1. Fixed Strike Asian Cap/Floor on CMS Rates
with Lognormal Approach
July 27, 2011
Issue 1.1
Prepared by
Ling Luo and Anthony Vaz
2. Issue 1.1
Summary
An analytic pricing methodology has been developed for Asian Cap/Floor with fixed strike rates
on CMS rates. The price of an Asian Cap/Floor depends on deal parameters such as reset and
payment structures, CMS rate Tenor and number of average CMS rates; and market data such as
discount curve, swaption volatility curve, realized CMS rates and correlations between projected
CMS rates.
This pricing methodology can price Vanilla Asian Cap/Floor (with fixed averaging periods), as well as
Plain Vanilla Cap/Floor.
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4. Issue 1.1
1 Introduction
1.1 Scope
It is desired to find a valuation formula for an Asian CMS Cap/Floor with a fixed strike rate. A CMS
(Constant Maturity Swap) rate is equal to the par swap rate for a swap that starts immediately
for a specified tenor. Denote R swap (t; T, M ) as the forward swap rate determined at time t for a
swap starting at time T with a tenor of M years, which implies the swap matures at time T + M .
Then the CMS rate with M -year tenor at T can be expressed as R CMS (T, M ) = R swap (T ; T, M ).
An Asian CMS Cap is a series of call options or caplets on the average CMS rate observed every
reset over a specified time period. An Asian CMS Floor is a series of put options or floorlets on the
average CMS rate observed every reset over a specified time period. The average number of CMS
rates could be fixed or varying for each caplet/floorlet. A Vanilla Asian Cap/Floor is an Asian
Cap/Floor with fixed averaging periods for each caplet/floorlet. When the averaging
the Vanilla Asian Cap/Floor becomes a Plain Vanilla Cap/Floor.
The price of a product at time t is the present value of all future cash flows generated by
the product. An Asian CMS Cap/Floor can be decomposed into a series of caplets/floorlets and
its price at time t equals to the sum of time t prices of the caplets/floorlets whose payments occur
after t. Denote TP (k), k = 1 , 2, . . . , as the scheduled payment times of a Cap/Floor and V (t; TP (k))
is the time t price of the correponding caplet/floorlet with payment at TP (k), then the price of the
Cap/Floor at time t is given by
P (t) =
k:TP (k)>t
V (t; TP (k)) . (1)
The prices of the caplets/floorlets can be calculated independently. For the rest of the document,
we focus on the analytic formula for the price of one caplet/floorlet.
Assume N is the average number of CMS rates for a caplet/floorlet and T1, T 2, . . . , T N are
the corresponding reset times, then the option payo at time TP (TP > T N ) for $1 notional is given
as follows
V (TP ) = τP × ω ×
1
N
N
i=1
R CMS (Ti , M ) − K
+
,
where we use the notation [·]+ to mean max (·, 0), K is the fixed strike rate, ω is the option ID
with ω = 1 for caplets and ω = − 1 for floorlets, Ti is the reset time for CMS rate R CMS (Ti , M ),
and τP denotes the year fraction between payment dates. The value of the payo is unknown until
time TN and is paid on TP . To compute the value of the option at time t (t < T P ), we compute an
expectation in the TP forward measure; that is,
V (t) = P (t, T P ) × τP × E TP
ω ×
1
N
N
i=1
R CMS (Ti , M ) − K
+
F t , t < T P (2)
where P (t, T P ) is the discount factor between t and TP , E TP {·} denotes the expectation under TP
forward measure and F t denotes the information filtration at time t. If t ≥ TN , all the CMS rates
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periods are
fixed at 1,
5. Issue 1.1
are known and we have
V (t) = P(t, TP ) × τP ×
[
ω ×
(
1
N
N∑
i=1
RCMS(Ti, M) − K
)]+
, TN ≤ t < TP .
The difficulty in evaluating the above expectation is that we need to have distributional assumptions
about the average CMS rate that are consistent with market observations on forward swap rates
at time t.
It is a standard practice1 to assume that forward swap rates follow lognormal distribution.
