Skip to content

Commit e4e265e

Browse files
author
jenkins
committed
git subrepo pull (merge) ore
subrepo: subdir: "ore" merged: "74cecf1a8e" upstream: origin: "git@gitlab.acadiasoft.net:qs/ore.git" branch: "master" commit: "c60faa741d" git-subrepo: version: "0.4.6" origin: "https://github.com/ingydotnet/git-subrepo" commit: "73a0129"
2 parents 1e853e1 + c60faa7 commit e4e265e

37 files changed

Lines changed: 8822 additions & 68 deletions

‎Docs/FormulaBasedCoupon/Digital_MC1K.pdf‎

122 KB
Binary file not shown.

‎Docs/FormulaBasedCoupon/Digital_MC1K_2.pdf‎

119 KB
Binary file not shown.

‎Docs/FormulaBasedCoupon/MC10K.pdf‎

114 KB
Binary file not shown.

‎Docs/FormulaBasedCoupon/MC1K.pdf‎

119 KB
Binary file not shown.

‎Docs/FormulaBasedCoupon/formulabasedcoupon.tex‎

Lines changed: 399 additions & 0 deletions
Original file line numberDiff line numberDiff line change
@@ -0,0 +1,399 @@
1+
\documentclass[12pt, a4paper]{article}
2+
3+
%\usepackage[urlcolor=blue]{hyperref}
4+
5+
\usepackage[disable]{todonotes}
6+
\usepackage{todonotes}
7+
8+
\usepackage{booktabs}
9+
\usepackage{pbox}
10+
11+
\usepackage{listings}
12+
\usepackage{amsmath}%
13+
\usepackage{amsfonts}%
14+
\usepackage{amssymb}%
15+
\usepackage{graphicx}
16+
\usepackage[miktex]{gnuplottex}
17+
\ShellEscapetrue
18+
\usepackage{epstopdf}
19+
\usepackage{longtable}
20+
\usepackage{floatrow}
21+
\usepackage{minted}
22+
\usepackage{textcomp}
23+
\usepackage{color,soul}
24+
\usepackage[font={small,it}]{caption}
25+
\floatsetup[listing]{style=Plaintop}
26+
\usepackage[utf8]{inputenc}
27+
28+
% Turn off indentation but allow \indent command to still work.
29+
\newlength\tindent
30+
\setlength{\tindent}{\parindent}
31+
\setlength{\parindent}{0pt}
32+
\renewcommand{\indent}{\hspace*{\tindent}}
33+
34+
\addtolength{\textwidth}{0.8in}
35+
\addtolength{\oddsidemargin}{-.4in}
36+
\addtolength{\evensidemargin}{-.4in}
37+
\addtolength{\textheight}{1.6in}
38+
\addtolength{\topmargin}{-.8in}
39+
40+
\usepackage{longtable,supertabular}
41+
\usepackage{footnote}
42+
\usepackage{listings}
43+
\lstset{
44+
frame=top,frame=bottom,
45+
basicstyle=\ttfamily,
46+
language=XML,
47+
tabsize=2,
48+
belowskip=2\medskipamount
49+
}
50+
51+
%\usepackage{float}
52+
\usepackage{tabu}
53+
\tabulinesep=1.0mm
54+
\restylefloat{table}
55+
56+
\usepackage{siunitx}
57+
\usepackage{hyperref}
58+
\hypersetup{
59+
colorlinks=true, %set true if you want colored links
60+
linktoc=all, %set to all if you want both sections and subsections linked
61+
linkcolor=blue, %choose some color if you want links to stand out
62+
}
63+
64+
%\usepackage[colorlinks=true]{hyperref}
65+
66+
\renewcommand\P{\ensuremath{\mathbb{P}}}
67+
\newcommand\E{\ensuremath{\mathbb{E}}}
68+
\newcommand\Q{\ensuremath{\mathbb{Q}}}
69+
\newcommand\I{\mathds{1}}
70+
\newcommand\F{\ensuremath{\mathcal F}}
71+
\newcommand\V{\ensuremath{\mathbb{V}}}
72+
\newcommand\YOY{{\rm YOY}}
73+
\newcommand\Prob{\ensuremath{\mathbb{P}}}
74+
\newcommand{\D}[1]{\mbox{d}#1}
75+
\newcommand{\NPV}{\mathit{NPV}}
76+
\newcommand{\CVA}{\mathit{CVA}}
77+
\newcommand{\DVA}{\mathit{DVA}}
78+
\newcommand{\FVA}{\mathit{FVA}}
79+
\newcommand{\COLVA}{\mathit{COLVA}}
80+
