What is Ultima

Ultima is the rate at which the聽vomma聽of an option聽reacts to volatility in the underlying asset. It is a third order derivative of the option value with respect to volatility. Ultima is a derivative of vomma, which is a derivative of vega. Ultima is part of the group of measures known as the "Greeks"聽which are used in option pricing and analysis. Other measures include delta, gamma, rho, and theta.聽

Breaking Down Ultima

Ultima is useful to investors who are making options trades and taking the vomma and vega into consideration, especially when implementing exotic options which may change format prior to expiry. Implied volatility, and its derivatives, are some of the inputs utilized in the Black-Scholes model. Other pricing models include the binomial聽pricing聽model, and put/call parity.

Understanding Ultima

To understand ultima, it is helpful to back up to vega. Vega is the rate of change between the underlying asset's implied volatility聽and the option's value. A vega聽of 0.2 means the price of the option will move $0.20 for each 1% change in implied volatility.

Vomma is the derivative of vega. Vomma聽tells a trader if vega will increase or decrease in relation to implied volatility. A positive vomma聽means if volatility increases the option's vega聽will increase, and if volatility falls then vega will decrease. A negative vomma聽indicates the opposite.聽

Ultima is thus a measure of how vomma will change as volatility changes.聽

Ultima Calculation

The formula for ultima is shown in the formula below.

Ultima=vomma=3V3\begin{aligned} \text{Ultima} = \frac{ \partial \text{vomma} }{\partial \sigma} = \frac{\partial ^ 3V}{\partial\sigma ^ 3} \end{aligned}Ultima=vomma=33V

The calculation is looking at the rate of change of vomma聽over the rate of change in volatility.

Assume that a call option has a vomma of three. This means vega聽will increase by three for each 1% change in implied volatility.聽

This is not static a static figure, though. As volatility rises or falls, the vega聽figure will also rise or fall. This is vomma. If vega is three but then increases to four on the next percentage rise in volatility, vomma is one.聽

Recall that vomma聽can be positive or negative. A positive figure will increase/decrease聽vega if volatility rises/falls. A negative figure will decrease/increase vega聽if volatility rises/falls.

As ultima聽relates to vomma, traders who own options look for high or increasing vomma, while those who are short聽will look for negative and decreasing vomma. Ultima can help determine聽if vomma is increasing or decreasing聽as volatility changes.聽