Abstract
We consider an economic model with a deterministic money market account and a finite set of basic economic risks. The real-world prices of the risks are represented by continuous time stochastic processes satisfying a stochastic differential equation of diffusion type. For the simple class of log-normally distributed instantaneous rates of return, we construct an explicit state-price deflator. Since this includes the Black-Scholes and the Vasicek (Ornstein-Uhlenbeck) return models, the considered deflator is called Black-Scholes- Vasicek deflator. Besides a new elementary proof of the Black-Scholes and Margrabe option pricing formulas a validation of these in a multiple risk economy is achieved.
Highlights
The first rigorous mathematical derivation of the BlackScholes formula in [1] relies on a dynamic delta-hedge portfolio and a risk-free argument of noarbitrage
We consider an economic model with a deterministic money market account and a finite set of basic economic risks
The real-world prices of the risks are represented by continuous time stochastic processes satisfying a stochastic differential equation of diffusion type
Summary
The first rigorous mathematical derivation of the BlackScholes formula in [1] (see [2]) relies on a dynamic delta-hedge portfolio and a risk-free argument of noarbitrage. Later on [3] introduced state-price deflators, which led to the insight that deflator based market valuation using the real-world probability measure is equivalent to market valuation based on a risk-neutral martingale measure. The real-world prices of these risks are represented by continuous time stochastic processes satisfying a stochastic differential equation of diffusion type. Besides a new elementary proof of the (slightly extended) Black-Scholes formula it provides a validation of it in a financial market with multiple economic risks. The same holds true for Margrabe’s formula for a European option to exchange one risky asset for another one
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