Convexity Adjustment: A User’s GuideYan Zeng

Version 1.0.1, last revised on 2015-02-14

Abstract

Elements of convexity adjustment.

Contents1 Introduction 3

2 Convexity adjusted interest rates 32.1 LIBOR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 LIBOR-in-arrears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 LIBOR paid at arbitrary time under the linear rate model . . . . . . . . . . . . . . . . 4

2.2 CMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.1 CMS paid at arbitrary time under the linear swap rate model . . . . . . . . . . . . . . 52.2.2 CMS paid at arbitrary time under Hagan’s model . . . . . . . . . . . . . . . . . . . . 7

2.3 Hull’s approach to convexity adjustment (LIBOR-in-arrears) . . . . . . . . . . . . . . . . . . 92.4 Option on interest rates paid at arbitrary time under linear rate model . . . . . . . . . . . . . 9

3 Convexity adjustment with volatility smile: pricing by replication 113.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Problem formulation and its solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Application to motivating examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.1 LIBOR and caplet paid at arbitrary time under linear rate model . . . . . . . . . . . . 143.3.2 CMS swap and CMS caplet paid at arbitrary time discounted by swap rate yield . . . 153.3.3 CMS swap and CMS caplet paid at arbitrary time under linear rate model . . . . . . 163.3.4 CMS caplet, floorlet, and floater paid at arbitrary time under Hagan’s model . . . . . 173.3.5 CMS digitals paid at arbitrary time under Hagan’s model . . . . . . . . . . . . . . . . 213.3.6 CMS fracl paid at arbitrary time under linear rate model . . . . . . . . . . . . . . . . 23

4 Comparison of various convexity adjusted prices 254.1 LIBOR-in-arrears swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 In-arrears caplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 CMS swaps and CMS caps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Variable interest rates in foreign currency 265.1 Multi-currency change of numeraire theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Quanto adjustments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.3 Quantoed options on interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

A Approximation of ratio of annuities under linear rate model 29

B Integral representation of convex functions 29

C Pricing of “trapezoidal” payoff by replication 30

1

D Computation of∫∞K0

C(K)dK and∫K0

0P (K)dK 30

2

1 IntroductionIn this note, we summarize various results on convexity adjustment. The exposition is based on Boenkost

and Schmidt [1], [2], Hagan [3], Hull [5], Hunt and Kennedy [4], Lesniewski [6], Pelsser [8], and Piza [9].Denote by P (t, T ) (0 ≤ t ≤ T ) the time-t value of a zero coupon bond with maturity T . τ(S, T ) is the

year fraction between time S and time T (S < T ). The simply-compounded forward interest rate F (t;S, T )is defined as

F (t;S, T ) =1

τ(S, T )

(P (t, S)

P (t, T )− 1

).

Suppose Tα < Tα+1 < · · · < Tβ is a set of future times such that the LIBOR rate is reset at Tα, · · · , Tβ−1

and is paid at Tα+1, · · · , Tβ for a floating-rate note. The forward swap rate Sα,β(t) at time t for the set oftimes T = {Tα, Tα+1, · · · , Tβ} (t ≤ Tα) and year fractions τ = {τα+1, · · · , τβ} (τi = τ(Ti−1, Ti)) is defined as

Sα,β(t) =P (t, Tα)− P (t, Tβ)∑β

i=α+1 τiP (t, Ti).

These two rates often appear as the underlyings in interest rate derivatives, and will serve as the prototypefor convexity adjustment.

2 Convexity adjusted interest rates2.1 LIBOR

The LIBOR rate L(S, T ) = F (S;S, T ) for the interval [S, T ] is given by

L(S, T ) =1

τ(S, T )

(1

P (S, T )− 1

).

Under the forward measure QT for which P (·, T ) is the numeraire, F (t;S, T ) is a martingale and thereforeEQT [L(S, T )] = F (0;S, T ). This leads to the pricing formula of a floater, which resets LIBOR at time S andmakes payment at time T .

2.1.1 LIBOR-in-arrears

For LIBOR-in-arrears, we need to evaluate EQS [L(S, T )], where QS is the forward measure for whichP (·, S) is the numeraire. The goal is to express EQS [L(S, T )] in terms of the forward rate F (0;S, T ) plussome “convexity” adjustment (recall EQT [L(S, T )] = F (0;S, T )):

EQS [L(S, T )] = EQT

[L(S, T )

P (S, S)/P (0, S)

P (S, T )/P (0, T )

]= EQT

[L(S, T ) · (1 + τ(S, T )L(S, T )) · P (0, T )

P (0, S)

]= EQT

[L(S, T ) · 1 + τ(S, T )L(S, T )

1 + τ(S, T )F (0;S, T )

]=

F (0;S, T ) + τ(S, T )EQT [L2(S, T )]

1 + τ(S, T )F (0;S, T )

Note EQT [L2(S, T )] = VarQT(L(S, T )) + (EQT [L(S, T )])2, we conclude

EQS [L(S, T )] = F (0;S, T ) +τ(S, T )VarQT

(L(S, T ))

1 + τ(S, T )F (0;S, T )(1)

3

Under the so-called market model which is the model underlying the market valuation for caps, theLIBOR L(S, T ) is lognormal under QT with volatility σ,

L(S, T ) = F (S;S, T ) = F (0;S, T ) exp{σWS − 1

2σ2S

},

where W is a standard Brownian motion. In this case, VarQT (L(S, T )) = F 2(0;S, T )(eσ2S − 1) and formula

(1) becomes

EQS [L(S, T )] = F (0;S, T )

[1 +

τ(S, T )F (0;S, T )(eσ2S − 1)

1 + τ(S, T )F (0;S, T )

]. (2)

2.1.2 LIBOR paid at arbitrary time under the linear rate model

Suppose the payment is made at an arbitrary time T ′ ∈ [S, T ]. This is the case of Asian floater, whereS and T are the starting time and ending time of a coupon period, respectively. Then

EQT ′ [L(S, T )] = EQT

[P (S, T ′)/P (0, T ′)

P (S, T )/P (0, T )L(S, T )

]The linear rate model assumes

P (S, T ′)

P (S, T )= a+ b(T ′)L(S, T ), ∀T ′ ∈ [S, T ]

which requires a = 1 by setting T ′ = T . This is effectively equivalent to assuming

L(T ′, T ) =b(T ′)

τ(T ′, T )L(S, T ), ∀T ′ ∈ [S, T ].

Moreover, the martingale property dictates

P (0, T ′)

P (0, T )= EQT

[P (S, T ′)

P (S, T )

]= a+ b(T ′)F (0;S, T ).

So we have b(T ′) =(

P (0,T ′)P (0,T ) − 1

)/F (0;S, T ) = τ(T ′,T )F (0;T ′,T )

F (0;S,T ) . In summary, the linear rate modelassumes

L(T ′, T )

L(S, T )=

F (0;T ′, T )

F (0;S, T ), ∀T ′ ∈ [S, T ] (3)

which can be summarized in words as

The ratio of LIBOR rates over the interval [T ′, T ] and [S, T ] is equal to the ratio of time-zero forwardrates over the same intervals.

Note the case of LIBOR-in-arrears, where T ′ = S, satisfies the assumption of linear rate model.Under the linear rate model assumption, we easily deduce that

EQT ′ [L(S, T )] = F (0;S, T )

[1 +

1− P (0, T )/P (0, T ′)

F 2(0;S, T )VarQT

(L(S, T ))

], ∀T ′ ∈ [S, T ] (4)

Remark 1. The original motivation for the linear rate model is probably the consideration that the “nat-ural rate” under T -forward measure QT is L(S, T ). So one would like to use L(S, T ) to approximateP (S, T ′)/P (S, T ), and linear function is obviously the simplest. This idea can be generalized to that ofmaking the Radon-Nikodym derivative a function of the payout rate.

4

Remark 2. For T ′ = S, formula (4) reduces to formula (1).

Under the market model where L(S, T ) is lognormal under QT with volatility σ,

L(S, T ) = F (0;S, T )eσWS− 12σ

2S

and formula (4) becomes more explicit:

EQT ′ [L(S, T )] = F (0;S, T )

[1 +

(1− P (0, T )

P (0, T ′)

)(eσ

2S − 1)

].

2.2 CMSFrom the definition of forward swap rate Sα,β(t), if we choose the annuity Nα,β

t =∑β

i=α+1 τiP (t, Ti)

as numeraire and denote by Qα,β the associated martingale measure (the “swap measure”), we have bymartingale property

EQα,β

[Sα,β(Tα)] = Sα,β(0).

If the payment is to be paid at some time T ′ > Tα, we need to compute under the T ′-forward measure QT ′

EQT ′ [Sα,β(Tα)] = EQα,β

[P (Tα, T

′)/P (0, T ′)

Nα,βTα

/Nα,β0

Sα,β(Tα)

]=

Nα,β0

P (0, T ′)EQα,β

[P (Tα, T

′)

Nα,βTα

Sα,β(Tα)

].

The goal is to express EQT ′ [Sα,β(Tα)] in terms of the time-zero swap rate Sα,β(0) plus some “convexity”adjustment.

2.2.1 CMS paid at arbitrary time under the linear swap rate model

Under the swap measure Qα,β associated with the annuity numeraire Nα,βt =

∑βi=α+1 τiP (t, Ti), the

entity most convenient for computation is the swap rate Sα,β(Tα). Therefore, a natural assumption for theso-called linear swap rate model is

P (Tα, T′)

Nα,βTα

= a+ b(T ′)Sα,β(Tα), T′ ≥ Tα.

To determine a and b, we first take expectation of both sides under the swap measure and use themartingale property to get

P (0, T ′)

Nα,β0

= a+ b(T ′)Sα,β(0).

This gives b(T ′) = 1Sα,β(0)

[P (0,T ′)

Nα,β0

− a]. To deduce the second equation for a and b, we note

1 =

∑βi=α+1 τiP (Tα, Ti)

Nα,βTα

=

β∑i=α+1

τi [a+ b(Ti)Sα,β(Tα)] = a

(1− Sα,β(Tα)

Sα,β(0)

) β∑i=α+1

τi +Sα,β(Tα)

Sα,β(0).

Therefore, we can solve for a: a = 1∑βi=α+1 τi

. In summary, the linear swap rate model makes theassumption

P (Tα,T ′)

Nα,βTα

= a+ b(T ′)Sα,β(Tα), T′ ≥ Tα

a = 1∑βi=α+1 τi

b(T ′) = 1Sα,β(0)

[P (0,T ′)

Nα,β0

− 1∑βi=α+1 τi

], T ′ ≥ Tα

(5)

5

and consequently

EQT ′ [Sα,β(Tα)] = Sα,β(0)

1 + 1− P (0,Tα)−P (0,Tβ)

Sα,β(0)P (0,T ′)∑β

i=α+1 τi

S2α,β(0)

VarN (Sα,β(Tα))

(6)

where VarN (Sα,β(Tα)) is the variance of the swap rate Sα,β(Tα) under the swap measure Qα,β .Under the so-called market model for swpation, it’s assumed the swap rate Sα,β(t) satisfies

dSα,β(t) = σα,βSα,β(t)dWα,βt , t ≤ Tα

where Wα,β is a standard Brownian motion under the swap measure Qα,β . The variance of the swap rateSα,β(Tα) under the swap measure is therefore S2

α,β(0)(eσ

2α,βTα − 1

). Then

EQT ′ [Sα,β(Tα)] = Sα,β(0)

[1 +

(1− P (0, Tα)− P (0, Tβ)

Sα,β(0)P (0, T ′)∑β

i=α+1 τi

)(eσ

2α,βTα − 1

)].

