BGM surfaces I
Suppose the current date is April 2011. On the triangular BGM volatility surface σ (t, T), 0 ≤ t ≤ T, highlight the areas of volatility upon which depend the prices of the following interest rate options.
(a) A 1yr-into-2yr swaption (that is, April 2012 option into a swap with start date April 2012 and end date April 2014).
(b) A 1yr-into-3yr swaption.
(c) A 2yr-into-1yr swaption.
(d) A 1yr-by-3yr cap.
(e) A one-year option into a swap with start date April 2013 and end date April 2014. (This is called a midcurve swaption.)
(f) A caplet on three-month libor LT, with T = 16 September 2013.
(g) A call option with exercise date 16 September 2013 on the underlying EDU3, the September 2013 Eurodollar contract.
(h) A midcurve call option on EDU3, that is, an option with exercise date 15 September 2011 on EDU3.
(i) A ‘1yr-into-1yr-1yr’ forward-starting swaption. The strike of the swaption is determined on April 2012 as the then forward swap rate (for a swap with dates April 2013 to April 2014), and the exercise date of the swaption is April 2013, into the swap with start date April 2013 and end date April 2014. (Derivatives of this form trade reasonably frequently in the US, and are sometimes referred to as ‘forward volatility’ trades.)
BGM surfaces I