Consider a commodity with constant volatility σand an expected growth rate that is a function so

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Consider a commodity with constant volatility σand an...

Consider a commodity with constant volatility $\sigma$ and an expected growth rate that is a function solely of time. Show that, in the traditional risk-neutral world, $$ \ln S_{T} \sim \phi\left[\left(\ln F(T)-\frac{1}{2} \sigma^{2} T, \sigma \sqrt{T}\right]\right. $$ where $S_{T}$ is the value of the commodity at time $T$ and $F(t)$ is the futures price at time 0 for a contract maturing at time $t$.

Consider a commodity with constant volatility $\sigma$ and an expected growth rate that is a function solely of time. Show that, in the traditional risk-neutral world,
$$
\ln S_{T} \sim \phi\left[\left(\ln F(T)-\frac{1}{2} \sigma^{2} T, \sigma \sqrt{T}\right]\right.
$$
where $S_{T}$ is the value of the commodity at time $T$ and $F(t)$ is the futures price at time 0 for a contract maturing at time $t$.

Key Concepts

Risk-Neutral Valuation

This concept refers to a method in financial mathematics where asset prices are calculated under a probability measure that assumes investors are indifferent to risk. Instead of using the actual expected return of an asset, the risk-neutral measure uses the risk-free rate as the drift. This simplifies the valuation, particularly in derivative pricing, as it allows future cash flows to be discounted at the risk-free rate, ensuring arbitrage-free prices in the market.

Log-Normal Distribution

In a risk-neutral framework, prices of underlying assets, such as commodities, are often modeled as following a log-normal distribution. This means that the natural logarithm of the asset price is normally distributed. The property is crucial because it implies that asset prices can never be negative and it also simplifies the analytical derivation of option prices and other financial derivatives.

Futures Pricing

Futures pricing is the process of determining the fair price of a contract that obligates the buyer to purchase, and the seller to sell, an asset at a predetermined date in the future at a specified price. Under risk-neutral pricing, the futures price is related to the expected future spot price adjusted for the cost of carry, and it serves as a key input in deriving the distribution of the logarithm of the asset value in the risk-neutral world.

Drift Adjustment with Volatility

In modeling the dynamics of asset prices, especially under the assumption of constant volatility, the drift term in the log-normal model is adjusted to account for volatility. Typically, when moving to a risk-neutral measure, the drift is modified by subtracting half of the variance (i.e., ½?²) from the natural logarithm of the futures price. This adjustment ensures that the expected return of the asset aligns with the risk-free rate and preserves the mathematical properties of the log-normal distribution.

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