Assessing Target Volatility Investment Strategies Using Stochastic Delayed Differential Models

Abstract

A target volatility strategy (TVS) is a risky asset-riskless bond dynamic portfolio allocation which makes use of the risky asset historical volatility as an allocation rule. High realized volatility means decreased equity exposure, reducing the total portfolio downside risk; conversely, lower volatility yields an increase in the equity position so that in safer market conditions the portfolio can benefit from the equity upside. If the historical volatility correctly predicts the market future volatility, a sequence of these adjustments should maintain the volatility of the investment constant over time around a target level specified by the investor. In a market with stochastic volatility, we present a stochastic continuous-time model for the value of a volatility targeting fund (TVF) using a system of stochastic delayed differential equations (SDDEs). Applying a multi-dimensional version of a theorem by Federico and Tankov (2015) we derive a finite-dimensional Markovian approximation for these equations, which we then implement in the the Heston variance model by developing an ad hoc Euler scheme. This framework allows an efficient numerical valuation of derivatives on TVFs, which is of critical importance for assessing the guarantee costs of such funds. Common wisdom is that if the strategy is correctly implemented, contingent claim valuations on the fund should be of Black-Scholes type. Under the present model, the numerical studies conducted by and large confirm this belief.