\newcommand{\FCA}{\mathit{FCA}}
81+
\newcommand{\FBA}{\mathit{FBA}}
82+
\newcommand{\KVA}{\mathit{KVA}}
83+
\newcommand{\MVA}{\mathit{MVA}}
84+
\newcommand{\PFE}{\mathit{PFE}}
85+
\newcommand{\EE}{\mathit{EE}}
86+
\newcommand{\EPE}{\mathit{EPE}}
87+
\newcommand{\ENE}{\mathit{ENE}}
88+
\newcommand{\PD}{\mathit{PD}}
89+
\newcommand{\LGD}{\mathit{LGD}}
90+
\newcommand{\DIM}{\mathit{DIM}}
91+
\newcommand{\bs}{\textbackslash}
92+
\newcommand{\REDY}{\color{red}Y}
93+
\newcommand\enforce{\,\raisebox{0.8 ex}{$\substack{!\\\displaystyle=}$} \,}
94+
95+
96+
\begin{document}
97+
98+
\title{ORE Formula Based Coupon Module}
99+
\author{Quaternion Risk Management}
100+
\date{20 Feburary 2019}
101+
\maketitle
102+
103+
\newpage
104+
105+
%-------------------------------------------------------------------------------
106+
\section*{Document History}
107+
108+
\begin{center}
109+
\begin{supertabular}{|l|l|l|p{7cm}|}
110+
\hline
111+
ORE+ Release & Date & Author & Comment \\
112+
\hline
113+
na & 29 August 2018 & Peter Caspers& initial version\\
114+
1.8.4.0 & 20 February 2019 & Sarp Kaya Acar & add examples \\
115+
\hline
116+
\end{supertabular}
117+
\end{center}
118+
119+
\vspace{3cm}
120+
121+
\newpage
122+
123+
\tableofcontents
124+
\vfill
125+
126+
\textcircled{c} 2019 Quaternion Risk Management Limited. All rights reserved.
127+
Quaternion\textsuperscript{\textregistered} is a trademark of Quaternion Risk Management Limited and is also registered
128+
at the UK Intellectual Property Office and the U.S. Patent and Trademark Office. All other trademarks are the property
129+
of their respective owners. Open Source Risk Engine\textsuperscript{\textcircled{c}} (ORE) is sponsored by Quaternion
130+
Risk Management Limited.
131+
132+
\newpage
133+
134+
135+
136+
%-------------------------------------------------------------------------------
137+
\section{Summary}
138+
139+
This document describes the formula based coupon implemented in ORE+.
140+
141+
142+
143+
\section{Methodology}
144+
\label{ssection_FormulaBasedLeg}
145+
\subsection{Introduction}
146+
147+
We consider a generalised structured coupon which at time $T$ pays
148+
149+
\begin{equation} \tau f(R_1(t), \ldots, R_n(t))
150+
\end{equation}
151+
152+
where $\tau$ is the year fraction of the coupon, $R_i$'s are IBOR and/or CMS rates fixed at time $t<T$ and $f$ is a
153+
payoff function. Moreover we assume that the currency of the coupon can be different than the currency of the rates,
154+
i.e. the quanto-payoffs are allowed. For instance, let us consider the below example
155+
%
156+
\begin{equation}
157+
f(R_1, R_2) = 1_{\{R_3>0.03\}}\max\{\min\{9\cdot (R_2-R_1)+0.02,0.08\},0.0\},
158+
\label{eq:FBCpayoff}
159+
\end{equation}
160+
%
161+
with $R_1=$GBP-CMS-2Y, $R_2=$EUR-CMS-10Y, $R_3=$USD-CMS-5Y. With the Formula-Based-Coupon module it is possible to price
162+
such kind of coupons by using the Monte-Carlo simulation technique. In the next section, we will present the underlying
163+
pricing model.
164+
165+
\subsection{Pricing Model}
166+
\label{ssection_FormulaBasedLegPricingModel}
167+
For non-callable structured coupons, we generalise the model in
168+
\cite{brigo}, 13.16.2. We assume the rates
169+
