Interest rate derivatives form part of the largest portion of traded financial instruments. Hence, it is important to have models that describe their dynamics accurately. This study aims at pricing quanto caps and floors using the multi-curve cross-currency LIBOR market model (MCCCLMM) dynamics. A Black Scholes MCCCLMM quanto caplet and floorlet formula is first derived. The MCCCLMM parameters are then calibrated to exactly match the USD and GBP cap market prices. The estimated model parameters are then used to price the quanto options in the Black MCCCLMM quanto caplet and floorlet formula. These prices are then compared to the quanto cap and floor prices estimated via Monte Carlo simulations so as to ascertain its pricing accuracy.
Interest rate modeling has been a major interest amongst researchers. This is mostly because the interest rate markets have grown to dominate the financial world due to its vast number of traded financial products flooding the markets. According to [
The term LIBOR stands for the London Interbank Offered rate. It is the benchmark rate at which major banks borrow from each other on a short term basis. This rate is monitored by ICE and is published daily for five different currencies and seven maturities. LIBOR is used widely in financial markets as the underlying in standard interbank products, commercial products, hybrid products and consumer related products. LIBOR equivalents also accepted globally are the European Interbank Offered Rate (EURIBOR), Tokyo Interbank Offered Rate (TIBOR), Shanghai Interbank Offered Rate (SHIBOR), and Mumbai Interbank Offered Rate (MIBOR).
Before 2007, the spread between the different LIBOR tenors was almost negligible. Hence, a single curve was sufficient for both discounting and in generating future cashflows. However, after the 2007 financial crisis, [
Under the multi-curve framework, one curve is used to generate future cashflows while the other one is used to discount the generated future cashflows. So far, many models have already been proposed in practice (see [
Our main interest in this paper is in pricing quanto caps and floors using the multi-curve cross-currency LIBOR market model (MCCCLMM) dynamics introduced in [
However, the calibration problem, over the decades, has not been an easy one. It has seen researchers resort to both parametric and non-parametric techniques of calibrating the LIBOR market model (LMM). For instance, [
In this paper, the instantaneous correlations are estimated from historical rates. The general piecewise constant assumption was assumed on the cap volatilities. The foreign exchange rate volatility was also estimated from historical data.
In this section, the relevant tools, models, methods and tests used are presented.
The datasets considered in this study were: Historical overnight and 6 month USD and GBP LIBORs ranging from 2/1/2008 to 2/1/2018 collected from [
R open software version 3.1.2 was used in simulating and analysing all the data in this study. Useful packages considered were “MASS”, “sde”, and “lmtest”.
Let X ( t ) be the spot foreign exchange rate at time t quoted as the ratio of units of United States Dollar (USD) to one unit of Pound Sterling (GBP). d be the US (domestic) economy and f be the UK (foreign) economy. D and L denote the risk-free and risky curves respectively. r d D and r f D be the risk-free short rates of interest associated with the US and UK economies respectively. In addition to this, P d D ( t , T ) and P f D ( t , T ) denote the risk-free zero coupon bonds in the domestic and foreign economies respectively. L d D ( t , T ) and L f D ( t , T ) denote the simply compounded overnight LIBORs associated with US and UK markets respectively. Last but not least, L d ( t , T ) and L f ( t , T ) denote the 6 month LIBOR associated with the US and UK markets.
The stochastic differential equations associated with the MCCCLMM dynamics that were considered in this study under the spot domestic risk neutral measure were given by:
{ d P d D ( t , T ) = r d D ( t ) P d D ( t , T ) d t − σ d D ( t , T ) P d D ( t , T ) d W d D ( t ) d L d ( t , T ) = ρ d D d L ∗ σ d D ( t , T + τ ) γ d L ( t , T ) L d ( t , T ) d t + γ d L ( t , T ) L d ( t , T ) d W d L ( t ) d L f ( t , T ) = ( ρ f D f L σ f D ( t , T + τ ) − ρ x x f D σ X ( t ) ) γ f L ( t , T ) L f ( t , T ) d t + γ f L ( t , T ) L f ( t , T ) d W f L ( t ) d X ( t ) = ( r d D ( t ) − r f D ( t ) ) X ( t ) d t + σ X ( t ) d W X X ( t ) (1)
where
σ k D ( t , T ) = ∑ j = 1 ⌊ τ − 1 ( T − t ) ⌋ τ L k D ( t , T − j τ ) 1 + τ L k D ( t , T − j τ ) γ k D ( t , T − j τ ) (2)
σ k L ( t , T ) = ∑ j = 1 ⌊ τ − 1 ( T − t ) ⌋ τ L k ( t , T − j τ ) 1 + τ L k ( t , T − j τ ) γ k L ( t , T − j τ ) (3)
⌊ τ − 1 ( T − t ) ⌋ denotes the greatest integer that is less than τ − 1 ( T − t ) .
And
W = [ W d D W d L W f D W f L W X X ] (4)
is a correlated Wiener process with a correlation matrix given by:
Σ = ( 1 ρ d D d L ρ d D f D ρ d D f L ρ x x d D ρ d D d L 1 ρ d L f D ρ d L f L ρ x x d L ρ d D f D ρ d L f D 1 ρ f D f L ρ x x f D ρ d D f L ρ d L f L ρ f D f L 1 ρ x x f L ρ x x d D ρ x x d L ρ x x f D ρ x x f L 1 ) (5)
where it is assumed that the Wiener processes are governed by constant correlation factors such that
ρ i j = ( ρ j i if j ≠ k ; j , k ∈ { d D , d L , f D , f L , x x } , | ρ j k | ≤ 1 1 if j = k (6)
Calibration is the process of estimating the model parameters such that they match the market prices.
