# On Deep Learning for computing the Dynamic Initial Margin and Margin Value
Adjustment

Joel P. Villarino1,2,111Corresponding author<EMAIL_ADDRESS>Álvaro Leitao2,3

(1Department of Mathematics, University of A Coruña, Spain
2CITIC Research Centre, University of A Coruña, Spain
3Universitat Oberta de Catalunya (UOC), Spain
)

ABSTRACT. The present work addresses the challenge of training neural networks
for Dynamic Initial Margin (DIM) computation in counterparty credit risk, a
task traditionally burdened by the high costs associated with generating
training datasets through nested Monte Carlo (MC) simulations. By condensing
the initial market state variables into an input vector, determined through an
interest rate model and a parsimonious parameterization of the current
interest rate term structure, we construct a training dataset where labels are
noisy but unbiased DIM samples derived from single MC paths. A multi-output
neural network structure is employed to handle DIM as a time-dependent
function, facilitating training across a mesh of monitoring times. The
methodology offers significant advantages: it reduces the dataset generation
cost to a single MC execution and parameterizes the neural network by initial
market state variables, obviating the need for repeated training. Experimental
results demonstrate the approach’s convergence properties and robustness
across different interest rate models (Vasicek and Hull-White) and portfolio
complexities, validating its general applicability and efficiency in more
realistic scenarios.

##  1 Introduction

Due to the financial crisis experienced in 2008, which led to the bankruptcy
of some major financial institutions, such as Lehman Brothers, the G8 World
Council promoted the regulation of stricter actions for financial markets,
specially in the over-the-counter (OTC) derivatives market. These are
bilateral transactions between (often large) institutions prevailed under
certain opacity, giving rise to a counterparty credit risk (CCR), a risk due
to the failure to fulfill contractual payment obligations, as a result of
outstanding cashflows that are not yet settled. Since then, financial
regulators have introduced the obligation of adopting measures to mitigate
CCR, which leads to a proliferation of models that take such risks into
account in derivatives pricing. They became globally known as _x_ -value
adjustment (xVA), where the _x_ accounts for the kind of risk to be mitigated.
The very first ones were credit and debit value adjustments (CVA and DVA),
whose foundations can be seen in, e.g., [8, 37, 7].

Following the regulatory requirements to mitigate the systemic risk associated
with non-cleared OTC derivatives, specially in the context of interest rate
derivatives, another key tool is the so-called collateralisation. It is based
on the periodic deposit of an amount of money (or some other security if it is
allowed) to serve as a protection against counterparty default. More
concretely, the Basel Committee on Banking Supervision (BCBS) and the
International Organization of Securities Commissions (IOSCO) mandate
institutions engaging in these transactions to post bilateral Variation Margin
(VM) and Initial Margin (IM) on a daily basis at a netting set level, [6]. VM
aims to cover the current exposure resulting from changes in the instrument’s
mark-to-market value by reflecting its current size, but has an associated
funding cost, known as funding value adjustment (FVA), which is reflected by
the expected exposure of the portfolio. Theoretical derivation is provided and
discussed in [37, 7]. Such a collateral is not sufficient to fully mitigate
the credit risk of the trades, which still arises from the time required for
the surviving party to close-out its position after default. This close-out
time is known as Margin Period of Risk (MPoR), typically of $10$ business
days. Then, IM aims to cover the potential future exposure that could arise in
such a period, acting as a protection against pronounced changes in the value
of the trades. The BCBS requires that the IM collateral for bilateral OTC
derivatives should be at least at the level of a $99\%$ value-at-risk (VaR)
changes over the MPoR. Given that the exchange of IM must take place over the
life of the transactions, it is to be expected that posting this collateral
will generate, as in the case of VM, associated funding costs. The total
expected cost is known as margin value adjustment (MVA) and was introduced for
the first time by the authors in [15]. The calculation of this adjustment
implies the necessity to accurately compute not only the spot IM value, but
also to forecast its future levels, a problem known as Dynamic Initial Margin
(DIM).

