exploiting volatility movements in the sydney futures exchange's bank bill contract

EXPLOITING VOLATILITY MOVEMENTS IN

THE SYDNEY FUTURES EXCHANGE’S BANK

BILL CONTRACT

B. F. HUNT, R. BHAR

ABSTRACT

An appropriate stochastic model was fitted to one year of data on the implied volatility of options

on 90 day bank accepted bill futures contracts traded in the Sydney Futures Exchange. The

model used was ARIMA augmented with day of the week variables, an option time to maturity

variable, and recent values of historic volatility. The high ex-post predictive accuracy of the

model was then employed as the central element of a strategy of buy low/sell high volatility.

We employed two trading schemes with suitably constructed Delta neutral portfolios com-

prising bill futures and call and put options on those futures over a period of six months, to test

whether speculative trading profit could be earned. The existence of trading profits before

transaction costs validated the potential of the buy low/sell high volatility strategies to generate

speculative profits. The absence of any such trading profits after transaction costs however,

showed that the market pricing of these securities is such that the dependencies within implied

volatility cannot be profitably exploited.

This result may be interpreted as evidence supporting an hypothesis of a semi-strong form

of market efficiency.

Direct all correspondence to: R. Bhar, Faculty of Business andTechnology, University of Western Sydney, Box 555, Can&&own, N.S.W., 2560, AUSTRALIA. l B. F. Hunt, University of Technology- Sydney, AUSTRALIA.

International Review of Economics and Finance, 2(4) 403-415 Copyright 0 1993 by JAI Press, Inc. ISSN: 1059-0560 All rights of remoduction in any form rese.rved

403

404 B. F. HUNT and R. BHAR

INTRODUCTION

This paper examines the efficiency of the bank Accepted Bill Futures market as traded in

the Sydney Futures Exchange @FE). The definition of efficiency applied here relates to the

possibility of obtaining abnormal profits by the appropriate use of available information.

The proposition of semi-strong market efficiency is that the market disseminates publicly

available information so rapidly that the prices already reflect this and no abnormal profit

can be earned. Thus, any publicly available information which can be used to produce

abnormal profits invalidates the proposition of semi-strong form market efficiency. Our

focus here is upon the information pertaining to volatility.

The growth of trading in Australian interest rate contracts has been rapid since their

introduction in 1979. While the Australian market is small in relation to world markets the

development of an Australian financial futures market has mirrored that of other countries.

Market practice in the SFE is closely related to that in the U.S. futures markets. While the

SFE was chosen as the focus of this study, the techniques used here, and perhaps the adduced

results, are directly applicable to other markets.

The SFE is the principal Australian interest rate hedge market. While there are two other

interest rate futures contracts, the Bank Accepted Bills contract is the most traded and liquid

contract.’ The SFE Bank Bill contract is for the delivery of 90 day Bank Accepted Bills to

the face value of AUD500,OOO. The contracts are for delivery on the second Friday of March,

June, September, and December. Both call and put options are traded on these futures

contracts. The prices of the nearest to the money call option on the closest to maturity futures

contract was used to construct a daily implied volatility series.

A forecasting model was fitted to the implied volatility series. We were encouraged in the

search for an efficacious forecasting model by the existence of previous studies, including French,

Schautz, and Stambaugh (1987) for equities and Bessembinder and Seguin (1990) for futures,

which had shown that volatilities showed persistent deviation from a random walk.*

Having obtained a suitable model, the forecasts of implied volatility were utilized in the

construction of delta neutral trading portfolios. These portfolios were designed to profit from

forecast changes in option implied volatility. Cumulative profit/loss from the implementa-

tion of the trading strategy over a six month period, both before and after transaction costs,

provided the measure of the strategy’s performance and thus evidence of the efficiency, or

otherwise, of the market.

Kemna (1989) and Wilson and Fung (1990) have previously examined the informational

content of implied volatility in equities and futures markets respectively. Both studies

examined the proposition that implied volatilities predicted future volatility. Our study goes

further in that we attempt to use the prediction of future volatility to predict and profit from

future changes in option prices.