The constant correlations ρ i j can be estimated using the Pearson’s correlation formula given by:
r = ∑ i = 1 n ( X i − X ¯ ) ( Y i − Y ¯ ) ∑ i = 1 n ( X i − X ¯ ) 2 ∑ i = 1 n ( Y i − Y ¯ ) 2 (7)
Given that the SDE of the foreign exchange rate is given by:
d X t X t = μ X d t + σ X d W x x (t)
then the SDE can be discretised as follows:
d X t X t = d ln X t = ln ( X t ) − ln ( X t − 1 ) = ln ( X t X t − 1 ) = μ X Δ t + σ X Δ W x x ( t ) (8)
and according to [
1) Calculate the logarithmic returns as follows:
l i = ln ( X i X i − 1 ) ; i ∈ { 1 , 2 , ⋯ , n } (9)
2) Calculate the mean of the logarithmic returns as follows:
l ¯ = ∑ i = 1 n l i (10)
3) Calculate the unbiased standard deviation of the logarithmic returns as follows:
s l = 1 n − 1 ∑ i = 1 n ( l i − l ¯ ) 2 (11)
and now σ X can be estimated as
σ ^ X = s l Δ t (12)
In this paper, the constant maturity-dependent volatility assumption under the general piecewise constant technique is considered. [
γ i ( t ) = S i ; 0 < t ≤ T i − 1 (13)
Calibrating the model to caplet amounts is equivalent to choosing deterministic LIBOR volatilities of forward rates γ 1 , γ 2 , ⋯ , γ m such that:
v T i − c a p 2 = 1 T i − 1 ∫ 0 T i − 1 γ i 2 ( t ) d t ; i = { 1 , 2 , ⋯ , m } (14)
The γ values are as summarised in
Under this assumption,
| Forward LIBOR | t ∈ ( 0, T 0 ] | ( T 0 , T 1 ] | ⋯ | ( T m − 2 , T m − 1 ] |
|---|---|---|---|---|
| L 1 ( t ) | S 1 | dead | ⋯ | dead |
| L 2 ( t ) | S 2 | S 2 | ⋯ | dead |
| ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
| L m ( t ) | S m | S m | ⋯ | S m |
T i − 1 v T i − c a p 2 = T i − 1 S i 2 for i ∈ { 1 , 2 , ⋯ , m } v T i − c a p 2 = S i 2 (15)
The advantage of using this assumption is that the S parameters fit the market cap volatilities.
It was assumed that in the U.S economy, the USD LIBOR is a domestic rate. In the same way, in the British economy, the GBP LIBOR is also a domestic rate. Hence under the domestic risk neutral measure, the MCCCLMM dynamics of the risky USD or GBP LIBOR in either domestic economies is given by:
d L k ( t , T ) L k ( t , T ) = ρ k D k L ∗ σ k D ∗ γ k L ∗ d t + γ k L ∗ d W k L ( t ) ; k ∈ { d , f } (16)
The Black-like formula for calculating USD or GBP caplets or floorlets in the respective domestic currencies is given by:
C k ( t ) = ω τ P k D ( t , T ) ( L k ( t , T ) ∗ e ρ k D k L ∗ σ k D ∗ γ k L ( T − t ) ∗ Φ ( ω d 11 ) − K k ∗ Φ ( ω d 21 ) ) (17)
where
ω = ( + 1 if it is a caplet contract − 1 if it is a floorlet contrac d 11 = ln ( L k ( T , T ) K k ) + ( ρ k D k L ∗ σ k D ∗ γ k L + 0.5 γ k L 2 ) ( T − t ) γ k L ∗ T − t d 21 = d 11 − γ k L ∗ T − t (18)
The MCCCLMM parameters fitted in the Black-like formula described in Equation (16), were minimized to exactly match the cap prices collected from [
Bootstrapping in finance is the process of constructing a zero coupon yield curve from a set of coupon bearing instruments by filling in the missing yields. In this research, the Nelson Siegel Svenssons (NSS) method is used. The NSS method given by
y ( t ) = β 0 + β 1 ( 1 − e − t λ 1 t λ 1 ) + β 2 ( 1 − e − t λ 1 t λ 1 − e − t λ 1 ) + β 3 ( 1 − e − t λ 2 t λ 2 − e − t λ 2 ) (19)
is used to bootstrap the yield curve from the USD treasury and UK gilt rates. The NSS model parameters β 0 , β 1 , β 2 , β 3 , λ 1 and λ 2 are estimated by minimizing the sum of squared errors (SSE):
min ∑ ( y A − y ) 2 (20)
where y A are the actual market rates and y are the rates estimated via the NSS method. The zero coupon bond prices are then estimated using the formula:
Z ( t ) = ( 1 + y ( t ) ) − t (21)
The option prices estimated using the Black’s formula were compared with the mean Monte Carlo simulated option prices using the mean absolute percentage error (MAPE) given by:
MAPE = 1 n ∑ i = 1 n | B P i − M P i | B P i ∗ 100 % (22)
where B P i is Black price and M P i is the Monte Carlo price.
