PUT‐CALL SYMMETRY: EXTENSIONS AND APPLICATIONS

Classic put‐call symmetry relates the prices of puts and calls at strikes on opposite sides of the forward price. We extend put‐call symmetry in several directions. Relaxing the assumptions, we generalize to unified local/stochastic volatility models and time‐changed Lévy processes, under a symmetry condition. Further relaxing the assumptions, we generalize to various asymmetric dynamics. Extending the conclusions, we take an arbitrarily given payoff of European style or single/double/sequential barrier style, and we construct a conjugate European‐style claim of equal value, and thereby a semistatic hedge of the given payoff.