The vast majority of studies of market efficiency have examined markets for the existence

or otherwise of profitable opportunities in terms of the price of the traded security. Our study

differs in that we did not examine market prices directly but rather indirectly through an

examination of one of the determinants of option market price, the option volatility. The

SFE Bank Bill Contract 405

90 Day Bank Bill Futures Yields 16.0%-

Jan 66 Apr 66 July 66 od66 Dee 66

Option Implied Volatility

0.90%

1 0.60%

0.70%

0.60%

0.50%

0.40%

0.30% m Jan 66 Apr 66 July 66 act 66 Dee 88

Figure 1. Sample period implied volatility and bill rates. The data points represent daily

values for the period January 1988-December 1988. The implied volatility series is com-

puted from the nearest to the money call option on the futures contract closest to maturity

using Black’s Model. The bill rates are derived as 100 minus the price of the futures contract

closest to maturity.


only other study which to our knowledge, has concentrated on implied volatility is the

Harvey and Whaley (1992) paper on the S&P 500 index.

If option volatility is forecastable it means that arbitrageurs can consistently generate

profits by forecasting future volatility and hence forecasting future option prices. It follows

that the existence of any structure other than a random walk in an option volatility series is

potentially indicative of market inefficiency if the non-randomness admits exploitable profit

opportunities after allowing for transaction costs. The SFE bank bill market is characterized

by many traders generating large trading volumes with market information displayed and

disseminated rapidly. One would not expect a priori a market like that for SFE bank bills to

be inefficient.

The fact that volatility, unlike price, is not directly observable may make the existence of

patterns in volatility more likely. One of the main contributions of this paper is that it provides

evidence that implied volatilities of bank bill futures options are forecastable and that this

forecastability implies trading strategies that are a profitable gross of transaction costs. The

fact that transactions costs eliminate any such profit infers that the market for bank bill futures

options is semi-strong form efficient.

METHODOLOGY

The study required data on the nearest bank bill futures contract and at the money call and

put options on this contract. The market prices used here are the closing (or settlement) prices

for the particular day. The data has been broken in to two periods. The in-sample model

estimation period runs from January 1988 to December 1988. Data from January 1989 to

June 1989 was used in an out-sample phase to test trading strategies and hence to make

inferences regarding market efficiency. Figure 1 sets out bank bill yield and implied

volatilities for the estimation sample period.

The principal unobserved variable in options pricing is the volatility of the underlying

security. As this volatility is unobservable market traders often focus upon the volatility

which is implied by the current market price of an option. This variable is termed the implied

volatility. Of course, the volatility implied by any pricing model is dependent upon the nature

of the pricing model. In this study we have used Black’s formula for options on futures.3

Support for the use of this model is given by Scott and Tucker (1989).4 There is of course

an implied volatility for each traded option series. A single implied volatility series was

constructed from the nearest to the money call option on the futures contract closest to

maturity. Implied volatility was computed using the Newton-Raphson algorithm applied to

Black’s model.

In the search for non-random patterns in the implied volatility series we used parametric

tests of correlation and non-parametric runs tests. Subsequently, the discovered nonrandom

patterns were modelled using a time series ARIMA model augmented with other instrumen-

tal variables. More specifically, variability in bill futures prices in the immediate past, option

length, and the day of the week were added to an ARIMA model to explain movements in

implied volatility. The model was developed in order to provide an implementable predictor


of future volatility. This model was used to forecast future volatility and to act as the basis

of a volatility trading strategy. Like all trading strategies, the aim of a volatility strategy is

to profit by buying low and selling high.

Since volatility is only one of a number of influences on the level of option prices, it is

necessary, in order to successfully implement a buy low/sell high volatility strategy, to

insulate any market position taken on the basis of forecast volatility from the effect of

changes in the value of other influential variables. In particular, it is necessary to isolate the

trading position from the effects of a change in the price of the underlying 90 day bills futures.

Here, this was achieved by constructing market positions which were sensitive to changes

in volatility but, by virtue of being delta neutral, were insensitive to changes in the bill

futures price. Two delta neutral, volatility sensitive market portfolios were designed to profit

by buying low/selling high volatility.

The first of these delta neutral portfolios were constructed using bill futures and call

options on those futures. The second, used call and put options on bill futures. A futures

contract is insensitive to volatility changes and is thus used in the first strategy simply to

render that strategy delta neutral. The value of both put and call options on bill futures is

positively related to the level of volatility. A delta neutral portfolio constructed from calls

and puts is thus more sensitive to volatility changes than one constructed from calls and

futures.