The lognormal forward swap model prices swaptions with Black swaption formula, which is the
standard formula employed in the swaption market. As discussed in the book by Haug [5], it is
not possible to find a closed-form solution for the valuation of options on an arithmetic average.
The main reason is that the sum of lognormal variables will not have a lognormal distribution.
Arithmetic average-rate options can be priced by analytical approximations or with Monte Carlo
simulation. We adapt the approximation developed by Turnbull and Wakeman [11] by adjusting the
mean and variance of a lognormal distribution so that they are consistent with the exact moments
of the arithmetic average. The adjusted mean and variance are then used as inputs in the Black
formula. The details of the approximation is discussed in Section 4. A convexity adjustment and a
timing adjustment are derived to account for the effect of valuing the CMS rates under a common
measure. The adjustments are implemented in terms of drift adjustments. Discussions are given
below and in Section 3 of this report.
1.2 Measure Change and Adjustment
The lognormal forward swap model assumes that a forward swap rate Rswap(t; T, M) is modelled
by following SDE
dRswap(t; T, M) = σswap(T, M) Rswap(t; T, M) dWA
(t) (3)
where 0 ≤ t ≤ Ti, σswap(T, M) is the swaption volatility corresponding to an option expiry T
and an underlying swap starting at T with tenor of M years (maturity T + M), and the pro-
cess WA is a Brownian motion under measure A. This assumption implies that the CMS rate
RCMS(Ti, M) = Rswap(Ti; Ti, M) is lognormally distributed under measure A. Conditional on the
market information at time t, the lognormal variable RCMS(Ti, M) can be expressed as
RCMS(Ti, M) = Rswap(t; Ti, M)×
exp
{
−
1
2
[σswap(Ti, M)]2
(Ti − t) + σswap(Ti, M) [WAi
i (Ti) − WAi
i (t)]
}
, (4)
where WAi
i is a Brownian motion corresponding to the forward swap rate Rswap(t; Ti, M) under
measure Ai.
To evaluate the Asian option in equation (2), all CMS rates need to be expressed under a
common probability measure. In particular, they must all be expressed in terms of a common
TP forward measure, where TP is the payoff time. The measure conversion is represented as a
correction to the expectation of the forward swap rate at future time s, expressed as the current
swap rate Rswap(t; Ti, M) plus a convexity adjustment and a timing adjustment; that is,
ETP
{Rswap(s; Ti, M)| Ft} ≈ Rswap(t; Ti, M) + Cnvxi(t, s) + Timingi(t, s) , t ≤ s ≤ Ti < TP .
1
See Brigo and Mercurio[1] and Hull[8].
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6. Issue 1.1
Denote ¯RCMS(t; Ti, M) as the expectation for CMS rate with s = Ti in the equation above, then
the CMS rate in equation (4) have the following expression under the common TP forward measure
and conditional on the market information at time t.
RCMS(Ti, M) ≈ ¯RCMS(t; Ti, M)×
exp
{
−
1
2
[σswap(Ti, M)]2
(Ti − t) + σswap(Ti, M) [WTP
i (Ti) − WTP
i (t)]
}
, (5)
where WTP
i is a Brownian motion corresponding to the forward swap rate Rswap(t; Ti, M) under
the common measure TP . Correlations between WTP
i and WTP
j can be introduced. The details of
these derivations are explained in Section 3 of this report.
1.3 Future Extensions
In future, it is desired to develop a valuation formula for an Asian CMS Cap/Floor with a floating
strike rate. Although the valuation analytics are beyond the scope of this report, its price can be
approximated by the similar methodology developed in this report for fixed strike Asian Cap/Floor.
The floating strike Asian CMS Caplet/Floorlet has a payoff at time TP (TP > TN ) given by
V (TP ) = τP ×
[
ω ×
(
1
N
N∑
i=1
RCMS(Ti, M)) − RCMS(TN , M)
)]+
.
The value of the payoff is unknown until time TN and is paid on TP . To compute the value of the
option at time t, we compute an expectation in the TP forward measure; that is,
V (t) = P(t, TP ) × τP × ETP
[
ω ×
(
1
N
N∑
i=1
RCMS(Ti, M) − RCMS(TN , M)
)]+
Ft
.
