# Black-Scholes: Risk and Randomness in Options

Reference is Basic Black-Scholes: Option Pricing and Trading by Timothy Falcon Crack.

The Black Scholes formula starts with defining a relationship via a partial differential equation between a stock price moving randomly with drift and a derivative price which derives from the stock price. By Ito’s Lemma, we can amount to a relationship between movements in the derivative price and the stock price. If we solve the partial differential equation we come up with an expression for the derivative price which is defined in terms of interest rate, the underlying and the time. How do we know this derivative is an option? By the manner in which the derivative derives its value from the stock price. We can manipulate this value to show put-call parity which is that owning a stock and a put gives the same outcomes as owning a call and some cash that pays an interest rate. The final expression for the derivative has Greeks which are how the derivative price moves in relation to the underlying “delta,” time “theta,” motion around the strike price “gamma” and interest rates. Notably when we hedge with options we use the underlying which moves more than the option so we use the proportion delta times the underlying to reduce the size of the hedge accordingly to delta hedge. If we hedge there is path dependency reflected in gamma as the stock may move around the option strike price and the optionality really pays off for the option if we are long the option and delta hedging it constantly as we will keep profiting from the hedge. If we are short a bunch of options we will be short gamma and be hurt by constant delta hedging added up if the stock moves around the strikes of the options.

In short, Black-Scholes is fundamentally a breakdown of the value of an option into the hedging profits or losses that result from using the option to maintain a bet on volatility while being neutral on stock direction. This is from the market maker’s perspective. From the speculator’s perspective it is just a bet but one that offers a trade that is path dependent and can be revised constantly though transaction costs prohibit constant trading at profitable levels, one does have to reform scenarios of options while holding them. From the hedger’s perspective an option gives interesting payoffs if you own a stock already but primarily an option if bought is a statement that you cannot forecast the path but you can still forecast volatility so you make a path dependent cash flow part of your calculation such that you become neutral to the path but your discount rate of the risky payoffs still takes into account volatility. We do assume volatility is an instantaneous measure of a stock’s trajectory as reflected by the partial differential equations which cast option prices as diffusing from a stock price that moves like a stochastic process known as Brownian motion.

The real question this all motivates is should you revise your holdings and trading positions constantly by watching the price to reflect optionality of different price moves? The answer is no. Even in active investing we don’t want to pay transaction costs for phantom benefits such as believing we can forecast the randomness that an option defines upon creation. As long as there is an options market on an underlying asset, you should not trade the underlying asset like it is an option on moves unless you are dealing with a situation like corporate bankruptcy where the price affects the underlying reality of what will change the stock price.