According to [
In this section, a brief introduction of the discretization scheme used in our simulations is done, how the correlated Wiener processes were generated, and finally how the cap and floor prices were simulated is given.
The Euler discretization scheme [
Consider an Itô process { X t } 0 ≤ t ≤ T with a stochastic differential Equation (SDE) given by:
d X t = μ ( t , X t ) d t + σ ( t , X t ) d W t (23)
and an initial deterministic value of X 0 = x 0 . Then the Euler approximation of X t for the interval t ∈ [ 0, T ] ; is a process Y such that 0 = t 0 < t 1 < ⋯ < t N = T satisfies the iterative scheme:
Y i + 1 = Y i + μ ( t i , Y i ) ( t i + 1 − t i ) + σ ( t i , Y i ) ( W i + 1 − W i ) ; i = 0 , 1 , 2 , ⋯ , N − 1 (24)
where Y 0 = X 0 .
According to [
W = A Z (25)
In our case, this is given by:
( d W d D d W d L d W f D d W f L d W x x ) = ( 1 0 0 0 0 a b 0 0 0 c d e 0 0 f g h i 0 j k l m n ) ( d Z 1 d Z 2 d Z 3 d Z 4 d Z 5 ) (26)
where:
Z i ; i ∈ { 1,2,3,4,5 } are independent standard normal variables.
a = ρ d D d L ; b = 1 − a 2 ; c = ρ d D f D
d = ρ d D f D − a c b ; e = 1 − c 2 − d 2 ; f = ρ d D f L − a c b
g = ρ d L f L − a f b ; h = ρ f D f L − c f − d g e ; i = 1 − f 2 − g 2 − h 2
j = ρ x x d D ; k = ρ x x d L − a j b ; l = ρ x x f D − c j − d k e
m = ρ x x f L − f j − g k − h l i ; n = 1 − j 2 − k 2 − l 2 − m 2
Remark: See proof in [
To price USD interest rate options, it was assumed that the USD LIBOR is a domestic rate in the US. The dynamics used to simulate the USD Forward risky LIBOR in the US economy were given by:
( a 1 = ρ d D d L d W d L = a 1 ∗ Z 1 + Z 2 1 − a 1 2 ln L d ( T i ) = ln L d ( T i − 1 ) + ρ d D d L ∗ σ d D ∗ γ d L ∗ Δ t + γ d L ∗ Δ t ∗ d W d L (27)
The formula expressed in Equation (28) was used to simulate USD cap or floor prices at time t.
U ( t ) = Δ t ∗ N d ∗ ∑ i = 1 n P d D ( t , T i ) ∗ max ( ω ( L d ( T i ) − K d ) , 0 ) (28)
where N d is the notional principal of the USD interest rate option.
To price GBP interest rate options, it was first assumed that the GBP LIBOR is a domestic rate in the U.K. The dynamics used to simulate the GBP Forward risky LIBOR in the U.K economy were given by:
( a 2 = ρ f D f L d W f L = a 2 ∗ Z 1 + Z 2 1 − a 2 2 ln L f ( T i ) = ln L f ( T i − 1 ) + ρ f D f L ∗ σ f D ∗ γ f L ∗ Δ t + γ f L ∗ Δ t ∗ d W f L (29)
The formula expressed in Equation (30) was used to simulate GBP cap or floor prices at time, t.
G ( t ) = Δ t ∗ N f ∗ ∑ i = 1 n P f D ( t , T i ) ∗ max ( ω ( L f ( T i ) − K f ) , 0 ) (30)
where N f is the notional principal of the GBP interest rate option.
To price GBP options in the US that remits payments in GBP, it is first assumed that the GBP LIBOR is a foreign rate in the US. The dynamics used to simulate the foreign Forward risky LIBORs under the spot domestic risk neutral measure, ℚ D , in the domestic economy was given by;
{ d W f L = f ∗ Z 1 + g ∗ Z 2 + h ∗ Z 3 + i ∗ Z 4 ln L f ( T i ) = ln L f ( T i − 1 ) + ( ρ f D f L ∗ σ f D − ρ x x f L ∗ σ x ) ∗ γ f L ∗ Δ t + γ f L ∗ Δ t ∗ d W f L (31)
The formula expressed in Equation (32) was used to simulate Quanto cap or floor prices at time, t.
Q ( t ) = Δ t ∗ X 0 ∗ N f ∗ ∑ i = 1 n P d D ( t , T i ) ∗ max ( ω ( L f ( T i ) − K f ) , 0 ) (32)
where N f is the notional principal of the GBP interest rate option.
A quanto is an interest rate derivative that allows the holder to receive payment in a currency different from that of the underlying. A caplet is a call optional derivative that offers payment to the holder whenever the interest rate exceeds the cap price at maturity and a cap is a series of caplets. A floorlet is the opposite of a caplet. It is a put optional derivative that offers payment to the holder whenever the put rate exceeds the interest rate at maturity and in the same way, a floor is a series of floorlets. In this section, it was assumed that there exists a domestic investor interested in hedging against foreign interest rate risk. It was also assumed that the investor prefers using their domestic currency in trading as opposed to using the foreign currency. Hence, the underlying in the quanto options are considered to be struck in foreign currency and payments converted into domestic currency using a fixed exchange rate.