Remark 3. The linear rate model for Libor and CMS can be generalized as follows. Write YS for a floatingrate which is set at time S. Let N , QN denote the natural (“market”) numeraire pair associated with YS

and all we need isEQN [YS ] = Y0,

where Y0 is known and a function of the yield curve P (0, ·) today.We are interested in today’s price of the rate YS to be paid at some time T ′ ≥ S,

P (0, T ′)EQT ′ [YS ] = N0EQN

[P (S, T ′)

NSYS

].

Assume a linear rate model of the form

P (S, T ′)

NS= a+ b(T ′)YS (7)

with some deterministic a, b(T ′) which have to be determined accordingly to make the model consistent. Wethen have

EQT ′ [YS ] = Y0

[1 +

b(T ′)

Y0(a+ b(T ′)Y0)VarQN (YS)

](8)

If in addition, the distribution of YS under QN is lognormal with volatility σY : YS = Y0eσY WS− 1

2σ2Y S, then

EQT ′ [YS ] = Y0

[1 +

b(T ′)Y0

a+ b(T ′)Y0(eσ

2Y S − 1)

].

Under the linear rate model for Libor, YS = L(S, T ) and NS = P (S, T ); under the linear rate mode forCMS, YS = Sα,β(Tα) and NS = Nα,β

Tα.

As a last comment, the linear approximation of linear rate model does seem very crude at first, but can bejustified by the following argument. Convexity corrections only become sizeable for large maturities. However,for large maturities the term structure almost moves in parallel. Hence, a change in the level of the long endof the curve is well described by the rate Y . Furthermore, for parallel moves in the curve, the ratio P (S,T ′)

NS

is closely approximated by a linear function of Y , which is exactly what the linear rate model does. Hence,exactly for long maturities the assumptions of the linear rate model become quite accurate. This leads to agood approximation of the convexity correction for long maturities.

6

2.2.2 CMS paid at arbitrary time under Hagan’s model

As seen in the previous section, the key to the convexity adjustment involving CMS rate, is to expressP (Tα,T ′)

Nα,βTα

as a function of swap rate Sα,β(Tα):

P (Tα, T′)

Nα,βTα

= G(Sα,β(Tα)).

In this section, we explain several such models proposed by Hagan [3].Model 1: Standard model. The standard method for computing convexity corrections uses bond

math approximations: payments are discounted at a flat rate, and the coverage (day count fraction) for eachperiod is assumed to be 1/q, where q is the number of periods per year. At any date t ≤ Tα, the annuity isapproximated by

Nα,βt = P (t, Tα)

β∑i=α+1

τiP (t, Ti)

P (t, Tα)≈ P (t, Tα)

β−α∑j=1

1/q

[1 + Sα,β(t)/q]j=

P (t, Tα)

Sα,β(t)

[1− 1

(1 + Sα,β(t)/q)n

]Here the forward swap rate Sα,β(t) is used as the discount rate, since it represents the average rate over thelife of the reference swap. In a similar spirit, the zero coupon bond for the pay date T ′ is approximated as

P (t, T ′) ≈ P (t, Tα)

(1 + Sα,β(t)/q)∆

where ∆ = T ′−Tα

Tα+1−Tα. Combined, the standard “bond math model” leads to the approximation

G(Sα,β(t)) =P (t, T ′)

Nα,βt

≈ Sα,β(t)

(1 + Sα,β(t)/q)∆· 1

1− 1(1+Sα,β(t)/q)n

Model 2: “Exact yield” model. We can account for the reference swap’s schedule and day countexactly by approximating

P (t, Ti)

P (t, Tα)≈

i∏j=α+1

1

1 + τjSα,β(t)

andP (t, T ′) ≈ P (t, Tα)

(1 + τα+1Sα,β(t))∆

where ∆ = T ′−Tα

Tα+1−Tα. Therefore

Nα,βt ≈ P (t, Tα)

β∑i=α+1

τi

i∏j=α+1

1

1 + τjSα,β(t)

=P (t, Tα)

Sα,β(t)

(1−

β∏i=α+1

1

1 + τiSα,β(t)

)

and

G(Sα,β(t)) =P (t, T ′)

Nα,βt

≈ Sα,β(t)

(1 + τα+1Sα,β(t))∆1

1−∏β

i=α+11

1+τiSα,β(t)

This approximates the yield curve as flat and only allows parallel shifts, but has the schedule right.Model 3: Parallel shifts. This model takes into account the initial yield curve shape, which can be

significant in steep yield curve environments. We still only allow parallel yield curve shifts, so we approximate

P (t, Ti)

P (t, Tα)=

P (0, Ti)

P (0, Tα)e−(Ti−Tα)s

7

where s is the amount of the parallel shift to be determined. To determine s, note

Nα,βt = P (t, Tα)

β∑i=α+1

τiP (t, Ti)

P (t, Tα)= P (t, Tα)

β∑i=α+1

τiP (0, Ti)

P (0, Tα)e−(Ti−Tα)s

and henceSα,β(t) =

P (t, Tα)− P (t, Tβ)

Nα,βt

=P (0, Tα)− P (0, Tβ)e

−(Tβ−Tα)s∑βi=α+1 τiP (0, Ti)e−(Ti−Tα)s

.

This equation implicitly determines s as a function of Sα,β(t). Therefore

Nα,βt

P (t, Tα)=

β∑i=α+1

τiP (0, Ti)

P (0, Tα)e−(Ti−Tα)s =

1− P (0,Tβ)P (0,Tα)e

−(Tβ−Tα)s

Sα,β(t)

and

G(Sα,β(t)) =P (t, T ′)/P (t, Tα)

Nα,βt /P (t, Tα)

=e−(T ′−Tα)s[

1− P (0,Tβ)P (0,Tα)e

−(Tβ−Tα)s]/Sα,β(t)

=Sα,β(t)e

−(T ′−Tα)s

1− P (0,Tβ)P (0,Tα)e

−(Tβ−Tα)s

where s is determined implicity in terms of Sα,β(t), by

Sα,β(t)

β∑i=α+1

τiP (0, Ti)e−(Ti−Tα)s = P (0, Tα)− P (0, Tβ)e

−(Tβ−Tα)s.

This model’s limitations are that it allows only parallel shifts of the yield curve and it presumes perfectcorrelation between long and short term rates.

Model 4: Non-parallel shifts. We can allow non-parallel shifts by approximating

P (t, Ti)

P (t, Tα)=

P (0, Ti)

P (0, Tα)e−[h(Ti)−h(Tα)]s

Then similar to Model 3, we have

G(Sα,β(t)) =Sα,β(t)e

−[h(T ′)−h(Tα)]s

1− P (0,Tβ)P (0,Tα)e

−[h(Tβ)−h(Tα)]s

where s is determined implicity in terms of Sα,β(t), by

Sα,β(t)

β∑i=α+1

τiP (0, Ti)e−[h(Ti)−h(Tα)]s = P (0, Tα)− P (0, Tβ)e

−[h(Tβ)−h(Tα)]s.

To complete the model, we need to select the function h(·) which determines the shape of the non-parallelshift. This is often done by postulating a constant mean reversion

h((T )− h(Tα) =1

κ

[1− e−κ(T−Tα)

].

Alternatively, one can choose h(·) by calibrating the vanilla swaptions which have the same start date Tα

and varying end dates to their market prices.

In either case, under the assumption P (Tα,T ′)

Nα,βTα

= G(Sα,β(Tα)), we have

EQT ′ [Sα,β(Tα)] = Sα,β(0) + EQα,β

{[G(Sα,β(Tα))

G(Sα,β(0))− 1

]Sα,β(Tα)

}(9)

8

2.3 Hull’s approach to convexity adjustment (LIBOR-in-arrears)This section is based on Hull [5], Chapter 20. We recall the relation between forward LIBOR rate

F (t;S, T ) and zero coupon bond price P (t, ·) is given by

P (t, T )

P (t, S)=

1

1 + τ(S, T )F (t;S, T ).

Write yt for F (t;S, T ) and define G(y) = 11+τ(S,T )y . Then Taylor expansion gives

P (t, T )

P (t, S)= G(yt) ≈ G(y0) +G′(y0)(yt − y0) +

1

2G′′(y0)(yt − y0)

2

Under the S-forward measure QS , EQS

[P (t,T )P (t,S)

]= P (0,T )

P (0,S) = G(y0). So taking expectation of both sides ofthe Taylor expansion, we have G(y0) ≈ G(y0) +G′(y0)

(EQS [yt]− y0

)+ 1

2G′′(y0)E

QS [(yt − y0)2]. This gives

EQS [yt] ≈ y0 −1

2

G′′(y0)

G′(y0)EQS [(yt − y0)

2].

Let t = S and approximate EQS [(yS − y0)2] by σ2y20S with σ the volatility of y. We then have

EQS [L(S, T )] ≈ F (0;S, T )

[1 +

τ(S, T )F (0;S, T )σ2S

1 + τ(S, T )F (0;S, T )

]This is the first order approximation of convexity adjustment formula (2). Note the approximation ofEQS [(yS − y0)

2] by σ2y20S is more or less equivalent to assuming yS is lognormally distributed.

2.4 Option on interest rates paid at arbitrary time under linear rate modelIn this section, we investigate European options on interest rates like LIBOR L(S, T ) for period [S, T ]

or CMS rates Sα,β(Tα) for tenure structure T = {Tα, Tα+1, · · · , Tβ}. The payment date of the option is anarbitrary time point T ′ with T ′ ≥ S or T ′ ≥ Tα, respectively.

Of particular interest are caps and floors or binaries. For standard caps and floors on LIBOR, we haveT ′ = T and the standard market model postulates a lognormal distribution of L(S, T ) under the forwardmeasure QT . For standard option on a swap rate Sα,β(Tα), i.e. swaptions, the market uses a lognormaldistribution for Sα,β(Tα) under the swap measure Qα,β . However in the general case, i.e. for options onLIBOR or CMS with arbitrary payment date T ′, a lognormal model would be inconsistent with the marketmodel for standard options. For example, if L(S, T ) is lognormal under QT , it cannot be lognormal underQS in general:

EQT [f(L(S, T ))] =P (0, S)

P (0, T )EQS

[f(L(S, T ))

1 + τ(S, T )L(S, T )

].

We follow the general setup of linear rate model. YS is a floating interest rate which is set at time S and(N,QN ) denotes the “market” numeraire pair associated with YS . We assume that the distribution of YS

under QN is lognormal with volatility σY ,

YS = Y0 exp{σY WS − 1

2σ2Y S

},

where W is a standard Brownian motion. For a payment date T ′ ≥ S, we further assume a linear rate modelof the form (7)

P (S, T ′)

NS= a+ b(T ′)YS .

Recall that for the case of YS = L(S, T ), NS = P (S, T ), and T ′ = S, i.e. LIBOR-in-arrears, the assumptionof a linear rate model is trivially satisfied and there is no restriction.

