Abstract
In this paper, we investigate the value of incorporating implied volatility from related option markets in dynamic hedging. We comprehensively model the volatility of all four S&P 500 cash, futures, index option and futures option markets simultaneously. Synchronous half-hourly observations are sampled from transaction data. Special classes of extended simultaneous volatility systems (ESVL) are estimated and used to generate out-of-sample hedge ratios. In a hypothetical dynamic hedging scheme, ESVL-based hedge ratios, which incorporate incremental information in the implied volatilities of the two S&P 500 option markets, generate profits from interim rebalancing of the futures hedging position that are incremental over competing hedge ratios. In addition, ESVL-based hedge ratios are the only hedge ratios that manage to generate sufficient profit during the hedging period to cover losses incurred by the physical portfolio.
JEL classification: G14, G28
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Notes
- 1.
S&P Depository Receipts are yet to exist during our sample period.
- 2.
Strictly speaking, V CI, t is not entirely exogenous in that a decision is required on the proportion of the underlying asset’s value to hedge against. However, this is a separate issue from the OHR determination.
- 3.
The unrestricted vector GARCH specification seldom leads to identifiable point estimates since doing so requires the inversion of the variance-covariance matrix at each sample point. When off-diagonal covariance terms are large relative to diagonal variance terms, the determinant of the matrix tends to zero, such that the inverse matrix can be unidentified. Such cases are very likely in the context of spot and the futures returns. This technical problem is commonly addressed by imposing a constant correlation and focus on modeling time varying spot and futures volatility, which somewhat defeats the purpose of dynamic hedging.
- 4.
The data was supplied by TICKDATA Inc but for various reasons, they stopped collection of options data streams.
- 5.
Individual stock prices are multiplied by the number of shares outstanding. The products are summed up and standardized by a pre-determined base value. Base values for the index are adjusted to reflect changes in capitalization due to mergers, acquisitions and rights issues etc.
- 6.
However, a trade may occur at a price of 0.025 index point if it is necessary to liquidate positions to allow for both parties to trade.
- 7.
For the March futures contract, there are option contracts expiring at the end of Jan, Feb, March etc up to May. Options expiring prior or during the March quarterly cycle are written on the March futures contracts. Else, they are written on the June futures contract.
- 8.
For missing observations in one market, corresponding observations from other markets are excluded. This gives a total of 3,186 out of a possible 3,289 observations. There are 13 half-hourly observations per day over 253 trading days in 1990. Details of the sampling procedure can be obtained from the authors upon request.
- 9.
For example, an option trading on 23rd April with May as the delivery month will have 27 trading days till maturity. Following conventions, this study uses 252 days for the denominator. Thus for that option, term-to-maturity is calculated as 23/252 = 0.10714.
- 10.
This is chosen over the 30 year Treasury bond rate, which may contain term structure premium.
- 11.
Although negative covariance is possible and should be allowed for, the spot and futures returns are expected to move in the same direction most of the time due to cost-of-carry. An examination of the data confirms this statement.
- 12.
Dum Close = 1 for the 3.00 p.m. closing return, and 0 otherwise.
- 13.
Tick volume is defined as the number of half hourly price changes. For example, CV t is the number of half-hourly price changes between time t-1 and t.
- 14.
Dum Open=1 for the 9.00 a.m. opening return, and 0 otherwise.
- 15.
While it can be shown that early exercise is never optimal for a futures option if the premium is subjected to futures-style margining, the premiums for FO are paid up front. With positive interest rates, early exercise remains a possibility.
- 16.
An alternative way to view the mapping of reduced form parameters back to the structural form parameters is to think about the systems in terms of the normalized matrix rank condition rather than the unnormalized matrix rank condition. In the former, there is an implied unity restriction imposed on endogenous variables in own structural equations. Then substitution between the structural and reduced forms is straightforward and in an identifiable system all structural parameters in every structural equation are identified. As such, the reduced-form equations provide unique projections of endogenous variables in the systems. However, there can be cases where alternative structural parameterizations provide non-nested competing sets of structural systems. In this case, artificial nested testing procedures need be employed to select between competing systems. In our paper, there are competing identifiable three and four equation systems. These are compared in terms of out of sample hedge ratio estimation and subsequently in terms of trade re-balancing performance.
- 17.
To note, utilizing ESVL-based hedge ratios may seem computationally tedious. However, as the forecasts are made one-step at a time, the majority of coefficients are stable when we sequentially expand the estimation period. Once the initial coefficient estimates are recorded, subsequent updating is computationally easy.
- 18.
The derivations are available upon request.
- 19.
As mentioned in the introduction, we assume the S&P 500 index as our physical portfolio.
- 20.
Note that ‘N’, which represents the number of futures contracts, is always positive. The ‘No. of contracts rebalanced at time t − 1’ is defined as \(({N}_{t-1} - {N}_{t-2})\). As such, if \({N}_{t-1} = 5\) and \({N}_{t-2} = 3\), then the ‘No. of contracts rebalanced at time t − 1’ is +2, which implies shorting an additional two futures contracts.
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Chng, M.T., Gannon, G.L. (2013). The Trading Performance of Dynamic Hedging Models: Time Varying Covariance and Volatility Transmission Effects. In: Lee, CF., Lee, A. (eds) Encyclopedia of Finance. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-5360-4_61
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