Theorem 1 (Black Scholes MCCCLMM Quanto-Caplet/Floorlet Formula). Assuming that the dynamics of the multi-curve cross currency LIBOR market model under the spot domestic martingale measure ℚ D is as defined in Equation (1) and further assuming the underlying L f ( . ) , is struck in foreign currency. Then the payoff at maturity of a quanto caplet or floorlet expressed in domestic currency is given by:
τ N f X 0 ( ω ∗ ( L f ( T , T ) − K f ) ) + (33)
and the arbitrage free price of the quanto caplet or floorlet at time t ≤ T is given by:
Q ( t , T , K f , X 0 , N f ) = τ ω N f X 0 P d D ( t , T ) [ L f ( t , T ) e ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L ∗ ( T − t ) Φ ( ω d 1 ) − K f ∗ Φ ( ω d 2 ) ]
where
ω = ( + 1 if it is a caplet contract − 1 if it is a floorlet contract
d 1 = ln ( L f ( t , T ) K f ) + ( ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L + 1 2 ∗ γ f L 2 ) ( T − t ) γ f L ∗ T − t
d 2 = d 1 − γ f L ∗ T − t
Φ ( . ) is the cdf of a standard normal distribution.
Remark See derivation of Black Scholes MCCCLMM Quanto Caplet formula in Appendix.
The MCCCLMM model parameters were calibrated to real world data using ATM cap prices and historical rates. The parameters were then used to price quanto caps and floors under the MCCCLMM.
Six months spaced ten year historical data was used in this study to estimate the model correlations and foreign exchange rate volatility. The data was taken for the period beginning from 2nd January 2008 to 2nd January 2018. The data consisted of the Overnight and 6 month GBP and USD LIBOR term structures obtained from [
| Data | Mean | Std. Dev |
|---|---|---|
| ON GBP LIBOR | 1.012771 | 1.483388 |
| 6M GBP LIBOR | 1.420781 | 1.611518 |
| ON USD LIBOR | 0.627407 | 1.040560 |
| 6M USD LIBOR | 1.091000 | 1.049520 |
| GBP/USD FX Rate | 1.560398 | 0.188801 |
Our MCCCLMM model was calibrated to match the cap market prices provided by [
The correlation parameters were estimated using Equation (7) using data collected from [
The foreign exchange rate volatility parameter was estimated using the formula expressed in Equation (12) from historical rates collected from [
Just as described in Section 2.5.3, we assumed that the S parameters fit the market cap volatilities. The S parameters for the GBP and USD markets are as summarized in
| Parameter | Estimate |
|---|---|
| ρ d D d L | 0.95207 |
| ρ f D f L | 0.98662 |
| ρ x x f L | 0.74244 |
| ρ d D f D | 0.85917 |
| ρ d L f D | 0.89546 |
| ρ d D f L | 0.79783 |
| ρ d L f L | 0.86769 |
| ρ x x f D | 0.74092 |
| ρ x x d D | 0.50421 |
| ρ x x d L | 0.44993 |
| Parameter | Estimate |
|---|---|
| σ X | 0.13260 |
| Term (Yrs) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| γ f L | 39.28 | 41.62 | 46.89 | 51.84 | 55.42 | 57.90 | 59. 43 | 60.18 | 60.18 | 59.76 |
| Term (Yrs) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| γ d L | 15.38 | 18.81 | 21.73 | 23.92 | 25.63 | 26.59 | 27.14 | 27.55 | 27.75 | 27.86 |
The USD LIBOR volatility parameter γ d L was also extracted directly from actual USD ATM cap prices dataset struck on 2/1/2018 and summarized in
The σ d D parameter was calibrated using Equation (17) to exactly match the actual USD ATM cap prices. The optimal values of σ d D were as summarized in
The σ f D parameter was calibrated using Equation (17) to exactly match the actual GBP ATM cap prices. The optimal values of σ f D were as summarized in
The US treasury and UK gilt yields collected from [
The USD and GBP yield curves extracted from [
The USD or GBP discount factors were estimated using the formula in Equation (21) where y(t) was taken to be the USD or GBP NSS yields. The estimated USD and GBP discount curves were then plotted in
Using the calibrated model parameters, the GBP and USD caps and floors were priced using the Black like formula expressed in Section 2.5.4.
| Term (Yrs) | σ d D |
|---|---|
| 1 | 0.90115 |
| 2 | 0.99222 |
| 3 | 0.73135 |
| 4 | 0.57991 |
| 5 | 0.48292 |
| 6 | 0.41020 |
| 7 | 0.36683 |
| 8 | 0.32919 |
| 9 | 0.30060 |
| 10 | 0.27847 |
| Term (Yrs) | σ f D |
|---|---|
| 1 | 0.69920 |
| 2 | 0.88745 |
| 3 | 0.72613 |
| 4 | 0.58257 |
| 5 | 0.48707 |
| 6 | 0.31340 |
| 7 | 0.36333 |
| 8 | 0.32514 |
| 9 | 0.29623 |
| 10 | 0.27347 |
| Parameter | USD Estimates | GBP Estimates |
|---|---|---|
| β 0 | 0.02801 | 0.02546 |
| β 1 | −0.01687 | −0.02149 |
| β 2 | 0.01250 | −0.01153 |
| β 3 | −0.02377 | −0.01152 |
| λ 1 | 0.56940 | 3.13672 |
| λ 2 | 1.22308 | 3.55075 |
USD ATM Cap and Floor prices for options struck on 2/1/2018, with a range of maturities, were calculated using the calibrated Black-like formula expressed in Equation (17).
