Interim-Base-case)      --      -24.2 *       --
Wilcoxon Statistic (Minimum-Interim)        --      -22.5 *       --

Table 5

Desriptive Statistics of the Risk-Neutral Probability of Rational
Exercise

This table sets out descriptive statistics of theretical risk-neutral
probalility of rational exercise for three VPOs conditional on the
stock price at each time t and based on the sample of VPO-stock pairs
matched to within one minute. Number represents the number of matched
VPO-stock pairs. The risk-neutral probablility of rational exercise is
calculated using equation (6).

Security   Number Market    Mean      Median

GPTO            387        0.98444    0.9983
GPTOA           778        0.9903     1.0000
WANO          2,480        0.9492     0.9953

Security    Maximum    Minimum   Stdev

GPTO        1.0000     0.8722    0.0244
GPTOA       1.0000     0.8895    0.0174
WANO        1.0000     0.6581    0.0865

(1.) The Australian Stock Exchange Listing Rules (appendix 6A.6 and previously Rule 3G(4)) specify standard timetables which listed companies must follow in relation to various matters including when the company has quoted options which are due to expire. In this regard the company must send a notice to each option holder at least 20 business days before the expiry date of the option. The notice must include inter alia the number of options held and the number of securities to be issued if they are exercised.

(2.) In this case the underlying state variable may be interpreted as the conditionally diluted stock price. See Handley (2002) for further details.

(3.) A dividend paid on the underlying stock during the life of the option will cause the stock price to fall on the ex-dividend date. If it is assumed that the amount and timing of the dividend and consequent drop off in the stock price is known with certainty, then Black (1975) suggests that the option be priced as a contingent claim on the stock price net of the escrowed dividend.

(4.) The price process [DELTA] = {[[DELTA].sub.t]; t [member of] [0, T]} for the present value of the escrowed future dividends to be paid on the underlying stock over the life of the VPO is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [d.sub.i]; i = 1,.., n are the known dividends with corresponding ex-dividend dates [t.sub.i] [member of] (t, T]; i = 1,.., n, T is the maturity date of the VPO and r is the instantaneous risk-free interest rate.

(5.) The VPO issue by West Australian Newspapers Limited commenced trading on the ASX on 12 August 1994 but transactions data was not available until 21 October 1994. The VPO issue by Westfield America Trust occurred in June 1998.

(6.) See for example Cox and Rubinstein (1985, p. 129).

(7.) The CGHO security has a strike of $0.40 per VPO and a maturity over the sample period ranging from 16 days (at the start) to 9 days (at the end). The model gives a theoretical price of around 7 cents. Given a minimum tick size of 1 cent, small absolute movements in the market price produce what appears to be non-trivial swings in the level of under/over pricing measured on a relative basis.

(8.) If the lower bound violations are excluded from the sample, then the mean (median) absolute relative pricing error falls to 16.85% (11.63%).

(9.) For example, for the WANO sample, the time between consecutive matched VPO-Stock pairs varies from zero seconds to several days with a mean of 0.52 days and a standard deviation of 1.05 days. The irregularly spaced nature of the data raises some difficult econometric issues. See for example the discussion in Engle and Russell (1998).

(10.) For example, HSBC Securities (1999).

(11.) An issue for consideration in empirical tests of arbitrage-free pricing models is whether any apparent mispricing can be explained by the existence of transactions costs (including commissions, bid-ask spreads, taxes and duties, short selling, market impact and opportunity costs), which are not taken into account if the pricing model assumes a frictionless market, as in this paper. Transaction costs imply that a range/bound of arbitrage-free prices may exist (rather than a single arbitrage-free price) as any apparent mispricing within the bound may not be able to be arbitraged away once these transactions costs are taken into account. Whilst this approach readily applies if the pricing model is based on a static hedging strategy, taking account of transaction costs in a dynamic hedge environment is less clear. In the case of continuous trading (as in this paper), hedging becomes prohibitively expensive no matter how small the transactions cost per trade. In the case of discrete trading, although it may be possible to derive a theoretical bound on the arbitrage-free price, in some cases the bound is too wide for any practical importance. See for example Martinelli (2000) for further details.

(12.) If the minimum pricing error is equal to zero, then the corresonding premium-volatility combination may be interpreted as an 'implied' premium-volatility estimate.

(13.) As a first approximation, if the cash repurchase right is European and the issuer follows a call strategy to minimise the value of the VPO then the model may readily be adjusted to include its effect. In the special case of a bounded VPO and a cash payment equal to the VPO discount (as applies in this case) then a sufficient condition for the rational exercise of the cash repurchase right by the issuer at maturity is [S.sub.T] > [x.sub.f]. Therefore, based on wealth considerations alone, a cash repurchase right eliminates the potential upside to the VPO investor in the event that the stock price finishes above the floor price at maturity. Since the GPTOA security is not subject to a floor on the number of shares per VPO, then the value of the cash repurchase right is likely to be very small and can reasonably be ignored.

(Date of receipt of final transcript: November, 2002. Accepted by Stephen Gray, Area Editor.)

References

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Australian Stock Exchange Limited, 1999, ASX Annual Stockmarket Summaries, Brisbane, Queensland.

Bakshi, G.C., Cao & Chen, Z. 1997, 'Empirical performance of alternative option pricing model,' Journal of Finance, vol. 52, pp. 2003-49.

Black, F. 1975, 'Fact and fantasy in the use of options', Financial Analysts Journal, vol. 31, July-August, pp. 36-41, 61-5, & 72.

Cox, J.C. & Rubinstein, M. 1985, Options Markets, Prentice-Hall Inc, Englewood Cliffs.

Engle, R.F. & Russell, J.R. 1998, 'Autoregressive conditional duration: A new model for irregularly spaced transaction data', Econometrica, vol. 66, pp. 1127-62.

Galai, D. 1977, 'Tests of market efficiency of the Chicago Board Options Exchange', Journal of Business, vol. 50, pp. 167-97.

Handley, J.C. 2002, 'On the valuation of warrants', Journal of Futures Markets, vol. 22, pp. 765-82.

Handley, J.C. 2000, 'Variable purchase options', Review of Derivatives Research, vol. 4, pp. 219-30.

HSBC Securities, 1999, Delta-to-Debt, Brokers Research Report, 24 June, pp. 1-27.

MacBeth, J.D. & Merville, L.J. 1979, 'An empirical examination of the Black-Scholes call option pricing model', Journal of Finance, vol. 34, pp. 1173-86.

Martinelli, L. 2000, 'Efficient option replication in the presence of transactions costs', Review of Derivatives Research, vol. 4, pp. 107-31.

Rubinstein, M. 1985, 'Nonparametric tests of alternative option pricing models using all reported trades and quotes on the 30 most active CBOE option classes from August 23, 1976 through August 31, 1978', Journal of Finance, vol. 40, pp. 455-80.

Whaley, R.E. 1982, 'Valuation of American call options on dividend paying stocks: Empirical tests', Journal of Financial Economics, vol. 10, pp. 29-58.

John C. Handley, Department of Finance, University of Melbourne, VIC 3010. Email: [e-mail]

Valuable comments from Tim Brailsford, Christine Brown, Kevin Davis, Richard Heaney, Avi Kamara, Krishnan Maheswaran, Barry Oliver, Sean Pinder, Dick Stapleton, two anonymous referees and seminar participants at the Australian National University, University of Melbourne and University of Sydney are gratefully acknowledged. Thanks also to Wendy Moir for programming assistance. The data used in this study was kindly provided by the Securities Industry Research Centre of Asia-Pacific (SIRCA).