This document discusses valuation methods for interest rate derivatives with unconventional payment dates, such as fixing in arrears swaps and caps/floors with upfront payments. It presents notation for describing these products, then develops a two-curve linear cash rate model to value them. Finally, it provides analytic approximations using normal models parameterized by basis point volatility.

1. Two Curves Upfront
Normal convexity corrections and two curve pricing
Gary J. Kennedy
Clarus Financial Technology
February 21, 2013
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2. Application
Consider simple products that depend on IBOR but have an
unnatural payment date
Fixing In-arrears swaps, caps, floors on IBOR
Payment upfront swaps, caps, floors on IBOR
Vanilla range accrual swaps on IBOR, where digitals are
replicated by call-spreads
Average swaps
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3. Notation
Follow notation of Hagan [Hag03, Hag04] as much as possible.
Consider an IBOR rate R(τ ), fixing at τ , with start date ts and
end date te . Let Z (t, T ) denote the value of a zero coupon bond
with maturity T at time t. For emphasis, D(T ) = Z (0, T ).
A caplet with payment date tp is then valued as
Z (τ, tp )
Vcaplet = D(te )E max(R(τ ) − K , 0) |F0
Z (τ, te )
Similarly,
Z (τ, tp )
Vswaplet = D(te )E R(τ ) |F0
Z (τ, te )
Z (τ, tp )
Vfloorlet = D(te )E max(K − R(τ ), 0) |F0
Z (τ, te )
We ignore the notional and accrual period length of the cashflow
for convenience. The accrual period length of the IBOR rate is α.
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4. Two curves within the payoff
Z (τ,tp )
How to relate Z (τ,te ) and R(τ ) in a two curve world?
Z R (t,u) Z (t,u)
Recall from [Hen10], Z R (t,v )
= βt (u, v ) Z (t,v ) , and assume
D R (u)
D(u)
βt (u, v ) = β0 (u, v ) = D R (v )
D(v ) .
In the special case that tp = ts (typical of in-arrears),
Z (τ, tp ) 1
= (1 + αR(τ ))
Z (τ, te ) βτ (ts , te )
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5. Linear Cash Rate Model
In the more general case of tp = ts we may use the linear cash rate
model considered in [HK00, Hag04]
Z R (τ, u)
≈ a + bu R(τ )
Z R (τ, te )
Z R (τ,te )
Since Z R (τ,te )
=1 for any R(τ ), bte must be zero, whence a = 1.
Z (τ,u)
Then since Z (τ,te ) is a martingale,
D(u) Z (τ, u) 1 Z R (τ, u) 1
=E =E R (τ, t )
≈ (1 + bu R(0))
D(te ) Z (τ, te ) βt (τ, te ) Z e β0
D R (u)
−1
D R (te )
Thus bu = R(0) . Notice that the boundary cases are captured
conveniently, for if u = ts , bu = α, whilst if u = te , bu = 0.
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6. Valuation
The payoffs are now expectations of quadratic functions of the
underlying IBOR rate, which can be fully replicated across the
smile, see for example, [HK00, Ope11, Hag03].
D(te )
Vcaplet = E [max(R(τ ) − K , 0)(1 + bu R(τ ))]
β0 (tp , te )
D(te )
Vswaplet = E [R(τ )(1 + bu R(τ ))]
β0 (tp , te )
D(te )
Vfloorlet = E [max(K − R(τ ), 0)(1 + bu R(τ ))]
β0 (tp , te )
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7. Basis point volatility valuation
For completeness we consider an analytic approximation using
normal model1 .
√
R(τ ) = R(0) + σ τ X
where X is a normal variable with mean zero and unit variance.
The volatility parameter, σ is referred to as basis point volatility, or
simply BpVol [Zho03].
D(tp )
Then since E[R 2 (τ )] = R(0)2 + σ 2 τ , and D(te ) = 1+bu R(0)
β0
bu σ 2 τ
Vswaplet = D(tp ) R(0) +
1 + bu R(0)
Should the volatility surface be parameterised by strike, choose the
σ = σ(R(0)).
1
Use of model which permitted negative rates became quite important in
Europe during the summer of 2011 when CHF became negative [Car12]
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8. Basis point volatility valuation
For caplet,
D(te )
E [max(R(τ ) − K , 0)(1 + bu R(τ ))]
β0 (tp , te )
bu σ 2 τ √
= D(tp ) R(0) + −K N(d) + σ τ n(d)
1 + bu R(0)
Where d = R(0)−K . Should the volatility surface be parameterised
√
σ τ
by strike, choose σ = σ(K ) when K > R(0). For K < R(0) use
put-call parity.
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9. Basis point volatility valuation
For floorlet,
D(te )
E [max(K − R(τ ), 0)(1 + bu R(τ ))]
β0 (tp , te )
bu σ 2 τ √
= D(tp ) K − R(0) − N(−d) + σ τ n(−d)
1 + bu R(0)
Should the volatility surface be parameterised by strike, choose
σ = σ(K ) when K < R(0). For K > R(0) use put-call parity.
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10. Bibliography I
L. Carver, Negative rates: Dealers struggle to price 0% floors,
Risk Magazine Nov. (2012).
P. Caspers, Normal libor in arrears, Available at SSRN
2188619 (2012).
P.S. Hagan, Convexity conundrums: pricing CMS swaps, caps
and floors, Wilmott magazine 1 (2003), 38.
P. Hagan, Accrual swaps and range notes, Working paper
(2004).
Marc Henrard, The irony in derivatives discounting part II: The
crisis, Wilmott Journal 2 (2010), no. 6, 301–316.
Phil Hunt and Joanne Kennedy, Financial derivatives in theory
and practice, Wiley, 2000.
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11. Bibliography II
A. Li and V. Raghavan, Libor-in-arrears swaps, Journal of
Derivatives, Spring (1996).
OpenGamma, Swap and cap/floors with fixing in arrears or
payment delay, www.opengamma.com (2011).
A. Pelsser, Mathematical foundation of convexity correction,
Quantitative Finance 3 (2003), no. 1.
Fei Zhou, Volatility skews, Lehman Brothers: Fixed Income
Research Sep (2003).
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