Abstract
We develop a new volatility measure: the volatility implied by price changes in option contracts and their underlying. We refer to this as price-change implied volatility. We compare moneyness and maturity effects of price-change and implied volatilities, and their performance in delta hedging. We find that delta hedges based on a price-change implied volatility surface outperform hedges based on the traditional implied volatility surface when applied to S&P 500 future options.
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Notes
The Black–Scholes model and Black model are the same model, but applied to different underlying. The underlying in our study are the S&P 500 futures options. Therefore, we use the term Black model instead of Black–Scholes model.
The Black formula for a European call option on a futures contract, F, is given by
$$ G(F,T,r,\sigma )=e^{-rT}\left( FN(d_{1})-KN(d_{2})\right) $$using standard notation and where N is the cumulative standard normal and
$$ \begin{aligned} d_{1} &=\frac{\ln (F/K)+\frac{\sigma ^{2}}{2}T}{\sigma \sqrt{T}}\\ d_{2} &=d_{1}-\sigma \sqrt{T.} \end{aligned} $$Although we use tick data, we do not investigate micro-structure issues since we use 1-, 2-, and 3-day price changes taken from trading hours with the greatest liquidity.
For constant σ pciv = σ, geometric Brownian motion and \(h\rightarrow 0, \) \(\Updelta F^{2}\rightarrow \sigma^{2}h\). In this case, Eq. (5) is exact and the returns on the portfolio are deterministic. Under standard assumptions, equating the right-hand side of the equation to the returns on a risk-free portfolio gives the Black partial differential equation for option price.
The t statistic is calculated as follows:
$$ t=\frac{0.2632-0.1837}{\sqrt{\frac{0.0906^{2}}{3061}+\frac{0.0543^{2}}{2274}} } $$Since t statistic assumptions are not met, the measure should be considered ad hoc.
VXO is defined as "the average over 8 near-the-money Black-Scholes implied volatilities at the two nearest maturities on S&P 100 index." As such, it is essentially an estimate of the 1-month, at-the-money implied volatility.
See also Blenman and Wang (2012) for a comparison of implied and realized volatilities.
Consider a scenario in which a financial institution wishes to lock in the profits from selling an overvalued (vis-à-vis Black’s value) option to a client. It does this by creating an equivalent synthetic long position in the option. Absent the synthetic position, the firm’s profit is at risk. For further discussion, see Hull (2008). Our application is slightly different in that we only investigate the 1-day portion of a hedged portfolio that would otherwise be maintained until option expiration.
In addition to estimates based on dollar values, we obtained estimates using percentage values, i.e., we used \(\frac{G_{j}^{\ast }-G_{j}(\sigma _{iv,j}(\user2{\theta }_{iv}))}{G_{j}(\sigma _{iv,j}(\user2{\theta }_{iv}))}\) instead of \(G_{j}^{\ast }-G_{j}(\sigma _{iv,j}(\user2{\theta }_{iv}))\) in the Eqs. (13 and 14).
These datasets are based on careful selection of data, preventing overlapping of observations and using trades in time of the highest liquidity (10:00 a.m.).
The only exception is the MSE-based loss function using percentages for calls (Table 9, panel A2), where price-level implied volatility has slightly lower RMSE than price-change and implied hedging volatilities.
Comparison of the impact of price-change implied volatility vis-à-vis the impact of other volatility measures on Greeks in a regression context can be found in Hilliard (2012).
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Hilliard, J., Li, W. Volatilities implied by price changes in the S&P 500 options and futures contracts. Rev Quant Finan Acc 42, 599–626 (2014). https://doi.org/10.1007/s11156-013-0354-z
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DOI: https://doi.org/10.1007/s11156-013-0354-z