9

We first consider the valuation of standard options on the rate YS but the option payout is at somearbitrary time T ′ ≥ S. The value of a call option with strike K is then

P (0, T ′)EQT ′[(YS −K)+

]= N0E

QN

[(YS −K)+

P (S, T ′)

NS

]= N0E

QN[(YS −K)+(a+ b(T ′)YS)

]= P (0, T ′)

Y0Φ(d1)(a− b(T ′)K)− aKΦ(d2) + b(T ′)Y 20 e

σ2Y SΦ(d1 + σY

√S)

a+ b(T ′)Y0

where Φ(·) is the c.d.f. of the standard normal distribution,

d1 =ln(Y0

K

)+ 1

2σ2Y S

σY

√S

, d2 =ln(Y0

K

)− 1

2σ2Y S

σY

√S

.

It is desirable to be able to use standard valuation formula (i.e. Black’s formula) also for options oninterest rates which are irregularly paid. For this purpose, we assume YS is lognormally distributed underQT ′ , with adjusted volatility.

Using formula (8), i.e. EQT ′ [YS ] = Y0

[1 + b(T ′)Y0

a+b(T ′)Y0(eσ

2Y S − 1)

], and moment matching, we can derive

YS ≈ EQT ′ [YS ] exp{σ∗Y WS − 1

2(σ∗

Y )2S

},

where W is a standard Brownian motion under QT ′ and

(σ∗Y )

2 = σ2Y + ln

[(a+ b(T ′)Y0)(a+ b(T ′)Y0e

2σ2Y S)

(a+ b(T ′)Y0eσ2Y S)2

]/S. (10)

This will give option price via Black’s formula.With the above lognormal approximation, we can consider an exchange option involving two interest

rates, Y1 and Y2. We assume Y1 and Y2 are set (fixed) at times S1 and S2, respectively, with S1 ≤ S2. Forexample, Y1 and Y2 could be LIBOR rates L(S1, T1) and L(S2, T2) referring to different fixing dates S1, S2

(e.g. LIBOR and LIBOR-in-arrears). One could also think of two CMS rates to be set at the same date butwith different tenors.

Suppose the option’s payoff is (Y2−Y1)+, paid at time T ′ ≥ max{S1, S2}. In view of the above lognormal

approximation, we can assume that both interest rates are lognormal under QT ′ (i = 1, 2)

Yi = Y 0i e

σiWiSi

− 12σ

2i Si , Y 0

i = EQT ′ [Yi],

with EQT ′ [Yi] given by formula (4), (6), or (8) and W i standard Brownian motion under QT ′ . Suppose theinstantaneous correlation between W 1 and W 2 is ρ, the fair price of the exchange option is then given by

P (0, T ′)[Y 02 N(b1)− Y 0

1 N(b2)],

where

b1 =ln(

Y 02

Y 01

)+ 1

2 (σ21S1 + σ2

2S2 − 2σ1σ2ρS1)√σ21S1 + σ2

2S2 − 2σ1σ2ρS1

, b2 =ln(

Y 02

Y 01

)− 1

2 (σ21S1 + σ2

2S2 − 2σ1σ2ρS1)√σ21S1 + σ2

2S2 − 2σ1σ2ρS1

For details of the computation, see Boenkost and Schmidt [1], Section 4.4, Proposition 7.

Remark 4. The market standard method of valuing options on convexity adjusted rates is to apply the Blackformula using the convexity adjusted rate as the forward rate. But to be conceptually correct, we should alsoconvexity adjust volatility by formula (10). These two adjustments combined is equivalent to assuming therate is lognormal under the T ′-forward measure QT ′ .

10

3 Convexity adjustment with volatility smile: pricing by replica-tion

As shown in Section 2 and Section 2.4, pricing irregular interest cash flows such as LIBOR-in-arrears orCMS requires a convexity correction on the corresponding forward rate. This convexity correction involvesthe volatility of the underlying rate as traded in the cap/floor or swaption market. In the same spirit, anoption on an irregular rate, such as an in-arrears cap or a CMS cap, is often valued by applying Black’sformula with a convexity adjusted forward rate and a convexity adjusted volatility. However, to a largeextent, this approach ignores the volatility smile, which is quite pronounced in the cap/floor market. Thissection solves this problem by replicating the irregular interest flow or option with liquidly traded optionswith different strikes. As an important consequence, one obtains immediately the respective simultaneousdelta and vega hedges in terms of liquidly traded options.

3.1 MotivationExample 1. (LIBOR and caplet paid at arbitrary time) Let Y = L(S, T ) be the LIBOR rate for

the period [S, T ]. Suppose our goal is to price an “exotic” payoff f(L(S, T )) at time S, or more generally,at time T ′ ≥ S. For example, f(L(S, T )) = τ(S, T )L(S, T ) is the payoff of a LIBOR, and f(L(S, T )) =τ(S, T )[L(S, T ) − K]+ is the payoff of a caplet. How can we use liquid instruments to hedge this exoticpayoff?

In the interest rate derivative market, caplets are quite liquid for (more or less) any strike K ≥ 0.Therefore, we intend to use caplets as hedging instruments. One difficulty is that by default, a caplet paysin arrears (i.e. at time T ).

To hedge the exotic payoff, we first need to “match” the payoff times. Since the exotic payoff is eitherpaid at time S or at time T ′ ≥ S, we discount everything to time S. First, the standard caplet paying attime T is equivalent to a payoff of

P (S, T )τ(S, T )[L(S, T )−K]+ =τ(S, T )

1 + τ(S, T )L(S, T )[L(S, T )−K]+

at time S. Second, under the linear rate model P (S,T ′)P (S,T ) = 1+

(P (0,T ′)P (0,T )

−1)

F (0;S,T ) L(S, T ), the exotic payoff f(L(S, T ))

at time T ′ is equivalent to a payoff of

P (S, T ′)f(L(S, T )) =1 +

(P (0,T ′)P (0,T )

−1)

F (0;S,T ) L(S, T )

1 + τ(S, T )L(S, T )f(L(S, T )) := f1(L(S, T ))

at time S. Here f1(y) :=1+

(P (0,T ′)P (0,T )

−1

)F (0;S,T )

y

1+τ(S,T )y f(y).

In summary, the exotic payoff can be written as f(L(S, T )), paid at time S; and the hedging instrumentscan be generalized to the form of g(L(S, T ))[L(S, T ) − K]+ (K ≥ 0), also paid at time S. Here g(y) :=

τ(S,T )1+τ(S,T )y . We are interested in finding a representation

f(y) = C +

∫ ∞

0

g(y)(y −K)+dµ(K)

with some locally finite signed measure µ on R+ and some constant C. If such a representation exists, thetime-zero fair price of the contingent claim f(L(S, T )) is given by

V exotic0 = C · P (0, S) +

∫ ∞

0

V caplet0 (K)dµ(K),

where V caplet0 (K) is the time-zero price of a caplet for the period [S, T ], with strike K.

11

Example 2. (CMS swap and CMS caplet paid at arbitrary time) Let Y = Sα,β(Tα) be theswap rate for the tenure structure T = {Tα, Tα+1, · · · , Tβ}. Suppose our goal is to price an “exotic” payofff(Sα,β(Tα)) at time Tα. For example, f(y) = y is a cash flow in a CMS swap (but paid in advance), andf(y) = (y −K)+ gives a CMS caplet.

In case the payoff is scheduled from a time T ′ ≥ Tα, we can approximately think of a derivative withpayoff

f(y) =

y

(1+y)T ′−Tαfor a CMS swap

(y−K)+

(1+y)T ′−Tαfor a CMS caplet;

at time Tα.How can we use liquid instruments to hedge this exotic payoff?In the interest rate derivative market, swaptions are quite liquid for any strike K ≥ 0. Therefore, we

intend to use swaptions as hedging instruments. There are two types of swaptions: the payoff of a cash-settledpayer swaption at time Tα is (

β∑i=α+1

τ(Ti−1, Ti)

[1 + Sα,β(Tα)]Ti−Tα

)(Sα,β(Tα)−K)+

and the payoff of a physically-settled payer swaption at time Tα is(β∑

i=α+1

τ(Ti−1, Ti)P (Tα, Ti)

)(Sα,β(Tα)−K)+.

The market does not make a significant difference in pricing a cash- or physically-settled swaption, and bothare priced under a lognormal Black model.

Example 3. (CMS swap and CMS caplet paid at arbitrary time under linear rate model)In Example 2, we approximated CMS swap rate and CMS caplet paid at arbitrary time by

Sα,β(Tα)

[1+Sα,β(Tα)]T ′−Tαfor a CMS swap

[Sα,β(Tα)−K]+

[1+Sα,β(Tα)]T ′−Tαfor a CMS caplet.

In the current example, we want to avoid this approximation but we then need the additional assumptionof linear rate model. The so-called linear swap rate model is

P (Tα,T ′)

Nα,βTα

= a+ b(T ′)Sα,β(Tα), T′ ≥ Tα

a = 1∑βi=α+1 τi

b(T ′) = 1Sα,β(0)

[P (0,T ′)

Nα,β0

− 1∑βi=α+1 τi

], T ′ ≥ Tα

where Nα,β· is the annuity numeraire Nα,β

t =∑β

i=α+1 τiP (t, Ti),Consequently, an exotic derivative paying f(Sα,β(Tα)) at some time T ′ ≥ Tα is equivalent to a payoff of

P (Tα, T′)f(Sα,β(Tα)) = Nα,β

Tα[a+ b(T ′)Sα,β(Tα)]f(Sα,β(Tα))

at time Tα. Under the swap measure Qα,β associated with the annuity numeraire Nα,β· , the price of the

contingent claim is given by

Nα,β0 EQα,β

[(a+ b(T ′)Sα,β(Tα))f(Sα,β(Tα))]

The price of physically-settled payer swaption can be written as

V swpt0 (K) = Nα,β

0 EQα,β [(Sα,β(Tα)−K)+

].

Comparing these two formulas leads us to looking for a representation of the form

(a+ b(T ′)Y )f(Y ) = C +

∫[0,∞)

(Y −K)+dµ(K).

12

3.2 Problem formulation and its solutionConsider a financial underlying with price Y at time T . Suppose there is a liquid market for plain vanilla

options on this underlying with all possible strikes K. Somewhat more generally, we suppose that for allK, the time-zero price V vanilla

0 (K) of the “plain vanilla” derivative with payoff g(Y )(Y −K)+ at time T isknown. Here g is a deterministic function.

Our goal is to price an exotic contingent claim with payoff f(Y ) at time T , where f is a deterministicfunction. The idea is to replicate the exotic payoff f(Y ) by a portfolio of traded derivatives g(Y )(Y −K)+

with different strikes K. If this is possible, then the replication is the key to incorporating the volatilitysmile of the liquid options into the pricing of the exotic derivative.

So we are looking for a representation

f(Y ) = C +

∫ ∞

0

g(Y )(Y −K)+dµ(K) (11)

with some locally finite signed measure µ on R+ and some constant C. By risk-neutral pricing, this repre-sentation will give the time-zero price of the contingent claim f(Y ):

V0 = C · P (0, T ) +

∫ ∞

0

V vanilla0 (K)dµ(K) (12)

Remark 5. The importance of formula (12) goes beyond the issue of just pricing a derivative with payofff(Y ), since it provides us with an explicit strategy for a simultaneous delta and vega hedge of the derivativef(Y ) in terms of liquidly traded products g(Y )(Y − K)+. In practice one would discretize the integralappropriately to get an (approximate) hedging strategy in a finite set of products g(Y )(Y − Ki)

+ withdifferent strikes Ki, i = 1, · · · .

