A target volatility (call) option is an option where the total underlying exposure is random and determined at maturity as the ratio of a predicted target volatility level with the volatility effectively realized by the underlying. This reduces the option price for a similar payoff if the volatility prediction was correct, thus allowing an affordable option position when vega hedging is too expensive or market implied volatilities are too high.
A target volatility strategy (TVS) is a bond-equity dynamic portfolio using the risky asset historical volatility as an allocation rule. High realized volatility decreases the equity exposure reducing the portfolio downside risk. Lower volatility increases the equity position as to benefit from the bearish market conditions. These adjustments over time should maintain the volatility of the investment constant around the investor’s desired target level.
In a market with stochastic volatility, we present a stochastic model for a TVS using a delayed differential system. We derive an approximate finite-dimensional Markovian approximation for the equations which we implement for the Heston model using a Euler scheme. This framework allows the valuation of guarantee costs of target volatility funds; if the constant volatility assumption is correct, such a value should be of Black-Scholes type. We investigate this claim within the presented model.