*The content of this article is excerpted from Chapter 4 "OTC Options" of "OTC Derivatives Knowledge Reader".
▍Background of the birth of the BSM model
Since the emergence of the organized OTC options market in the 19th century, how to determine the value of an option has gradually attracted the attention of the industry and academia. For a stock that is in circulation, we can easily observe its price from the market, and if the liquidity is good enough, the price we observe will also be close to the true value. However, for derivatives, especially over-the-counter derivatives, their prices will be difficult to observe. The role of the model is to provide a valuation method that allows people to extrapolate the value of other illiquid securities from more liquid securities at a known price.
Option pricing models have been studied for a long time. Since Brownian motion was proposed in the 20s of the 19th century, the financier Louis Bachelier, a financier in ·France 1900, introduced Brownian motion into his option pricing model, and then more and more scientists have modeled asset value based on Brownian motion. It wasn't until 1973 that Myron Scholes, ·a professor at Stanford University, and his colleague, the late mathematician Fisher Black, collaborated to develop a relatively complete model ·of option pricing. At the same time, Harvard Business School professor Robert · Merton found the same formula and many other valid conclusions about options. Both papers were published in different journals almost simultaneously. Therefore, the Black-Scholes model (BS model) can also be called the Black-Scholes-Merton model (BSM model). Merton expands the connotation of the original model and enriches the application scenarios of the BSM model.
The classical BSM model, which assumes that asset prices obey geometric Brownian motion, was widely accepted by the industry after it was proposed, and from then until the United States financial crisis in 1987, the prices of options in the market almost coincided with the BSM model, with only a slight deviation due to the adjustment of volatility at different maturities. Even after the 1987 financial crisis, the market still used the implied volatility of the BSM model to describe the price of options.
▍BSM model assumptions
(1) The option is a European-style option exercised at the end of the period;
(2) There is no dividend on the underlying asset during the duration of the option;
(3) Ability to go long and short on any number of underlying assets;
(4) the price of the underlying asset obeys the geometric Brownian motion;
(5) the market is frictionless, that is, there are no taxes and transaction costs;
(6) The short-term risk-free interest rate is constant and knowable during the duration of the option;
(7) The stock price return obeys a normal distribution, and the volatility of the return is constant and knowable.
Cash and bonds are the most basic assets, and their prices affect almost all derivatives prices. Under the premise that the default rate is very low, bank savings account interest, treasury bond interest rate, LIBOR, etc. can all be used as an approximation of the return on risk-free assets. Based on the model abstraction of continuous compounding, we can obtain the most basic risk-free asset price model.
In general, the dynamic process of a money market account can be written as:
But in the BSM model, assumption (6) sets the risk-free rate as a constant, thus:
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