Proposition 3.1. The exotic payoff f(Y ) allows for a replication (11) with some locally finite signed measureµ on [0,∞) and some constant C if and only if(i) limx↓0 f(x) = C;(ii) the function f−C

g , extended to the domain of definition [0,∞) by setting f(0)−Cg(0) = 0, is a difference of

convex functions on [0,∞).The measure µ is then generated by the following right continuous generalized dsitrbution function (functionof locally bounded variation)

dµ(y) = dD+

(f(y)− C

g(y)

)with D+ denoting the right derivative and defining D+

(f−Cg

)(0−) = 0.

Proof. For necessity, note

f(y)− C

g(y)=

∫ ∞

0

(y −K)+dµ+(K)−∫ ∞

0

(y −K)+dµ−(K),

where µ+ and µ− are the positive and negative parts of µ, respectively. For sufficiency, we apply the integralrepresentation of convex functions (Proposition B.1)

f(y)− C

g(y)− f(0)− C

g(0)=

∫ y

0

µ(K)dK = µ(y)y −∫ y

0

Kdµ(K) =

∫ y

0

(y −K)dµ(K) =

∫ ∞

0

(y −K)+dµ(K)

for some function µ(·) of locally finite variation.

Remark 6. By risk-neutral pricing under the T -forward measure, the time-zero price of the contingent claimis also given by the formula

V0 = P (0, T )EQT [f(Y )] = P (0, T )

∫ ∞

0

f(x)dFY (x),

13

where FY (·) is the cumulative distribution function of Y under QT . Since

V vanilla0 (K) = P (0, T )EQT [g(Y )(Y −K)+],

we have

D+Vvanilla0 (x) = P (0, T )EQT [g(Y )D+(Y − x)+] = −P (0, T )EQT [g(Y )1{x<Y }] = −P (0, T )

∫ ∞

x

g(y)dFY (y)

anddD+V

vanialla0 = P (0, T )g(x)dFY (x).

Therefore by integrating by parts twice, we have (define C = limx↓0 f(x))

V0 = C · P (0, T ) +

∫ ∞

0

f(x)− C

g(x)dD+V

vanilla0 (x)

= C · P (0, T )−∫ ∞

0

D+Vcall0 (x)df(x)

= C · P (0, T ) +

∫ ∞

0

V call0 (x)d2f(x).

This approach is consistent with formula (12).

3.3 Application to motivating examples3.3.1 LIBOR and caplet paid at arbitrary time under linear rate model

We consider a coupon period [S, T ] and denote by τ the day count τ(S, T ) of [S, T ] in year fraction.Consider a LIBOR paid at time T ′ ≥ S. Under the linear rate model, the payoff is equivalent to a payoff of

1 + b(T ′)L(S, T )

1 + τL(S, T )τL(S, T )

at time S, where b(T ′) =

(P (0,T ′)P (0,T )

−1)

F (0;S,T ) . Therefore f(y) = 1+b(T ′)y1+τy τy and C = limx↓0 f(x) = 0. The standard

caplet paying at time T is equivalent to a payoff ofτ

1 + τL(S, T )[L(S, T )−K]+

at time S. Therefore, g(y) = τ1+τy . Combined, for LIBOR paid at time T ′ ≥ S, we have

f(y)− C

g(y)= y[1 + b(T ′)y],

which is a convex function. Accordingly,

µ(y) = D+

(f(y)− C

g(y)

)=

{1 + 2b(T ′)y y ≥ 0

0 y < 0.

In view of dµ({0}) = µ(0)− µ(0−) = 1, we obtain the formula for the price of a LIBOR expressed in termsof caplet prices V caplet

0 (K) with different strikes K:V0(τL(S, T ) paid at time T ′ ≥ S) = V caplet0 (0) +

∫∞0

V caplet0 (K)2b(T ′)dK,

b(T ′) =

(P (0,T ′)P (0,T )

−1)

F (0;S,T )

(13)

14

Now consider a caplet paid at time T ′ ≥ S. Under the linear rate model, the payoff is equivalent to apayoff of

1 + b(T ′)L(S, T )

1 + τL(S, T )τ [L(S, T )−K]+

at time S. Therefore f(y) = 1+b(T ′)y1+τy τ(y −K)+ and C = limx↓0 f(y) = 0. We then have

f(y)− C

g(y)= [1 + b(T ′)y](y −K)+,

which is a convex function. Accordingly,

µ(y) = D+

(f(y)− C

g(y)

)=

{1− b(T ′)K + 2b(T ′)y y ≥ K

0 y < K.

In view of dµ({K}) = µ(K)−µ(K−) = 1+ b(T ′)K, we obtain the formula for the price of a caplet expressedin terms of caplet prices V caplet

0 (K) with different strikes K:V0(τ [L(S, T )−K]+ paid at time T ′ ≥ S) = V caplet0 (K)(1 + b(T ′)K) +

∫∞K

V caplet0 (K)2b(T ′)dK

b(T ′) =

(P (0,T ′)P (0,T )

−1)

F (0;S,T )

(14)

Remark 7. In case the cap market quotes no smile, it is easy to verify that formula (13) is reduced toformula (2).

3.3.2 CMS swap and CMS caplet paid at arbitrary time discounted by swap rate yield

Consider a CMS swap rate Sα,β(Tα) paid at time T ′ ≥ Tα. The payoff is approximated by a payoff of

Sα,β(Tα)

(1 + Sα,β(Tα))T′−Tα

at time Tα. Therefore, f(y) = y

(1+y)T ′−Tαand C = limx↓0 f(x) = 0.

The payoff of a cash-settled payer swaption at time Tα is(β∑

i=α+1

τ(Ti−1, Ti)

[1 + Sα,β(Tα)]Ti−Tα

)(Sα,β(Tα)−K)+

So, using cash-settled payer swaption as hedging instruments, we have

g(y) =

β∑i=α+1

τi(1 + y)Ti−Tα

We then havef(y)− C

g(y)=

y∑βi=α+1

τi(1+y)Ti−T ′

=y

DV01(y) ,

where DV01(y) :=∑β

i=α+1τi

(1+y)Ti−T ′ , the so-called present (or dollar) value of one basis point factor. It iskind of hard to determine the convexity of f(y)−C

g(y) , or to find the two convex functions whose difference isf(y)−Cg(y) . But we can easily see that f(y)−C

g(y) is absolutely continuous and its derivative is of finite variationlocally. Therefore, by Proposition B.2, Proposition 3.1 applies. Accordingly,

µ(y) = D+

(f(y)− C

g(y)

)=

DV01(y)−yDV01′

(y)

DV012(y)

y ≥ 0

0 y < 0

15

In view of dµ({0}) = µ(0) − µ(0−) = 1

DV01(0) = 1∑βi=α+1 τi

, we obtain the formula for the price of a CMS

swap rate expressed in terms of swaption price V swpt0 (K) with different strikes K:V0(Sα,β(Tα) paid at time T ′ ≥ Tα) =

V swpt0 (0)∑βi=α+1 τi

+∫∞0

V swpt0 (K)

(DV01(K)−KDV01′(K)

DV012(K)

)′dK

DV01(y) =∑β

i=α+1τi

(1+y)Ti−T ′

(15)

Now consider a CMS caplet [Sα,β(Tα) − K]+ paid at time T ′ ≥ Tα. The payoff is approximated by apayoff of

[Sα,β(Tα)−K]+

[1 + Sα,β(Tα)]T′−Tα

at time Tα. Therefore f(y) = (y−K)+

(1+y)T ′−Tαand C = limx↓0 f(x) = 0.

Like the case of CMS swap rate, by using cash-settled payer swaption as hedging instruments, we have

f(y)− C

g(y)=

(y −K)+∑βi=α+1

τi(1+y)Ti−T ′

=(y −K)+

DV01(y) .

Accordingly,

µ(y) = D+

(f(y)− C

g(y)

)=

DV01(y)−(y−K)DV01′

(y)

DV012(y)

y ≥ K

0 y < K

In view of dµ({K}) = µ(K)− µ(K−) = 1

DV01(K)= 1∑β

i=α+1τi

(1+K)Ti−T ′, we obtain the formula for the price

of a CMS caplet expressed in terms of swaption price V swpt0 (K) with different strike K:V0([Sα,β(Tα)−K]+ paid at time T ′ ≥ Tα) =V swpt0 (K)

DV01(K)+∫∞K

V swpt0 (K) · h(K) · dK

DV01(y) =∑β

i=α+1τi

(1+y)Ti−T ′

(16)

where

h(y) =

(DV01(y)− (y −K)DV01′(y)

DV012(y)

)′

= (y −K)

[2(DV01′(y))2

DV013(y)− DV01′′(y)

DV012(y)

]− 2

DV01′(y)DV012(y)

.

Formula (16) is widely used by sophisticated practitioners to value CMS caps.

3.3.3 CMS swap and CMS caplet paid at arbitrary time under linear rate model

Consider a CMS swap rate Sα,β(Tα) paid at time T ′ ≥ Tα. As explained in Example 3 of Section 3.1, weare looking for integral representation of the form

[a+ b(T ′)Sα,β(Tα)]Sα,β(Tα) = C +

∫[0,∞)

(Sα,β(Tα)−K)+dµ(K).

Using notation of Proposition 3.1, f(y) = [a+ b(T ′)y]y, g(y) = 1, and C = limx↓0 f(x) = 0. Therefore

µ(y) = D+

(f(y)− C

g(y)

)=

{a+ 2b(T ′)y y ≥ 0

0 y < 0

16

In view of dµ({0}) = µ(0) − µ(0−) = a, we obtain the formula for the price of a CMS swap rate expressedin terms of swaption price V swpt

0 (K) with different strikes K:V0(Sα,β(Tα) paid at time T ′ ≥ Tα) = aV swpt

0 (0) +∫∞0

V swpt0 (K) · 2b(T ′)dK

a = 1∑βi=α+1 τi

b(T ′) = 1Sα,β(0)

[P (0,T ′)

Nα,β0

− 1∑βi=α+1 τi

], T ′ ≥ Tα

(17)

Now consider a CMS caplet [Sα,β(Tα) − K]+ paid at time T ′ ≥ Tα. We are looking for integral repre-sentation of the form

[a+ b(T ′)Sα,β(Tα)][Sα,β(Tα)−K]+ = C +

∫[0,∞)

(Sα,β(Tα)−K)+dµ(K).