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7. Issue 1.1
2 Background
2.1 Notation
Consider a swap that starts at time T0 ≥ 0 and ends at time T0 + M with a tenor of M years, it
has n payments per year at
T1 < . . . < TnM , where TnM = T0 + M.
The first cash flow exchange takes place at T1. A plain vanilla swap contract consists of two legs.
One leg pays a floating rate (such as, LIBOR or CDOR) plus a margin spread and the other leg
pays a fixed rate called the swap rate. The swap rate is determined prior to the start of the swap.
Let P(t, T) be the discount factor between t and T, then the value of the floating leg at time t
(t ≤ T0), Vfloat(t), is given by
Vfloat(t) = P(t, T0) − P(t, TnM ).
The value of the fixed leg at time t, Vfixed(t), is given by
Vfixed(t) = Rswap ×
nM∑
j=1
τjP(t, Tj) ,
where τj is the year fraction between Tj−1 and Tj. The par swap rate is determined so that fixed
and floating legs have the same value. Thus the t-time par swap rate is determined as follows:
Rswap(t; T0, M) =
P(t, T0) − P(t, TnM )
∑nM
j=1 τjP(t, Tj)
.
The notation Rswap(t; T0, M) indicates that the par swap rate is determined at time t for a swap
starting at time T0 with a tenor of M years, which implies the swap matures at time TnM .
A CMS (constant maturity swap) is a swap contract where one leg pays the M-year swap rate
(and possibly plus some margin) while the other leg usually pays a floating rate (such as LIBOR
or CDOR). The CMS rates are usually set in advance. In particular, the payment on Ti+1 depends
on the CMS rate RCMS(Ti, M) which is calculated at Ti for the swap starting at Ti and ending at
Ti + M, that is, RCMS(Ti, M) = Rswap(Ti; Ti, M).
2.2 Black Swaption Formula
A swaption is an option granting its owner the right, but not the obligation to enter into an
underlying interest rate swap. There are two types of swaption contracts.
1. A payer swaption gives the owner of the swaption the right to enter into a swap, in which he
would pay the fixed leg at the strike rate and receive the floating leg.
2. A receiver swaption gives the owner of the swaption the right to enter into a swap, in which
he would receive the fixed leg at the strike rate and pay the floating leg.
A swaption is characterized by the following:
1. strike rate (equal to the fixed rate of the underlying swap),
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8. Issue 1.1
2. length of the option period,
3. term of the underlying swap,
4. notional amount, and
5. frequency of payments on the underlying swap.
Fisher Black developed a simple formula for pricing swaptions based on the assumption that
swap rates are lognormally distributed. The Black formula for swaption pricing ([1], [8]) can be
formally derived by using an annuity factor as a numeraire to define a probability measure in which
the swap rate has a lognormal distribution. Define the annuity factor A(t; Ti, M) as follows.
A(t; Ti, M) =
i+nM∑
j=i+1
τjP(t, Tj).
As a shorthand notation, we use Ai to denote the probability measure induced by the numeraire
A(t; Ti, M). The swap rate Rswap(t; T0, M) can be expressed as follows
Rswap(t; T0, M) =
P(t, T0) − P(t, TnM )
A(t; T0, M)
.
Under probability measure A0, the swap rate is a martingale; that is,
Rswap(t; T0, M) = EA0
{Rswap(T; T0, M)| Ft} , t ≤ T ≤ T0.
This implies that the stochastic differential equation (SDE) of the swap rate can be written as
follows
dRswap(t; T0, M) = σswap(T0, M) Rswap(t; T0, M) dWA0
0 (t), 0 ≤ t ≤ T0.
where σswap(T0, M) is the swaption volatility corresponding to an option expiry T0 and an under-
lying swap starting at T0 with tenor of M years, and the process WA0
0 is a Brownian motion under
measure A0.
