Given a dateT, we define therisky payer swaptionwith expiryTas the single claim with maturityT and payoff 1{D>T}CT(FT −K)+, whereF is a forward swap rate andCits natural numeraire76,Kis a constant (called the strike) and D is the time of default. Note that the effective date of the underlying swap (F, C) must be greater than the expiry date T, but need not be equal to it.
A risky payer swaption is equivalent to the right to enter into a forward payer swap, provided no default has occurred by the time of the expiry. GivenT < T0, we calldelayed risky payer swaptionwith observation dateT and expiryT0, the single claim with maturityT0 and payoff 1{D>T}CT0(FT0−K)+. A delayed risky swaption is equivalent to the right to enter into a forward payer swap on the expiry dateT0, provided no default has occurred by the time of the observation dateT. Note that a long position in a delayed risky swaption with observation date T and expiry T0, together with a short position in a risky swaption with expiryT0, is equivalent to the right to enter into a forward payer swap on the expiry dateT0provided defaulthasoccurred, in the time interval ]T, T0]. Risky swaptions and delayed risky swaptions will be seen to play an important role in the next section, where we study the impact of possible default, on the pricing of an interest rate swap transaction.
In this section, we concentrate on the question of non-arbitrage pricing of risky swaptions and delayed risky swaptions. More generally, we consider the single claim with maturity T and payoffBTCTg(FT), where g is an arbitrary payoff function, andBis the collapsing process representing the price process of
75Strictly speaking, its continuous partY =X∗/V.
76i.e. the underlying annuity, delta, pv01 , pvbp. . .
the risky zero with maturityT. The case of a risky payer swaption corresponds tog(x) = (x−K)+, whereas a delayed risky payer swaption is clearly equivalent tog(x) being the undiscounted price at time T of a payer swaption with strike K and expiry T0, given an underlying forward swap rate ofx.77
We denoteV the price process of the default-free zero with maturityT. The four processesC,CF,V andBare assumed to be tradable. An investor entering into a strategyθandψ(up to timeD) relative toCF andCrespectively, funding his position in CF and C with V, having chosen the collapsing process B as numeraire, has a wealth processπsatisfying the SDE:
dπt=θtd(CF)Dt +ψtdCtD−θtCtFt+ψtCt
Vt
dVtD+πt−
Bt∗dBt (51) whereB∗ is the continuous part ofB. The associated terminal wealth is:78
πT =BT
π0
B0
+ Z T
0
θˆtdXt+ Z T
0
ψˆtdYt
!
(52) where the continuous semi-martingalesX,Y are defined asX =C0F e−[C0F,P] andY =C0e−[C0,P], the processes ˆθand ˆψare defined as ˆθ= (θe[C0F,P])/P and ψˆ= (ψe[C0,P])/P, the process C0 =C/V is the forward annuity of the under- lying swap, andP =B∗/V is the continuous part of the survival probability processB/V. A sufficient condition for replication is:
π0
B0
+ Z T
0
θˆtdXt+ Z T
0
ψˆtdYt=CTg(FT) (53) SinceVT = 1 we haveCT =YTe[C0,P]T, and from [C0F, P]T = [C0, P]T+ [F, P]T
we see thatFT = (XT/YT)e[F,P]T. Hence, provided both brackets [C0, P] and [F, P] are assumed deterministic, the quantity CTg(FT) can be viewed as a function of the history of X and Y between time 0 and T, and the martin- gale representation theorem79 will be successfully applied, for a wide range of distributional assumptions on C0 and F. When this is the case, our claim is replicable, and we have:
π0=B0EQ0[CTg(FT)] (54)
77e.g.g(x) =xN(d)−KN(d−u) whered= (ln(x/K)+u2/2)/u, anduis the non-annualized total volatility ofF, betweenT andT0. More generally,g(x) =EQ[(FT0−K)+|FT=x] where F is a martingale underQ.
78This is yet another SDE! However, the fact that equation (52) is indeed the terminal wealth associated with (51) can be seen by applying (22) separately to:
dπt1=θtd(CF)Dt −θtCtFt
Vt
dVtD+π1t− Bt∗dBt
and:
dπt2=ψtdCtD−ψtCt
Vt
dVtD+π2t− Bt∗dBt
whereπ=π1+π2, and arbitraryπ01andπ02such thatπ0=π10+π20.
79Strictly speaking, a two-dimensional version of it.
where Q0 is a measure under which the semi-martingales C0F e−[C0F,P] and C0e−[C0,P] are in fact martingales.