Traditionally, the use of methodologies based on VaR metrics generates
disputes between the parties because of their dependency on the model under
consideration. The IM calculation was not exempt, so in order to simplify the
reconciliation between counterparties, the International Swaps and Derivatives
Association (ISDA) has promoted a method for computing spot IM, called
Standard Initial Margin Model (SIMM) [24]. This model is designed to estimate
a $99\%$ VaR level by leveraging first order portfolio sensitivities (delta
and vega) to specific risk factors, with the appropriate weighting based on
parameter calibration conducted during stressful periods. Hence, the
computation of the MVA under ISDA SIMM approach boils down to the challenge of
computing the forward portfolio sensitivities to a large set of underlying
risk factors. A detailed explanation of the spot IM-SIMM and the extension to
the forward computation can be found at [25, 38], respectively. In both of
them, it is presented the approach commonly known as brute-force simulation.
Basically, it is a Monte Carlo (MC) framework where we need to compute, for
each MC path, the forward portfolio sensitivities with respect to each SIMM
risk factor. This leads to a nested MC simulation which has a high
computational cost, specially for exotic derivatives that have no closed-form
pricing formula.

In practice, where portfolios can be made up of hundreds or thousands of
products, the high computational cost of the aforementioned brute-force
approach means that may not be feasible in many production systems. So, in
recent years, new solutions have emerged to approximate such quantities. On
the one hand, a great deal of work has focused on numerical regression
techniques based on the modification and improvement of the least-squares MC
approach presented in [31]. The authors in [28] employ such a numerical method
to efficiently estimate the sensitivities required in the IM-SIMM methodology,
whilst the authors in [33] have proposed a statistical method in which they
fit DIM conditional moments, estimated using least-squares MC, via Johnson-
type distributions. Other statistical approaches can be found in, e.g., [2].
On the other hand, much effort has been put into improving the computing speed
of forward sensitivities. In this direction, we can highlight works such as
[42], where the authors present a method based on Chebyshev tensors to compute
dynamic sensitivities of financial instruments within a MC simulation; or [11,
4, 26], where it has been shown how adjoint algorithmic differentiation (AAD)
can be used to calculate sensitivities (even along the MC paths) reliably and
orders of magnitude faster than with finite-difference approaches.

In recent years, new methodologies, based on the advances in machine learning
and deep learning, have arisen in the quantitative finance field (see [21, 22,
29, 30], for example). Accounting for CCR and xVA in general, we highlight
works like [40], where the authors employ deep learning to solve partial
differential equations for pricing options subject to CCR; [1], where they
compute xVA quantities combining forward/backward stochastic differential
equation (SDE) formulations with deep regression techniques; or [14], where a
computational framework for portfolio-wise risk management founded on
recursive application of a neural network-based BSDE solver is presented. In
the particular case of DIM and MVA we also find works using deep learning
tools. For example, in [5] a neural network-based method to approximate the
Bermudan pricing function and sensitivities is proposed; in [32] the authors
achieve a neural network-based IM simulation of a portfolio consisting of $50$
trades; or [20], where a static replication algorithm for interest rate
options with early-exercise features is combined with neural networks to get
conditional prices and sensitivities, which serve as an input to SIMM-driven
MVA quantification.

However, many of the these deep learning alternatives to compute DIM-SIMM and
consequent MVA depend on the production of computationally expensive training
datasets, even when taking advantage of the statistical methods presented
above. In this sense, our proposal is based on using a multi-output neural
network as a proxy for DIM at monitoring times (which can be integrated to get
MVA), incorporating one of the main ideas presented in [22] (and also
exploited, in a different financial context, in [17]) for its training.
Basically, they prove that, if you want to train a network to learn a quantity
defined in terms of a conditional expectation on an initial state, you can use
a training set made up of noisy estimations of such a quantity. Following this
principle, we build training datasets for interest rate portfolios in which
the IM-SIMM risk factors are encapsulated in interest rate model parameters
and parsimonious representations of term structures (that serves as our
network inputs, leading to a significant reduction in the number of model
inputs), and whose labels (DIM targets) are highly noisy, simulated with a
single MC sample. This strategy leads to two competitive advantages for the
implementation in a real CCR engine. First, for the price of a single MC
execution, we get a whole training dataset, and second, once the network has
been fitted, it can predict DIM trajectories for a subset of interest rate
model parameters (it parameterizes DIM trajectories). This is a major
contribution since it allows, in a few minutes, to obtain precise DIM
estimations at monitoring times of interest throughout the life of the desired
portfolio, ready for use wherever needed.

The outline of the paper is as follows. In Section 2 we start by introducing
the theoretical background for the DIM and MVA computation in the interest
rate framework. Section 3 is devoted to the methodology followed in the
elaboration of the training and validation datasets, as well as to the
presentation of our proposal. In Section 4 we present numerical experiments to
validate our methodology, essentially convergence and error analysis on a
simple portfolio, consisting of a single swap; and the application to a more
realistic portfolio. Finally, a summary of the work and conclusions are
presented in Section 5.

