From Time Series of Returns to Option Prices: The Stochastic Discount Factor Approach

摘要: In the perfect and unrealistic Black Scholes (J Polit Econ 81:637–659, 1973) world, dynamics \((S_{t})_{t\in [0,T]}\) of risky asset, under historical probability \(\mathbb{P}\), is given by following stochastic differential equation: $$\displaystyle{ dS_{t} =\mu S_{t}dt +\sigma S_{t}dW_{t} }$$ where \((W_{t})_{t\in a standard Brownian motion \(\mathbb{P}\). this case, there no ambiguity in definition arbitrage-free price any European contingent claim with maturity T. fact, complete market which set continuous time, value none other than replicating portfolio. Moreover, prices may be expressed terms conditional expectations unique equivalent martingale measure Q whose density respect to Girsanov theorem \frac{dQ} {d\mathbb{P}} = e^{-\frac{\mu -r} {\sigma } W_{T}-\left (\frac{\mu \right )^{2} \frac{T} {2} r constant continuously compound risk-free rate. Unfortunately, as we have seen Sect. 2.1, restrictive underlying hypotheses (constant volatility, independent increments, Gaussian log-returns, etc…) are questioned many empirical studies GARCH models appear excellent alternative solutions potentially overcome some well-documented systematic biases associated model.