The Black-Scholes model is a mathematical model for pricing options that is based on the assumption of continuous trading and the use of risk-neutral pricing.
All models have faults, that doesn't mean you can't use them as tools for making decisions. - Myron Scholes
The Black-Scholes model is one of the financial instruments that give the holder the right, but not the obligation, to buy or sell an underlying asset at a specified price on or before a certain date. It is used to determine the theoretical price of European call and put options, which are options that can be exercised only at the expiration date of the option.
Imagine that you are a trader who is considering buying a European call option on a stock. You believe that the stock will increase in value over the next month, and you want to purchase an option that will allow you to buy the stock at a fixed price (the strike price) if it does increase in value.
To determine the price at which you should buy the option, you can use the Black-Scholes model to calculate the theoretical value of the option based on certain input variables such as the current stock price, the strike price, the time until expiration, the risk-free interest rate, and the expected volatility of the stock.
By plugging these variables into the Black-Scholes formula, you can determine the fair price for the option given your expectations about the future movement of the stock. You can then use this information to decide whether or not to go ahead with the trade.
The Black-Scholes formula for the price of a call option is given by:
Call price = S * N(d1) - X * exp (-r * T) * N(d2)
where:
S is the current price of the underlying asset
X is the strike price of the option
T is the time until expiration, in years
r is the risk-free interest rate
N(d1) and N(d2) are the standard normal cumulative distribution functions of d1 and d2, respectively, which are defined as:
d1 = (ln(S/X) + (r + s^2/2) * T) / (s * sqrt(T))
d2 = d1 - s * sqrt(T)
s is the annualized volatility of the underlying asset
The Black-Scholes formula for the price of a put option is similar, with the main difference being that the call price is replaced with the put price and d1 is replaced with -d1.
Keep in mind that the Black-Scholes model is just one tool among many that traders use to make decisions about options trading. It is based on certain assumptions about market conditions and behaviour, and these assumptions may not always hold true in the real world. As such, it is important to consider other factors and use a variety of tools and techniques when making investment decisions.
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