##  2 Problem formulation

This section covers the introduction of relevant quantities when discussing
CCR and collateralization, the basics of IM and the introduction of the
standard methodology for its computation, as well as the theoretical
background for the generation of prices and sensitivities of the proposed
products and portfolios. Once these topics are presented, we consider its
application in the calculation of DIM and MVA, which play an important role in
producing the datasets used in network training and validation.

###  2.1 Model assumptions

From now on, we consider a frictionless and arbitrage-free continuous-time
financial market with a finite time-horizon $T_{f}>0$. In addition, let
$\left(\Omega,\mathcal{F},\mathbb{Q}\right)$ be the probabilistic space with
the market filtration $\mathcal{F}$, given by $(\mathcal{F}_{t})_{0\leq t\leq
T_{f}}$, and the risk neutral probability measure $\mathbb{Q}$. Such measure
is associated with the money market account $M_{t}$ as the numéraire, ensuring
that all the attainable claims denominated by the numéraire are martingales
under $\mathbb{Q}$, [19].

Given that we work with interest rate products, the short-rate dynamics we
choose for the simulation of these products is of particular relevance. In
particular, we restrict ourselves to work with Gaussian affine term-structure
models since they provide closed-form formulas which allow the tractability of
the computations, [9]. Thus, we consider the Markovian state variables
$x_{t}=\left(x_{1}(t),\dots,x_{d}(t)\right)\in\mathbb{R}^{d}$, and the smooth
functions $\mu:[0,T_{f}]\times\mathbb{R}^{d}\to\mathbb{R}^{d}$,
$\sigma:[0,T_{f}]\to\mathbb{R}^{d\times d}$. The state variables are assumed
to verify the stochastic differential equation (SDE),

$\text{d}x_{t}=\mu(t,x_{t})\text{d}t+\sigma(t)\text{d}W_{t},$ (2.1)

where $W_{t}=\left(W_{1}(t),\dots,W_{d}(t)\right)$ is a $d$-dimensional
Brownian motion under $\mathbb{Q}$ and $\mu$ is an affine map in the $x_{t}$
components. Under these assumptions, we are able to define the short-rate
$r_{t}$ as an affine map of the state variables $x_{t}$, leading to the
advantage of being able to express the continuously compounded spot rate,
commonly known as zero-rate, as an affine function of such a short-rate
$r_{t}$. This relationship is exploited through the definition of zero-coupon
bonds. Recall that a zero-coupon bond is a contract that guarantees its holder
the payment of one unit of currency at time $T$, with no intermediate payments
[9], and usually its value at time $t\leq T$ is denoted by $P(t,T)$. Under
affine term-structure models the zero-coupon bond prices are exponential
affine in $x_{t}$, i.e., they can be written in the form

$P(t,T)=B(t,T)e^{-C(t,T)r(t)},$

with $B$ and $C$ deterministic functions of time, which allows us to get the
zero-rates via

$Y(t,T)=-\dfrac{\log\left(P(t,T)\right)}{\tau\left(t,T\right)},$

where $\tau(t,T)$ denotes the year fraction between $t$ and $T$. For the sake
of simplicity, we adopt a single-curve framework in which the same short-rate
is used for pricing and discounting.

###  2.2 Pricing approach and instruments

Under the market assumptions established above, the general pricing formula of
a financial derivative (or a portfolio of derivatives) under perfect
collateralization with payoff $V(T)$ paid at time $T>t$ is given by

$V(t)=\mathbb{E}^{\mathbb{Q}}\left[e^{-\int_{t}^{T}r(u)\text{d}u}V(T)|\mathcal{F}_{t}\right].$
(2.2)

This general pricing formula can be clearly simplified in the case of the
product we discuss, namely interest rate swaps (IRS). It should be noted that,
if a portfolio of derivatives is considered, this can be reduced to the sum of
the value of the individual contracts by means of (2.2).

####  2.2.1 Interest rate swaps

An IRS is a contract where one party agrees to pay the fixed cash flows that
are equal to the interest at a predetermined, fixed rate on a notional amount
(fixed leg) and where the other party pays a floating interests on the same
notional amount (floating leg). We provide an informal approach to explain the
swap pricing formula. Details of how to formally derive the expression from
(2.2) can be found, for example, in [35, 36].

