## Brownian Motion Form

Suppose a commodity—let's take copper—has a price governed by geometric Brownian motion as dS = ¡iSdt + oSd~ (13.25)

where z is a standard Wiener process. If an investor buys copper and holds it, there is a proportional storage cost that is paid at the rate of cS per unit time. If at any moment / the investor holds copper with total value W(t), the holding cost can be paid at the rate of cW(/)d/ by selling copper at this rate The process for the value of copper holdings is therefore dW = pWdt + aW d~ — cW di

where IY(0) = 5(0) Equation (1.3.26) can now be regarded as that governing the value of a security with the holding costs accounted for. We might term W the value of net copper, since it is the net value after holding costs,

If we consider an investment opportunity that involves copper, such as an option on copper futures or a real option on a project that involves copper as a commodity (such as a copper mining operation or an electrical equipment project), we can value this opportunity by risk-neutral techniques We change the process for net copper to risk-neutral form since it is net copper that can be used in constructing a replication of other securities. Specifically, in a risk-neutral setting with interest rate /, net copper is governed by dW = i Welt + aW dz (1.3.27)

where £ is a standard Wiener process

The appropriate transformation embodied in the foregoing is that from (13.26) to (13 27), which boils down to the change n~~c r. This is equivalent to ¡i —► / +c Hence the original copper price in a risk-neutral world satisfies dS = (/■ + c)S d/ + ct5 dz. (13.28)

This is the equation that should be used for risk-neutral valuation of copper-related investments.

## Lessons From The Intelligent Investor

If you're like a lot of people watching the recession unfold, you have likely started to look at your finances under a microscope. Perhaps you have started saving the annual savings rate by people has started to recover a bit.

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