Optimal-Hedging

In this project we implement optimal delta hedging on S&P 500 index options under the industry-famous stochastic volatility model, the SABR model. The SABR model is calibrated on SPX option time series, which is dynamically hedged using SABR delta and Bartlett's delta.

References

Most implementations in this project are based on Bartlett's delta in the SABR model by Hagan and Andrew (2019) and Optimal Delta Hedging by Hull and White (2016).

Scripts

• data: SPX 500 options data from 2023-02-01 to 2023-02-28 from WRDS
• papers: a list of papers used for this project
• presentation: presentation slides for the project
• sabr_calibration.ipynb: an example of SABR model implementation and calibration
• optimal_hedging.ipynb: main notebook that calibrates SABR model and optimal delta hedging

Overview

SABR Model

SABR model is a stochastic volatility model given by

$$ \begin{align*} dF(t) &= \sigma(t)(F(t)+\theta)^{\beta}dW_1(t), F(0)=f\ d\sigma(t) &= \nu\sigma(t)dW_2(t), \sigma(0)=\sigma_0\ \end{align*} $$

where $W_1(t)$ and $W_2(t)$ are two correlated Wiener processes with correlation $\rho$, namely, $dW_1(t)dW_2(t) = \rho dt$.

• F: forward rate
• $\sigma$: volatility of forward rate
• $\nu$: volatility of volatility (volvol)
• $\theta$: shift parameter to avoid negative rates

The SABR model gains its popularity due to its ability to capture the volatility smile observed in the market. It is a common practice to set $\beta=0.5$ for interest rate derivatives and $\beta=1$ for equity options.

Optimal Delta Hedging

The SABR delta is given by $$\Delta^{\text{SABR}} = \frac{\partial B}{\partial F} + \frac{\partial B}{\partial \sigma} \frac{\partial \sigma_{\text{imp}}}{\partial F} $$

The Bartlett's delta further incorporates the adjustment for the implied volatility skew

$$\Delta^{\text{Bartlett}} = \frac{\partial B}{\partial F} + \frac{\partial B}{\partial \sigma} \left( \frac{\partial \sigma_{\text{imp}}}{\partial F} + \frac{\partial \sigma_{\text{imp}}}{\partial \sigma} \frac{\rho \alpha}{C(F_t)}\right) $$

Bartlett's delta is the optimal delta for hedging in the SABR model, which can be approximated by $$\Delta^{mod}\approx \Delta^{BS}+\text{Vega}^{BS}\times \eta$$

Hedging Performance Evaluation

In Hull and White (2016), the effectiveness of a hedge is measured by the $Gain$ metric, defined as the percentage reduction in the sum of squared residuals resulting from the hedge, i.e. $$\text{Gain} = 1- \cfrac{\sum(\Delta f - \delta_{\text{SABR}}\Delta S)^2}{\sum(\Delta f - \delta_{\text{BS}}\Delta S)^2}$$

Results

Implied volatility Smile Calibration

BS Delta (Blue), SABR Delta (Red), Bartlett's Delta (Green)

Bartlett's Delta for Different Maturities

Hedging Parameters Evolution

Hedging Gain for Bartlett's Delta

SABR Delta vs. Bartlett's Delta

Short Summary

• SABR model calibrates the implied volatility smile of SPX 500 options data extremely well.
• Both SABR delta and Bartlett’s delta are effective in hedging the options, much better than Black-Scholes delta.
• Bartlett’s delta performs slightly but consistently better than SABR delta.