OPTION VALUATION FOR INTEREST RATE CAPLETS: RATE VOL CORRELATION AND FAIR SKEW

Abstract

The Black Scholes model naively assumes that volatility is constant across strikes. To correct for this, we make volatility a function. To illustrate this with numbers, we recognize that Vcall = BS(f,K, , t) where f is the forward rate, K is the strike, is the volatility function, and t is the time to maturity. Differentiating with respect to forward rate now yields the revised delta risk. By the chain rule, we get: @V @f @BS @f + @BS @ @ @f = BS + BS @ @f As you can see, we can calculate the Black Scholes and Black Scholes , so the only term we are missing is the @ @f . The goal of this paper is two-fold. First we want to explore if there is actually a correlation between interest rates and volatility and to quantify the correlation. This correlation can be used for bookkeeping purposes and to determine the amount of hedging correction needed. We hypothesize that there is a negative correlation between rate move and volatility move. This makes sense theoretically since if rates were 0.01%, a rate change of 1bp would be a 100% increase. In this scenario, volatility would be high. However, if rates were 20%, a 1bp rate change would have little effect, so the volatility would be low. If this correlation exists, Black delta would be insufficient for trading purposes since it would overestimate how much we need to hedge for caplets (explained in more detail in the next section) and underestimate how much we need to hedge for floorlets. For example, let’s say we are long a high strike caplet and the rate rises. Here the delta will increase due to a favorable change in spot. If volatility remained the same, the Black delta would give an accurate estimate of our delta. However, we theorize that a downward volatility move often accompanies this favorable rate shift. This fall in volatility is unfavorable for us, and the delta should actually be lower than the Black delta. abstract 6 In the second part of the paper, we extend these results further. By using the rate/vol correlation obtained in the first part, we can backsolve for a variable we call . We use and introduce the historical vol of vol for our Monte Carlo simulation to get the fair skew of the implied volatility across different strikes. This ultimately allows us to see some discrepancies in the market possibly due to supply and demand effect.