2 The Volatility Structure Implied by Options on the SPI Futures Contract by Christine A. Brown † Abstract: The Asay (1982) option pricing model prices options on futures contracts where the premia are margined. The model assumes that the volatility of the underlying futures contract is constant over the life of the option. However it is an empirical observation in many markets that options on the same underlying futures contract with the same maturity, but at different strikes, trade at different implied volatilities. Since the 1987 crash, it has been documented that in many markets the volatility implied by out-of-the-money put options is higher than that implied by out-of-the-money call options. This phenomenon has become known as the ‘volatility skew’. This paper examines the volatility structure for options on the SPI futures contract over the period June 1993 to June 1994, and provides theoretical explanations consistent with its shape. Keywords: ASAY MODEL; VOLATILITY SMILE; VOLATILITY SKEW. † Department of Accounting and Finance, University of Melbourne, Parkville, Victoria 3052. Email: [email protected] I am grateful to Rob Brown, Kevin Davis and James Pickles for helpful comments, and to Raymond Liu for research assistance. Australian Journal of Management, Vol. 24, No. 2, December 1999, © The Australian Graduate School of Management – 115– AUSTRALIAN JOURNAL OF MANAGEMENT December 1999 1. Introduction he Sydney Futures Exchange (SFE) is one of the few futures exchanges to trade Toptions on futures where the options are subject to futures style margining.1 An initial margin2 is deposited and the option contract is marked to market at the end of each day, where the loss on a long option position is limited to the size of the initial premium. The margining applied to options traded on the SFE implies that option pricing models, developed for the situation where option premia are paid up front, are inappropriate. The standard option pricing model used to price options on futures contracts is the Black (1976) model for pricing options on commodity futures. Asay (1982) and Lieu (1990) have modified the Black (1976) model,3 to account for the daily margining of the option contract. Using the Asay model the call premium, C, for a European option on the underlying futures, F, satisfies: C = FN(d1) – XN(d2), (1) ln (FX/t ) + 1σ 2 where : d1 = 2 , σ t d2 = d1 – σ√t, and F = futures price; X = exercise price; C = call price; t = time to maturity; and σ = instantaneous volatility. The put premium, P, is given by: P = XN(–d2) – FN(–d1) (2) The model given by equations 1 and 2 is similar to Black’s (1976) model for pricing options on commodity futures, the difference being the absence of the interest rate term in the above equations. Therefore the price of an option where the premium is margined, relative to one where the premium is paid up front, will be higher. When the option premium is paid up front the buyer is committing funds and the writer receives those funds, while for options which are margined the option premium no longer flows from the buyer to the writer. It can be shown that under certain conditions it is optimal to exercise a put option early when the premium is charged up front. The Black and Scholes (1973) model and the Black (1976) model which are developed for European options are 1. Another exchange which also margins its futures options is the LIFFE. 2. The SFE introduced the SPAN margining system in 1994. This system accounts for the overall risk of a position (containing futures and options on the SPI) as both the futures price moves and the volatility changes. 3. This modified model will be referred to as the Asay model throughout the paper. –116– Vol. 24, No. 2 Brown: VOLATILITY STRUCTURE IMPLIED BY OPTIONS not immediately applicable to pricing American options because these models do not price the early exercise flexibility of American put options (or American call options where the underlying asset pays a dividend).4 However, Lieu (1990) shows that under futures-style margining it is never optimal to exercise a call option or a put option early, and therefore the model given by equations 1 and 2 above applies to the American style options on the Share Price Index (SPI) futures contract traded on the SFE. The assumptions underpinning the Asay (1982) model are the same as for the Black (1976) and the Black and Scholes (1973) models. The underlying futures price is assumed to be lognormally distributed, markets are assumed to be frictionless with trading taking place continuously, and the short term interest rate is assumed to be known with certainty. The process that drives changes in the futures price is assumed to have two components: an expected drift rate and an uncertain or stochastic component scaled by the volatility parameter, which is