# Options

The value of options depends upon a number of parameters, as shown originally in the Black-Scholes model: the value of the underlying, the horizon to maturity, the volatility of the underlying, the risk-free interest rate. The model is in line with intuition. The option value increases with the underlying asset value, with its volatility, with the maturity, and decreases with the interest rate. Options sensitivities are known as the "Greek letters."

The sensitivity with respect to the underlying asset is the "delta" (8). The formula for the delta of stock options is simply A call (put) *— SAS.* The delta is low if the option is "out-of-the-money" (asset price below strike) because we do not get any money by exercising unless the asset value changes significantly. However, when exercise provides a positive payoff, 8 gets closer to 1 because the payoff of the option increases by the same amount as the underlying asset.

The sensitivity can be anywhere in the entire range between 0 and 1, and the highest change of the sensitivity occurs when underlying value is close to the strike price. The variation of 8 is the "convexity" of the option. Gamma (y) is the change of the delta when the underlying changes. It is the change of the slope of the curve representing the option value as a function of the underlying. The option is sensitive to the time-to-maturity because a longer horizon increases the chances that the stock moves above the strike price. The sensitivity with respect to residual maturity is theta (8). The Greek letter 8 measures the "time decay" of the option value. The higher the volatility of the underlying asset, the higher is the chance that the value moves above the strike during a given period. Hence, the option has also a positive sensitivity to the underlying asset volatility, which is the "Vega" (v). Since any payoff appears only in the future, today's value requires discounting, which implies a negative sensitivity to the level of the risk-free interest rate. "Rho" (p) is the change due to a variation of the risk-free rate.

Other options on interest rates or exchange rates follow similar principles. There are closed-form formulas for plain vanilla options. Accordingly, there are sensitivities similar to the delta of stock options and others.

# Forward Contracts and Interest Rate Swaps

For all instruments that can be replicated by a portfolio of other instruments, the sensitivities are derived from those of the components of the replicating portfolio. For the forward family of contracts, the replicating portfolio is static. This makes it easy to derive the sensitivities of these instruments.

All forward contracts can be replicated with static portfolios. Hence, they have a closed-form valuation formula. The sensitivities are derived either analytically or numerically using such formulae. For example, a forward exchange contract depends on three risk factors, the spot rate and the two interest rates in the two currencies. Similarly a forward interest rate contract is replicated from lending and borrowing the same amount in the same currency for differing maturities. It has a long and a short leg, each with a single final cash flow. The sensitivities of each leg are the durations of those cash flows with the appropriate signs.

Interest rate swaps (IRSs) can be seen as borrowing and lending the same amount in a single currency for the same maturity, with different interest rates. Under the "horizontal" view, the sensitivity with respect the fixed rate is that of the fixed rate leg. Note that the sensitivity of an IRS is entirely due to the fixed leg since the variable rate leg has zero duration.