Properties of American Volatility Options in the Mean-Reverting 3/2 Volatility Model

Abstract

This paper investigates problems associated with the valuation of callable American volatility put options. Our approach involves modeling volatility dynamics as a mean-reverting 3/2 volatility process. We first propose a pricing formula for the perpetual American knock-out put. Under the given conditions, the value of perpetual callable American volatility put options is discussed.

On American VIX options under the generalized 3/2 and 1/2 models

“…As X is not the price of a traded asset, one should allow for the possibility of a nonzero market price of risk associated with the VIX. Following papers by Stein and Stein () and Grünbichler and Longstaff (), we assume that the market price of risk is such that the risk‐neutral process for X is of the same form as the real process . For this, one chooses as .…”

“…They also compute European VIX option prices under this model. Relying on this evidence, Liu (2015) formulates a free-boundary problem for the valuation of the American VIX put option under the 3/2 model and shows monotonicity properties of the option price function and optimal exercise boundary. Although the evidence provided in Goard and Mazur (2013) shows that the 3/2 model dominates the alternatives considered, the analysis performed tests for overidentifying restrictions relative to a specific benchmark.…”

“…Valuation formulas have developed around a set of well‐known models for the evolution of the underlying volatility. Formulas for European volatility options can be found in Whaley () under the assumption of a geometric Brownian motion process (GBMP) and in Grünbichler and Longstaff () for a mean‐reverting square root volatility process (MRSRP), also known as CIR process (see Cox, Ingersoll, & Ross, ). For American‐style volatility contracts, Detemple and Osakwe () provide formulas for GBMP, mean‐reverting Gaussian process (MRGP), mean‐reverting square root process (MRSRP), and mean‐reverting log process (MRLP).…”

Finite horizon portfolio selection problems with stochastic borrowing constraints

“…According to Hopf's boundary point lemma (See Liu [21]), we obtain that ∂ z (∂ t Q) > 0, which contradicts the ∂ z (∂ t Q)(t, z 1 ) = 0, ∀ t ∈ [t 2 , t 1 ]. Therefore, the free boundary z is strictly decreasing.…”

“…The proof is given in Appendix D. First, Figure 1 plots the free boundary z (t) defined in (30). By using the recursive integration method proposed by Huang et al [16], we numerically solved the integral equation of time-reversed free boundary z (t) in (21).…”