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.
Path dependency is interesting for an option as standard ways of discounting cash flows for an option would have to reflect in an interest rate that changes with time and the path of the underlying. Normally we only have to account for risk, discounting riskier cash flows at a higher interest rate. But this is not possible because the various states of the world in which the option pays off are quite different from one another and while one may think we can take an expectation across all the states and discount at an appropriate interest rate, the expectation is of a random process and thus the interest rate is time varying because as time goes on the expectation could be wildly different from where the process started from, where the underlying started from, and is consequently riskier and moreover the interest rate is path dependent because if the stock price moves a lot the payoff becomes riskier because the stock price moving a lot reflects probably a change in volatility of the underlying. I’m using a non rigorous explanation because I believe the essence of understanding Black-Scholes is to understand why the model is used instead of intuition which would treat an option simply as a levered bet on a stock. The answer is we don’t know where the stock will go so every option contains a certain optionality value which derives from the path a stock will take so the value of an option is not necessarily path dependent but your outcome from owning an option certainly is path dependent. Compare to owning a stock with implied optionality like a distressed company, you will certainly care about the journey that the distressed company takes in recovering from near bankruptcy as that is what you paid for, the optionality to endure that journey while if you just wanted to bet on the company you could go long or short an etf containing the company that would dilute the impact of the implied optionality and make it more a directional bet on prospects. Put in plain speech, when I buy an option it is like I bring an umbrella. I care if it rains. If I just wanted to bet on whether it rains, I only care if it rains not the anticipation of the rain which could allow me to sell my umbrella for a better price in a downpour that may or may not last for a long time. In short we can’t value an option in traditional methods of evaluating cash flow because that miscasts the nature of randomness which tells you that the option is exactly replicable by a stock and underlying that is bought and sold constantly and is self funded so there is no risk in the option compared to pursuing that strategy in replicating the option by buying and selling the stock. In short, an option has risky payoffs but these risky payoffs are identical to that of a stock portfolio constructed in a dynamic fashion and so we evaluate the value of the option with regards to the intrinsic random characteristics of the underlying stock portfolio which implies the appropriate discount rate of the option payoffs which is in fact the interest rate if the option is being evaluated in a risk neutral framework as it should because there is any amount of risk but we don’t care about the risk of the option because it is compensated for in the price of the option insofar as volatility proxies for risk because the replicating portfolio only sees the volatility of a stock as a risk to be expensively hedged for in reconstructing the option. This is all non rigorous and it’s not meant to help your intuition of option Greeks for example but to show you the Black-Scholes model is like a candle and the burning wick is the option Greeks. Light is cast on how to price options relative to one another and the stock. A model is not just right in aggregate and averages but also for each individual option it is the best guess. The basic line we learn from the candle of Black-Scholes is it is about volatility. One might have thought in classical logical intuition that an option is a directional levered bet with a shield or protection against making a mistake and the optionality to act upon the bet with optimal timing. That is incorrect. An option is a bet primarily on volatility as reflected in the path of a stock because an option is primarily a reflection of path dependent risk. I will illustrate with an example: suppose I buy some shares of AT&T and hope for a spike up after a dividend increase after which I would sell the stock for a hefty annualized return on what I call event-driven trading. Have I in any sense bought an option by buying the stock at this time? No. The risk is not path dependent. At any given time we are hoping for an event to happen and the risk is that the event doesn’t happen. In corporate bankruptcy there is implied optionality because the risk is path dependent: how the stock does influences whether real bankruptcy proceedings can be negotiated in a way that benefits or hurts the stock.
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.
Suppose I want to ask a girl out. If I don’t know if she will keep interacting with me I can’t predict the path and I buy an option by being extra polite and graciously doing whatever she asks me to do because I can always drop her like a hot potato if she has certain irascible habits and all I afforded was my patience by paying her the option of saying I was her suitor. If I know she will keep interacting with me because we share the same friends circle for whatever reason or she is dependent on me at work, I can act like a flirtatious buffoon and still hold her regard as it is rational to bluff if you hold nothing but some people are also rational to concede gracefully if they hold nothing. Everyone has their own wisdom. In other words, because I can predict the path of where she is going I don’t need to buy an option I can just speculate on her intentions. If an options market exists on the girl, that is she has a career for example and so derivatives of the girl exist in her personalities at work and in other business settings, I can rest assured I don’t need to predict her path, I never have to say I’m her suitor unless I want to buy an option and most importantly if I start joking with her, I never have to worry about leaving the flat price market and have to think about optionality because I would be overpaying to bet on randomness when her career as an options market already exists for me to make path independent bets such as worrying about whether she likes my salary. The flat price market is simply whether I can make her happy while the optionality market is how she will respond if I do this or that, but we know if she has a career or an options market based on her, I don’t need to bet on her path I always can bet on her volatility and so if I do bet on her path I don’t need to worry about optionality because the options market has converted all optionality to volatility bets and all that’s left for me is to bet on her path on the flat market of making her happy or barring that to return to the options market and bet on her volatility by thrilling her with a sword dance of mathematics. For math is not logic, it is closer to physics. But before we understand math let us remember the options market is for trading like stocks, it is not for forming mindless strategies which wouldn’t work there as they wouldn’t work in stocks because in the options market you too trade against someone and just because an option is harder to value doesn’t mean they won’t know they are getting a good deal if you are always wrong about direction. Part of path is direction. Part of volatility is path. So when we bet on volatility and not path with options we are still betting on the part of the path which can be summarized by differential equations and thus volatility in a Wiener process and that is if you say to someone on New Year’s Eve in Boston he drank your soda after you buy him a drink, he probably told no one, as the path is remarkable from Boston to Chicago but volatility is when the same night at a Japanese restaurant making jokes about shared values between China and Japan, you see Chicago style. To new beginnings and old friends as a stock is like a new beginning and an option is like an old friend, the more volatility the more interesting your old friend but it’s more about how the volatility changes, it’s more about his sense of humor or her accent.