First, we introduce some notation relative to the payment schedules for the
fixed and floating leg of the swap. Let be

$\begin{split}\mathcal{T}^{P}&=\\{T_{0}^{P},\dots,T_{i}^{P},\dots,T_{n}^{P}\\},\\\
\mathcal{T}^{R}&=\\{T_{0}^{R},\dots,T_{j}^{R},\dots,T_{m}^{R}\\},\end{split}$
(2.3)

the fixed and floating schedules, respectively. Without loss of generality, we
assume that $T_{0}^{P}=T_{0}^{R}$, $T_{n}^{P}=T_{m}^{R}$.

Regarding the fixed leg, at every instant $T_{i}^{P}\in\mathcal{T}^{P},i>0,$
it pays out the amount $NK\tau_{P}(T_{i-1}^{P},T_{i}^{P})$ corresponding to a
fixed interest rate $K$, a nominal value $N$ and a year fraction convention
$\tau_{P}$. In order to know the present value of the cash flows, it is
necessary to apply the corresponding discounts. Thus, the discounted value at
time $t$ is

$PV_{\text{fixed}}(t;\mathcal{T}^{P},K)=N\sum_{k=\eta_{P}(t)}^{n}\tau_{P}(T_{k-1}^{P},T_{k}^{P})P(t,T_{k}^{P})K,$
(2.4)

with $\eta_{P}(t)=\min\\{{i\in\\{1,\dots,n\\}:T_{i}^{P}>t}\\}$.

The floating leg pays at every instant $T_{j}^{R}\in\mathcal{T}^{R},j>0$ the
amount $N\tau_{R}(T_{i-j}^{R},T_{j}^{R})L(T_{j-1}^{R},T_{j}^{R})$, with
$L(T_{j-1}^{R},T_{j}^{R})$ the underlying spot float rate resetting at
previous instant $T_{j-1}^{R}\in\mathcal{T}^{R}$. Discounting future cash
flows, the present value of this leg at time $t$ is

$PV_{\text{float}}(t;\mathcal{T}^{R})=N\sum_{k=\eta_{R}(t)}^{m}\tau_{R}(T_{k-1}^{R},T_{k}^{R})P(t,T_{j}^{R})F_{k}(t),$
(2.5)

where $\eta_{R}(t)=\min\\{{j\in\\{1,\dots,m\\}:T_{j}^{R}>t}\\}$ and $F_{j}(t)$
is the forward rate observed at time $t$, which can be written as

$F_{j}(t)=\begin{cases}\dfrac{1}{\tau_{R}(T_{j-1}^{R},T_{j}^{R})}\left(\dfrac{P(t,T_{j-1}^{R})}{P(t,T_{j}^{R})}-1\right),&t<T_{j}^{R},\\\
L(T_{j-1}^{R},T_{j}^{R}),&T_{j-1}^{R}\leq t\leq T_{j}^{R}.\end{cases}$

Thus, the price of the IRS at time $t$ is given by the difference between the
present values of such legs (2.4)-(2.5), i.e.,

$V_{\text{S}}(t;\mathcal{T}^{P},\mathcal{T}^{R},K,w)=wN\left[\sum_{k=\eta_{R}(t)}^{m}\tau_{R}(T_{k-1}^{R},T_{k}^{R})P(t,T_{j}^{R})F_{k}(t)-\sum_{k=\eta_{P}(t)}^{n}\tau_{P}(T_{k-1}^{P},T_{k}^{P})P(t,T_{k}^{P})K\right],$
(2.6)

with $w=\in\\{1,-1\\}$ for payer or receiver swap, respectively. Typically,
the swap is contractually configured to be at-the-money (ATM) at inception,
i.e., the initial value of the swap is zero for both parties. The strike value
for which the swap equals zero is called swap rate,
$S(t;\mathcal{T}^{P},\mathcal{T}^{R})$, and is given by

$S(t;\mathcal{T}^{P},\mathcal{T}^{R})=\dfrac{\sum_{k=\eta_{R}(t)}^{m}\tau_{R}(T_{k-1}^{R},T_{k}^{R})P(t,T_{j}^{R})F_{k}(t)}{A(t;\mathcal{T}^{P})}.$
(2.7)

where
$A(t;\mathcal{T}^{P})=\sum_{k=\eta_{P}(t)}^{n}\tau_{P}(T_{k-1}^{P},T_{k}^{P})P(t,T_{k}^{P})$
is known as the annuity factor, which represents the present value of a basis
point of the swap.

