Valuation of Quanto Caps and Floors in a Calibrated Multi-Curve Cross-Currency LIBOR Market Model

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 mul-ti-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.


Introduction
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 [1], interest rate products form the largest portion of traded instruments in the financial markets. Initially, interest rates were modeled using 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 [11] [14]- [27] and so on).
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 [27]. The model parameters are first calibrated to exactly match the market observable cap prices. The estimated parameters from the calibration process are then used to essentially price quanto caps and floors in a Black MCCCLMM quanto caplet or floorlet formula derived in this paper. The quanto cap and floor prices are also estimated using Monte Carlo simulations and a comparison between the two models is done. The discretization scheme used to discretize the forward LIBORs is the Euler scheme.
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, [28] developed a fast at the money (ATM) calibration of the LMM using Lagrange multipliers. He calibrated his model using ATM caps, swaptions and historical correlations. [29] came up with a numerical technique for calibrating financial models that essentially solves an inverse problem associated with some partial differential equations. [30] calibrated the LMM using cap and swaption price data collected on 16 th May 2010. They calibrated the cap volatilities using the Separable piecewise constant (SPC) parameterization technique and Linear-Exponential (L-E) formulation both under the general piecewise constant assumption. They estimated their model correlations using swaptions. [31] explains the different types of assumptions that can be made on the general piecewise constant technique for cap volatilities. [32] explains the concept behind the general piecewise constant. He goes further ahead to explain that the main problem in calibrating the LMM is in finding a volatility function that accurately reproduces a sample of market derivative prices e.g. cap and swaption prices.
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.

Materials and Methods
In this section, the relevant tools, models, methods and tests used are presented.

Data Analysis Tool
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".

The Multi-Curve Cross-Currency LIBOR Market Model (MCCCLMM)
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:  denotes the greatest integer that is less than ( ) is a correlated Wiener process with a correlation matrix given by: fD  fL  dD  dD  dD  dD  xx  dL  fD  fL  dL  dD  dL  dL  xx  fD  fD  fL  fD  dD  dL  fD  xx  fL  fL  fL  fL  dD  dL  fD  xx  dD  dL  fD  fL  xx  xx  xx where it is assumed that the Wiener processes are governed by constant correlation factors such that

Calibration of MCCCLMM Parameters to Market Data
Calibration is the process of estimating the model parameters such that they match the market prices.

Estimation of MCCCLMM Correlations
The constant correlations j i ρ can be estimated using the Pearson's correlation formula given by: then the SDE can be discretised as follows: and according to [37], for one to estimate the volatility parameter from historical rates, the following steps should be considered: 1) Calculate the logarithmic returns as follows: 2) Calculate the mean of the logarithmic returns as follows: and now X σ can be estimated as

Parameterization of the Forward LIBOR Volatility
In this paper, the constant maturity-dependent volatility assumption under the general piecewise constant technique is considered. [31] explains in detail the possible assumptions that can be considered under this technique. It is assumed that Calibrating the model to caplet amounts is equivalent to choosing deterministic LIBOR volatilities of forward rates 1 2 , , , m γ γ γ  such that: The γ values are as summarised in Table 1.
Under this assumption, The advantage of using this assumption is that the S parameters fit the market cap volatilities.

Calibration via the Black MCCCLMM Formula
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: The Black-like formula for calculating USD or GBP caplets or floorlets in the respective domestic currencies is given by: 11 21 , , e kL kD kL kD where ( ) ( )( ) The MCCCLMM parameters fitted in the Black-like formula described in Equation (16), were minimized to exactly match the cap prices collected from [36].

Yield Curve Bootstrapping
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 is used to bootstrap the yield curve from the USD treasury and UK gilt rates. The NSS model parameters 0 1 2 3 1 , , , , β β β β λ and 2 λ are estimated by minimizing the sum of squared errors (SSE): (20) where A y 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:

Mean Error Analysis
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: where i BP is Black price and i MP is the Monte Carlo price.
According to [38], if MAPE 10% ≤ , then the model is considered to be highly accurate. If 10% MAPE 20% < ≤ , then the model is considered to be a good model. If 20% MAPE 50% < ≤ , then the model is considered to be reasonable, and if MAPE 50% > then the model is taken to be inaccurate.

MCCCLMM Simulation
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.

Discretization Scheme
The Euler discretization scheme [39] was used to discretize our calibrated model dynamics so as to enable us to simulate the risky forward LIBORs in discrete time.
Consider an Itô process { } 0 t t T X ≤ ≤ with a stochastic differential Equation (SDE) given by: and an initial deterministic value of 0 0 X x = . Then the Euler approximation of ; is a process Y such that where 0 0 Y X = .

Generation of Correlated Wiener Processes
According to [40], the correlated Wiener process, W can be simulated by applying Cholesky decomposition as follows: In our case, this is given by: where: Remark: See proof in [27].

Simulation of USD Cap and Floor Prices
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: The formula expressed in Equation (28) where d N is the notional principal of the USD interest rate option.

Simulation of GBP Cap and Floor Prices.
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: The formula expressed in Equation (30) where f N is the notional principal of the GBP interest rate option.

Simulation of Quanto Cap and Floor Prices
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, D  , in the domestic economy was given by; The formula expressed in Equation (32) where f N is the notional principal of the GBP interest rate option.

Black MCCCLMM Quanto Caplet or Floorlet Formula
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.
Assuming that the dynamics of the multi-curve cross currency LIBOR market model under the spot domestic martingale measure D  is as defined in Equation (1) and further assuming the underlying ( ) τ ω + * − (33) and the arbitrage free price of the quanto caplet or floorlet at time t T ≤ is given by: is the cdf of a standard normal distribution. Remark See derivation of Black Scholes MCCCLMM Quanto Caplet formula in Appendix.

