المدربThe lognormal volatility of a rates option is the volatility at which Black’s formula, applied to the forward rate, reproduces its price; it is a relative volatility, a percentage of the rate, and exists only for positive forward and strike.
أمثلة
Example 4.8 (Five years into ten)
On chapter 2’s curves the ten-year swap starting in five years has a forward par rate of 3.098% against six-month Euribor and a (discount-curve) annuity of 7.740. At 80 basis points of normal volatility (a lognormal volatility of 26.2%), the at-the-money physical payer on EUR 100 million is worth EUR 5 525 150. The traditional cash formula, option, gives EUR 5 410 452, 2.1% less: , because the cash annuity discounts at the Euribor swap rate, above the €STR rates of the collateralised annuity. The par-yield payoff is also not a function of the annuity measure’s numeraire, so the formula is itself an approximation; the collateralised cash price removes both problems by settling at the physical value.