Note that the derived IRS pricing formula (2.6) is model-independent (due to
its linear payoff), i.e., it does not require any interest rate model to
compute its net present value. Furthermore, future swap prices can be computed
with any model that allows the valuation of zero-coupon bonds, justifying the
adoption of affine term-structure models.

###  2.3 Collateral and Initial Margin

As discussed in the introduction, one of the major concerns in the financial
world today is to be able to assess and mitigate the potential losses that an
institution may incur in the event of a counterparty default. The most
commonly used metrics for quantifying expected losses in the counterparty
default are based on the concept of exposure, which represents the value that
may be at risk. In particular, we focus on the expected positive exposure
(EPE), which quantifies the expected losses in the event of a counterparty
default; and on the potential future exposure (PFE), which indicates what the
worst-case scenario for losses (with respect to a certain confidence level)
would be at a future time, [16].

There are several mechanisms for reducing the institution’s exposure to open
positions with its counterparty, including credit/debit value adjustments,
netting strategies or collateralisation tools, such as VM and IM, being the
later the one addressed here. IM can be seen as an amount that covers PFE for
the expected time between the last VM exchange and the liquidation of
positions on the default counterparty, [10]. The BCBS-IOSCO have jointly
established rules for the calculation of IM. According to these rules, IM must
be computed as a netting set $99\%$ confidence level PFE over a MPoR; and
market participants have some freedom to develop internal models for the
computation of said quantity, subject to validation and backtesting by
regulators. Nevertheless, in the face of potential disputes between entities
due to discrepancies in the calculated value, ISDA has introduced SIMM, a
sensitive-based approach based on the risk profile of a position in terms of
delta, vega and curvature sensitivities, defined across risk factors by tenor,
expiry and asset class [23]. We adopt this model to calculate IM values
required for our approach.

Following [25], this methodology distinguishes between four product classes:
Interest Rates and Foreign Exchange (FX), Credit, Equity and Commodity, in
such a way that every trade is assigned to an individual bucket. The total IM
is given by the sum of IM-SIMM of each individual product class, i.e.,

$\text{IM}=\text{SIMM}_{\text{RatesFX}}+\text{SIMM}_{\text{Credit}}+\text{SIMM}_{\text{Equity}}+\text{SIMM}_{\text{Commodity}},$

with $\text{SIMM}_{X}$ the IM value of the product class $X$. In addition,
every trade could be affected by a class-specific risk from the risk classes
provided, namely: interest rate, qualifying credit, non-qualifying credit,
equity, commodity and foreign exchange. Following this order, the SIMM for
each product class $X$ is given by the variance-covariance formula

$\text{SIMM}_{X}=\sqrt{\sum_{r}\text{IM}_{r}^{2}+\sum_{r}\sum_{r\neq
s}\psi_{rs}\text{IM}_{r}\text{IM}_{s}},$

where $\text{IM}_{r}$ refers to the margin associated to each risk class and
$\psi_{rs}$ is a correlation matrix between them given by ISDA. Lastly, the
margin for each risk class is given by the sum of the delta, vega and
curvature margins, i.e.,

$\text{IM}_{r}=\text{DeltaMargin}_{r}+\text{VegaMargin}_{r}+\text{CurvatureMargin}_{r},$

where each component represents a PFE estimation with respect to the
underlying risk factors. Details on how to calculate the delta, vega and
curvature margins can be found in [25] and our choice is described in Section
3.1.

###  2.4 DIM and MVA

The use of different forms of collateral is associated with different funding
costs. Since IM is a form of collateralisation that cannot be netted and must
be posted into a segregated account, it is not rehypothecatable, leading to a
considerable cost for the entities involved in the exchange. The risk
associated with this funding cost is quantified by the MVA, introduced in
[15]. Assessing MVA involves determining the price associated with holding IM
throughout the lifetime of the transaction in question, which entails the need
to calculate the IM at future times. Due to the stochastic nature of the
underlying dynamics, the future IM value inherently represents a stochastic
process [20], whose valuation is performed by taking its conditional
expectation. Thus, given the future IM, $\text{IM}(t),\,0\leq t\leq T$, DIM is
understood as the expected IM collateral that has to be posted by the party at
time $t$, and it is mathematically defined as

