## Wednesday, March 20, 2013

### The Correct Way to Spot Relative Value in Volatility Surface

While looking at the swaption grid of at-the-money volatility, and comparing them for relative richness, failure to take in to account the current yield curve and volatility surface term structure can be very mis-leading. This is especially true given the current steep slopes, where this simple ommission can be disastrous. For example if 3m2y options look very cheap compared to 3m10y, it does not mean buying the former and selling the later is profitable, even if the market does not move against us. The 3m options will become 2m options in 1 months time. And if 2m2y is even more cheaper to 2m10y, compared to the cheapness of 3m2y to 3m10y, then this position can bleed very seriously. Also as the time decays, the strike of the options changes from ATMF and slides along the skew, and this may also distort the trade dynamics. Therefore to compare apple to apple, it is utmost important to convert the ATM vol numbers to an equivalent number taking in account this time decay and skew slide. Below is a robust method for that

The concept of the method is quite stratight forward: that the total PnL loss from a long straddle position arising out of time decay in the real world (chaging volatility term structure and yield curve term structure) should be equivalent to the pure time decay loss in a world where volatility and yield curve term structure are flat. This method then converts each point to a equivalent number in a locally flat vol and curve scenario, which can then be compared directly - bps to bps - to any other point.

One way to get the equivalent volatility is by setting variance loss on both this scenario equal, as below

The same can be obtained from the basic options equation. Consider the PnL for a delta-hedged equation, ignoring cross-greeks. Then the pnl from a equivalent vol position should be equal to the same on the surface
Expressing all the greeks in terms of Gamma, assuming low enough cost of carry
Substituting, that gives
Which is exactly the expression above