What is Ultima

Ultima is the rate at which the vomma of an option 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" which are used in option pricing and analysis. Other measures include delta, gamma, rho, and theta

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

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

Ultima Calculation

The formula for ultima is shown in the formula below.

Ultima=vommaσ=3Vσ3\begin{aligned} \text{Ultima} = \frac{ \partial \text{vomma} }{\partial \sigma} = \frac{\partial ^ 3V}{\partial\sigma ^ 3} \end{aligned}Ultima=σvomma=σ33V

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

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

This is not static a static figure, though. As volatility rises or falls, the vega 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. 

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

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