$\text{DIM}(t)=\mathbb{E}^{\mathbb{Q}}\left[e^{-\int_{0}^{t}r_{u}\text{d}u}\text{IM}(t)\left|\right.\mathcal{F}_{0}\right].$
(2.8)

Once the DIM is known for each future instant of time, the MVA is computed by
integrating the product of DIM by the spread of the collateral rate with
respect to the risk-free rate until maturity, i.e., if we denote by $f(t)$ the
funding spread at time $t>0$,

$\text{MVA}=\int_{0}^{T}f(s)\text{DIM}(s)\text{d}s.$ (2.9)

For the numerical experiments presented below, we assume a simplified
expression for the funding spread given by the authors in [15] under certain
funding strategy and contractual conditions. It leads to the following closed-
form,

$f(s)=\left(\left(1-R_{B}\right)\lambda_{B}(s)-s_{I}(s)\right)e^{-\int_{t_{0}}^{s}\left(\lambda_{B}(u)+\lambda_{C}(u)\right)\text{d}u},\quad
s>t_{0},$ (2.10)

where $R_{B}\in[0,1]$ is the recovery rate of the seller B if its counterparty
C defaults, $\lambda_{B}(s),\lambda_{C}(s)$ are the default intensities for
each party at time $s$, and $s_{I}(s)$ is the spread on IM at time $s$.

##  3 Methodology

First, we define the kind of products/portfolios we focus on this work, mainly
interest rate products without optionality, and the simplifications it entails
in the IM-SIMM computation. Next, we show the nested MC scheme for computing
DIM by brute force. This strategy is used to simulate our validation dataset.
In addition, its presentation serves to highlight the problems it presents,
mainly the computational cost due to the required nested simulations and re-
evaluations. Then, we introduce the methodology we propose to approximate DIM
using neural networks and the strategy followed to reduce the high training
cost due to the dataset production.

###  3.1 Simplifications IM-SIMM

As stated above, we only deal with interest rate products and/or portfolios,
so the total IM is entirely represented by the IM-SIMM of the interest rate
product class and is only affected by interest rate risk factors, mainly
market interest rates (delta margin) and market implied volatilities (vega and
curvature margin). Furthermore, since we work with products that are not
subject to optionality, we can forget about implied volatilities (as the vega
of such derivatives is zero) and work only with the interest rate risk
factors. These considerations allow us to ensure that the IM of the described
portfolios is entirely defined by the interest rate delta margin, i.e.,
$\text{IM}=\text{DeltaMargin}_{\text{IR}}$222It should be noted that, although
we take these considerations into account in this work, the methodology
described is of general applicability, and not only particular to the
simplified case..

The interest rate risk factors are identified as the spine points of the
market yield curve at the tenors
$\mathcal{T}=\\{2\text{W},1\text{M},3\text{M},6\text{M},1\text{Y},2\text{Y},3\text{Y},5\text{Y},10\text{Y},15\text{Y},20\text{Y},30\text{Y}\\}$,
and the sensitivity associated with each risk factor is given by the PV01 of
the instrument [25], i.e., if we denote by $s_{k}$ the sensitivity of the
instrument $V_{t}$ associated with the tenor $\tau_{k}\in\mathcal{T}$, and
$Y_{k}$ the yield value at that tenor, then
$s_{k}=V_{t}(Y_{k}+1\text{bp})-V_{t}(Y_{k})$, where $1\text{bp}=0.01\%$. Once
the $12$-dimensional sensitivity vector is calculated333Following the ISDA-
SIMM requirements, if other tenors are used to compute the sensitivities, they
have to be linearly allocated into the ISDA buckets., its entries are weighted
by parameters provided by ISDA and then aggregated to provide the IM.

###  3.2 Monte Carlo simulation of DIM

The simulation of collateralized exposure and XVA has a number of
particularities associated with its application in the real world. We try to
simplify these particularities as much as possible given the illustrative
purpose of this article. We consider an interest rate derivative, or a
portfolio of interest rate derivatives, subject to the assumptions stated in
the previous section, and maturing at time $T>0$. We define a equispaced time
grid $0=t_{0}<t_{1}<\dots<t_{N}=T$ with $N+1$ monitoring times and time step
$h_{t}$, which ideally would be equal to the MPoR. We select the short-rate
dynamics in line with the features set out in Section 2.1 and assume its
parameters are known via calibration procedure. We set as an initial condition
the market state at time $t_{0}$, i.e., the spine points of the yield curve at
the ISDA-SIMM tenors,
$Y^{0}=\left(Y_{1}^{0},Y_{2}^{0},\dots,Y_{12}^{0}\right)$, which are obtained
from the market rates for the same tenors,
$R^{0}=\left(R_{1}^{0},R_{2}^{0},\dots,R_{12}^{0}\right)$, via bootstrap
procedure, [38]. Throughout the simulation, we need to interpolate discount
values from the yield curve points. We apply a linear interpolation in yield
between shock maturities given its simplicity, but any other method presented
in [18] could be applied.