It may appear from (54) that our objective of pricing the claim with payoff BTCTg(FT) has been achieved. However, although it is probably fair to say that a lot of work has been done (in particular, showing that the claim is replicable under reasonable assumptions), equation (54) is not very satisfactory: CT being insidethe expectation, the relationship between (54) and the standard price of a European swaption or related claim (of the form C0EQ[g(FT)]), is not very clear. Equation (54) is also misleading, as it indicates that the distributional assumption made on C (or C0) could play an important role, when in fact, the following will show that the distribution of C0 only matters in as much as the terminal bracket [C0, P]T is concerned: definingZT = (V0e−[C0,P]T/C0)CT, usingVT = 1, and the fact thatC0e−[C0,P] is aQ0-martingale, we have :
EQ0[ZT] = V0
C0
EQ0[CT0e−[C0,P]T] = V0
C0
C00 = 1 (55) SoZT is a probability density underQ0, and ifdQ=ZTdQ0, from (54):
π0=B0
C0
V0
e[C0,P]TEQ0[ZTg(FT)] =P0C0e[C0,P]TEQ[g(FT)] (56) whereP0=B0/V0 is the current survival probability with maturityT, and C0
the current annuity of the underlying forward swap rate. The attractiveness of (56) is obvious: the non-arbitrage price π0 of a risky swaption (or related claim), appears to be the standard priceC0EQ[g(FT)] multiplied by a survival probabilityP0 (not a big surprise, the payoff being conditional on no default), with an additional (and by now fairly common), convexity adjustmente[C0,P]T. The problem with equation (56), is that despite its remarkable appeal to intu- ition it is pretty useless, unless the distribution ofF underQis known.80 When we said that C0EQ[g(FT)] was the standard price, we were being economical with the truth: it is indeedthe standard price, providedF is a martingale un- derQ. As far equation (56) is concerned, there is no reason why this should be the case. In fact, as shown in appendix A.6, the processF e−[F,P] (and not F itself) is a martingale underQ.
So it seems that equation (56), with the knowledge that the pricing measure Q is such that F e−[F,P] is a martingale, is a far better answer to our pricing problem than equation (54). And so it is. However, the road to (56) was long and tedious, making the whole argument somewhat unconvincing, with the belief that a more elegant and direct route should exist. The reason we obtained (54) instead of (56), was our choice of numeraire B: if the process BC had been a tradable process, we could have chosenBC as collapsing numeraire instead ofB, giving us a terminal wealthπT with BTCT as a common factor (instead of just BT). The replicating condition would have involved g(FT) (instead of CTg(FT)) and it is believable that (56) would have been derived without much more effort. . . The problem is thatBC is not a tradable process.81
80Very often, knowing the distribution ofF underQ, amounts to knowing itsdriftunderQ.
81We can always assume anything to be tradable, but it would not make sense to do so.
One solution to the problem is to consider the SDE:
dπt=θtd(CF)Dt −θtFtdCtD+ dBt
Bt∗ +dCtD
Ct −dVtD Vt
πt− (57) The financial interpretation of (57) could be phrased asthe SDE governing the wealth process of an investor entering into a strategyθ (up to time D) relative toCF, using C to fund his position in CF, investing his total wealth once in the collapsing numeraireB, and once in the numeraire C, using V to fund his position in B and C. In appendix A.6, we show that the solution to (57) is given by:
πt=BtCt
Vt
π0V0
B0C0
+ Z t
0
θˆsdFˆs
e−[C0,P]t (58) where the continuous semi-martingale ˆF is defined as ˆF =F e−[F,P], the process θˆis defined as ˆθ= (θe[F,P]+[C0,P])/P, the processC0 =C/V is the forward annu- ity of the underlying swap, andP=B∗/V is the continuous part of the survival probability processB/V. SinceVT = 1, a sufficient condition for replication is:
π0V0
B0C0
+ Z T
0
θˆtdFˆt=g(FT)e[C0,P]T (59) and we see that the non-arbitrage priceπ0 is indeed given by (56), whereQis a measure under which, the semi-martingale ˆF is indeed a martingale. In the case of a risky swaption, we finally have:
π0=P0C0e[C0,P]TEQ[(FT −K)+] (60) whereP0is the current survival probability with maturityT, andC0the current annuity of the underlying swap. This price is exactly the naive price, except for the adjustments e[C0,P]T and e[F,P]T required on C0 and F0 respectively. The case of the delayed risky swaption is handled by applying (56) to the function g(x) =EQ[(FT0−K)+|FT =x]. We obtain:82
π0=P0C0e[C0,P]TEQ[(FT0−K)+] (61) whereQis such thatFis aQ-martingale, with adjusted initial valueF0e[F,P]T.83 This is also very close to the naive valuation, i.e. the standard price of the European swaption with expiryT0, multiplied by the survival probability with maturity T. The only difference is the presence of the convexity adjustments e[C0,P]T ande[F,P]T, required onC0 andF0 respectively.
82Having adjusted the initial valueF0toF0e[F,P]T, we can assume that the pricing measure Qin (56) is such thatF is aQ-martingale. We have:
EQ[g(FT)] =EQ[EQ[(FT0−K)+|FT]] =EQ[(FT0−K)+]
83Strictly speaking,Fte−[F,P]t∧T is aQ-martingale.