Time-dependent Volatility Multi-stage Compound Real Option Model and Application

The simple compound option model has many limitations when applied in practice. The research on compound option theory mainly focuses on two directions. One is the extension from two-stage to multi-stage, and the other is the modification of the stochastic difference equations which describe the movement of underlying asset value. This paper extends the simple compound option model in both two directions and proposes the Time-dependent Volatility Multi-stage Compound Real Option Model. Due to the introduction of time-dependent volatility, it is difficult to derive the closed-form solution by the traditional analytical approach. This paper presents the pricing governing partial differential equation, proposes the boundary conditions and terminal conditions, and then gets the numerical solution by Finite Differential Methods. Finally this paper applies Time-dependent Volatility Multi-stage Compound Real Option Model to evaluate venture capital investment. The numerical results show that the Fixed Volatility Multi-stage Compound Real Option Model underestimates significantly the intrinsic value of venture capital investment as well as exercise threshold of later stages, but overestimates the exercise threshold of earlier stages.