| Maturity Date | Strike Price (%) | Cap Price ($) | Floor Price ($) |
|---|---|---|---|
| 2/1/2019 | 1.961 | 14.4 | 7.4 |
| 2/1/2020 | 2.126 | 60.2 | 15.9 |
| 4/1/2021 | 2.199 | 129.0 | 28.7 |
| 3/1/2022 | 2.242 | 218.2 | 46.0 |
| 2/1/2023 | 2.273 | 325.3 | 67.7 |
| 2/1/2024 | 2.303 | 442.8 | 93.9 |
| 2/1/2025 | 2.334 | 571.4 | 123.4 |
| 2/1/2026 | 2.365 | 706.5 | 156.4 |
| 4/1/2027 | 2.394 | 847.0 | 192.4 |
| 3/1/2028 | 2.420 | 992.4 | 230.7 |
GBP ATM Cap and Floor prices for options struck on 2/1/2018, with a range of maturities, were calculated using the calibrated Black-like formula expressed in Equation (17).
ATM Quanto Cap and Floor prices for options struck on 2/1/2018 with a range of maturities were calculated using the calibrated Black MCCCLMM Quanto Caplet or Floorlet Formula expressed in Theorem 1.
Where X 0 was taken to be the GBP/USD foreign exchange rate closing price on 2/1/2018 given by 1.3588. The time evolution of the quanto option prices is as shown in
From
The domestic and foreign LIBOR model dynamics were first discretized using the Euler scheme.
The correlation matrix of our observed data was found to be:
Σ = ( 1.00000 0.95207 0.85917 0.79783 0.50421 0.95207 1.00000 0.89546 0.86769 0.44993 0.85917 0.89546 1.00000 0.98662 0.74092 0.79783 0.86769 0.98662 1.00000 0.74244 0.50421 0.44993 0.74092 0.74244 1.00000 ) (34)
| Maturity Date | Strike Price (%) | Cap Price (£) | Floor Price (£) |
|---|---|---|---|
| 2/1/2019 | 0.662 | 11.8 | 7.1 |
| 2/1/2020 | 0.800 | 50.2 | 16.1 |
| 4/1/2021 | 0.903 | 118.9 | 30.0 |
| 3/1/2022 | 0.982 | 216.6 | 50.9 |
| 2/1/2023 | 1.052 | 341.2 | 79.3 |
| 2/1/2024 | 1.111 | 485.5 | 115.5 |
| 2/1/2025 | 1.164 | 647.5 | 158.5 |
| 2/1/2026 | 1.214 | 822.4 | 207.8 |
| 4/1/2027 | 1.261 | 1006.0 | 262.2 |
| 3/1/2028 | 1.305 | 1195.4 | 320.9 |
| Maturity Date | Strike Price (%) | Cap Price ($) | Floor Price ($) |
|---|---|---|---|
| 2/1/2019 | 0.662 | 14.1 | 10.6 |
| 2/1/2020 | 0.800 | 57.0 | 24.6 |
| 4/1/2021 | 0.903 | 128.2 | 46.8 |
| 3/1/2022 | 0.982 | 221.3 | 79.9 |
| 2/1/2023 | 1.052 | 330.5 | 124.4 |
| 2/1/2024 | 1.111 | 446.9 | 180.2 |
| 2/1/2025 | 1.164 | 567.5 | 245.6 |
| 2/1/2026 | 1.214 | 688.2 | 319.4 |
| 4/1/2027 | 1.261 | 806.3 | 400.0 |
| 3/1/2028 | 1.305 | 920.5 | 486.1 |
The lower triangular matrix defined in Equation (25) was then calculated and found to be:
A = ( 1.00000 0.00000 0.00000 0.00000 0.00000 0.95207 0.30587 0.00000 0.00000 0.00000 0.85917 0.25325 0.44462 0.00000 0.00000 0.79783 0.35339 0.47601 0.10949 0.00000 0.50421 − 0.09844 0.74816 0.17190 0.38312 ) (35)
The correlated Wiener process was then estimated as:
( d W d D ( t ) d W d L ( t ) d W f D ( t ) d W f L ( t ) d W x x ( t ) ) = 0.5 ( 1.00000 0.00000 0.00000 0.00000 0.00000 0.95207 0.30587 0.00000 0.00000 0.00000 0.85917 0.25325 0.44462 0.00000 0.00000 0.79783 0.35339 0.47601 0.10949 0.00000 0.50421 − 0.09844 0.74816 0.17190 0.38312 ) ( Z 1 Z 2 Z 3 Z 4 Z 5 ) (36)
where Z i ; i ∈ { 1 , 2 , 3 , 4 , 5 } are independent standard normal random variables.