Using notation of Proposition 3.1, f(y) = [a+b(T ′)y](y−K)+, g(y) = 1, and C = limx↓0 f(x) = 0. Therefore

µ(y) = D+

(f(y)− C

g(y)

)=

{[a− b(T ′)K] + 2b(T ′)y y ≥ K

0 y < K

In view of dµ({K}) = µ(K) − µ(K−) = a + b(T ′)K, we obtain the formula for the price of a CMS capletexpressed in terms of swaption price V swpt

0 (K) with different strikes K:V0([Sα,β(Tα)−K]+ paid at time T ′ ≥ Tα) = [a+ b(T ′)K]V swpt

0 (K) +∫∞K

V swpt0 (K) · 2b(T ′)dK

a = 1∑βi=α+1 τi

b(T ′) = 1Sα,β(0)

[P (0,T ′)

Nα,β0

− 1∑βi=α+1 τi

], T ′ ≥ Tα

(18)

3.3.4 CMS caplet, floorlet, and floater paid at arbitrary time under Hagan’s model

In Section 2.2.2, we explained several models for the ratio

P (t, T ′)/Nα,βt = G(Sα,β(t))

andP (0, T ′)/Nα,β

0 = G(Sα,β(0)),

where G(x) could be

G(x) =

x(1+x/q)∆

· 11− 1

(1+x/q)nModel 1

x(1+τα+1x)∆

· 1

1−∏β

i=α+11

1+τix

Model 2xe−(T ′−Tα)s

1−P (0,Tβ)

P (0,Tα)e−(Tβ−Tα)s

Model 3

xe−[h(T ′)−h(Tα)]s

1−P (0,Tβ)

P (0,Tα)e−[h(Tβ)−h(Tα)]s

Mdoel 4

with s determined implicitly in terms of x by

x

β∑i=α+1

τiP (0, Ti)e−(Ti−Tα)s = P (0, Tα)− P (0, Tβ)e

−(Tβ−Tα)s

for Model 3, and s determined implicity in terms of x by

x

β∑i=α+1

τiP (0, Ti)e−[h(Ti)−h(Tα)]s = P (0, Tα)− P (0, Tβ)e

−[h(Tβ)−h(Tα)]s.

17

for Model 4.We first consider CMS caplet. The payoff of [Sα,β(Tα)−K]+ at time T ′ ≥ Tα is equivalent to a payoff

ofP (Tα, T

′)[Sα,β(Tα)−K]+.

at time Tα. By risk-neutral pricing,

V0([Sα,β(Tα)−K]+ paid at time T ′ ≥ Tα)

= Nα,β0 EQα,β

[P (Tα, T

′)[Sα,β(Tα)−K]+

Nα,βTα

]

= P (0, T ′)EQα,β

[P (Tα, T

′)/Nα,βTα

P (0, T ′)/Nα,β0

[Sα,β(Tα)−K]+

]

= P (0, T ′)EQα,β {[Sα,β(Tα)−K]+

}+ P (0, T ′)EQα,β

[(P (Tα, T

′)/Nα,βTα

P (0, T ′)/Nα,β0

− 1

)[Sα,β(Tα)−K]+

]

where we observe that(

P (t,T ′)

Nα,βt

)0≤t≤T ′

is a martingale under swap measure Qα,β and

EQα,β

[P (Tα, T

′)

Nα,βTα

]=

P (0, T ′)

Nα,β0

The first term is exactly the price of a European swaption with notional P (0, T ′)/Nα,β0 , regardless of

how the swap rate Sα,β(t) is modeled. The second term is the “convexity correction”. Since Sα,β(t) is amartingale and P (Tα,T ′)/Nα,β

Tα

P (0,T ′)/Nα,β0

− 1 is zero on average, this term goes to zero linearly with the variance of theswap rate Sα,β(t) and is much smaller than the first term.

Under a general rate model, we assume P (t, T ′)/Nα,βt = G(Sα,β

t ). Then for f(x) := [G(x)/G(Sα,β(0))−1](x−K), we have the property

f ′(K)(x−K)+ +

∫ ∞

K

(x− ζ)+f ′′(ζ)dζ =

{f(x) for x ≥ K

0 for x < K

So[G(x)/G(Sα,β(0))− 1](x−K)+ = f ′(K)(x−K)+ +

∫ ∞

K

(x− ζ)+f ′′(ζ)dζ

and

EQα,β

[(P (Tα, T

′)/Nα,βTα

P (0, T ′)/Nα,β0

− 1

)[Sα,β(Tα)−K]+

]= EQα,β

{[G(Sα,β

Tα)/G(Sα,β(0))− 1](Sα,β

Tα−K)+

}= f ′(K)EQα,β

[(Sα,βTα

−K)+] +

∫ ∞

K

EQα,β

[(Sα,β(Tα)− ζ)+]f ′′(ζ)dζ

Therefore, the convexity correction is equal to

cc =P (0, T ′)

Nα,β0

[f ′(K)C(K) +

∫ ∞

K

C(ζ)f ′′(ζ)dζ

]and the time zero price of a CMS caplet paid at time T ′ ≥ Tα is

V0([Sα,β(Tα)−K]+ paid at time T ′ ≥ Tα) =P (0, T ′)

Nα,β0

{[1 + f ′(K)]C(K) +

∫ ∞

K

C(ζ)f ′′(ζ)dζ

}

18

Here C(K) = Nα,β0 EQα,β

[(Sα,β(Tα)−K)+] is the time zero value of a vanilla payer swaption with strike K.Repeating the above argument shows that the value of a CMS floorlet is

V0([K − Sα,β(Tα)]+ paid at time T ′ ≥ Tα) =

P (0, T ′)

Nα,β0

{[1 + f ′(K)]P (K)−

∫ K

−∞P (ζ)f ′′(ζ)dζ

}

where f(x) is the same as before and P (K) = Nα,β0 EQα,β

[(K − Sα,β(Tα))+] is the value of the receiver

swaption with strike K.By the call-put parity and by setting K = Sα,β(0), we have C(Sα,β(0)) − P (Sα,β(0)) = 0. Hence the

value of a CMS floater is

V0(Sα,β(Tα) paid at time T ′ ≥ Tα)

=P (0, T ′)Sα,β(0) +P (0, T ′)

Nα,β0

[∫ ∞

Sα,β(0)

C(ζ)f ′′atm(ζ)dζ +

∫ Sα,β(0)

−∞P (ζ)f ′′

atm(ζ)dζ

]

wherefatm(x) = [G(x)/G(Sα,β(0))− 1](x− Sα,β(0))

As a last word, by call-put parity, we can price an in-the-money caplet or floorlet as a swaplet plus anout-of-the-money floorlet or caplet.

Analytical formulas: The function G(x) is smooth and slowly varying, regardless of the model used toobtain it. Since the probable swap rates Sα,β(·) are heavily concentrated around around Sα,β(0), it makessense to expand G(x) as

G(x) ≈ G(Sα,β(0)) +G′(Sα,β(0))(x− Sα,β(0)) + · · ·

If we take the expansion to the linear term, f(x) becomes a quadratic function

f(x) ≈ G′(Sα,β(0))

G(Sα,β(0))(x− Sα,β(0))(x−K).

Thenf ′(K) =

G′(Sα,β(0))

G(Sα,β(0))[K − Sα,β(0)] , f

′′(ζ) = 2G′(Sα,β(0))

G(Sα,β(0)).

Consequently, we have (note G(Sα,β(0)) =P (0,T ′)

Nα,β0

)

V0([Sα,β(Tα)−K]+ paid at time T ′ ≥ Tα)

=P (0, T ′)

Nα,β0

C(K) +G′(Sα,β(0))

[(K − Sα,β(0))C(K) + 2

∫ ∞

K

C(ζ)dζ

]

V0([K − Sα,β(Tα)]+ paid at time T ′ ≥ Tα)

=P (0, T ′)

Nα,β0

P (K) +G′(Sα,β(0))

[(K − Sα,β(0))P (K)− 2

∫ K

−∞P (ζ)dζ

]

and

V0(Sα,β(Tα) paid at time T ′ ≥ Tα) = P (0, T ′)Sα,β(0) + 2G′(Sα,β(0))

[∫ ∞

Sα,β(0)

C(ζ)dζ +

∫ Sα,β(0)

−∞P (ζ)dζ

]

19

We note 1∫ ∞

K

C(ζ)dζ = Nα,β0

∫ ∞

K

EQα,β

[(Sα,β(Tα)− ζ)+]dζ =1

2Nα,β

0 EQα,β {([Sα,β(Tα)−K]+)2

}Therefore, an alternative form of the formulas is

V0([Sα,β(Tα)−K]+ paid at time T ′ ≥ Tα)

=P (0, T ′)

Nα,β0

C(K) +G′(Sα,β(0))Nα,β0 EQα,β

{[Sα,β(Tα)− Sα,β(0)] [Sα,β(Tα)−K]

+}

for the CMS caplets,

V0([K − Sα,β(Tα)]+ paid at time T ′ ≥ Tα)

=P (0, T ′)

Nα,β0

P (K)−G′(Sα,β(0))Nα,β0 EQα,β

{[Sα,β(0)− Sα,β(Tα)] [K − Sα,β(Tα)]

+}

for CMS floorlets, and

V0(Sα,β(Tα) paid at time T ′ ≥ Tα)

= P (0, T ′)Sα,β(0) +G′(Sα,β(0))Nα,β0 EQα,β

{[Sα,β(Tα)− Sα,β(0)]

2}

for floater.To finish the calculation, one needs an explicit model for the swap rate Sα,β(t). There are two simple

models one can use. The first model is Black’s model, which assumes that the swap rate Sα,β(Tα) is lognormal with a volatility σB :

Sα,β(Tα) = Sα,β(0)eσBWTα− 1

2σ2BTα .

With this model, one obtains (see Appendix D)

V floater0 = P (0, T ′)Sα,β(0) +G′(Sα,β(0))N

α,β0 (Sα,β(0))

2(eσ

2BTα − 1

)for CMS floaters,

V caplet0

=P (0, T ′)

Nα,β0

C(K) +G′(Sα,β(0))Nα,β0

[(Sα,β(0))

2eσ2BTαΦ(d3/2)− Sα,β(0)(Sα,β(0) +K)Φ(d1/2)

+Sα,β(0)KΦ(d−1/2)]

for CMS caplets, and

V floorlet0

=P (0, T ′)

Nα,β0

P (K)−G′(Sα,β(0))Nα,β0

[(Sα,β(0))

2eσ2BTαΦ(−d3/2)− Sα,β(0)(Sα,β(0) +K)Φ(−d1/2)

+Sα,β(0)KΦ(−d−1/2)]

for CMS floorlets. Here

dλ =ln(

Sα,β(0)K

)+ λσ2

BTα

σB

√Tα

.

1The interpretation of∫∞K C(ζ)dζ = 1

2Nα,β

0 EQα,β {([Sα,β(Tα)−K]+)2

}is that the value of the right side is obtained

through the value of the left side: C(K)’s are directly given by market quotes for various K’s, and we don’t know the distributionof Sα,β(Tα) without further assumptions.

20

The second model is the normal, or absolute model, which assumes that the swap rate follows

dSα,β(t) = σNdWt.

This yieldsV floater0 = P (0, T ′)Sα,β(0) +G′(Sα,β(0))N

α,β0 σ2

NTα

for the CMS floaters,

V caplet0 =

P (0, T ′)

Nα,β0

C(K) +G′(Sα,β(0))Nα,β0 σ2

NTαΦ

(Sα,β(0)−K

σN

√Tα

)for CMS caplets, and

V floorlet0 =

P (0, T ′)

Nα,β0

P (K)−G′(Sα,β(0))Nα,β0 σ2

NTαΦ

(K − Sα,β(0)

σN

√Tα

)for CMS floorlets. We can obtain the normal vol σN by noting that if σB is the lognormal volatility for aswaption with forward rate Sα,β(0) and strike K, then the normal volatility σN of this swaption is

σN = σBSα,β(0)−K

ln(

Sα,β(0)K

) 1

1 + 124σ

2BTα + 1

5760σ4BT

2α

Near the money ((Sα,β(0)−K)/K is less than 20% or so), we can replace this formula with

σN = σB

√Sα,β(0)K ·

1 + 124 ln2

(Sα,β(0)

K

)+ 1

1920 ln4(

Sα,β(0)K

)1 + 1

24σ2BTα + 1

5760σ4BT

2α

N.B. The key concern with Black’s model is that it does not address the smiles and/or skews seen inthe marketplace. This can be partially mitigated by using the correct volatilities. For CMS floaters, thevolatility σatm for at-the-money swpations should be used, since the expected value includes high and lowstrike swaption equally. For out-of-the-money caplets and floorlets, the volatility σK for strike K shouldbe used, since the swap rate Sα,β(Tα) near K provide the largest contribution to the expected value. Forin-the-money options, the largest contributions come from swap rates Sα,β(Tα) near the mean value Sα,β(0).Accordingly, call-put parity should be used to evaluate in-the-money caplets and floorlets as a CMS swappayment plus an out-of-the-money floorlet or caplet.

