Implied volatility and the risk-free rate of return in options markets

We numerically solve systems of Black-Scholes formulas for implied volatility and implied risk-free rate of return. After using a seemingly unrelated regressions (SUR) model to obtain point estimates for implied volatility and implied risk-free rate, the options are re-priced using these parameters. After repricing, the difference between the market price and model price is increasing in time to expiration, while the effect of moneyness and the bid-ask spread are ambiguous. Our varying risk-free rate model yields Black-Scholes prices closer to market prices than the fixed risk-free rate model. In addition, our model is better for predicting future evolutions in model-free implied volatility as measured by the VIX.