Black - Scholes model
Black - Scholes option pricing model (Eng.
Black-Scholes Option Pricing Model, OPM) - a model that defines the theoretical
price of European options, implying that if the underlying asset is traded in
the market, the option price on it has implicitly positioned itself in the
market. This model is widespread in practice and, among other things, can
also be used for the valuation of all derivatives, including warrants,
convertible securities, and even for the valuation of the equity capital of
financially dependent companies.
According to the Black-Scholes model, a key
element in determining the value of an option is the expected volatility of the
underlying asset. Depending on the fluctuation of the asset, the price for
it increases or decreases, which directly affects the value of the
option. Thus, if the value of the option is known, it is possible to
determine the level of volatility expected by the market
·
Securities (the underlying asset) are traded continuously, and
their price behavior follows a geometric Brownian model of motion with known
parameters (in particular, these parameters are constant throughout the option
period).
·
In the option base asset, dividends are not paid over the entire
term of the options.
·
There are no transaction costs associated with buying or selling
shares or options.
·
The short-term risk-free interest rate is known and is constant
over the life of the option.
·
Any buyer of security can obtain a loan at a short-term
risk-free interest rate to pay a portion of its price.
·
Short selling is allowed without restrictions, and the seller
will immediately receive all cash for the securities that were sold short at
today's price.
·
There is no possibility of arbitration.
The
derivation of the model is based on the concept of risk-free hedging. By
buying shares as well as selling call options on those shares, an investor can
build a risk-free position where the earnings from the shares will accurately
offset the losses on the options and vice versa.
The risk-free hedge position must return at
the same rate as the risk-free interest rate, otherwise, there will be an opportunity
for arbitrage profit, and the investor, trying to take advantage of this
opportunity, will bring the option price to the equilibrium level determined by
the model.
The Black-Scholes model directly follows the
Merton model, which allows modeling the value of the company's equity capital
based on the value of the firm's value and its debt, which is represented as a
bond without coupon [4]. In this case, we represent equity capital S in
the form of a long call option over the total value of firm V at the strike
price in face value of the zero-coupon bond F:
Debt D, in turn, can be represented as a
portfolio either with a long position with a bond without coupon F and a sell
option on company V's capital at the strike price F, or a long position on
company V's capital and a short call option at V with F strike:
D
Comments