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Option Portfolios

Equities that have a very straightforward exposure to idiosyncratic and systematic risk. Options, on the other hand, have exposure to not only the underlying asset, but also interest rates, time, and volatility. These exposures are inputs to the Black-Scholes option pricing model(see Deriving the Black-Scholes Model). Since these inputs affect the value of the option in question, the partial derivative of the function can tell us how the option value changes when one of these exposures changes holding the others constant. Let’s first discuss all of the partial derivatives of this option pricing model, then code an algorithmic hedging system using Python.

The Greeks

Using a Taylor series expansion we can derive all of the greeks. The greeks tell us how we can expect an option or portfolio of options to change when a change occurs in one or more of the option exposures. Something important to note is that all first-order approximations are linear, and the option pricing function is non-linear. This means the more the underlying parameter deviates from the initial partial-derivative calculation the less accurate it will be. This is why the greeks are updated, generally, in real-time, so we can constantly have a new set of expectations for option or portfolio value when _something _changes (for more see Option Greeks in Python).

Note: In the following scenarios we are generally considering calls and puts, some more advanced spreads may be long or short with different exposure parity.

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