## 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.

## Retirement Planning For The Golden Years

If mutual funds seem boring to you, there are other higher risk investment opportunities in the form of stocks. I seriously recommend studying the market carefully and completely before making the leap into stock trading but this can be quite the short-term quick profit rush that you are looking for if you am willing to risk your retirement investment for the sake of increasing your net worth. If you do choose to invest in the stock market please take the time to learn the proper procedures, the risks, and the process before diving in. If you have a financial planner and you definitely should then he or she may prove to be an exceptional resource when it comes to the practice of 'playing' the stock market.

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