to evolve with a shifted lognormal dynamics under the $T$-forward
170+
measure in the respective currency of $R_i$ as
171+
172+
\begin{equation}\label{lndyn} dR_i = \mu_i (R_i + d_i) dt + \sigma_i
173+
(R_i + d_i) dZ_i
174+
\end{equation}
175+
176+
with displacements $d_i>0$, drifts $\mu_i$ and volatilities $\sigma_i$
177+
or alternatively with normal dynamics
178+
179+
\begin{equation}\label{ndyn} dR_i = \mu_i dt + \sigma_i dZ_i
180+
\end{equation}
181+
182+
for $i=1,\ldots,n$ where in both cases $Z_i$ are correlated Brownian
183+
motions
184+
185+
\begin{equation} dZ_i dZ_j = \rho_{i,j} dt
186+
\end{equation}
187+
188+
with a positive semi-definite correlation matrix $( \rho_{i,j})$. The
189+
drifts $\mu_i$ are determined using given convexity adjustments
190+
191+
\begin{equation} c_i = E^T(R_i(t)) - R_i(0)
192+
\end{equation}
193+
194+
where the expectation is taken in the $T$-forward measure in the
195+
currency of the respective rate $R_i$. Slightly abusing notation $c_i$
196+
can be computed both using a model consistent with \ref{lndyn}
197+
resp. \ref{ndyn} (i.e. a Black76 style model) or also a different
198+
model like e.g. a full smile TSR model to compute CMS
199+
adjustments. While the latter choice formally introduces a model
200+
inconsistency it might still produce more market consistent prices at
201+
the end of the day, since it centers the distributions of the $R_i$
202+
around a mean that better captures the market implied convexity
203+
adjustments of the rates $R_i$ entering the structured coupon payoff.
204+
205+
Under shifted lognormal dynamics the average drift is then given by
206+
207+
\begin{equation} \mu_i = \frac{\log( (R_i(0)+d_i+c_i) / (R_i(0)+d_i)
208+
)}{t}
209+
\end{equation}
210+
211+
and under normal dynamics
212+
213+
\begin{equation} \mu_i = c_i / t
214+
\end{equation}
215+
216+
The NPV $\nu$ of the coupon is now given by
217+
218+
\begin{equation}\label{struccpnnpv} \nu = P(0,T) \tau E^T ( f(R_1(t),
219+
\ldots, R_n(t)) )
220+
\end{equation}
221+
222+
where $P(0,T)$ is the applicable discount factor for the payment time
223+
in the domestic currency and the expectation is taken in the
224+
$T$-forward measure in the domestic currency. To adjust \ref{lndyn}
225+
resp. \ref{ndyn} for the measure change between the currency of $R_i$
226+
and the domestic currency (if applicable), we apply the usual Black
227+
quanto adjustment
228+
229+
\begin{equation}
230+
\mu_i \rightarrow \mu_i + \sigma_i \sigma_{i,X} \rho_{i,X}
231+
\end{equation}
232+
233+
where $\sigma_{i,X}$ is the volatility of the applicable FX rate and
234+
$\rho_{i,X}$ is the correlation between the Brownian motion driving
235+
the FX process and $Z_i$.
236+
237+
To evaluate the expectation in \ref{struccpnnpv} a Monte Carlo
238+
simulation can be used to generate samples of the distribution of
239+
$(R_1, \ldots, R_n)$, evaluate $f$ on these samples and average the
240+
results.
241+
242+
\section{Parametrisation}
243+
244+
%\subsection{Trade Components}
245+
246+
\subsection{Formula Based Leg Data}
247+
248+
The formula based leg data allows to use complex formulas to describe coupon payoffs. Its {\tt LegType} is {\tt
249+
