The Mathematics of Derivatives: Tools for Designing Numerical Algorithms

CHAPTER 4

Applications of Stochastic Calculus to Finance

4.1. RISK PREMIUM DERIVATION

The following is a very surprising result. For a single Wiener process driving a market, there is a universal risk premium (or price of risk). Consider two derivatives of this driver that trade in this market,

dS₁ = S₁µ₁ dt + S₁σ₁ dz(t),

dS₂ = S₂µ₂ dt + S₂σ₂ dz(t),

where the drift and volatilities may be arbitrary functions of time and their respective stocks. We may think of security 1 as an option and security 2 as a stock. Note that they are perfectly correlated because the Wiener process is the same for each security (admittedly, this is not a very realistic model of a market). There may be an infinite number of different securities from which to choose the two securities.

Construct a portfolio that is long one security and short Δ shares of the other:

Π = S₁ − ΔS₂.

LONG AND SHORT

Long is a finance term meaning to own an amount. Short means to effectively own a negative amount. The latter is realized in the capital markets by borrowing a fungible security and immediately selling it—and then buying it back at a later date and returning this under the borrow agreement. Fungible means securities for which this transaction is allowed.

Now consider the process for this portfolio:

dΠ = dS₁ − ΔdS₂

     = (µ₁S₁ − Δµ₂S₂) dt + (σ₁S₁ − Δσ₂S₂) dz(t).

Carefully note that the number of shares of security 2 that we are short—that is, Δ —may be chosen as a function of time and stock price to ensure that ...