assumed constant over the life of the option. In practice, the volatility parameter is the only unobservable parameter in the model. Because the option price, the futures price, time to maturity, interest rate and strike price are all observable (or measurable), we can substitute these parameters into the equations, and solve for the only unknown parameter, volatility.5 This volatility derived from the known parameters is called the implied volatility and is obtained by backing the volatility out of equations 1 and 2. Using this procedure to solve for an implied volatility assumes that market participants are using the Asay model to price options. If the model were correctly specified and all its assumptions valid, then the implied volatility for all options (observed at the same time) with the same maturity but different strikes, should be equal. Indeed, under the assumption that the market for stock index futures options is efficient then empirical observation of implied volatility varying across strike prices, contrasts the option valuation with the Black (1976) or the Asay (1982) formulae. In many markets prior to the 1987 stock market crash, there appeared to be a symmetry around the zero moneyness, where out-of-the-money and in-the-money options traded at higher implied volatilities than the implied volatilities for at-the- money options. This dependence of implied volatility on the strike, for a given maturity became known as the smile effect, although the exact structure of volatility varied across markets and even within a particular market from day to day. However, since the 1987 stock market crash the smile has changed shape in many markets, particularly for traded options on stock indexes, where the function has gone from a smile shape to more of a ‘sneer’. The idea of the volatility ‘smile’ had its genesis in the early papers documenting the systematic pricing biases of the Black and Scholes (1973) option pricing model. Black (1975) suggests that the non-stationary behaviour of volatility would lead the Black-Scholes model to overprice or underprice options. Other authors have confirmed the existence of systematic biases in the model.6 4. It may be optimal to exercise an American call option just prior to a dividend payment. Merton (1973) shows that this will not occur in theory but it could occur in practice due to factors outside the option pricing model. 5. This requires a numerical search procedure because the equations are not immediately solvable for volatility. 6. See, for example, Macbeth and Merville (1979), Whaley (1982), Emanuel and Macbeth (1982), Rubinstein (1985), Brown and Taylor (1997). – 117 – AUSTRALIAN JOURNAL OF MANAGEMENT December 1999 Following Heynen, Kemna and Vorst (1994), explaining the implied volatility structure may lead to the conclusion that option prices are better described by an alternative underlying asset price process. Taylor and Xu (1993) study the smile effect of implied volatilities and show that the existence of stochastic volatility is a sufficient reason for smiles to exist. They show that an approximation to the theoretical implied volatility is a quadratic function of ln(F/X) where F is the forward price and X is the strike price, and that this approximate function has a minimum when X = F. This theoretical result requires that asset price and volatility differentials are uncorrelated and that volatility risk is not priced. Using currency option data obtained from the Philadelphia Stock Exchange, over the period from 1984 to 1992, and regressing a function of theoretical and observed implied volatilities on moneyness, they find little evidence of asymmetry in implied volatilities. However, the empirical smile pattern is about twice the size predicted by the theory. Figure 1 Implied Volatility Smiles for German Mark/USD Call Options (adapted from Xu & Taylor (1993)) 2 1.5 1 10-Day 30-Day 0.5 Implied Volatility Ratio 0 -0.1 -0.05 0 0.05 0.1 Moneyness Figure 1 reproduces the nature of the volatility smile for foreign exchange options as presented by Taylor and Xu (1993). On the vertical axis the ratio is the implied volatility at the relevant strike divided by the implied volatility when the strike price is exactly equal to the forward price of the currency.7 Thus, a ratio of 1.5 for a 5% out-of-the-money option implies that the out-of-the-money option has an implied volatility that is 1.5 times the implied volatility of an at-the-money option. Shastri and Wethyavivorn (1987) find similar results for foreign currency options traded in 1983 and 1984, while Sheik (1991) has argued that a similar pattern occurred for S&P 500 options during 1983 to 1985. 7. Because it may not be possible to observe an option trading exactly at-the-money Xu and Taylor (1994) give a method to estimate the at-the-money implied volatility.