The pricing of commodity contracts

Propuesta metodológica para la valoración de opciones sobre tasa de cambio USD-COP

“…Con el fin de resolver este problema, Garman-Kohlhagen desarrollan una propuesta de valoración de opciones sobre divisas, la cual relacionan a los trabajos obtenidos por Black (1976) en su modelo de valoración de opciones sobre commodities, y Samuelson-Merton (Merton, 1976) en su modelo de dividendos proporcionales.…”

Variance Risk Premia in Energy Commodities

“…Suppose at time t we have a range of options expiring at time T on a futures contract maturing at time T 1 , and let σ denote the Black (1976a) implied volatility of the option that is closest to at the money (ATM). In a Black (1976a) log-normal setting, for an option with strike X, moneyness defined as…”

“…The integrals are evaluated with "Simpson's rule" using 999 integration points for each integral. On a given day, options prices corresponding to the required strikes in the integration rule are obtained by first linearly interpolating between the available Black (1976a) implied volatilities and then converting from implied volatilities to prices. For strikes below the lowest available strike, we use the implied volatility at the lowest strike.…”

Investigating unusual changes in real-dollar exchange rate

“…Using analogous rationale, Black (1976) demonstrates that future prices present a stochastic behavior similar to stocks paying a continuous dividend yield and derives a pricing formula for options on currency futures. Hull (2003) shows that futures and spot options prices with similar features should be equally priced whenever future options mature at the same time as its underlying future contract.…”

“…In this work, we use Black (1976) model for European options on futures, which may be similar to Garman and Kohlhagen (1983) whenever future values are priced at their fair values. The implied volatilities are obtained by making market prices equal to the one obtained by using the option pricing formula:…”