RESULTS

Model Estimation

Prior to modelling the levels implied volatility series was first differenced to produce a

stationary series. The first phase of our investigation involved a search for predictable

dependencies in the constructed implied volatility series. Autocorrelations were calculated

for 20 lags. Each of these correlations were significantly different from zero at the five

percent level. Further, the Box-Pierce Q-statistic for the 20 lags of 1771.9 was highly

significant. While these parametric statistics indicate that the implied volatility series did not

follow a random walk they are dependent upon distributional assumptions. Nonparametric

run tests are more robust tests of the random walk hypothesis.5

The expected total number of runs of all signs and the variance of this statistic can be

estimated (Mood, 1940). This provided the basis for an extremely robust test of inde-

pendence. In our sample the total observed number of runs was 155 while the expected

number of runs of all signs was 117.33. The standardized normal Z-statistic was 4.88, which

was significant at the 0.10 percent level. The results of analysis of the total number of runs

suggest a lack of independence in the volatility series. A Chi-squared test indicated that the

series contained significantly more (at the 1 percent level) short runs than would be expected

under an assumption of random implied volatility changes. This result of excessive short

runs is indicative of a reversal pattern in the implied volatility series. Such patterns in respect

of prices in volatile markets have been previously found to occur.6

B. F. HUNT and R. BHAR

The results of the correlation and runs analysis suggest that the implied volatility series

contains a degree of dependence. On that basis, the next step was to identify a suitable model

to account for this dependence. Our emphasis was upon modelling the implied volatility

using a stochastic ARIMA time series model augmented with appropriate instrumental

variables.7 The instrumental variables tested to provide a further explanation of the implied

volatility series were:

1) the time to the expiration of the option;

2) the day of the week; and

3) a measure of daily volatility of the underlying futures contract.

Our reason for using time to maturity as an explanatory variable was the existence of a

number of studies which show that the Black model’s theoretical prices have been at

systematic variance with market prices.8 Thus, we hypothesized that the implied volatilities

derived from market prices using the Black model might well vary systematically with time

to maturity. The effect of the influence of time to maturity was included in the ARIMA model

through the use of a square root of days to maturity explanatory variable.’

The existence of a day of the week effect was tested using daily dummy variables. The

rationale for the inclusion of these variables was provided by the many studies which have

shown that volatile markets often have systematic day of the week effect. (See for example

French (1980) and Gibbons and Hess (1981)). Although these studies have concentrated on

returns it is reasonable to extend the argument to the possibility of day of the week effect on

volatilities.

A measure of daily volatility in the underlying futures contractlo was used in an attempt

to explain some of the variation in the implied volatility series.’ ’ Reason would suggest that

both implied volatility and historical volatility ought to be predictors of future volatility. On

that premise, increases in market volatility, as indicated by large changes in the price of the

underlying futures, ought to be reflected in both the level of the implied volatility and the

most recent historical volatility as given by measures of daily futures volatility. If the implied

volatility series were to react instantly to changes in market volatility there would be no

relationship between implied volatility and past daily observed volatility. However, where

there is a delayed adjustment or overreaction of implied volatility to actual volatility, implied

volatility will be statistically related to the lagged values of the daily futures volatility. In

that case the lagged daily volatility will be of assistance in forecasting future changes in

implied volatility.

One approach to the selection of a best forecasting model is to include all variables which

improve the forecast without justification as to the variables’ theoretical appropriateness. Our

approach, however has been to include only those variables which can be justified in terms of

the market structure and to select a preferred model on the statistical diagnostic criteria.

The preferred model set out in Table 1 was ARIMA( 1 ,l ,1) augmented with lagged values

of daily futures price volatility, square root of days to option maturity and day of the week

dummy variables.


Table 1. Model Estimation. The implied volatility model was estimated, using TSP 6.0, from a sample of daily implied volatilities over the period January 1988 to December 1988.

In order to produce a stationary series the implied volatility was first differenced. This differencing required that a number of the explanatory variables were also first differenced.

Model Structure

The estimated model consisted of an ARIMA (l,l,l) augmented with three classes of instrumental variables.

1) Day of the week dummies.