The payoff from a payer swaption at the option expiry T0, which corresponds to the start
time of the underlying swap, is given by
V payer
swaption(T0) =
nM∑
j=1
P(T0, Tj) × τj × [Rswap(T0; T0, M) − K]+
= A(T0; T0, M) × [Rswap(T0; T0, M) − K]+
.
From the Fundamental Theorem of Arbitrage Free Pricing, it follows that
V payer
swaption(t) = A(t; T0, M) × EA0
{
V payer
swaption(T0)
A(T0; T0, M)
Ft
}
= A(t; T0, M) × EA0
{
A(T0; T0, M) × [Rswap(T0; T0, M) − K]+
A(T0; T0, M)
Ft
}
= A(t; T0, M) × EA0
{
[Rswap(T0; T0, M) − K]+
Ft
}
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9. Issue 1.1
At time t < T0, the swap rate Rswap(T0; T0, M) is has not be realized. Under the lognormal
assumption of Rswap(T0; T0, M), the time t swaption value is computed by
V payer
swaption(t) = A(t; T0, M) × Black
(
K, Rswap(t; T0, M), σswap(T0, M)
√
T0 − t, +1
)
,
where the Black() formula is defined in the Section A in Appendix.
Similarly, the payoff from a receiver swaption at the option expiry T0, which corresponds to
the start time of the underlying swap, is given by
V receiver
swaption(T0) =
nM∑
j=1
P(T0, Tj) × τj × [K − Rswap(T0; T0, M)]+
= A(T0; T0, M) × [K − Rswap(T0; T0, M)]+
.
Under the lognormal assumption of Rswap(T0; T0, M), the time t swaption value is computed by
V receiver
swaption(t) = A(t; T0, M) × Black
(
K, Rswap(t; T0, M), σswap(T0, M)
√
T0 − t, −1
)
.
2.3 Forward Measure
The probability measure that results from using a zero coupon bond P(t, T) as a numeraire is
called the T forward measure [1][8]. Consider a tradable security S, under T forward measure; the
following quotient is a martingale
S(t)
P(t, T)
;
that is,
S(t)
P(t, T)
= ET
{
S(s)
P(s, T)
Ft
}
, t ≤ s ≤ T.
Consider the forward swap rate Rswap(t; T0, M), the following martingale property holds
under T0 forward measure since a forward swap rate is derived from a tradeable security and its
value is known at T0.
Rswap(t; T0, M)
P(t, T0)
= ET0
{
Rswap(s; T0, M)
P(s, T0)
Ft
}
, t ≤ s ≤ T0.
With s = T0, we have
Rswap(t; T0, M) = P(t, T0) ET0
{Rswap(T0; T0, M)| Ft} .
It implies that the expected future swap rate is related to its t-time value by a simple discount
factor under T0 forward measure. We shall use this result in the next section.
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10. Issue 1.1
3 Modelling Swap Rates Under a Common Forward Measure
From our discussion in Section 2.2, it is standard practice to assume that a swap rate Rswap(t; Ti, M),
Rswap(t; Ti, M) =
P(t, Ti) − P(t, Ti+nM )
A(t; Ti, M)
,
is modelled by (3), that is
dRswap(t; Ti, M) = σswap(Ti, M) Rswap(t; Ti, M) dWAi
i,t ,
where 0 ≤ t ≤ Ti, and σswap(Ti, M) is the swaption volatility corresponding to an an option expiry
Ti and an underlying swap starting at Ti with tenor of M years, and the process WAi
i is Brownian
motion under measure Ai. Under probability measure Ai, the swap rate is a martingale; that is,
Rswap(t; Ti, M) = EAi
{Rswap(s; Ti, M)| Ft} , for t ≤ s ≤ Ti.
A CMS rate with M-year tenor at Ti can be expressed as RCMS(Ti, M) = Rswap(Ti; Ti, M).
In the case of a fixed strike call option, we wish to compute the expectation of the following
[
1
N
N∑
i=1
RCMS(Ti, M) − K
]+
,
and in the case of a fixed strike put option, we compute the expectation of the following
[
K −
1
N
N∑
i=1
RCMS(Ti, M)
]+
.