The formula expressed in Equation (28) was used to simulate USD cap or floor prices at time, t. N d = $ 10000 . P d D ( t ,. ) is the risk-free discount factor associated with the USD discount curve. The mean simulated USD at the money (ATM) Cap and Floor prices stuck on 2/1/2018 were as summarised in
The formula expressed in Equation (30) was used to simulate GBP cap or floor prices at time, t. N f = £ 10000 , P f D ( t ,. ) is the GBP risk-free discount factor associated with the U.K discount curve, Δ t = 0.5 . The mean simulated GBP at the money (ATM) Cap and Floor prices stuck on 2/1/2018 were as summarised in
| Maturity Date | Strike Price (%) | Cap Price ($) | Floor Price ($) |
|---|---|---|---|
| 2/1/2019 | 1.961 | 14.4 | 7.4 |
| 2/1/2020 | 2.126 | 60.1 | 15.9 |
| 4/1/2021 | 2.199 | 129.0 | 28.7 |
| 3/1/2022 | 2.242 | 219.0 | 46.0 |
| 2/1/2023 | 2.273 | 325.6 | 67.7 |
| 2/1/2024 | 2.303 | 443.8 | 93.7 |
| 2/1/2025 | 2.334 | 572.8 | 123.4 |
| 2/1/2026 | 2.365 | 706.4 | 157.0 |
| 4/1/2027 | 2.394 | 847.3 | 192.5 |
| 3/1/2028 | 2.420 | 992.0 | 230.5 |
| Maturity Date | Strike Price (%) | Cap Price (£) | Floor Price (£) |
|---|---|---|---|
| 2/1/2019 | 0.662 | 11.7 | 7.1 |
| 2/1/2020 | 0.800 | 50.4 | 16.1 |
| 4/1/2021 | 0.903 | 119.1 | 30.0 |
| 3/1/2022 | 0.982 | 216.6 | 50.9 |
| 2/1/2023 | 1.052 | 341.4 | 79.2 |
| 2/1/2024 | 1.111 | 484.7 | 115.6 |
| 2/1/2025 | 1.164 | 648.0 | 158.6 |
| 2/1/2026 | 1.214 | 823.7 | 207.9 |
| 4/1/2027 | 1.261 | 1008.4 | 262.3 |
| 3/1/2028 | 1.305 | 1188.0 | 320.8 |
The formula expressed in Equation (32) was used to simulate quanto cap or floor prices at time, t. N f = £ 10000 , P d D ( t ,. ) is the USD risk-free discount factor associated with the US discount curve, Δ t = 0.5 . The mean simulated at the money (ATM) quanto Cap and Floor prices stuck on 2/1/2018 were as summarised in
The Black MCCCLMM cap and floor prices were compared to those estimated via the Monte Carlo simulation technique so as to ascertain the pricing accuracy of the Black MCCCLMM formulas using the mean absolute percentage error (MAPE) technique. The MAPE results were as given in
Using the calibrated parameters, different USD, GBP and Quanto Cap and Floor prices were estimated under different strike price assumptions. The cap volatility
| Maturity Date | Strike Price (%) | Cap Price ($) | Floor Price ($) |
|---|---|---|---|
| 2/1/2019 | 0.662 | 14.1 | 10.6 |
| 2/1/2020 | 0.800 | 57.0 | 24.6 |
| 4/1/2021 | 0.903 | 128.6 | 46.8 |
| 3/1/2022 | 0.982 | 220.0 | 80.1 |
| 2/1/2023 | 1.052 | 330.5 | 124.4 |
| 2/1/2024 | 1.111 | 448.0 | 180.3 |
| 2/1/2025 | 1.164 | 569.4 | 245.6 |
| 2/1/2026 | 1.214 | 685.2 | 319.4 |
| 4/1/2027 | 1.261 | 804.0 | 400.1 |
| 3/1/2028 | 1.305 | 919.1 | 485.8 |
| Quanto Type | MAPE (%) |
|---|---|
| Cap | 0.2354 |
| Floor | 0.0393 |
surfaces of the USD, GBP and GBP/USD Quanto options were as shown in
The Floor volatility surfaces of the USD, GBP and GBP/USD Quanto options were as shown in
This study aimed at pricing quanto caps and floors using the multi-curve cross-currency LIBOR market model dynamics. It focused on calibrating the model to real world USD and GBP cap market prices. The MCCCLMM correlation and foreign exchange rate volatility parameters were estimated from 10 years historical market rates. The generated payoffs were later on discounted using USD treasury rates. The estimated MCCCLMM parameters were then used to price quanto cap and floor options both under the derived Black MCCCLMM quanto caplet and floorlet formula and also via 100,000 Monte Carlo simulations. The two methods were then compared using the mean absolute percentage error (MAPE) to ascertain the pricing accuracy of the derived model. The MAPEs between the two models were found to lie below 1% hence
deeming the two models as similar implying that the derived MCCCLMM quanto caplet or floorlet formula is highly accurate. The volatility surfaces of the quanto options were similar to those of the GBP and USD options.
It should be noted that due to data availability constraints, the MCCCLMM parameters in this paper were neither calibrated using swaptions nor discounted using overnight indexed swap (OIS) rates. Hence in this paper we were constrained on pricing our caps and floors on a single tenor (6 months). Also, the model was discounted using treasury rates as they are also proxies for the risk-free rate. In the future, it is advisable for the model to be discounted using the OIS rates and compare between the two methods.
We thank the Editor and the reviewers for their comments. The first author acknowledges financial support from Pan African University Institute of Basic and Applied Sciences Scholarship. This support is greatly appreciated.
The authors declare that they have no conflicts of interest.