Numerical Results
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.

Data Description
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 [12], [13], and the GBP/USD foreign exchange rate obtained from [33]. The descriptive statistics of the data was as illustrated in Table 2 below.

Calibration of the MCCCLMM Parameters
Our MCCCLMM model was calibrated to match the cap market prices provided by [36]. The parameters estimated were as summarised in the subsections of this section. For presentability purposes, in this paper, the parameter estimates stated were rounded off to the nearest 5 decimal places. However, the values used while pricing were not rounded off.

Estimation of the MCCCLMM Correlation Parameters
The correlation parameters were estimated using Equation (7) using data collected from [12] [13]. These parameter estimates were as summarized in Table  3.

Estimation of the Foreign Exchange Rate Volatility Parameter
The foreign exchange rate volatility parameter was estimated using the formula expressed in Equation (12) from historical rates collected from [33]. The X σ estimate was as summarised in Table 4. We assumed that 0.5. t ∆ =

Forward LIBOR Volatility Estimates
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 Table 5 and Table 6 respectively. The GBP LIBOR market volatility parameter fL γ was extracted directly from actual GBP ATM cap prices dataset struck on 2/1/2018 and summarized in Table 5.    The USD LIBOR volatility parameter dL γ was also extracted directly from actual USD ATM cap prices dataset struck on 2/1/2018 and summarized in Table 6.

Parameter
The dD σ parameter was calibrated using Equation (17) to exactly match the actual USD ATM cap prices. The optimal values of dD σ were as summarized in Table 7.

Parameters
The fD σ parameter was calibrated using Equation (17) to exactly match the actual GBP ATM cap prices. The optimal values of fD σ were as summarized in Table 8.

The Discount Curve
The US treasury and UK gilt yields collected from [35] and [34] respectively were assumed to be the risk free rates. The USD and GBP yield curves were then estimated using the NSS method.

Bootstrapping of the USD and GBP Yield Curves
The USD and GBP yield curves extracted from [35] and [34] respectively were bootstrapped using the NSS method described in Section 2.6. The estimated NSS model parameters of the two curves were as summarised in Table 9. Figure 3 and Figure 4 below show how the NSS yield curve matches the actual yields.

Estimation of the Discount Curve
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 Figure 5 and Figure 6.

Valuation of Caps and Floors in the Black MCCCLMM Formula
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. Journal of Mathematical Finance

USD Cap and Floor Prices
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). Table 10 gives the results of the USD ATM Cap and Floor prices in basis points (bp). Journal of Mathematical Finance

GBP Cap and Floor Prices
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). Table 11 gives the results of the GBP ATM Cap and Floor prices in basis points (bp).

Valuation of Quanto Caps and Floors in the Black MCCCLMM Formula
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. Table 12 gives the results of the Quanto ATM Cap and Floor prices in basis points (b.p). Where 0 X 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 Figure 7.
From Figure 7 it can be seen that for the 10 year period, the quanto caps are expected to be sold at a higher price compared to the quanto floors. However, it was noted that the rate of appreciation of the floor prices is quite high such that it is expected that they will eventually overtake the cap prices as time goes by.

Simulation of Cap and Floor Prices
The domestic and foreign LIBOR model dynamics were first discretized using the Euler scheme.

Generation of the Correlated Wiener Processes
The correlation matrix of our observed data was found to be:    The lower triangular matrix defined in Equation (25) was then calculated and found to be: The correlated Wiener process was then estimated as: where { } ; 1, 2,3, 4,5 i Z i∈ are independent standard normal random variables.

Simulation of USD Cap and Floor Prices
The formula expressed in Equation (28)  ( ) ,.

D d P 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 Table  13.

Simulation of GBP Cap and Floor Prices
The formula expressed in Equation (30)   ,.

D f P t
is the GBP risk-free discount factor associated with the U.K discount curve, 0.5 t ∆ = . The mean simulated GBP at the money (ATM) Cap and Floor prices stuck on 2/1/2018 were as summarised in Table 14.

Simulation of Quanto Cap and Floor Prices
The formula expressed in Equation (32) was used to simulate quanto cap or floor prices at time, t.
is the USD risk-free discount factor associated with the US discount curve, 0.5 t ∆ = . The mean simulated at the money (ATM) quanto Cap and Floor prices stuck on 2/1/2018 were as summarised in Table 15.

Comparison between the Black MCCCLMM Formulas and the Monte Carlo Simulations
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 Table 16. According to [38]

Cap and Floor Volatility Surfaces
Using the calibrated parameters, different USD, GBP and Quanto Cap and Floor prices were estimated under different strike price assumptions. The cap volatility  different options had roughly the same shape. It was also noted that in general over the three options, when the strike prices increase, the CAP prices reduced.
In addition to this, cap prices increased with maturity.
The Floor volatility surfaces of the USD, GBP and GBP/USD Quanto options were as shown in Figure 9 below: From Figure 9, we noticed that the floor volatility surfaces of the 3 different options also had roughly the same shape. It was also noted that in general, when the strike prices increase, the floor prices also increase. In addition to this, floor prices increased with maturity.

Conclusions
This study aimed at pricing quanto caps and floors using the multi-curve   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.
where A I is an indicator function such that Solving for M: We saw that  2π 1 ln ,

Data Availability
The historical datasets used are freely available on the websites [11], [12], [33]. [34], [35] as at the date last accessed. The USD and GBP ATM market cap prices used (provided by [36]) are given in Appendix B.