The difficulty is that all the CMS rates are modelled using different probability measures. Hence to
compute the expectation, we must model all the CMS rates under a common probability measure.
The CMS rates RCMS(Ti, M) is the swap rate observed at time Ti for i = 1, 2, . . . , N, while
the related payment is made at a later time TP . In the following subsections, the swap rates
will be first modelled under Ti forward measure and then modelled using a common TP forward
measure. An implied convexity adjustment (from Ai swap measure to Ti forward measure) and
an implied timing adjustment (from Ti forward measure to TP forward measure) will be derived.
These adjustments will be used to adjust the expected value of CMS rate in (5). For simplicity,
we assume that the accrual periods for swap rates are constant and equal to 1/n, where n is the
number of coupon payments per year.
3.1 Changing from Ai Swap Measure to Ti Forward Measure
Suppose that Gi(yi) is the price of a future M-year bond at time Ti that pays c/n coupon at the
end of each payment period, n is the number of coupon payments per year and y is its yield with 2
Gi(y) =
i+nM∑
j=i+1
c
n
×
(
1 +
y
n
)−n(Tj−Ti)
.
2
The price formula neglects the final payment of the notional amount under the consideration that there is generally
no exchange of principal in the CMS swap.
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11. Issue 1.1
It can be approximated by its second order Taylor series expanded about yt as follows, where yt is
the forward bond yield at time t (t < Ti).
Gi(y) ≈ Gi(yt) + G′
i(yt)(y − yt) +
1
2
G′′
i (yt)(y − yt)2
, (6)
where G′
i(x) and G′′
i (x) are the first and second derivatives of Gi with respect to x. Under the Ti
forward measure, the expected future bond price equals the forward bond price, that is
ETi
{Gi(y)| Ft} = Gi(yt) . (7)
Taking the expectation of (6) under the Ti forward measure with the identity (7) yields
G′
i(yt) ETi
{y − yt| Ft} +
1
2
G′′
i (yt) ETi
{
(y − yt)2
Ft
}
≈ 0 .
The expression ETi
{
(y − yt)2 Ft
}
is approximately y2
t σ2
y(Ti − t), where y is assumed to follow a
lognormal distribution with volatility σy. Hence it is approximately true that
ETi
{y| Ft} ≈ yt −
1
2
G′′
i (yt)
G′
i(yt)
y2
t σ2
y (Ti − t) ,
with
ETi
{y| Ft} = 0 if yt = 0 .
The CMS rates RCMS(Ti, M) can be considered as the yield at time Ti on a M-year bond
with a coupon equal to today’s forward swap rate. Let Ri
t = Rswap(t; Ti, M) and σi
R = σswap(Ti, M)
for i = 1, 2, . . . , N, and then the expected swap rate Ri
Ti
under Ti forward measure equals to
ETi
{
Ri
Ti
Ft
}
= Ri
t + Cnvxi(t, Ti), where
Cnvxi(t, Ti) = −
1
2
G′′
i (Ri
t)
G′
i(Ri
t)
(Ri
t)2
(σi
R)2
(Ti − t)
Note that Cnvxi(t, Ti) = 0 if Ri
t = 0.
3.2 Changing from Ti Forward Measure to TP Forward Measure
To change the measure from Ti to TP , the Radon-Nikodym derivative is used as follows.
ETP
{
Ri
Ti
Ft
}
= ETi
{
Ri
Ti
×
dQTP
dQTi
Ti
Ft
}
= ETi
{
Ri
Ti
×
P(Ti, TP )/P(t, TP )
P(Ti, Ti)/P(t, Ti)
Ft
}
= ETi
{
Ri
Ti
× P(Ti, TP ) Ft
}
×
P(t, Ti)
P(t, TP )
. (8)
Suppose that fi
t := ffwd(t; Ti, TP ) is the forward rate (with compounding frequency n) during
future time period from Ti to TP at time t with 0 ≤ t ≤ Ti < TP , and Hi(fi
t ) is the forward price
of a future zero-coupon bond at time Ti that pays 1 at time TP , then
Hi(fi
t ) =
(
1 +
fi
t
n
)−n×(TP −Ti)
=
P(t, TP )
P(t, Ti)
.