2) Time to Maturity.

3) Measure of Daily Futures.

Volatility

Five dummies were included to identify any day of the week effects.

This variable was included to capture the influence, if any, of the

length of the option on implied volatilities.

A series of daily volatility in the underlying contract was induded to

proxy for the effect of recent dhanges in historical volatility on

implied volatility.

R-Squared 0.26

Model Parameters Coef t-Stat F-test

Ma(l) -0.560 -3.12**

Ar(l) 0.150 0.92

Days 0.004 1.68

Monday 0.012 2.33**

Tuesday -0.008 -1.54

Wednesday -0.001 -0.31

Thursday -0.113 -2.17

Friday 0.009 0.17

Dvol(0) 0.020 3.56**

Dvol(-1) 0.016 2.14**

Dvol(-2) 0.008 0.72

Dvol(-3) 0.003 0.44

Dvol(4) 0.007 1.34

*represents significance at the 5% level, ** represents signifi-

cance at the 1% level

F-statistic 6.44

Where, days to maturity. = fust difference of the square root of the time

Dvol(i) = The first difference of the daily futures price volatility lagged i days (see Note 7)

MA(l) = First order moving average.

AR(l) = First order autocorrelation.

Autocorrelations

The series were checked for stationatity using the Q statistic computed over 20 lags

Implied Volatility series 1771.85

First differenced Implied volatility 49.87

Mode1 residuals 18.49

Residuals in the forecast 32.50

Forecast Correlation

The correlation of the forecast implied volatility with actual volatility was 0.8755 The forecast mean absolute percentage error was 4.49%.

-


The preferred model contains the contemporaneous daily volatility variable, dvol(0). This

variable, while included for model estimation purposes, was set equal to zero for the forecast

period as it is not observable at the time of forecast. Although the model was selected on the

basis of its fit over the sample period, of interest is its fit over the forecast period. Figure 2

plots the actual and model forecast implied volatility, for both one and two day forecasts,

over the forecast period January to June 1989. The correlation of the one day forecasts with

the actual implied volatility was 0.8733 and the mean absolute percentage error was 2.81

percent. The same statistics for the two day forecasts were 0.7569 and 3.97 percent

respectively. One deficiency of correlation statistics in the analysis of forecasts is that they

do not identify consistent bias in forecasts. For example, a forecast which consistently

over- or under-estimates a target series may well have a high correlation with the target

series. In contrast, Theil’s U statistic, inter aliu, takes account of forecast bias. Theils’s U

statistics, which are presented in Table 2, indicates that the model has some power.

Market Strategy

Two distinct trading portfolios were constructed and used in an attempt to profit from the

preferred model’s forecast of implied volatility. Both portfolios were constructed to be delta

neutral. The first portfolio (call/futures) was constructed from call options and futures. The

second portfolio (call/put) consisted of call options and put options. The two portfolios

differed in their sensitivity to volatility. The call/put was approximately twice as sensitive

to volatility. As the call/put portfolio was more highly geared to volatility changes it was

expected to generate greater trading profits but with greater risk/error than the call/futures

portfolio.

The basis of our volatility trading strategy was to buy volatility when the forecast model

indicated that volatility was relatively cheap and to sell volatility when the model indicated

volatility was relatively dear. For example, if the model forecast that volatility would

increase in the future, i.e., that the forecast future volatility exceeded the current level of

Table 2. Measurement of Forecast Accuracy by Theil’s U Statistic

Statistic

U Urn

US

UC

I-Day Forecast

0.0394

0.0065

0.0006

0.9929

2-Day Forecast

0.0550

0.0398

0.0008

0.9594

Theil’s U statistic can be decomposed into three. component parts Urn, US, UC. Where,

Urn is the percentage of error due to bias, ideally 0%,

US is the percentage of error due to variance, ideally 0%

UC is the percentage of error due to covariance, ideally 100%.