FormulaBased}, and it has the data section {\tt FormulaBasedLegData}. It supports IBOR and CMS based payoffs with
250+
quanto and digital features. The following examples shows the definition of a coupon paying a capped / floored cross
251+
currency EUR-GBP CMS Spread contingent on a USD CMS barrier.
252+
253+
254+
255+
The {\tt Index} field supports operations of the following kind:
256+
\begin{itemize}
257+
\item indices like IBOR and CMS indices, and constants as factors,
258+
spreads and/or cap/floor values;
259+
\item basic operations: $+$, $-$, $\cdot$, $/$;
260+
\item operators gtZero() (greater than zero) and geqZero() (greater than or equal zero) yielding $1$ if the argument is
261+
$>0$ (resp. $\geq 0$) and zero otherwise
262+
\item functions: abs(), exp(), log(), min(), max(), pow()
263+
\end{itemize}
264+
%
265+
In listing \ref{lst:FBLegdata}, we present the {\tt FormulaBasedLegData} of the payoff in equation \ref{eq:FBCpayoff}.
266+
%
267+
\begin{listing}
268+
\begin{minted}[fontsize=\footnotesize]{xml}
269+
<LegData>
270+
<LegType>FormulaBased</LegType>
271+
<Payer>true</Payer>
272+
<Currency>EUR</Currency>
273+
...
274+
<FormulaBasedLegData>
275+
<Index>gtZero({USD-CMS-5Y}-0.03)*
276+
max(min(9.0*({EUR-CMS-10Y}-{GBP-CMS-2Y})+0.02,0.08),0.0)</Index>
277+
<IsInArrears>false</IsInArrears>
278+
<FixingDays>2</FixingDays>
279+
</FormulaBasedLegData>
280+
...
281+
</LegData>
282+
\end{minted}
283+
\caption{FormulaBasedLegData configuration.}
284+
\label{lst:FBLegdata}
285+
\end{listing}
286+
%
287+
\section{Pricing Engine settings}
288+
289+
The configuration of the pricing engine for the model introduced in section \ref{ssection_FormulaBasedLegPricingModel}
290+
is given in listing \ref{lst:FBCpricer}.
291+
292+
\begin{itemize}
293+
\item The formula based coupon pricer uses the pricing engine configuration for { \tt CapFlooredIborLeg} and {\tt CMS}.
294+
For the configuration of these coupon pricers we refer to sections 7.3, 7.7.3 and 7.74 in \cite{oreug}.
295+
\item The parameter \verb+FXSource+ specifies the FX index tag to be used to look up FX-Ibor or FX-CMS correlations, see
296+
also 7.10.2 in \cite{oreug}.
297+
\end{itemize}
298+
%
299+
\begin{listing}
300+
\begin{minted}[fontsize=\footnotesize]{xml}
301+
<!-- Formula Based Coupons -->
302+
<Product type="FormulaBasedCoupon">
303+
<Model>BrigoMercurio</Model>
304+
<ModelParameters>
305+
<Parameter name="FXSource">ECB</Parameter>
306+
</ModelParameters>
307+
<Engine>MC</Engine>
308+
<EngineParameters>
309+
<Parameter name="Samples">1000</Parameter>
310+
<Parameter name="Seed">42</Parameter>
311+
<Parameter name="Sobol">Y</Parameter>
312+
<Parameter name="SalvageCorrelationMatrix">Y</Parameter>
313+
</EngineParameters>
314+
</Product>
315+
\end{minted}
316+
\caption{Pricing engine configuration for formula based coupon pricer.}
317+
\label{lst:FBCpricer}
318+
\end{listing}
319+
%
320+
\section{Market Data}
321+
322+
The relevant market data for the formula based coupon pricer encompasses
323+
324+
\begin{enumerate}
325+
\item rate curves (for index projection and cashflow discounting)
326+
\item cap / floor volatilities (for Ibor coupon pricers in the relevant currencies)
327+