Wamwea, C., Ngare, P., Mbele Bidima, M.L.D. and Mwelu, S. (2019) Valuation of Quanto Caps and Floors in a Calibrated Multi-Curve Cross-Currency LIBOR Market Model. Journal of Mathematical Finance, 9, 698-725. https://doi.org/10.4236/jmf.2019.94036
Assuming the parameters are constant such that the dynamics of the foreign risky LIBOR under the spot domestic risk-neutral measure can be expressed as follows:
d L f ( t , T ) L f ( t , T ) = ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L ∗ d t + γ f L ∗ d W f L ( t ) (37)
Then the price at time t of a quanto caplet struck on this LIBOR is given by:
Q C p l t ( t , T , K f , X 0 , N f ) = τ N f X 0 P d D ( t , T ) E ℚ D [ ( L f ( T , T ) − K f ) + | F t ] = τ N f X 0 P d D ( t , T ) E ℚ D [ ( L f ( T , T ) − K f ) I A | F t ] = τ N f X 0 P d D ( t , T ) E ℚ D [ L f ( T , T ) I A | F t ] ︸ M − τ N f X 0 P d D ( t , T ) E ℚ D [ K f I A | F t ] ︸ N (38)
where I A is an indicator function such that
I A = ( 1 if L f ( T , T ) − K f > 0 0 elsewhere (39)
M and N were then solved separately and the results inserted back into Equation (38) at the end.
The risky foreign LIBOR defined in Equation (37) under the spot domestic risk neutral measure ℚ D has a solution given by:
L f ( T , T ) = L f ( t , T ) e ( ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L − 1 2 ∗ γ f L 2 ) ( T − t ) + γ f L ( W f L ( T ) − W f L ( t ) ) (40)
Solving for M: We saw that
M = E ℚ D [ L f ( T , T ) I A | F t ] = E ℚ D [ L f ( t , T ) e ( ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L − 1 2 ∗ γ f L 2 ) ( T − t ) + γ f L ( W f L ( T ) − W f L ( t ) ) I A | F t ] = L f ( t , T ) e ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L ∗ ( T − t ) ∗ E ℚ D [ e − 1 2 ∗ γ f L 2 ∗ ( T − t ) + γ f L ( W f L ( T ) − W f L ( t ) ) I A | F t ] (41)
Defining a new probability measure V given by the Radon Nikodym derivative
d V d Q D = e − 1 2 ∗ γ f L 2 ∗ ( T − t ) + γ f L ( W f L ( T ) − W f L (t) )
such that
d W f L V ( t ) = d W f L ( t ) − γ f L ∗ d t (42)
then
M = L f ( t , T ) e ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L ∗ ( T − t ) * E ℚ D [ e 1 2 ∗ γ f L 2 ∗ ( T − t ) + γ f L ( W f L ( T ) − W f L ( t ) ) I A | F t ] = L f ( t , T ) e ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L ∗ ( T − t ) E V [ I A | F t ] = L f ( t , T ) e ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) * γ f L * ( T − t ) ∗ 0 ∗ P r V ( L f ( T , T ) − K f ≤ 0 | F t ) + L f ( t , T ) e ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L ∗ ( T − t ) ∗ 1 ∗ P r V ( L f ( T , T ) − K f > 0 | F t ) = L f ( t , T ) e ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L ∗ ( T − t ) ∗ 1 ∗ P r V ( L f ( T , T ) − K f > 0 | F t ) (43)
Now solving for P r V ( L f ( T , T ) − K f > 0 | F t )
Under the V-measure, the dynamics of L f ( t , T ) is given by:
d L f ( t , T ) L f ( t , T ) = ( ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L + γ f L 2 ) ∗ d t + γ f L ∗ d W f L V ( t ) (44)
Solving this using Itô formula, we get:
L f ( T , T ) = L f ( t , T ) e ( ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L + 1 2 ∗ γ f L 2 ) ( T − t ) + γ f L ( W f L V ( T ) − W f L V ( t ) ) = L f ( t , T ) e R (45)
where
R = ( ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L + 1 2 ∗ γ f L 2 ) ( T − t ) + γ f L ( W f L V ( T ) − W f L V ) (t)
E [ R ] = μ R = ( ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L + 1 2 ∗ γ f L 2 ) ( T − t )
V ( R ) = σ R 2 = γ f L 2 ∗ ( T − t )
such that R ~ N ( μ R , σ R 2 ) . Therefore:
P r V ( L f ( T , T ) − K f > 0 | F t ) = P r ( L f ( t , T ) e R − K f > 0 | F t ) = P r ( e R > K f L f ( t , T ) | F t ) = P r ( R > ln ( K f L f ( t , T ) ) | F t ) = ∫ L * ∞ 1 σ R 2 π e − 1 2 ( R − μ R σ R ) 2 d R = ∫ Z 1 * ∞ 1 2 π e − 1 2 Z 1 2 d Z 1 = ∫ − ∞ d 1 1 2π e − 1 2 Z 1 2 d Z 1 ︸ Φ (d1)