Implied Volatility Observed vs 1 Day Forecast

- Observed l 1 Day Forecast

Observed vs 2 Day Forecast

0.9

0.8

0.7

0.6

0.5

0.4

Jan 89 Feb 69 Mar 89 Apr 89 Apr 69 May 89 June 89

- Observed l 2 Day Forecast

Figure 2. Actual and forecast implied volatility. The forecast implied volatility was

derived from an augmented ARIMA model. These series and the actual implied volatility is

plotted against time for the forecast period January 1989-June 1989. Theil’s U-statistic was

0.0394 and 0.0550 for the one day and the two day forecast respectively.


option implied volatility we would “buy” volatility through the purchase of a delta neutral

portfolio whose value was positively correlated to volatility changes. Such a portfolio was

said to be long volatility. Specifically we chose either a long volatility future/call portfolio

(sold future, bought call) or a long volatility call/put portfolio (bought call, bought put

option). If the model forecast of an increase in volatility was vindicated by the passage of

time then the value of the long volatility portfolio would increase thus yielding a profit for

the trading strategy. Conversely if the forecast model indicated that volatility was to fall we

would sell volatility by taking a short volatility portfolio, either a short futures/call portfolio

(bought futures, sold call) or a short call/put (sold call, sold put). The specific details of the

long and short volatility positions are set out in Table 3. Here it can be seen in that each

position is rendered delta neutral using the computed call option delta from Black’s model.

The trading strategy involved the taking of a position on the basis of forecast volatility on

one day and the liquidation of that position at a later date. In the first instance the strategy

involved the taking of a position at the close of trade on one day and the liquidation of that

position on the following day. However such a strategy was open to the legitimate criticism

that it was not implementable. Here we assumed we could use the closing price on a particular

day for both the generation of the forecast and as the entry price for trading. Accordingly we

designed another strategy which was implementable. Namely, we used a two day forecast

to take a position based on forecast produced yesterday in order to undertake a one day

trading strategy entered into today and liquidated tomorrow. Here there is a one day lag

between the observation of prices and the selection of positions and the resulting trades. We

report the results for both the one day and the two day forecast strategies.

The trading processes generated daily profit/loss series which were accumulated to give

an overall profit/loss for the entire forecast period. In order to account for transaction costs

the net change in futures and options positions from one day to the next was calculated. The

transaction costs in a deregulated market like the SFE depend upon the nature of transactor.

We have used the per unit transaction cost, supplied by Barclays de Zoete Wedd, applicable

to a large institutional client.

The success of the market strategy was dependant upon both the ability of the model to

forecast changes in the implied volatility and upon the ability of the constructed portfolio to

be insensitive to changes in variables other than volatility. In particular, if changes in the

Table 3. Market Strategy Positions

Portfolio Call/Future CalVPut

Long Volatility Buy 1 Call Buy 1 Call

Sell d Futures BUY d/(1-d) Puts

Short Volatility Sell Call Sell 1 Call

Buy d Futures Sell d/(1-d) Puts

Now Each of the strategies are standardized to 1 call option. “d” is the Black

model delta of a call option.


Table 4. Trading Profits

Panel A One Day Forecast Horizon

Portfolio Perjiect Foresight Forecast

Without Transaction Costs

Call/Puture

Call/Put

With Transaction Costa

Call/Puture

call/Put

Panel B Two Day Forecast

Portfolio

Without Transaction Costs

ChWFUtUE

Call/Put

With Transaction Costs

Call/Ptmue

Call/Put

162,632 (-0.66%)

367,208 (-3.22%)

71,968 (-22.71%)

222,804 (-18.93%)

Perfect Foresight

18,157 (-2.66%)

55,152 (-12.68%)

-41,756 (-24.78%)

-48,464 (-28.40%)

56,328 (-0.24%)

87,844 (1.63%)

-28,732 (-20.56%)

47,512 (-14.21%)

Forecast

19,936 (-0.32%)

34,066 (4.35%)

-37,279 (-20.04%)

65,961 (-11.37%)

Note: The values in the body of the table represent the cumulative profit resulting from trading

in portfolios standardized to include 100 calls. The specific transaction cost used in

computing the trading profit/loss was $4.34 per trade. The percentage figures in paren-

theses associated with the two day forecasts with transaction cost included refer to the mean of the daily return on funds invested in the implementing the call/future portfolio.

price of bank bills futures, unaccompanied by a change in volatility, were able to significantly

alter the value of our call/futures or call/put portfolios this would obscure the results of

volatility trading.