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12. Issue 1.1
At time Ti, the zero coupon bond has value
Hi(fi
Ti
) =
P(Ti, TP )
P(Ti, Ti)
= P(Ti, TP ) .
The expected future zero-coupon bond price equals the forward bond price under the Ti forward
measure, that is
ETi
{
Hi(fi
Ti
) Ft
}
= Hi(fi
t ) =
P(t, TP )
P(t, Ti)
. (9)
Apply the martingale property in equation (9) to obtain the following.
ETi
{
(Ri
Ti
− Ri
t) × [Hi(fi
Ti
) − Hi(fi
t )] Ft
}
= ETi
{
Ri
Ti
Hi(fi
Ti
) − Ri
Ti
Hi(fi
t ) − Ri
t Hi(fi
Ti
) + Ri
t Hi(fi
t ) Ft
}
= ETi
{
Ri
Ti
× Hi(fi
Ti
) Ft
}
− ETi
{
Ri
Ti
Ft
}
× Hi(fi
t ) .
On the other hand, Hi(fi
Ti
) can be approximated by its first order Taylor series expanded about
fi
t . Hence,
ETi
{
(Ri
Ti
− Ri
t) × [Hi(fi
Ti
) − Hi(fi
t )] Ft
}
≈ ETi
{
(Ri
Ti
− Ri
t) × [H′
i(fi
t ) × (fi
Ti
− fi
t )] Ft
}
≈ H′
i(fi
t ) Ri
t σi
R fi
t σi
f ρi
R,f (Ti − t) ,
under the assumption that fi
t follows a lognormal distribution with volatility σi
f . Note that H′
i(x)
is the first derivative of Hi with respect to x and ρi
R,f measures the correlation between Ri and fi.
By comparing the two equations above, we obtain
ETi
{
Ri
Ti
× Hi(fi
Ti
) Ft
}
≈ ETi
{
Ri
Ti
Ft
}
× Hi(fi
t ) + H′
i(fi
t ) Ri
t σi
R fi
t σi
f ρi
R,f (Ti − t) .
Thus equation (8) becomes
ETP
{
Ri
Ti
Ft
}
= ETi
{
Ri
Ti
× Hi(fi
Ti
) Ft
}
×
1
Hi(fi
t )
≈ ETi
{
Ri
Ti
Ft
}
+
H′
i(fi
t )
Hi(fi
t )
Ri
t σi
R fi
t σi
f ρi
R,f (Ti − t)
≈ Ri
t + Cnvxi(t, Ti) + Timingi(t, Ti)
with
Cnvxi(t, Ti) = −
1
2
G′′
i (Ri
t)
G′
i(Ri
t)
(Ri
t)2
(σi
R)2
(Ti − t) (10)
Timingi(t, Ti) = −
TP − Ti
1 + fi
t /n
Ri
t σi
R fi
t σi
f ρi
R,f (Ti − t) (11)
The convexity and the timing adjustments on CMS rate are similar to those given in Hull[8],
Equation (32.2).
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13. Issue 1.1
4 Analytic Pricing of Fixed Strike CMS Asian Cap/Floor
This section discusses the analytic formula for pricing fixed strike CMS Asian caplet/floorlet based
on the lognormal assumption of CMS rates. Let Tk−1 ≤ t < Tk, then the CMS rates reset before t
have been realized and the remaining unknown CMS rates are RCMS(Ti, M) for i ≥ k. The price
of the caplet/floorlet in Equation (2) can be written as
V (t) = P(t, TP ) × τP × ETP
{
[ω × (X − ˆK)]+
Ft
}
, where
X =
1
N
N∑
i=k
RCMS(Ti, M),
ˆK = K −
1
N
k−1∑
i=1
RCMS(Ti, M). (12)
ˆK is called the adjusted strike. We adapt the approximation developed by Turnbull and Wake-
man [11] on equity Asian option, that is, the sum of correlated lognormal variables, X, can be
approximated by a single lognormal variable by matching their first and second moments.
