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Risk-Neutral Portfolio Management
Applications of the fundamental partial differential equation
You’re a portfolio manager and you have a portfolio with a quantity of q financial derivative contracts and p shares of the underlying asset. Mathematically, the portfolio at any point in time can be described by the following equation.
Where…
- π — Portfolio value
- p — Number of shares of the underlying asset
- S — Price of the underlying asset at t
- q — Number of financial derivative contracts
- D — Price of the financial derivative contract
Your team is preparing research to assist in determining your net position in the underlying asset (bullish/bearish). Unfortunately, their research has been delayed, and you have exposure to changes in the underlying asset. Fortunately, this problem can be framed as a partial differential equation, and a solution in terms of a portfolio that mitigates directional risk can be found. As it turns out, every pricing model for every financial derivative must satisfy this partial differential equation. Let’s get started modelling this equation.
Geometric Brownian Motion
I don’t want to prolong the explanation for Geometric Brownian motion so I will offer a concise mathematical definition and additional resources. The best way to mathematically model the underlying asset is by Geometric Brownian motion.
Where…
- S — Price of the underlying asset
- μ — Drift term
- dt — Change in time
- σ — Volatility term
- dW — Change in Brownian motion
This equation effectively models the underlying asset, accounting for the expected returns (μdt) and the random shocks to those returns (σdW). It’s okay of the equation doesn’t resonate with you, we are just going to use it to identify the source of randomness in our portfolio later on. It also may be more intuitive to look at it graphically…
