# 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.