Pricing A Payer Swaption in a BDT Model

In this lesson, we used the BDT model to price a payer swaption.

We are pricing a 2-8 payer swaption.  The 2 and 8 mean that the option is an option to enter an 8 year swap in 2 years time, and since swaps have payments made in arrears, the payments would be made in years 3-10.  The "payer" part of the swaption simply means that if the option is exercised, the exerciser pays the fixed rate and receives the floating rate.  This option would have been extremely profitable if bought at the bottom of the recession, when interest rates were near 0%.   This means we will be using a 10-period lattice.

We will also assume that b(i) = b = 0.005 for all i.  Remember that b(i) is the volatility of the short-rate.  By assuming b=0.005 for all i, we are assuming a constant volatility.  We will change this assumption later.

We are going to assume a notional principal of $1 million.  Let S(2) denote the value of the swap at time t=2.  We can compute this price by starting at the value of the swap at time t=10 and discounting backward from t=10 to t=2.  Once we have the values at time t=2, we determine whether the option will be exercised at t=2 by the value of the swaption: max(0,S(2)).  We then discount these values back to find the swaption price at t=0.

Assuming that we have calibrated the zcb according to the steps above with a b =0.005, we find that the swaption price is $13,339.  If we then doubled b to b=0.010, we find a swaption price of $19,497.  This is about 50% higher than our original price.  This is a very significant difference in the swaption prices.  Swaption prices clearly depend on the volatility of the market.  This is apparent because increasing volatility means that there is increasing upside that the short rate will be higher, and therefore the swaption will be worth more.  However, with increasing volatility there is not increasing downside since the swaption is worth the max(0,S(2)) so if the short-rate is negative then the swaption will simply not be exercised.

Here we can see how important it is to calibrate the BDT model according to different observations of volatility.  We want the calibration to be "close" to the securities we want to price with the calibrated model.  For example a zcb does not depend much on volatility, while caplets and floorlets are much more dependent on the volatility in the model.  More of this will be discussed in later lessons.