Threefold duality and the Eightfold path of Option Trading
It is a truth universally acknowledged that the mere whiff of duality is catnip to the mathematician.
Given any asset X, like a stock, sold for a price P means there is a duality between buying and selling: one party buys X for P while another sells X for P.
Implicitly there is another duality: instead of interchanging the buy and sell actions, one can interchange the asset X and the price P, treating money as an asset and the asset as a medium of exchange.
A European call option at strike price S gives one the option to buy the underlying asset X at price S on the expiration date. [1]
There is a dual option—called a (European) put option that gives one the right [option] to sell the underlying on the expiration date.
Optionality is beautiful. Optionality is brilliant. The pursuit of optionality writ large is the great purpose of the higher limbic system. An option is optionality incarnate, offering limited downside yet unlimited upside potential.
There are much more classic financial instruments, but options represent a powerful abstraction layer on top of direct asset ownership.
Rk. Any security’s payoff profile can be theoretically replicated using only put and call options. This is known as the Options Replication Theorem or sometimes the Fundamental Theorem of Asset Pricing in its broader form.
Options themselves can be bought and sold.
There is now a fourfold of options:
Buy a Call option
Sell a Call option
Buy a Put option
Sell a Put option
A single transaction might offer all four in some combination at the same time.
Creating options
There is a third hidden duality in option trading—that between owning and creating an option. So far we have regarded as the existence of options as given—options are bought and sold and they can be exercised (extinguished) but where do options come from? But options have to be created.
Creating an option is called a ‘writing an option’. Writing an option is an inherent dangerous activity, only done by the largest, strongest, most experienced economical actors. Writing an option exposes its scribe to unlimited downside yet limited upside.
Why is this interesting?
Options are closely related to infraBayesianism and Imprecise Probability
Duality is always interesting.
There has been some speculation about a Logic of Capitalism—intrinstic Grammatica of Cash. Perhaps the eightfold duality of option trading is the skeleton key to its formulation.
^ There is a minor difference between so-called ‘American’ style and ‘European’ style options. The former allows exercising the options up to the expiry date while the latter only allows exercising at the date. I’m taking European options by default, mostly because the Black-Scholes equation[2] is more easily expressed for European options—they admit a closed form solutions while there is no closed form solution for American options in general.
^ Note that the Black-Scholes equation is a special case of the Hamilton-Jacobi-Bellman equation, the central equation in both physics and reinforcement learning & control theory.
Eightfold path of option trading
Threefold duality and the Eightfold path of Option Trading
It is a truth universally acknowledged that the mere whiff of duality is catnip to the mathematician.
Given any asset X, like a stock, sold for a price P means there is a duality between buying and selling: one party buys X for P while another sells X for P.
Implicitly there is another duality: instead of interchanging the buy and sell actions, one can interchange the asset X and the price P, treating money as an asset and the asset as a medium of exchange.
A European call option at strike price S gives one the option to buy the underlying asset X at price S on the expiration date. [1]
There is a dual option—called a (European) put option that gives one the right [option] to sell the underlying on the expiration date.
Optionality is beautiful. Optionality is brilliant. The pursuit of optionality writ large is the great purpose of the higher limbic system. An option is optionality incarnate, offering limited downside yet unlimited upside potential.
There are much more classic financial instruments, but options represent a powerful abstraction layer on top of direct asset ownership.
Rk. Any security’s payoff profile can be theoretically replicated using only put and call options. This is known as the Options Replication Theorem or sometimes the Fundamental Theorem of Asset Pricing in its broader form.
Options themselves can be bought and sold.
There is now a fourfold of options:
Buy a Call option
Sell a Call option
Buy a Put option
Sell a Put option
A single transaction might offer all four in some combination at the same time.
Creating options
There is a third hidden duality in option trading—that between owning and creating an option. So far we have regarded as the existence of options as given—options are bought and sold and they can be exercised (extinguished) but where do options come from? But options have to be created.
Creating an option is called a ‘writing an option’. Writing an option is an inherent dangerous activity, only done by the largest, strongest, most experienced economical actors. Writing an option exposes its scribe to unlimited downside yet limited upside.
Why is this interesting?
Options are closely related to infraBayesianism and Imprecise Probability
Duality is always interesting.
There has been some speculation about a Logic of Capitalism—intrinstic Grammatica of Cash. Perhaps the eightfold duality of option trading is the skeleton key to its formulation.
^ There is a minor difference between so-called ‘American’ style and ‘European’ style options. The former allows exercising the options up to the expiry date while the latter only allows exercising at the date. I’m taking European options by default, mostly because the Black-Scholes equation[2] is more easily expressed for European options—they admit a closed form solutions while there is no closed form solution for American options in general.
^ Note that the Black-Scholes equation is a special case of the Hamilton-Jacobi-Bellman equation, the central equation in both physics and reinforcement learning & control theory.