Statistics 简明教程

Statistics - Black-Scholes model

Black Scholes 模型是一个数学模型，用于查看股票等金融工具的价格随时间的变化，它可用于计算欧式看涨期权的价格。该模型假设大量交易的资产的价格遵循具有恒定漂移和波动的几何布朗运动。对于股票期权，Black Scholes 模型包含基础股票的恒定价格变化、金钱的时间价值、期权执行价格及其到期时间。

The Black Scholes model is a mathematical model to check price variation over time of financial instruments such as stocks which can be used to compute the price of a European call option. This model assumes that the price of assets which are heavily traded follows a geometric Brownian motion having a constant drift and volatility. In case of stock option, Black Scholes model incorporates the constant price variation of the underlying stock, the time value of money, strike price of the option and its time to expiry.

Black Scholes 模型由 Fisher Black、Robert Merton 和 Myron Scholes 于 1973 年开发，并在欧元金融市场中广泛使用。它提供了确定期权公平价格的最佳方法之一。

The Black Scholes Model was developed in 1973 by Fisher Black, Robert Merton and Myron Scholes and is still widely used in euporian financial markets. It provides one of the best way to determine fair prices of options.

Inputs

Black Scholes 模型需要五个输入。

The Black Scholes model requires five inputs.

  1. Strike price of an option

  2. Current stock price

  3. Time to expiry

  4. Risk-free rate

  5. Volatility

Assumptions

Black Scholes 模型假定以下几点。

The Black Scholes model assumes following points.

  1. Stock prices follow a lognormal distribution.

  2. Asset prices cannot be negative.

  3. No transaction cost or tax.

  4. Risk-free interest rate is constant for all maturities.

  5. Short selling of securities with use of proceeds is permitted.

  6. No riskless arbitrage opportunity present.

Formula

其中——

Where −

  1. ${C}$ = Value of Call Option.

  2. ${P}$ = Value of Put Option.

  3. ${S}$ = Stock Price.

  4. ${K}$ = Strike Price.

  5. ${r}$ = Risk free interest rate.

  6. ${T}$ = Time to maturity.

  7. ${\sigma}$ = Annualized volatility.

Limitations

Black Scholes 模型有以下限制。

The Black Scholes model have following limitations.

  1. Only applicable to European options as American options could be exercised before their expiry.

  2. Constant dividend and constant risk free rates may not be relistic.

  3. Volatility may fluctuate with the level of supply and demand of option thus being constant may not be true.