A BRIEF NOTE ON THE RELATIONSHIP BETWEEN VOLATILITY AND TIME

It is common in finance for the core essence of an idea to be hidden behind complex language and the liberal use of mathematics. One of the objectives of the derivatives section of the Financial Pipeline is to make the revolution in financial engineering accessible to everyone so that people can make informed decisions about the opportunities derivative instruments present and the pitfalls they create.

In previous articles we have compared products with linear payoff profiles to instruments with non-linear payoff profiles and we learned that only non-linear products have time value. We talked about the true meaning of time value to a derivatives professional and the question he must ask himself when evaluating a particular structure.

"If I buy this structure, will I be able to make more money trading the underlying cash instrument than I will pay in time decay over the life of the instrument?" (Conversely, he could have asked himself, "If I sell this product, will the losses I sustain trading the underlying cash instrument against this structure be less than the premium I am paid at inception?").

Let us restrict ourselves to the question facing the prospective buyer of this non-linear instrument. Think of it as a call option on a particular stock, say Dollar.com. For the purpose of argument, assume that the current stock price is $100, our option's notional amount is 100 shares, the options matures in 3 months and the strike price is $100.

In evaluating this central question of value, there are two important factors that stand out: volatility and time. We will consider them one at a time.

The higher the implied volatility of this product, the higher the premium will be and the more difficult it will be to pay for the option. However, if we expect actual volatility to be higher than the implied volatility, it may pay for us to own this option and to trade GE stock against it.

The key here is our expectation of what volatility will actually turn out to be as it relates to the implied volatility.

For example, if implied volatility on our Dollar.com call is 15% (on an annualized basis) in the marketplace but we think that actual volatility will be closer to 25%, we should buy the option. We will make more money delta-hedging the option (or rebalancing the delta in response to market movements) than we will pay in premium. See previous articles for an explanation of delta-hedging.

Now, what does it mean if implied volatility for our option in the marketplace jumped from 15% to 25% immediately after we bought our option. This might happen if there was an unexpected announcement from a takeover company, Savage LBO LLC, that they were going to make an unfriendly bid for Dollar.com. The outcome is uncertain.

First, we will see the premium jump higher for the option we own. We will own something that has increased substantially in value. Because the option is at-the-money spot (i.e. its strike is equal to the current spot rate), this effect is at a maximum. Recall that the change in the option's value due to a change in implied volatility, all other things being equal, is its vega.

Second, we can see that the 3 month call we bought at an implied volatility of 15% is now worth what a 6 month call with the same strike before volatilities shot up in response to the announcement. Therefore, the move higher in implied volatilities is like an extension of our 3 month option into a 6 month option. It is as if we got 3 months for free.

Time value is the measure of how much money we should make if the stock turns out to be as volatile as the implied volatility says it should be. Changes in implied volatility necessarily mean changes in time value (and therefore premiums). We could make the same absolute amount of money delta hedging a short-dated option on a very volatile underlying as we could on a long-dated option on a calm underlying.