where
L * = ln ( K f L f ( t , T ) )
Z 1 = R − μ R σ R Z 1 * = 1 σ R ( ln ( K f L f ( t , T ) ) − μ R )
d 1 = − Z 1 * = 1 σ R ( μ R + ln ( L f ( t , T ) K f ) )
Hence M becomes:
M = L f ( t , T ) e ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L ∗ ( T − t ) Φ ( d 1 ) (46)
where
d 1 = ln ( L f ( t , T ) K f ) + ( ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L + 1 2 ∗ γ f L 2 ) ( T − t ) γ f L ∗ T − t (47)
Solving for N:
N = E ℚ D [ K f I A | F t ] = K f * E ℚ D [ I A | F t ] = K f ∗ 0 ∗ P r ℚ D [ L f ( T , T ) − K f ≤ 0 | F t ] + K f ∗ 1 ∗ P r ℚ D [ L f ( T , T ) − K f ≤ 0 | F t ] = K f ∗ P r ℚ D [ L f [ T , T ] − K f ≤ 0 | F t ] (48)
Solving for P r ℚ D [ L f ( T , T ) − K f ≤ 0 | F t ]
We saw that under the ℚ D -measure,
L f ( T , T ) = L f ( t , T ) e ( ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L − 1 2 ∗ γ f L 2 ) ( T − t ) + γ f L ( W f L ( T ) − W f L ( t ) ) = L f ( t , T ) e S (49)
where
S = ( ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L − 1 2 ∗ γ f L 2 ) ( T − t ) + γ f L ( W f L ( T ) − W f L ( t ) ) (50)
E [ S ] = μ S = ( ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L − 1 2 ∗ γ f L 2 ) ( T − t ) (51)
V ( S ) = σ S 2 = γ f L 2 ∗ ( T − t ) (52)
such that S ~ N ( μ S , σ S 2 ) . Therefore:
P r ℚ D ( L f ( T , T ) − K f > 0 | F t ) = P r ( L f ( t , T ) e S − K f > 0 ) = P r ( e S > ( K f L f ( t , T ) ) ) = P r ( S > ln ( K f L f ( t , T ) ) ) = ∫ L * ∞ 1 σ S 2 π e − 1 2 ( S − μ S σ S ) 2 d S = ∫ Z 2 * ∞ 1 2 π e − 1 2 Z 2 2 d Z 2 = ∫ − ∞ d 2 1 2 π e − 1 2 Z 2 2 d Z 2 ︸ Φ (d2)
where
L * = ln ( K f L f ( t , T ) ) ; Z 2 = S − μ S σ S
Z 2 * = 1 σ S ( ln ( K f L f ( t , T ) ) − μ S )
d 2 = − Z 2 * = 1 σ S ( μ S + ln ( L f ( t , T ) K f ) )
Hence N becomes:
N = K f ∗ Φ ( d 2 ) (53)
where
d 2 = ln ( L f ( t , T ) K f ) + ( ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L − 1 2 ∗ γ f L 2 ) ( T − t ) γ f L ∗ T − t = d 1 − γ f L ∗ T − t (54)
Hence, the quanto caplet price at time t, with payments given at maturity in domestic currency whenever the foreign risky LIBOR exceeds the foreign strike price, K f is given by:
Q C p l t ( t , T , K f , X 0 , N f ) = τ N f X 0 P d D ( t , T ) ∗ [ L f ( t , T ) e ( ρ f D f L ∗ σ d D − ρ x x f L ∗ σ X ) ∗ γ f L ∗ ( T − t ) Φ ( d 1 ) − K f ∗ Φ ( d 2 ) ] (55)
where d 1 and d 2 is as expressed in Equations (47) and (54) respectively.
Remark: The Black MCCCLMM quanto floor let formula can be derived in the same way.
B. ATM CAP Market Price DataCap prices struck on 2/1/2018 on the 6 month USD and GBP LIBORs with maturities ranging from 1 year to 4 years were used to calibrate some of the MCCCLMM parameters to market. The USD and GBP cap price datasets provided by [
| Maturity Date | Strike Price (%) | Lognormal Volatility (%) | Cap Price (£) in bp |
|---|---|---|---|
| 2/1/2019 | 0.662 | 39.28 | 11.8 |
| 2/1/2020 | 0.800 | 41.61 | 50.3 |
| 4/1/2021 | 0.903 | 46.89 | 118.9 |
| 3/1/2022 | 0.982 | 51.84 | 216.2 |
| 2/1/2023 | 1.052 | 51.84 | 341.2 |
| 2/1/2024 | 1.111 | 57.90 | 485.5 |
| 2/1/2025 | 1.164 | 59.43 | 647.5 |
| 2/1/2026 | 1.214 | 60.18 | 822.4 |
| 4/1/2027 | 1.261 | 60.18 | 1006.0 |
| 3/1/2028 | 1.305 | 59.76 | 1195.4 |
| Maturity Date | Strike Price (%) | Lognormal Volatility (%) | Cap Price ($) in bp |
|---|---|---|---|
| 2/1/2019 | 1.961 | 15.38 | 14.4 |
| 2/1/2020 | 2.126 | 18.81 | 60.3 |
| 4/1/2021 | 2.199 | 21.73 | 128.5 |
| 3/1/2022 | 2.242 | 23.92 | 218.5 |
| 2/1/2023 | 2.273 | 25.63 | 325.3 |
| 2/1/2024 | 2.303 | 26.59 | 442.8 |
| 2/1/2025 | 2.334 | 27.14 | 571.4 |
| 2/1/2026 | 2.365 | 27.55 | 706.5 |
| 4/1/2027 | 2.394 | 27.75 | 847.0 |
| 3/1/2028 | 2.420 | 27.86 | 992.4 |
Data Availability
The historical datasets used are freely available on the websites [