3.3.5 CMS digitals paid at arbitrary time under Hagan’s model

To compute the time-zero value of a CMS digital call/put, we note

V0(1{Sα,β(Tα)>K} paid at time T ′ ≥ Tα)

= P (0, T ′)EQT ′[1{Sα,β(Tα)>K}

]= − d

dKP (0, T ′)EQT ′

[(Sα,β(Tα)−K)+

]= − d

dKV0([Sα,β(Tα)−K]+ paid at time T ′ ≥ Tα)

Similarly,

V0(1{Sα,β(Tα)<K} paid at time T ′ ≥ Tα) =d

dKV0([K − Sα,β(Tα)]

+ paid at time T ′ ≥ Tα).

Therefore, from the integral representation of CMS caplet/floorlet price, we have

V0(1{Sα,β(Tα)>K} paid at time T ′ ≥ Tα)

=− P (0, T ′)

Nα,β0

d

dKC(K) +G′(Sα,β(0))

[(Sα,β(0)−K)

d

dKC(K) + C(K)

]

21

V0(1{Sα,β(Tα)<K} paid at time T ′ ≥ Tα)

=P (0, T ′)

Nα,β0

d

dKP (K)−G′(Sα,β(0))

[(Sα,β(0)−K)

d

dKP (K) + P (K)

]Under the lognormal model,

C(K) = Nα,β0 [Sα,β(0)Φ(d1)−KΦ(d2)]

where d1 = d1(σK) =ln(

Sα,β(0)

K

)+ 1

2σ2KTα

σK

√Tα

and d2 = d2(σK) = d1(σK)− σK

√Tα. Since

d

dK[Sα,β(0)Φ(d1)−KΦ(d2)] =

∂

∂K[Sα,β(0)Φ(d1)−KΦ(d2)] +

∂

∂σ[Sα,β(0)Φ(d1)−KΦ(d2)]

dσK

dK

= − 1

KσK

√Tα

[Sα,β(0)Φ′(d1)−KΦ′(d2)]− Φ(d2) + Sα,β(0)Φ

′(d1)√

TαdσK

dK

and

Sα,β(0)Φ′(d1) = Sα,β(0) ·

1√2π

e−(d2+σK

√Tα)2

2 = Sα,β(0)e−d2σK

√Tα− 1

2σ2KTαΦ′(d2) = KΦ′(d2),

we conclude ddKC(K) = Nα,β

0

[−Φ(d2) + Φ′(d2)K

dσK

dK

√Tα

]. Therefore, the value of a digital call under

lognormal model is

V0(1{Sα,β(Tα)>K} paid at time T ′ ≥ Tα)

= P (0, T ′)

[Φ(d2)− Φ′(d2)K

dσK

dK

√Tα

]+G′(Sα,β(0))N

α,β0 Sα,β(0) {Φ(d1)− Φ(d2)

−[Φ′(d1)− Φ′(d2)]KdσK

dK

√Tα

}By the call-put parity, the value of a digital put under lognormal model is

V0(1{Sα,β(Tα)<K} paid at time T ′ ≥ Tα)

= P (0, T ′)− V0(1{Sα,β(Tα)>K} paid at time T ′ ≥ Tα)

= P (0, T ′)

[Φ(−d2) + Φ′(d2)K

dσK

dK

√Tα

]+G′(Sα,β(0))N

α,β0 Sα,β(0) {Φ(d2)− Φ(d1)

−[Φ′(d2)− Φ′(d1)]KdσK

dK

√Tα

}Under the normal model,

C(K) = Nα,β0

[(Sα,β(0)−K)Φ(d1) + Φ′(d1)σ

√Tα

]where d1 =

Sα,β(0)−K

σ√Tα

and σ is the normal vol. Since

d

dK

[(Sα,β(0)−K)Φ(d1) + Φ′(d1)σ

√Tα

]=

∂

∂K

[(Sα,β(0)−K)Φ(d1) + Φ′(d1)σ

√Tα

]+

∂

∂σ

[(Sα,β(0)−K)Φ(d1) + Φ′(d1)σ

√Tα

] dσ

dK

= −Φ(d1) + Φ′(d1)√Tα

dσ

dK,

we conclude ddKC(K) = Nα,β

0

[−Φ(d1) + Φ′(d1)

√Tα

dσdK

]. Therefore, the value of a digital call under normal

model is

V0(1{Sα,β(Tα)>K} paid at time T ′ ≥ Tα)

= P (0, T ′)

[Φ(d1)− Φ′(d1)

√Tα

dσ

dK

]+G′(Sα,β(0))N

α,β0 Φ′(d1)

√Tα

[(Sα,β(0)−K)

dσ

dK+ σ

]

22

By the call-put parity, the value of a digital put under normal model is

V0(1{Sα,β(Tα)<K} paid at time T ′ ≥ Tα)

= P (0, T ′)

[Φ(−d1) + Φ′(d1)

√Tα

dσ

dK

]+G′(Sα,β(0))N

α,β0 Φ′(d1)

√Tα

[(K − Sα,β(0))

dσ

dK− σ

]3.3.6 CMS fracl paid at arbitrary time under linear rate model

A floating CMS range accrual pays out[leverage · S1(t

10) + spread

]1{Rmin<S2(t20)<Rmax} + fixRate

at time tp ≥ t20 ≥ t10, where S1(·) is a swap rate with schedule t10 < t11 < · · · < t1n1and corresponding coverage

τ11 < · · · < τ1n1, and S2(·) is a swap rate with schedule t20 < t21 < · · · < t2n2

and corresponding coverageτ21 < · · · < τ2n2

. The time-zero value of the payoff is

V0 = P (0, tp)(

leverage · EQtp

[1{Rmin<S2(t20)<Rmax}S1(t

10)]+ spread · EQtp

[1{Rmin<S2(t20)<Rmax}

])+P (0, tp) · fixRate

Denote by L1 and L2 the annuities corresponding to S1 and S2, respectively. Then by formula (5) fromthe linear rate model

EQtp

[1{Rmin<S2(t20)<Rmax}S1(t

10)]

= EL2

[P (t20, tp)/P (0, tp)

L2(t20)/L2(0)· 1{Rmin<S2(t20)<Rmax}S1(t

10)

]= EL2

[(1 + a+ bS2(t

20)) · 1{Rmin<S2(t20)<Rmax}S1(t

10)]

where a = L2(0)

P (0,tp)∑n2

i=1 τ2i

− 1, b = 1S2(0)

[1− L2(0)

P (0,tp)∑n2

i=1 τ2i

]. Assume EL2 [S1(t

10)|S2(t

20)] is a linear function

of S2(t20):

EL2 [S1(t10)|S2(t

20)]− EL2 [S1(t

10)] ≈ C

(S2(t

20)− EL2 [S2(t

20)])= C[S2(t

20)− S2(0)]

By requiring EL2[EL2 [S1(t

10)|S2(t

20)] · S2(t

20)]= EL2

[S1(t

10)S2(t

20)], we obtain

C = corrL2(S1(t10), S2(t

20))

√varL2S1(t10)

varL2S2(t20)

Consequently, we have

EQtp

[1{Rmin<S2(t20)<Rmax}S1(t

10)]

= EL2

[(1 + a+ bS2(t

20)) · 1{Rmin<S2(t20)<Rmax}

(EL2 [S1(t

10)] + C

[S2(t

20)− S2(0)

])]To compute EL2 [S1(t

10)], we apply the linear rate model L2(t)

L1(t)= a1 + b1S1(t)

EL2 [S1(t10)] = EL1

[L2(t

10)/L2(0)

L1(t10)/L1(0)S1(t

10)

]= EL1

[(a1 + b1S1(t

10))S1(t

10)]

= a1S1(0) + b1S21(0) + b1var

L1S1(t10)

where a1 =L1(0)

∑n2i=1 τ2

i

L2(0)∑n1

j=1 τ1j

and b1 = 1S1(0)

[1− L1(0)

∑n2i=1 τ2

i

L2(0)∑n1

j=1 τ1j

](see Appendix A). Define

E0 = EQtp

[1{Rmin<S2(t20)<Rmax}

],

23

then

V0 = P (0, tp)(

leverage · EQtp

[1{Rmin<S2(t20)<Rmax}S1(t

10)]+ spread · E0

)+ P (0, tp) · fixRate

Define

E0 = EL2

[1{Rmin<S2(t20)<Rmax}

]E1 = EL2

[1{Rmin<S2(t20)<Rmax}S2(t

20)]

E2 = EL2

[1{Rmin<S2(t20)<Rmax}S

22(t

20)]

E0cc = aE0 + bE1 = EL2

[1{Rmin<S2(t20)<Rmax}(a+ bS2(t

20))]

E1cc = aE1 + bE2 = EL2

[1{Rmin<S2(t20)<Rmax}S2(t

20)(a+ bS2(t

20))]

EA1 = EL1 [(a1 + b1S1(t10))S1(t

10)] = EL2 [S1(t

10)]

We can write EQtp

[1{Rmin<S2(t20)<Rmax}S1(t

10)]

as

EQtp

[1{Rmin<S2(t20)<Rmax}S1(t

10)]

= EL2

[(1 + a+ bS2(t

20)) · 1{Rmin<S2(t20)<Rmax}

(EL2 [S1(t

10)] + C

[S2(t

20)− S2(0)

])]= EL2

[1{Rmin<S2(t20)<Rmax}

(EL2 [S1(t

10)] + C

[S2(t

20)− S2(0)

])]+EL2

[(a+ bS2(t

20)) · 1{Rmin<S2(t20)<Rmax}

] (EL2 [S1(t

10)]− C · S2(0)

)+EL2

[(a+ bS2(t

20)) · 1{Rmin<S2(t20)<Rmax}C · S2(t

20)]

= [EA1 − C · S2(0)] · E0 + C · E1 + [EA1 − C · S2(0)] · E0cc+ C · E1cc

= E + cc

where E = [EA1 − C · S2(0)] · E0 + C · E1 and cc = [EA1 − C · S2(0)] · E0cc+ C · E1cc. Therefore

V0 = P (0, tp)[leverage · (E + cc) + spread · E0

]+ P (0, tp) · fixRate (19)

The following table shows how to compute the various quantities in the above derivation

Formula Black Model (S > 0 always)S1(t

10) = S1(0)e

σ1BW

t10− 1

2 (σ1B)2t10 ,

S2(t20) = S2(0)e

σ2BW

t20− 1

2 (σ2B)2t20

E0 = EQtp

[1{Rmin<S2(t20)<Rmax}

]replication (Appendix C)

E0 = EL2 [1{Rmin<S2(t20)<Rmax}] Φ

(ln Rmax

S2(0)

σ2B

√t20

+ 12σ

2B

√t20

)− Φ

(ln Rmin

S2(0)

