Summary of South African Agricultural Commodity Derivatives

## Introduction Options are financial contracts that give the buyer the right, but not the obligation, to buy or sell an underlying asset at a specified price before or at expiration. Pricing an option involves separating its **intrinsic value** from its **time (extrinsic) value** and understanding how several variables affect the premium. The sensitivity of an option's premium to changes in these variables is commonly measured by the option "Greeks." > **Definition:** An option's **premium** is the price paid by the buyer to the seller for the rights conveyed by the option. ## Components of an Option Premium ### Intrinsic vs Time (Extrinsic) Value - **Intrinsic value**: For a call, max $$\text{Intrinsic} = \max\left(0, S - K\right)$$ For a put, $$\text{Intrinsic} = \max\left(0, K - S\right)$$ where $S$ is the underlying futures price and $K$ is the strike price. - **Time (extrinsic) value**: The portion of the premium above intrinsic value; driven by the probability the option will become more valuable before expiration. > **Definition:** **Moneyness** describes the relationship between $S$ and $K$: in-the-money (ITM), at-the-money (ATM), out-of-the-money (OTM). ### Time Value vs Moneyness (visual concept) - Time value is highest near ATM and falls as option becomes deep ITM or deep OTM. - ITM options have higher total premiums because of intrinsic value, while OTM options consist mainly of time value. ## The Greeks — Sensitivities That Drive Option Prices Greeks quantify how the option premium responds to small changes in key variables. ### Delta ($\Delta$) - **Meaning:** Measures the rate of change in the option premium for a small change in the underlying futures price. - For a call: $\Delta > 0$ (premium increases when $S$ rises). For a put: $\Delta < 0$ (premium increases when $S$ falls). - **Behavior with moneyness:** $|\Delta|$ increases as the option goes deeper ITM and decreases as it moves deeper OTM. > **Definition:** **Delta** is the partial derivative of the option price with respect to the underlying price: $\Delta = \partial C / \partial S$ for calls (or $\partial P / \partial S$ for puts). ### Gamma ($\Gamma$) - **Meaning:** Measures the rate of change of delta with respect to the underlying price: $\Gamma = \partial \Delta / \partial S$. - High gamma implies delta changes quickly as $S$ moves — important for hedging because it indicates how often a delta hedge must be adjusted. > **Definition:** **Gamma** captures convexity of the option price relative to $S$. ### Theta ($\Theta$) - **Meaning:** Measures the sensitivity of the option premium to the passage of time (time decay). - Options lose time value as expiration approaches; longer time to expiration increases time value. > **Definition:** **Theta** is the partial derivative of the option price with respect to time: $\Theta = \partial C / \partial t$ (typically negative for long option positions). ### Vega - **Meaning:** Measures sensitivity of the option premium to changes in volatility of the underlying: $\text{Vega} = \partial C / \partial \sigma$. - **Effect by moneyness:** ATM options are most sensitive to volatility changes; ITM options have intrinsic value less affected by volatility; OTM options depend heavily on volatility to become profitable. > **Definition:** **Vega** quantifies how much the premium changes for a given change in implied volatility $\sigma$. ### Rho ($\rho$) - **Meaning:** Measures sensitivity of the option premium to changes in the risk-free interest rate: $\rho = \partial C / \partial r$. - For options on futures or commodities, the practical effect of interest rates on premiums is often small, but conceptually higher interest rates reduce call premiums because the buyer faces greater opportunity cost of paying upfront. > **Definition:** **Rho** captures the impact of interest-rate changes on option prices. ## Other Determinants of Option Premiums 1. **Time to expiration**: More time raises