Our market trading strategy was predicated on an ability to forecast future implied

volatility and to profit from these forecasts using a specific volatility strategy. A logical

benchmark for measuring the success, if any, of our strategy was the result of a similar

strategy based on a perfect foresight prediction model. In a perfect foresight strategy future

volatility changes are anticipated without fail. While the results of this strategy are clearly

unattainable, they nevertheless provide an upper limit against which one can judge the results

of other attainable strategies. That is, the results of the perfect foresight model provided an

upper limit to the trading profits possible under our strategy of buying low volatility selling

high volatility. Other simple benchmark strategies such as buying and holding either calls,

puts, or futures are inappropriate in the context of volatility forecasting as they are not delta

neutral and are consequently effected by factors other than volatility.

It was of course important that the perfect foresight volatility strategy be able to produce

positive profits after allowing for transaction costs. If this were not so then the inability of


a strategy of this type to produce profits would indicate the inappropriateness of our chosen

strategy. It would not necessarily indicate market efficiency.

Table 4 sets out the profit and loss for both strategies. As stated earlier we employed both

a one day and a two day forecast strategy. It can be seen that ignoring transaction costs both

forecast portfolios based upon predicted implied volatility have positive returns. The trading

profits generated by these portfolios however, are much less than those which would be

generated by the Perfect Foresight Model. The call/future strategy returned some 35 percent

of the profits available to this trading strategy (as given by the perfect foresight model). The

put/call strategy while generating a greater absolute level of profit provides a smaller

percentage, 23 percent of the available profits.

When transaction costs are accounted for it renders both of our forecast implied volatility

strategies unprofitable. The perfect foresight profits remain however, strongly positive. The

after transaction results support a proposition of semi-strong form efficiency.

The two day forecast trading produced different results. Generally the profit or loss

attached to the two day forecast strategy is less than the one day equivalent strategy. This is

not unexpected as we would expect two day forecasts to be less accurate than the one day

forecasts. While the one day forecast strategy can be criticized as being difficult to implement

the two day forecast strategy can be criticized as being unrealistic. No trader would wait, as

the two day forecast strategy requires, until the close of trading next day to undertake a

transaction based on a buy or sell signal generated 24 hours earlier.

CONCLUSION

The objective of our study was to test the proposition that the market for bank bill futures

options was semi-strong form efficient with respect to the information contained within

option implied volatilities. The methodology employed showed that implied volatilities were

non-random and predictable at least to some extent. Thus, the volatility trading strategies

employed in the study had the potential to generate profits (as shown by the perfect foresight

results) however, the implied volatility series did not contain sufficient dependencies, after

accounting for transaction costs, to allow for profitable exploitation.

NOTES

1. The SFE trades three interest rate contracts. These are the three and 10 year 12 percent coupon Treasury Bonds contract and the 90 day bank accepted discount bills contract.

2. Predicting volatility is also central to studies under the heading of ARCWGARCH models. 3. See Hull (1989, p. 147). The fact that the SFE use Black’s model to compute theoretical prices

and option deltas provides further support for the use of the model in this study. 4. Scott Tucker (1989) found in investigating currency options that a stationary parameter models,

like Black’s futures model, were as accurate as more sophisticated models such as C.E.V. models.

5. A run in a data series is defined as the number of successive changes of the same sign.

6. See Cootner, Alexander, and Alexander in Cootner (1964)


7. Various ARIMA models, augmented with instrumental variables, were estimated and diagnos-

tically checked using the Micro TSP 6.0 package.

8. See for example Kemna (1989).

9. Arguments contained in Milonas (1986) would justify the use a linear measure of days to expiry.

However we chose to use a non-linear (square root) measure on the ground that this was more consistent

with the way in which time is incorporated into the standard option pricing models.

10. Daily volatility of the underlying futures contract is constructed as the standard deviation of

a single day’s futures return. The use of standard deviation rather than variance in measuring volatility

is consistent with our definition of implied volatility, which is more accurately described as implied

standard deviation. The daily volatility of futures series was constructed as the absolute value of the

logarithm of the ratio of today’s futures price to yesterday’s futures price, multiplied by the square

root of 250 to annualize the measure.

11. The inclusion of daily volatility as an explanatory variable was expected to account for the

reversal pattern in the implied volatility series.

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exploiting volatility movements in the sydney futures exchange's bank bill contract

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