\(\)It’s well-known that vanilla option valuation does not depend on the term structure of volatility and interest rates. This means that the price depends only on the *average* volatility and *average* interest rate between the valuation date and maturity, *not* on how those quantities are distributed within the interval.

A way to visualize this and understand it intuitively is as follows. Consider a large set of paths of the underlying which have been generated by a Monte Carlo routine. The value of the option is the average over all paths of the quantity \(Max(S(T) – K, 0)\). Now, imagine stretching and compressing the paths in different places as if they were plasticine, corresponding to concentrating volatility more in some places than others. It’s as if the underlying were moving faster in some regions, and slower in others, yet \(S(T)\) remains the same for each path. Thus, the price remains the same.

Interest rates affect the underlying’s drift term. Yet, as for volatility, \(S(T)\) depends only on the total proportional increase that the drift term bestows on the underlying, not on where in the interval this increase occurs.

What about barrier options? There are a few cases to consider.

First, we consider the case of a full barrier option. This means that the barrier is monitored for the full length of the deal from the valuation date to maturity, as opposed to only being monitored for a subset of it. We also assume that the underlying’s drift term is zero (this typically occurs when interest rates are zero, for example). In this case, valuation is actually still independent of volatility term structure. This can be understood by realizing that stretching or compressing the paths in different places does not change whether they breach the barrier, but only *when* they breach the barrier. Thus whether a given path has knocked-in or knocked-out remains unchanged.

Next, we consider the case of a partial or window barrier option. This means that the barrier is only monitored some of the time, with the monitoring period starting after the valuation date and/or ending before maturity. We still assume that the underlying drift is zero. As mentioned above, while a different volatility term structure does not change whether a path breaches the barrier, it does change when it does. Thus, it can affect whether the path breaches the barrier inside the monitoring window or outside, thus changing whether the path knocks in/out or not. Thus, for partial and window barrier options, valuation is *not* independent of volatility term structure.

Finally, let’s consider the case of a non-zero drift term. In this case, valuation is not independent of volatility or interest rate term structure regardless of whether it is a full barrier option or a partial/window barrier option. To understand this, consider that the movements in the underlying due to volatility are proportional to the current underlying price. If the underlying is monotonically drifting upwards throughout the monitoring window, then volatility applied early on will cause smaller changes in the underlying than if they were applied towards the end of the monitoring window. Thus, if the volatility term structure concentrates volatility towards the end of the interval after the underlying has had time to drift upwards, they are more likely to cause the underlying to rise above an upper barrier. Thus, volatility term structure and interest rate term structure affect knock out / knock in probability and thus affect valuation.