To simplify the notation, we rewrite the following
µi(t) = ¯RCMS(t; Ti, M) = ETP
{Rswap(Ti; Ti, M)| Ft}
as the CMS rate mean conditional on market information on time t. Then Equation (5) becomes
to
RCMS(Ti, M) ≈ µi(t) exp
{
−
1
2
(σi
R)2
(Ti − t) + σi
R [WTP
i (Ti) − WTP
i (t)]
}
, where
µi(t) = Rswap(t; Ti, M) + Cnvxi(t, Ti) + Timingi(t, Ti). (13)
The convexity adjustment and the timing adjustment of the expected forward swap rate are defined
in Equation (10) and (11) respectively. Denote µx as the expected value of X, σx as the standard
deviation of ln(X) and ρij as the correlation between WTP
i and WTP
j . Then conditional on market
information at time t, the first and the second moments of X can be derived as
E[X| Ft] = µx =
1
N
N∑
i=k
µi(t), (14)
E[X2
Ft] = (µx)2
exp{(σx)2
} =
1
N2
N∑
i=k
N∑
j=k
µi(t) µj(t) exp{ρij σi
R σj
R [min(Ti, Tj) − t]} (15)
In this report, the instantaneous correlation between projected CMS rates is modelled by a two-
parameter function as follows,
ρi,j = β1 + [1 − β1] × exp{−β2 × |i − j|}, with 0 ≤ β1 ≤ 1 and β2 ≥ 0. (16)
It starts at 1 when i = j and decreases to β1 as |i − j| → ∞. The parameter β1 can be interpreted
as the limit of correlation that is approached for rates far separated in time, while the parameter
β2 measures the speed of decay which describes how fast the correlation decreases and approaches
β1. β1 could have negative value, however we restrict it to be nonnegative here.
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14. Issue 1.1
By applying the assumptions above, the fixed strike CMS Asian option price can be calculated
by the Black formula described in Appendix A.
V (t) ≈
P(t, TP ) × τP × Black( ˆK, µx, σx, ω), if ˆK > 0 and µx > 0;
P(t, TP ) × τP × [ω × (µx − ˆK)]+, if ˆK ≤ 0 or µx = 0.
(17)
Note that V (t) = 0 if N = 0. Once the prices of all caplets/floolets are known, the price of an
Asian CMS Cap/Floor can be calculated as the sum of the caplets/floorlets prices, as shown in
Equation (1).
5 Parameter Determination in Analytic Pricing Formula
An Asian CMS Cap/Floor can be decomposed into a series of caplets/floorlets. The price of each
caplet/floorlet with $1 notional is calculated based on the analytic pricing formula (17). The
parameters in the formula are defined as follows, where the current time (time 0) corresponds to
the curve date.
• t is the valuation time, t ≥ 0.
• TP is the payment time of the caplet/floorlet, TP > t.
• P(t, TP ) is the discount factor between t and TP .
• τP is the accrual period between payments with the first τP starts from the effective date.
• ω = 1 for caps and ω = −1 for floors.
• ˆK is the adjusted strike in Equation (12), where
– K is the fixed strike rate.
– N is the average number of CMS rate in the caplet/floorlet.
– T1, . . . , TN are the reset times of the caplet/floorlet with TP = TN+1.
– M is the tenor of the underlying CMS rate.
– RCMS(Ti, M), i = 1, . . . , k − 1, are realized CMS rates, Tk−1 ≤ t.
• µx and σx are derived from Equation (14) and (15), where
– µi(t) is defined in (13), which is the expected value of RCMS(Ti, M) at time t.
– ρi,j is defined in (16), which is the correlation between projected CMS rates.
– σi
R is the volatility of forward swap rate Rswap(t; Ti, M).
The valuation time, the reset time and the payment time are determined by the fraction of year
between their corresponding dates and the curve date based on the input day-count basis.
For parameters in Equation (10) and (11), Ri
t and fi
t are derived from the input discount
curve based on the following formulas, where n is the number of coupon payments per year for the
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15. Issue 1.1
fixed leg of the swap.