\item swaption volatilities (for CMS coupon pricers in the relevant currencies)
328+
\item correlation curves for the relevant Ibor-Ibor, Ibor-CMS, CMS-CMS, Ibor-FX, CMS-FX pairs
329+
\end{enumerate}
330+
331+
See \cite{oreug} for details on the setup and configuration of this market data.
332+
333+
\section{Examples}
334+
335+
\subsection{CMS Spread}
336+
We demonstrate a single currency Swap (currency EUR, maturity 20y, notional 10m, receive fixed $0.011244\%$ annual, pay
337+
$\max \left\{ \text{CMS-EUR-10Y}- \text{CMS-EUR-1Y}, 0.0 \right \}$ semi-annual).
338+
%
339+
We simulate the exposure with 1000 Monte-Carlo paths by using the formula based coupon pricer and the analytical
340+
CMS-Spread pricer [section 7.10.8, \cite{oreug}]. The number of Monte-Carlo paths in formula based coupon pricer is
341+
1000. EPE profiles with both runs are given in figure \ref{fig:MC1K}. We observed that the run with the formula based
342+
coupon pricer is approximately $2.5$ times slower than the analytical one.
343+
%
344+
\begin{figure}
345+
\includegraphics[scale=0.55]{MC1K.pdf}
346+
\caption{20Y EUR interest rate swap with floored CMS spread leg, i.e.
347+
$\max(\text{EUR-CMS-10Y} - \text{EUR-CMS-1Y},\, $0.5\%$) $ }
348+
\label{fig:MC1K}
349+
\end{figure}
350+
351+
\subsection{Digital CMS Spread}
352+
As a second example, we demonstrate a single currency Swap (currency EUR, maturity 20y, notional 10m, receive fixed
353+
$0.011244\%$ annual, pay
354+
$\left(\text{CMS-EUR-10Y}- \text{CMS-EUR-1Y} \right)+ 0.01 * 1_{\left\{\text{CMS-EUR-10Y}-
355+
\text{CMS-EUR-1Y}>0.0\right\}} $ semi-annual).
356+
357+
\begin{figure}
358+
\includegraphics[scale=0.55]{Digital_MC1K_2.pdf}
359+
\caption{20Y EUR interest rate swap with digital CMS spread leg, i.e.
360+
$\left(\text{CMS-EUR-10Y}- \text{CMS-EUR-1Y} \right)+ 0.01 * 1_{\left\{\text{CMS-EUR-10Y}-
361+
\text{CMS-EUR-1Y}>0.0\right\}} $ }
362+
\label{fig:Digital:MC1K}
363+
\end{figure}
364+
365+
366+
367+
\pagebreak
368+
%\appendix
369+
%\section*{Appendices}
370+
%\addcontentsline{toc}{section}{Appendices}
371+
\renewcommand{\thesubsection}{\Alph{subsection}}
372+
%\section{Appendix}
373+
374+
%\subsection{Test}\label{ss_LGM}
375+
376+
377+
378+
\begin{thebibliography}{1}
379+
380+
\bibitem{oreug} ORE User Guide,
381+
\url{http://www.opensourcerisk.org/documentation/}
382+
% \bibitem{AP}Andersen, Leif B. G., Piterbarg, Vladimir V.: Interest
383+
% Rate Modeling -- Vol. 3: Products and Risk Management, Atlantic
384+
% Financial Press, 2010
385+
\bibitem{brigo}Brigo, Damiano; Mercurio, Fabio: Interest Rate Models - Theory and
386+
Practice, 2nd Edition, Springer, 2006
387+
% \bibitem{Hagan}Hagan, Patrick: Convexity Conundrums, Wilmott, 2003
388+
% \bibitem {Hfb}Heidorn, Schmidt, \textit{LIBOR in Arrears}, 1998, Frankfurt School of Finance
389+
% \bibitem{LSG}Lichters, Roland; Stamm, Roland; Gallgher, Donal: Modern
390+
% Derivatives Pricing, Palgrave Macmillan, 2015
391+
% \bibitem {Papa}Papaioannou Denis, \textit{Applied Multidimensional
392+
% Girsanov Theorem}, Electronic copy available at:
393+
% http://ssrn.com/abstract=1805984
394+
395+
396+
397+
\end{thebibliography}
398+
399+
\end{document}

0 commit comments

Comments
 (0)