Abstract
In this study, using the method of discounting the terminal expectation value into its initial value, the pricing formulas for European options are obtained under the assumptions that the financial market is risk-aversive, the risk measure is standard deviation, and the price process of underlying asset follows a geometric Brownian motion. In particular, assuming the option writer does not need the risk compensation in a risk-neutral market, then the obtained results are degenerated into the famous Black–Scholes model (1973); furthermore, the obtained results need much weaker conditions than those of the Black–Scholes model. As a by-product, the obtained results show that the value of European option depends on the drift coefficient μ of its underlying asset, which does not display in the Black–Scholes model only because μ = r in a risk-neutral market according to the no-arbitrage opportunity principle. At last, empirical analyses on Shanghai 50 ETF options and S&P 500 options show that the fitting effect of obtained pricing formulas is superior to that of the Black–Scholes model.
Highlights
Introduction e option pricing theory began in 1900 when the French mathematician Louis Bachelier deduced an option pricing formula under the assumption that underlying asset prices follow a Brownian motion with zero drift
Bernarda and Czadob [6] investigate the pricing of basket options and more generally of complex exotic contracts depending on multiple indices. eir approach assumes that the underlying assets evolve as dependent GARCH(1, 1) processes. e dependence among the assets is modeled using a copula based on pair copula constructions
We take the Shanghai 50 ETF options, the first floor option in the Chinese financial market, and S&P 500 options as samples to compare the fitting effect. e empirical analyses show that the fitting effect of our pricing formulas is superior to that of the Black–Scholes model
Summary
We will investigate European option pricing and compare our results with those of Black and Scholes [1]. Black and Scholes [1] present nine assumptions in the market for the security and for the option and obtain their famous option pricing formula. It is possible to borrow any fraction of the price of a security to buy it or to hold it, at the short-term interest rate. When the above assumptions all hold, Black and Scholes [1] derived the pricing formula for European options, which is the Black–Scholes model. Where S0 is the initial price of underlying asset, K is the strike price of option, r is the short-term interest rate, σ is the diffusion coefficient of underlying asset, τ is the left expiration time of option, Φ(·) is the cumulative density function of standard normal distribution, and d1.
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