ABSTRACT
The most important risk measures are deltas (sensitivity to movements of the underlying yield curve) and vegas (sensitivity to changes in the implied volatilities of the instruments to which the model is calibrated); in this chapter we show how to compute pathwise deltas along the lines of Glasserman and Zhao [44]. Denote by P (numeraire Nt expectation E) whichever measure, spot P0 or
terminal Pn, we are using. Let C0 be the present value of the discounted payoff stream ‘payoff C (·)’, and Dc = ∂∂K(0,Tc) denote partial differentiation with respect to the initial value K (0, Tc) of the forward K (t, Tc). Then
C0 = N0 Epayoff C ⇒ DcC0 = Dc µ N0
¶ giving
DcC0 = N0Dc
µ C0 N0
¶ + C0Dc lnN0 = N0DcEpayoff C + C0Dc lnN0,
with Dc lnN0 = 0 for P0 and Dc lnN0 = − δc
1 + δcK (0, Tc) for Pn.
