ABSTRACT
The vegas of an option, that is, its sensitivity to changes in the implied volatilities of the instruments to which the model is calibrated, are as important a risk measure as the option’s deltas. But because in the shifted version of BGM, the shift a (T ) and volatility ξ (t, T ) functions are jointly fitted to swaption values during the volatility part of the calibration, the vega hedge must comprise both volatility and shift components. In this chapter we show, along the lines of Pelsser et al [89], how to compute vegas (including shift hedges) by perturbing the underlying BGM shift a (T ) and volatility ξ (t, T ) functions in such a way that only swaptions pSwpn (t, κ, Tj , TN ) of a particular maturity Tj (but different strikes) are affected. The corresponding changes in value of the exotic option that we wish to hedge, then yield the required hedge parameters. Denote by P (numeraire Nt expectation E) whichever of the spot P0 or
terminal Pn measures we are using, and let C0 be the present value of the discounted payoff stream ‘payoff C (·)’ comprising our exotic option. From Chapter-7, two swaptions say pSwpn (t, κ1, Tj , TN ) and pSwpn (t, κ2, Tj , TN ) at different strikes κ1 and κ2 suffice to fix first the shift and then the zeta at a particular exercise time Tj , so we assume two such swaptions will figure in the hedge. Note that these swaptions have exactly the same implied volatility and shift, differences in their values arise only from the strikes. Perturbing by a small amount ∆θ the shift α (Tj , TN ) of just the jth swap-
tion pSwpn (t, κ, Tj , TN ) changes its value by ∆θ pSwpn (0, κ, Tj, TN ) where
α (Tj , TN ) → (1 +∆θ)α (Tj , TN ) ⇒
∆θ pSwpn (0, κ, Tj , TN ) = ½ pSwpn (0, κ, (1 +∆θ)α (Tj , TN ))
−pSwpn (0, κ, α (Tj , TN ))
¾ =
∂ ∂α
pSwpn (0, κ, α (Tj , TN )) × α (Tj , TN ) ∆θ.
