In the Q& Bank, part 3, there is a question wrt pricing a bond option where the yield volatility is given. To calculate the price volatility we take the initial yield (yo) and multiply it by the modified duration and the yield volatility. In the example the bond has a term of 4 years, with the option expiring at the end of the first year. Why is the price volatility calculated using the yield at the end of the year and the modified duration at the same point in time? I would have thought that initial yield refers to t=0. This really puzzles me!
Throwing a guess here (haven't looked at the question) - the option is on a 3-year bond, so you want the volatility and related stats of 3-year bonds, not of 4-year bonds. The bond is currently a 4-year bond, but the option is on the bond in a year's time, which will be a 3-year bond.
Thank you so much for replying. Making the analogy to share price options though (can I even do this? ) when we price the share option we are looking at current share price volatility (because anything else would be assumed) or implied vols currently, even though the option is actually on the share price in, say, a year's time. Thus in the case of a bond option- should we not use current yields to calculate CURRENT price volatilities. Although, it might then be that in the case of bonds, we can calculate yields at the time of exercise and hence it is more accurate to use volatility then? Hmmmm. What do you think?
The problem with the share analogy is that the share's properties don't change. So if you spoke about the volatility of the share price or the volatility of the forward share price, you're essentially talking about the same thing. In my mind, that is not the case with a bond. The true underlying is actually the forward bond, which is a 3-year bond in your case. So if you wanted to estimate empirically the volatility of the option you're talking about - a currently 4-year bond with expiration in 1 years' time - you would need to collate information about the prices over time of 3-year (forward) bonds (or yields). Think about it this way - you have an option on a 3-year bond and the option has a term of 1 year. The underlying is a 3-year bond, not a 4-year bond. And since the yield curve is not flat, and neither is the term-structure of yield volatilities, you cannot equate the 3-year yield volatility to the 4-year yield volatility. Shorter term rates in general are more volatile than longer term rates. It makes sense that you would need to use the duration and vols of the underlying at option maturity as this is where the option is based. Thinking of an example -> consider a boundary case -> so consider a call option on a 4-year zero coupon bond with redemption at 100, with strike at 98, and the option expires in 4-years (I'm specifically creating a boundary case). So the option expires when the ZCB expires. According to what you were saying, you would estimate volatilities on a 4-year ZCB, and then proceed to apply whatever option pricing formula. But we know with 100% certainty that the bond will be worth 100 when the option expires and hence the payout is a guaranteed 100-98 = 2. So in fact, we don't even have an option here. It's a guaranteed future payment. As I was getting at above, the true underlying is the forward bond which in this case is a 0-year bond. And we know that a 0-year bond has 0% volatility, which matches the case. Hope this helps?