σ2B

√t20

+ 12σ

2B

√t20

)E1 = EL2 [1{Rmin<S2(t20)<Rmax}S2(t

20)] S2(0)

[Φ

(ln Rmax

S2(0)

σ2B

√t20

− 12σ

2B

√t20

)− Φ

(ln Rmin

S2(0)

σ2B

√t20

− 12σ

2B

√t20

)]E2 = EL2 [1{Rmin<S2(t20)<Rmax}S

22(t

20)] S2

2(0)e(σ2

B)2t20

[Φ

(ln Rmax

S2(0)

σ2B

√t20

− 32σ

2B

√t20

)−Φ

(ln Rmin

S2(0)

σ2B

√t20

− 32σ

2B

√t20

)]E0cc = aE0 + bE1

E1cc = aE1 + bE2

EA1 = EL1 [(a1 + b1S1(t10))S1(t

10)] a1S1(0) + b1S

21(0)e

(σ1B)2t10

E = [EA1 − C · S2(0)] · E0 + C · E1

cc = [EA1 − C · S2(0)] · E0cc+ C · E1cc

24

By noting Φ′′(x) = −xΦ′(x), it’s easy to derive∫ y

−∞ xΦ′(x)dx = −Φ′(y) and∫ y

−∞ x2Φ′(x)dx = Φ(y) −yΦ′(y). Therefore we have

Formula Normal Model (S could be negative)S1(t

10) = S1(0) + σ1

NWt10, S2(t

10) = S2(0) + σ2

NWt20

E0 = EQtp

[1{Rmin<S2(t20)<Rmax}

]replication (Appendix C)

E0 = EL2 [1{Rmin<S2(t20)<Rmax}] Φ

(Rmax−S2(0)

σ2N

√t20

)− Φ

(Rmin−S2(0)

σ2N

√t20

)E1 = EL2 [1{Rmin<S2(t20)<Rmax}S2(t

20)] σ2

N

√t20

[Φ′(

Rmin−S2(0)

σ2N

√t20

)− Φ′

(Rmax−S2(0)

σ2N

√t20

)]+S2(0)

[Φ

(Rmax−S2(0)

σ2N

√t20

)− Φ

(Rmin−S2(0)

σ2N

√t20

)]E2 = EL2 [1{Rmin<S2(t20)<Rmax}S

22(t

20)] (σ2

N )2t20 [Φ (y)− yΦ′ (y)]|(Rmax−S2(0))/(σ2N t20)

(Rmin−S2(0))/(σ2N t20)

+2S2(0)E1 − S22(0)E0

E0cc = aE0 + bE1

E1cc = aE1 + bE2

EA1 = EL1 [(a1 + b1S1(t10))S1(t

10)] a1S1(0) + b1S

21(0) + b1(σ

1N )2t10

E = [EA1 − C · S2(0)] · E0 + C · E1

cc = [EA1 − C · S2(0)] · E0cc+ C · E1cc

4 Comparison of various convexity adjusted prices4.1 LIBOR-in-arrears swaps

We compare pricing formulas of LIBOR-in-arrears swaps under replication (formula (13)), convexitycorrection with adjusted forward rate (formula (2)), and convexity correction with adjusted forward rate(formula (2)) and adjusted volatility (formula (10))

V0(τ(S, T )L(S, T ) paid at time S)

=

V caplet0 (0) + 2τ(S, T )

∫∞0

V caplet0 (K)dK replication

τ(S, T )P (0, S)F (0;S, T )[1 +

τ(S,T )F (0;S,T )(exp{σ2atmS}−1)

1+τ(S,T )F (0;S,T )

]adj Fwd

τ(S, T )P (0, S)F (0;S, T )[1 + τ(S,T )F (0;S,T )(exp{(σ∗)2S}−1)

1+τ(S,T )F (0;S,T )

]adj Fwd & Vol

where σatm is the at-the-money volatility of the caplet for the period [S, T ] and

(σ∗)2 = (σatm)2 + ln{[a+ b(S)F (0;S, T )][a+ b(S)F (0;S, T )e2σ

2atmS ]

[a+ b(S)F (0;S, T )eσ2atmS ]2

}/S

= (σatm)2 + ln{[1 + τ(S, T )F (0;S, T )][1 + τ(S, T )F (0;S, T )e2σ

2atmS ]

[1 + τ(S, T )F (0;S, T )eσ2atmS ]2

}/S

The case of “adj Fwd & Vol” is equivalent to assuming L(S, T ) is lognormal under QS , while the case of“adj Fwd” is kind of halfway from “lognormal under QT ” to “lognormal under QS”.

Numerical experiments show results by these formulas are quite close and the two naive approaches(adjusted forward rates without and with additional adjusted volatilities) yield results whose difference tothe correct one from replication are negligible in practice.

4.2 In-arrears capletsWe compare pricing formulas of in-arrears caplets under replication (formula (14)), Black formula with

adjusted forward rate (formula (2)), and Black’s formula with adjusted forward rate (formula (2)) and

25

adjusted volatility (formula (10))

V0(τ(S, T )[L(S, T )−K]+ paid at time S)

=

V caplet0 (K)[1 + τ(S, T )K] + 2τ(S, T )

∫∞K

V caplet0 (K)dK replication

τ(S, T )P (0, S) [Fadj(σstrike)Φ(d1)−KΦ(d2)] Black formula with adj Fwdτ(S, T )P (0, S) [Fadj(σ

∗)Φ(d∗1)−KΦ(d∗2)] Black formula with adj Fwd & Vol

where Fadj(σ) = F (0;S, T )[1 + τ(S,T )F (0;S,T )(exp{σ2S}−1)

1+τ(S,T )F (0;S,T )

]is the convexity adjusted forward rate, σstrike is

the volatility taken from the smile according to the strike rate of the caplet, Φ(·) is the c.d.f. of a standardnormal distribution,

d1 =ln(Fadj(σstrike)/K) + σ2

strikeS/2

σstrike

√S

, d2 = d1 − σstrike

√S,

and

(σ∗)2 = (σstrike)2 + ln

{[1 + τ(S, T )F (0;S, T )][1 + τ(S, T )F (0;S, T )e2σ

2strikeS ]

[1 + τ(S, T )F (0;S, T )eσ2strikeS ]2

}/S

d∗1 =ln(Fadj(σ

∗)/K) + (σ∗)2S/2

σ∗√S

, d∗2 = d∗1 − σ∗√S.

The case of “Black formula with adj Fwd & Vol” is the market standard method of valuing options onconvexity adjusted rates: assume the underlying rate is lognormal under QS and apply the Black formula.

Numerical experiments show results by these formulas are quite close and the two naive approaches(adjusted forward rates without and with additional adjusted volatilities) yield results whose difference tothe correct one from replication are negligible in practice.

4.3 CMS swaps and CMS capsFormula (16) and formula (18) give prices by replication. For prices in terms of adjusted swap rate, the

adjusted swap rate is given by

EQT ′ [Sα,β(Tα)] = Sα,β(0)

[1 +

(1− P (0, Tα)− P (0, Tβ)

Sα,β(0)P (0, T ′)∑β

i=α+1 τi

)(eσ

2atmTα − 1

)].

For price in terms of adjusted forward swap rate and adjusted volatility, the adjusted volatility is given by

(σ∗)2 = (σatm)2 + ln

[a+ b(T ′)Sα,β(0)][a+ b(T ′)Sα,β(0)e

2(σatm)2Tα

][a+ b(T ′)Sα,β(0)eσ

2atmTα ]2

/Tα.

Again, price in terms of adjusted forward rate and adjusted volatility is obtained by assuming theunderlying rate is lognormal under QT ′ and applying Black formula.

The naive approaches, although taking into account the smile by taking the caplet volatility from theswaption smile, show a significant mispricing relative to the correct valuations based on replication. Com-paring the results of the two replication formulas, it turns out that the replication based on the idea ofcash-settled swaptions consistently leads to slightly higher CMS caplet prices.

5 Variable interest rates in foreign currencyWe are interested in the price of a domestic interest rate to be paid in foreign currency units at some

time T > 0.

26

5.1 Multi-currency change of numeraire theoremSuppose we have a domestic economy d and a foreign economy f , together with the exchange rate X that

expresses the value of one unit foreign currency in terms of domestic currency. Let Nd denote a numeraireprocess with associated martingale measure Qd for the domestic economy, and Nf a numeraire process withassociated martingale measure Qf for the foreign economy.

The domestic value today of a domestic payoff ZdT to be paid in foreign units and at time T is by the

general theory

X0Nf0 E

Qf

[ZdT

NfT

].

On the other hand, the same payoff translated back into domestic currency with the exchange rate at timeT should trade at the same domestic price, therefore

X0Nf0 E

Qf

[ZdT

NfT

]= Nd

0EQd

[ZdTXT

NdT

].

This gives us the multi-currency change of numeraire theorem

dQd

dQf

∣∣∣∣FT

=Nd

T /Nd0

NfT /N

f0

· X0

XT,dQf

dQd

∣∣∣∣FT

=Nf

T /Nf0

NdT /N

d0

· XT

X0. (20)

5.2 Quanto adjustmentsWe consider a domestic variable interest rate Y d

S set at time S and paid at time T ′ ≥ S in foreign currencyunits. Let P f (·, T ) denote the time-T maturity foreign zero coupon bond and Qf

T the associated foreignT -forward measure. The time-zero value of Y d

S in foreign currency is

P f (0, T ′)EQf

T ′ [Y dS ] = P f (0, T ′)EQd

[Y dS

P f (S, T ′)/P f (0, T ′)

NdS/N

d0

· XS

X0

]=

Nd0

X0EQd

[Y dS

XSPf (S, T ′)

P d(S, T ′)

P d(S, T ′)

NdS

]By definition the ratio Pd(S,T ′)

NdS

is a Qd-martingale in the time variable S ≤ T ′. The expression XSP f (S,T ′)Pd(S,T ′)

is the time-S forward foreign exchange rate for delivery at time T ′ ≥ S.

Assumption 1: We assume the domestic numeraire Nd is the natural numeraire associated with Y dS

such that Y d0 = EQd

[Y dS ], and

Y dS = Y d

0 eσY WS− 1

2σ2Y S

where W is a standard Brownian motion under Qd.Assumption 2: We assume the linear rate model for Pd(S,T ′)

NdS

:

P d(S, T ′)

NdS

= a+ b(T ′)Y dS , T

′ ≥ S.

Assumption 3: The forward foreign exchange rate is lognormally distributed:

XSPf (S, T ′)

P d(S, T ′)= X∗

0eσfxW

fxS − 1

2σ2fxS

where W fx is a standard Brownian motion under Qd and X∗0 = EQd

[XSP f (S,T ′)Pd(S,T ′)

]. Here σfx is identified

with the implied volatility of a foreign exchange rate option with maturity S, which is a crucial simplificationas long as the payment date T ′ is not close to S.

27

To calculate X∗0 , we note

X0Pf (0, T ′)

Nd0

= EQd

[XSP

f (S, T ′)

NdS

]= EQd

[XSP

f (S, T ′)

P d(S, T ′)

P d(S, T ′)

NdS

]= X∗

0 [a+ b(T ′)Y d0 e

ρσfxσY S ],

with ρ as correlation between the driving Brownian motions W fx and W . Therefore

X∗0 =

X0Pf (0, T ′)

Nd0 [a+ b(T ′)Y d

0 eρσfxσY S ]

.

