Description: This article shows a different form of the Black-Scholes formula for European calls and puts under risk-neutral assumptions, that permits a comfortable verbal interpretation. The derivation of this alternative form appears at the end of the article.

Description: This article explores the effectiveness of hedging delta and rho of equity options. This provides insight into the frequency and severity of losses due to not hedging volatility risk or other higher order risks. 10yr-equity put option strategies were chosen to represent the risk of hedging guaranteed benefits attached to variable annuities, while 1-yr put and call strategies were modeled to investigate the risk of hedging equity index interest credited to fixed indexed annuities. In both cases, the value of the option was compared to the value of delta/rho hedges in the tail of both actual historical and simulated scenarios for equity returns and volatilities. The historical path of interest rates and equities was generally used to highlight the hedge impact for different implied volatility assumptions in each example.

Description: By necessity, attributions of financial processes require approximations. Tools used for approximations include Aumann-Shapley, time step, step-through, average step-through, partial derivative, and average partial derivative. Significantly non-linear paths can cause problems during attribution. Time (theta) and market factors must be handled concurrently. Typically contractual factors can be handled separately. Average Step-Through, Aumann-Shapley and Aumann-Shapley-3 points perform the best given the same time period. Attribution should be done with fairly small time steps as opposed to using a lot of points in the Aumann-Shapley calculation, but using Aumann-Shapley-3 points as a last step has some advantages.