Ri
0 = Rswap(0; Ti, M) = n ×
P(0, Ti) − P(0, Ti + M)
∑nM
k=1 P(0, Ti + k/n)
fi
0 = ffwd(0; Ti, TP ) = n ×
[(
P(0, TP )
P(0, Ti)
)−1/[n(Tp−Ti)]
− 1
]
, TP > Ti
Here Ti is the reset time for Rswap(0; Ti, M). Discount factors are log-linearly interpolated from the
input discount curve with flat extension of continuous compounded zero rates at both ends of the
curve. The volatilities σi
R and σi
f are linearly interpolated from the corresponding input volatility
curve with flat extension of volatilities at both ends of the curve. The correlations ρi
R,f are linearly
interpolated from the input correlation curve with flat extension of volatilities at both ends of the
curve.
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16. Issue 1.1
A Black Formula for Lognormal Random Variables
Proposition A.1 Suppose V is a lognormally distributed random variable such that
E[V ] = m
and the standard deviation of ln(V ) is s. Then
E{max(V − K, 0)} = m Φ (d1(K, m, s)) − KΦ (d2(K, m, s))
and
E{max(K − V, 0)} = −m Φ (−d1(K, m, s)) + KΦ (−d2(K, m, s))
where Φ() is the standard Gaussian cummulative distribution function
Φ(x) =
1
√
2π
∫ x
−∞
e−1
2
λ2
dλ,
d1(K, m, s) =
ln(m/K) + s2/2
s
,
and
d2(K, m, s) =
ln(m/K) − s2/2
s
.
Proof See [8] pages 307-308.
From the structure of the formulae in Proposition A.1, the Black formula [1] can be defined
as follows.
Black(K, m, s, a) = a m Φ (ad1(K, m, s)) − a K Φ (ad2(K, m, s)) (18)
Hence,
E{max(V − K, 0)} = Black(K, m, s, +1)
and
E{max(K − V, 0)} = Black(K, m, s, −1).
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17. Issue 1.1
References
[1] Damiano Brigo and Fabio Mercurio, Interest Rate Models - Theory and Practice, Springer
Verlag, New York, 2001.
[2] H. Geman, N. El Karoui and J. C. Rochet, “Change of Numeraire, Changes of Probability
Measures and Pricing of Options”, Journal of Applied Probability, Vol. 32, 443-458, 1995.
[3] Patrick Hagan, “Convexity conundrums: Pricing CMS Swaps, Caps and Floors”, Wilmott
Magazine, March 2003, p.38-44.
[4] Michael Harrison and Stanley Pliska, “Martingales and Stochastic integrals in the theory of
continuous trading”, Stochastic Processes and their Applications, Volume 11, No.3, pp. 215260,
1981.
[5] Espen Gaarder Haug, The Complete Guide to Option Pricing Formulas, 2nd edition, McGraw-
Hill, New York, 2006.
[6] Vicky Henderson and Rafal Wojakowski, “On the Equivalence of Floating and Fixed Strike
Asian Options”, Journal of Finance, Vol. 52, No. 3, pp.923-973, 2001.
[7] Vicky Henderson, David Hobson, William Shaw, and Rafal Wojakowski, “Bounds for in-
progress floating-strike Asian options using symmetry”, Annals of Operations Research, Vol.
151, No. 1, pp. 81 - 98, 2007.
[8] John Hull, Options, Futures, and Other Derivative Securities, 7th edition, Prentice Hall, New
Jersey, 2009.
[9] F. A. Longstaff and E. S. Schwartz, “Valuing American Options by Simulation: A Simple
Least-Squares Approach”, Review of Financial Studies, Vol. 14, No. 1, pp. 113147, 2001.
[10] Marek Musiela and Marek Rutkowski, Martingale Methods in Financial Modelling, Springer
Verlag, New York, 1997.
[11] Stuart Turnbull and Lee Wakeman, “A Quick Algorithm for Pricing European Average Op-
tions”, Journal of Financial and Quantitative Analysis, Vol. 26, No. 3, September 1981, pp.
377-389.
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