Through tedious calculation, we can conclude

EQf

T ′ [Y dS ] = Y d

0

eρσfxσY S [a+ b(T ′)Y d0 e

ρσfxσY S+σ2Y S ]

a+ b(T ′)Y d0 e

ρσfxσY S

5.3 Quantoed options on interest ratesUnder Assumption 1-3, the call option on the domestic variable rate Y d

S set at time S and paid at timeT ′ ≥ S in foreign currency units is given by

P f (0, T ′)EQf

T ′ [(Y dS −K)+]

=P f (0, T ′)

a+ b(T ′)Y d0 e

ρσfxσY S

[Y d0 e

ρσfxσY SΦ(d1 + ρσfx

√S)(a− b(T ′)K)+

b(T ′)(Y d0 )

2e(σ2Y +2ρσfxσY )SΦ(d1 + (ρσfx + σY )

√S)− aKΦ(d2 + ρσfx

√S)]

with

d1 =ln(

Y d0

K

)+ 1

2σ2Y S

σY

√S

, d2 =ln(

Y d0

K

)− 1

2σ2Y S

σY

√S

.

The corresponding put formula is

P f (0, T ′)EQf

T ′ [(Y dS −K)+]

=P f (0, T ′)

a+ b(T ′)Y d0 e

ρσfxσY S

[−Y d

0 eρσfxσY SΦ(−d1 − ρσfx

√S)(a− b(T ′)K)−

b(T ′)(Y d0 )

2e(σ2Y +2ρσfxσY )SΦ(−d1 − (ρσfx + σY )

√S) + aKΦ(−d2 − ρσfx

√S)]

Finally, the formula for a digital is

P f (0, T ′)EQf

T ′ [1{Y dS >K}] =

P f (0, T ′)

a+ b(T ′)Y d0 e

ρσfxσY S

[b(T ′)Y d

0 eρσfxσY SΦ(d1 + ρσfx

√S) + aΦ(d2 + ρσfx

√S)].

Remark 8. In the special case of Y d being the domestic LIBOR rate Ld(S, T ) for the period of [S, T ] andT ′ = T , we have Nd(·) = P d(·, T ) and a = 1, b ≡ 0 in Assumption 2. The price of a call option on Ld(S, T )paid at time T in foreign currency units is therefore

P f (0, T )EQfT [(Ld(S, T )−K)+] = P f (0, T )[Y d

0 eρσfxσY SΦ(d1 + ρσfx

√S)−KΦ(d2 + ρσfx

√S)]

= P f (0, T )[Y d0 Φ(d1)−KΦ(d2)]

where

d1 =ln(

Y d0

K

)+ 1

2σ2Y S

σY

√S

, d2 =ln(

Y d0

K

)− 1

2σ2Y S

σY

√S

.

This is the market standard method of valuing options on convexity corrected rates: apply the Black formulausing the convexity corrected rate as the forward rate.

28

A Approximation of ratio of annuities under linear rate modelIn pricing floating rate CMS range accrual, we need to approximate the ratio of two annuities by swap

rate. More formally, suppose t10 < t11 < · · · < t1n1and t20 < t21 < · · · < t2n2

are two schedules with theirrespective day counting conventions {τ11 , τ12 , · · · , τ1n1

} and {τ21 , τ22 , · · · , τ2n2}. For k = 1, 2, annuity Lk(t) is

defined as

Lk(t) =

nk∑i=1

τki P (t, tki ), t ≤ tk0

and forward swap rate Sk(t) is defined as

Sk(t) =P (t, tk0)− P (t, tknk

)∑nk

i=1 τki P (t, tki )

=P (t, tk0)− P (t, tknk

)

Lk(t), t ≤ tk0 .

We want to find out the linear approximation L2(t)L1(t)

= α+ βS1(t). Indeed, by formula (5)

L2(t)

L1(t)=

n2∑i=1

τ2i P (t, t2i )

L1(t)=

n2∑i=1

τ2i[a+ b(t2i )S1(t)

]=

∑n2

i=1 τ2i∑n1

j=1 τ1j

+

n2∑i=1

τ2i · 1

S1(0)

[P (0, t2i )

L1(0)− 1∑n1

j=1 τ1j

]· S1(t)

Therefore, the linear swap rate model impliesL2(t)L1(t)

= α+ βS1(t)

α =∑n2

i=1 τ2i∑n1

j=1 τ1j

β = 1S1(0)

·[L2(0)L1(0)

−∑n2

i=1 τ2i∑n1

j=1 τ1j

] (21)

B Integral representation of convex functionsThis section is based on Niculescu and Persson [7], Section 1.6.It is well known that the differentiation and the integration are operations inverse to each other. A

consequence of this fact is the existence of a certain duality between the class of convex functions on an openinterval and the class of nondecreasing functions on that interval.

Given a nondecreasing function φ : I → R and a point c ∈ I, we can attach to them a new function f ,given by

f(x) =

∫ x

c

φ(t)dt.

It is easy to verify that f is a convex function and f is differentiable at each point of continuity of φ withf ′ = φ at such points.

On the other hand, the subdifferential allows us to state the following generalization of the fundamentalformula of integral calculus:

Proposition B.1. Let f : I → R be a continuous convex function and let φ : I → R be a function such thatφ(x) ∈ ∂f(x) for every x ∈ intI. Then for every a < b in I we have

f(b)− f(a) =

∫ b

a

φ(t)dt.

Proof. Clearly, we may restrict ourselves to the case where [a, b] ⊂ intI. If a = t0 < t1 < · · · < tn = b is apartition of [a, b], then

f ′−(tk−1) ≤ f ′

+(tk−1) ≤f(tk)− f(tk−1)

tk − tk−1≤ f ′

−(tk) ≤ f ′+(tk)

29

for every k. As f(b)− f(a) =∑n

k=1[f(tk)− f(tk−1)], a moment’s reflection shows that

f(b)− f(a) =

∫ b

a

f ′−(t)dt =

∫ b

a

f ′+(t)dt.

As f ′− ≤ φ ≤ f ′

+, this forces the equality in the statement.

A property very useful when joint with Proposition B.1 is the following

Proposition B.2. If f : I → R is absolutely continuous and its derivative φ is of finite variation locally,then f can be written as the difference of two convex functions.

Proof. By Fundamental Theorem of Calculus for absolutely continuous functions, we have

f(b)− f(a) =

∫ b

a

φ(t)dt.

By properties of functions of finite variation, φ can be written as the difference of two monotone increasingfunctions: φ = φ+ − φ−. Then

f(x) = f(a) +

∫ x

a

φ+(t)dt−∫ x

a

φ−(t)dt.

This proves the claim.

C Pricing of “trapezoidal” payoff by replicationThe payoff of a “trapezoidal” payoff is 1{Rmin<Y<Rmax}(Y · leverage + spread). It can be decomposed as

1{Rmin<Y<Rmax}(Y · leverage + spread)= leverage · [(Y −Rmin)

+ − (Y −Rmax)+] + 1{Y >Rmin}(Rmin · leverage + spread)

−1{Y >Rmax}(Rmax · leverage + spread)

Therefore, we have the following pricing formula via replication

V0 = leverage · [V caplet0 (Rmin)− V caplet

0 (Rmax)] + V digital_call0 (Rmin)(Rmin · leverage + spread)

−V digital_call0 (Rmax)(Rmax · leverage + spread)

D Computation of∫∞K0

C(K)dK and∫ K0

0 P (K)dK

We recall C(K) = FΦ(d1/2(K)) − KΦ(d−1/2(K)) and P (K) = −FΦ(−d1/2(K)) + KΦ(−d−1/2(K)),where dλ(K) = ln(F/K)

σ√T

+ λσ√T . We shall verify that

∫ ∞

K

C(x)dx =1

2F 2eσ

2TΦ(d3/2(K))− FKΦ(d1/2(K)) +1

2K2Φ(d−1/2(K))

and ∫ K

0

P (x)dx =1

2F 2eσ

2TΦ(−d3/2(K))− FKΦ(−d1/2(K)) +1

2K2Φ(−d−1/2(K))

We first note by Φ′(d−1/2(K)) = FKΦ′(d1/2(K)),

d

dKC(K) = FΦ′(d1/2(K))

(− 1

Kσ√T

)− Φ(d−1/2(K))−KΦ′(d−1/2(K)) ·

(− 1

Kσ√T

)= −Φ(d−1/2(K)).

30

So by integration-by-parts formula, we have∫ ∞

K

C(x)dx = xC(x)|∞K −∫ ∞

K

xd

dxC(x)dx = −KC(K) +

∫ ∞

K

xΦ(d−1/2(x))dx

= −KC(K) +x2

2Φ(d−1/2(x))

∣∣∣∣∞K

−∫ ∞

K

x2

2Φ′(d−1/2(x))

(− 1

xσ√T

)dx

= −KC(K)− K2

2Φ(d−1/2(K)) +

1

2σ√T

∫ ∞

K

xΦ′(d−1/2(x))dx

= −KFΦ(d1/2(K)) +K2

2Φ(d−1/2(K)) +

1

2σ√T

∫ ∞

K

xΦ′(d−1/2(x))dx

By the change-of-variable y = ln xF , we have∫ ∞

K

xΦ′(d−1/2(x))dx =

∫ ∞

ln KF

F 2e2ye− 1

2

(−y

σ√

T− 1

2σ√T)2

√2π

dy = σ√T · F 2eσ

2TΦ(d3/2(K))

Combined, we have obtained∫ ∞

K

C(x)dx =1

2F 2eσ

2TΦ(d3/2(K))− FKΦ(d1/2(K)) +1

2K2Φ(d−1/2(K))

By the put-call parity C(K)− P (K) = F −K and∫ ∞

0

C(x)dx = limK↓0

∫ ∞

K

C(x)dx =1

2F 2eσ

2T ,

we conclude∫ K

0

P (x)dx =

∫ K

0

[C(x)− F + x]dx =

∫ ∞

0

C(x)dx−∫ ∞

K

C(x)dx− FK +K2

2

=1

2F 2eσ

2T − FK +1

2K2 − 1

2F 2eσ

2TΦ(d3/2(K)) + FKΦ(d1/2(K))− 1

2K2Φ(d−1/2(K))

=1

2F 2eσ

2TΦ(−d3/2(K))− FKΦ(−d1/2(K)) +1

2K2Φ(−d−1/2(K))

References[1] W. Boenkost and W. Schmidt. Notes on convexity and quanto adjustments for interest rates and related

options. Working paper. October 2003. 3, 10

[2] W. Boenkost and W. Schmidt. Interest rate convexity and the volatility smile. Working paper. May2006. 3

[3] P. Hagan. Convexity conundrums: Pricing CMS swaps, cpas, and floors. Wilmott Magazine, March,p.38-44. 3, 7

[4] P. J. Hunt and J. E. Kennedy. Financial derivatives in theory and practice. Revised Edition, Wiley,2004. 3

[5] J. Hull. Options, futures, and other derivatives. Fourth Edition. Prentice-Hall, 2000. 3, 9

[6] A. Lesniewski. Convexity. 3

[7] Constantin P. Niculescu and Lars-Erik Persson. Convex functions and their applications: A contempo-rary approach. Springer. 29

[8] A. Pelsser. Mathematical foundation of convexity correction. Wroking paper. April 8, 2001. 3

[9] M. Piza. Convexity correction. January 18, 2007. 3

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