Consistent Pricing of Options on Leveraged ETFs

Abstract

We consider the problem of pricing options on a leveraged ETF (LETF) and the underlying ETF in a model-consistent manner. We show that if the underlying ETF has Heston dynamics, then the LETF also has Heston dynamics so that options on both the ETF and the LETF can be priced analytically using standard transform methods. If the underlying ETF has tractable jump-diffusion dynamics, then the dynamics of the corresponding LETF are generally intractable in that we cannot compute a closed-form expression for the characteristic function of the $\log$-LETF price. In that event we either (i) evaluate the appropriate transform numerically or (ii) propose tractable approximations based on either moment-matching techniques or saddlepoint approximations to the LETF price dynamics under which the transform can be found in closed form. In a series of numerical experiments including both low- and high-volatility regimes, we show that the resulting LETF option price approximations are very close to the true prices which we calculate via Monte Carlo. Because approximate LETF option prices can be computed very quickly, our methodology should be useful in practice for pricing and risk-managing portfolios that contain options on both ETFs and related LETFs. Our numerical results also demonstrate the model dependency of LETF option prices, and this is particularly noticeable in high-volatility environments.