Traditionally delta has been calculated as the partial derivative of the value of the portfolio with respect to the underlying asset. Implied volatilities and other variables are kept constant. It is now recognized that this is not the best way of proceeding for equities, and possibly for other underlying assets as well. When an equity price changes there is a tendency for volatility to change as well. Specifically, when the equity’s price increases, its volatility tends to decrease, and vice versa. There are two competing theories as to why this happens. One involves leverage. The amount of debt that a company has is fairly static so that when a company’s equity increases (decreases) its leverage decreases (increases) causing volatility to decrease (increase). This leverage argument suggests that the causality is from the equity price to the volatility. The other theory is known as the “volatility feedback effect.” In this the causality is the other way round. When volatility increases (decreases) the stock becomes more (less) risky as an investment so investors require a higher (lower) return than previously and the stock price therefore decreases (increases).
A delta measure that takes account of the negative correlation between an equity’s price and its volatility is known as a “minimum variance delta.” In this blog we look at two approaches for calculating it.
Stochastic Volatility Models
One approach for calculating a minimum variance delta is to replace the Black-Scholes model by a stochastic volatility model. The model for valuing a portfolio dependent on a particular stock or an equity index or equity index then takes the form:
\[ \mbox{Value} = G \left(S,\sigma,\dots\right) \]
where $S$ is the value of the underlying asset (stock or stock index), and $\sigma$ the (current) value of the asset’s volatility. The other arguments of the function $G$
are interest rates, the dividend yield and parameters driving the process for the volatility. The Wiener process determining the evolution of the volatility is assumed to be negatively correlated with the Wiener process driving the evolution of the asset price.
It might be tempting to set
\[\delta_{MV} = \frac{\partial G}{\partial S} \]
where $\delta_{MV}$ denotes the minimum variance delta. However this would be incorrect as it would be considering the effect of a change in the asset price without any change in volatility—even though a stochastic volatility model is being used. The correct approach is to set
\[\delta_{MV} = \frac{ G \left( S+\Delta S, \sigma + \mathbb{E} \left( \Delta \sigma \big| \Delta S \right),\dots \right) -G \left( S, \sigma ,\dots \right) }{\Delta S} \]
where $ \mathbb{E}$ denotes expected value. The term $ \mathbb{E} \left( \Delta \sigma \big| \Delta S \right)$ is the expected change in the volatility as a result of the stock price change. The value of the expected change depends on the stochastic volatility model being used. For example, if the process for the asset price is
\[dS = \cdots dt + \sigma S dz_1\]
and the process for its volatility is
\[d\sigma = \cdots dt + \xi \sigma dz_2\]
then
\[\mathbb{E} \left( \Delta \sigma \big| \Delta S \right) = \frac{\xi \rho \Delta S}{S}\]
where $ \rho $ is the correlation between the two Wiener processes $\dz_1$ and $ \dz_2$.
A number of researchers such as Bakshi et al (1997, 2000), Bartlett (2006), Alexander and Nogueira (2007), Alexander et al (2009), and Poulsen et al (2009) have followed this approach. The general finding is that stochastic volatility models produce better deltas than Black-Scholes for equity indices.
Vega-Based Models
The value of an option can be written as
\[ \mbox{Option Price} = f_{BS}\left(S, \sigma_{imp}, \dots \right)\]
where $f_{BS}$ is the Black-Scholes function, $\sigma_{imp}$ is the implied volatility, and the other arguments of $f_{BS}$ are the interest rate and dividend yield which can, without too much of an approximation, be assumed to be constant.
It is then true that
\[ \delta_{MV} = \frac{\partial f_{BS}}{\partial S} + \frac{\partial f_{BS}}{\partial \sigma_{imp}} \frac{\partial \mathbb{E}\left(\sigma_{imp} \right)}{\partial S} = \delta_{BS}+\nu_{BS}\frac{\partial \mathbb{E}\left(\sigma_{imp} \right)}{\partial S}\]
where $\delta_{BS}$ is the Black-Scholes delta as it is normally calculated, $\nu_{BS}$ is the Black-Scholes vega (partial derivative with respect to implied volatility), and $\mathbb{E}\left(\sigma_{imp} \right)$ is the expected value of the implied volatility as a function of $S$. If an assumption is made about the partial derivative:
\[ \frac{\partial \mathbb{E}\left(\sigma_{imp} \right)}{\partial S}\]
then equation (1) can be used to calculate $\delta_{MV}$ from $\delta_{BS}$ local volatility model proposed by Derman and Kani (1994) and Dupire (1994) suggests that the partial derivative is close to the slope of the volatility smile (see Derman et al (1995) and Coleman et al (2001). Crépey (2004) and Vähämaa (2004) test this. Hull and White (2016) try a different approach. They attempt to determine the partial derivative empirically. They find that the assumption that it is a quadratic in $\delta_{BS}$ divided by the product of the square root of the option life and the asset price works well.
Conclusions
Delta hedging is straightforward. A trader usually incurs minimal transaction costs when taking a position in the underlying asset to manage delta. This is in contrast to vega and gamma hedging where positions in options or other non-linear products are required to effect changes. It makes sense to get as much mileage as possible from delta hedging. Moving from the traditional Black-Scholes delta to the minimum variance delta does this. Indeed, it has the dual advantage of reducing vega risk and tackling the risk associated with movements in the underlying asset more precisely.
REFERENCES
Alexander, C., A. Kaeck, and L.M. Nogueira, “Model risk adjusted hedge ratios” Journal of Futures Markets 29, 11 (2009):1021-1049.
Alexander, C. and L.M. Nogueira, “Model-free hedge ratios and scale invariant models,” Journal of Banking and Finance, 31 (2007): 1839-1861.
Bakshi, G., C. Cao, and Z. Chen, “Empirical performance of alternative option pricing models,” Journal of Finance, 52, 5 (December 1997): 2003-2049.
Bakshi, G., C. Cao, and Z. Chen, “Pricing and hedging long-term options,” Journal of Econometrics, 94 (2000): 277-318.
Bartlett, B, “Hedging Under SABR Model,” Wilmott Magazine, July / August (2006): 2-4.
Coleman, T, Y. Kim, Y. Li, and A. Verma, “Dynamic hedging with a deterministic local volatility model,” Journal of Risk, 4, 1 (2001): 63-89.
Crépey, S., “Delta-hedging vega risk,” Quantitative Finance, 4 (October 2004): 559-579.
Derman, E. and I. Kani, “Riding on a Smile,” Risk, 7 (February 1994): 32-39.
Derman, E., I. Kani, and J.Z. Zou “The Local Volatility Surface: Unlocking the Information in Index Option Prices,” Goldman Sachs Selected Quantitative Strategies Reports (December 1995).
Dupire, B. “Pricing with a smile,” Risk, 7 (February 1994):18-20.
Hull, J. and A. White, “Optimal Delta Hedging for Options,” Working Paper, University of Toronto.
Poulsen, R., K., R. Schenk-Hoppé, and C.-O Ewald, “Risk minimization in stochastic volatility models: model risk and empirical performance,” Quantitative Finance, 9, 6 (September 2009): 693-704.
Vähämaa, S., “Delta hedging with the smile,” Financial Markets and Portfolio Management, 18, 3 (2004): 241-255.