In this post, I use `R`

packages `RQuantLib`

and `ESGtoolkit`

for the calibration and simulation of the famous Hull and White short-rate model.

`QuantLib`

is an open source C++ library for quantitative analysis, modeling, trading, and risk management of financial assets. `RQuantLib`

is built upon it, providing `R`

users with an interface to the library .

`ESGtoolkit`

provides tools for building Economic Scenarios Generators (ESG) for Insurance. The package is primarily built for research purpose, and comes with no warranty. For an introduction to `ESGtoolkit`

, you can read this slideshare, or this blog post. A development version of the package is available on Github with, I must admit, only 2 or 3 commits for now.

The Hull and White (1994) model was proposed to address Vasicek’s model poor fitting of the initial term structure of interest rates. The model is defined as:

Where and are positive constants, and is a standard brownian motion under a risk-neutral probability. , which is constant in Vasicek’s model, is a function constructed so as to correctly match the initial term structure of interest rates.

An alternative and convenient representation of the model is:

where

and are market-implied instantaneous forward rates for maturities .

In insurance market consistent pricing, the model is often calibrated to swaptions, as there are no market prices for embedded options and guarantees.

Two parameters and are surely not enough to get back the whole swaptions volatility surface (or even ATM swaptions). But a *perfect* calibration to market-quoted swaptions isn’t vital and may lead to unnecessary overfitting. A more complex model may fit more precisely the swaptions volatility surface, but could still be proved to be more wrong for the purpose: insurance liabilities do not even match *exactly* market swaptions’ characteristics.

It’s worth mentioning that the yield curve bootstrapping procedure currently implemented in `RQuantLib`

, makes the implicit assumption that LIBOR is a good *proxy* for risk-free rates, and collateral doesn’t matter. These assumptions are still widely used in insurance, **along with** simple (parallel) adjustments for credit/liquidity risks, but were abandoned by markets after the 2007 subprime crisis.

For more details on the new multiple-curve approach, see for example http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2219548. In this paper, the authors introduce a multiple-curve bootstrapping procedure, available in `QuantLib`

.

# Cleaning the workspace rm(list=ls()) # RQuantLib loading suppressPackageStartupMessages(library(RQuantLib)) # ESGtoolkit loading suppressPackageStartupMessages(library(ESGtoolkit)) # Frequency of simulation and interpolation freq <- "monthly" delta_t <- 1/12 # This data is taken from sample code shipped with QuantLib 0.3.10. params <- list(tradeDate=as.Date('2002-2-15'), settleDate=as.Date('2002-2-19'), payFixed=TRUE, dt=delta_t, strike=.06, method="HWAnalytic", interpWhat="zero", interpHow= "spline") # Market data used to construct the term structure of interest rates # Deposits and swaps tsQuotes <- list(d1w =0.0382, d1m =0.0372, d3m = 0.0363, d6m = 0.0353, d9m = 0.0348, d1y = 0.0345, s2y = 0.037125, s3y =0.0398, s5y =0.0443, s10y =0.05165, s15y =0.055175) # Swaption volatility matrix with corresponding maturities and tenors swaptionMaturities <- c(1,2,3,4,5) swapTenors <- c(1,2,3,4,5) volMatrix <- matrix( c(0.1490, 0.1340, 0.1228, 0.1189, 0.1148, 0.1290, 0.1201, 0.1146, 0.1108, 0.1040, 0.1149, 0.1112, 0.1070, 0.1010, 0.0957, 0.1047, 0.1021, 0.0980, 0.0951, 0.1270, 0.1000, 0.0950, 0.0900, 0.1230, 0.1160), ncol=5, byrow=TRUE) # Pricing the Bermudan swaptions pricing <- RQuantLib::BermudanSwaption(params, tsQuotes, swaptionMaturities, swapTenors, volMatrix) summary(pricing) # Constructing the spot term structure of interest rates # based on input market data times <- seq(from = delta_t, to = 5, by = delta_t) curves <- RQuantLib::DiscountCurve(params, tsQuotes, times) maturities <- curves$times marketzerorates <- curves$zerorates marketprices <- curves$discounts ############# Hull-White short-rates simulaton # Horizon, number of simulations, frequency horizon <- 5 # I take horizon = 5 because of swaptions maturities nb.sims <- 10000 # Calibrated Hull-White parameters from RQuantLib a <- pricing$a sigma <- pricing$sigma # Simulation of gaussian shocks with ESGtoolkit set.seed(4) eps <- ESGtoolkit::simshocks(n = nb.sims, horizon = horizon, frequency = freq) # Simulation of the factor x with ESGtoolkit x <- ESGtoolkit::simdiff(n = nb.sims, horizon = horizon, frequency = freq, model = "OU", x0 = 0, theta1 = 0, theta2 = a, theta3 = sigma, eps = eps) # I use RQuantlib's forward rates. With the low monthly frequency, # I consider them as being instantaneous forward rates fwdrates <- ts(replicate(nb.sims, curves$forwards), start = start(x), deltat = deltat(x)) # alpha t.out <- seq(from = 0, to = horizon, by = delta_t) param.alpha <- ts(replicate(nb.sims, 0.5*(sigma^2)*(1 - exp(-a*t.out))^2/(a^2)), start = start(x), deltat = deltat(x)) alpha <- fwdrates + param.alpha # The short-rate r <- x + alpha # Stochastic discount factors (numerical integration currently is very basic) Dt <- ESGtoolkit::esgdiscountfactor(r = r, X = 1) # Monte Carlo prices and zero rates deduced from stochastic discount factors montecarloprices <- rowMeans(Dt) montecarlozerorates <- -log(montecarloprices)/maturities # RQuantLib uses continuous compounding # Confidence interval for the difference between market and monte carlo prices conf.int <- t(apply((Dt - marketprices)[-1, ], 1, function(x) t.test(x)$conf.int)) # Viz par(mfrow = c(2, 2)) # short-rate quantiles ESGtoolkit::esgplotbands(r, xlab = "maturities", ylab = "short-rate quantiles", main = "short-rate quantiles") # monte carlo vs market zero rates plot(maturities, montecarlozerorates, type='l', col = 'blue', lwd = 3, main = "monte carlo vs market \n zero rates") points(maturities, marketzerorates, col = 'red') # monte carlo vs market zero-coupon prices plot(maturities, montecarloprices, type ='l', col = 'blue', lwd = 3, main = "monte carlo vs market \n zero-coupon prices") points(maturities, marketprices, col = 'red') # confidence interval for the price difference matplot(maturities[-1], conf.int, type = 'l', main = "confidence interval \n for the price difference")