Fixed a number $M$ of MC scenarios, for each scenario $1\leq j\leq M$ and
monitoring time $t_{i}$,

  1. 1.

We calculate the price of the instrument/portfolio, $V_{j}^{i}$, based on the
forward market risk factors
$Y_{j}^{i}=\left(Y_{1,j}^{i},\dots,Y_{12,j}^{i}\right)$. These are directly
dependent on the short-rate $r_{j}^{i}$, generated from the chosen dynamics
and the value in the previous time $r_{j}^{i-1}$.

  2. 2.

Once the future price, $V_{j}^{i}$, has been calculated, we compute the
sensitivity of the instrument/portfolio with respect to the $k$-th forward
risk factor, $s_{k,j}^{i}$, by shifting $Y_{k,j}^{i}$ one basis point and re-
evaluating the product.

  3. 3.

The vector of sensitivities $\left(s_{1,j}^{i},\dots,s_{12,j}^{i}\right)$ is
then fed into the SIMM-IM calculation engine to produce the IM value
$\text{IM}_{j}^{i}$.

Once we have simulated all the IM values for time $t_{i}$, we apply the MC
estimator to (2.8) for approximating the DIM value at time $t_{i}$,
$\text{DIM}_{i}$, i.e.,

$\text{DIM}_{i}=\dfrac{1}{M}\sum_{j=1}^{M}\text{IM}_{j}^{i}e^{-\sum_{l\leq
i}r_{j}^{i}h_{t}}.$

A summary of the DIM calculation for a given monitoring time can be found in
Algorithm 1. It should be noted that the error in the MC estimate is
approximately normally distributed with mean $0$ and standard deviation
$\sigma_{i}/\sqrt{M}$, [12], where $\sigma_{i}$ is the standard deviation of
DIM at time $t_{i}$, portfolio dependent and usually unknown. Thus, the method
presents a square-root convergence rate and adding one decimal place of
precision requires $100$ times as many paths considered, making accurate DIM
estimates very costly to obtain.

Finally, once the DIM has been calculated for all monitoring times, we can
approximate the MVA of the instrument/portfolio. The standard approach we
follow consists of applying a simple quadrature rule to (2.9),

$\text{MVA}\approx\sum_{i=1}^{N}f_{i}\text{DIM}_{i}h_{t},$ (3.1)

where $f_{i}$ is the funding spread at time $t_{i}$.

Algorithm 1 Nested MC simulation of DIM

1:Simulation times $0=t_{0}<t_{1}<\dots<t_{N}=T$ with $h_{t}$ time step. Yield
curve points at the ISDA-SIMM tenors for time $t_{0}$,
$\\{Y^{0}_{k}\\}_{k=1}^{12}$. Model parameters $\Theta$. Number of MC paths,
$M$. Samples of the standard normal distribution i.i.d.,
$(W)_{j}^{i},\,j=0\dots,M-1,\,i=0\dots N-1$.

2:Compute $\text{DIM}_{i}$, the dynamical IM at time $t_{i}$.

3:$x^{0}\leftarrow Y^{0}$ $\triangleright$ Get the initial state variables
$x_{0}$ from the initial yield curve $Y^{0}$.

4:for $i=1$, $i<N+1$, $i++$ do

5: for $j=0$, $j<M$, $j++$ do

6: $x_{j}^{i}\leftarrow\left(x_{j}^{i-1},W_{j}^{i-1},\Theta\right)$
$\triangleright$ Get state variable $x_{j}^{i}$ from the previous state via IR
dynamics.

7: $\\{Y_{k,j}^{i}\\}_{k=1}^{12}\leftarrow\left(x_{j}^{i},\Theta\right)$
$\triangleright$ Obtain yield curve points from the current state vars. and
model.

8: $V_{j}^{i}\leftarrow\\{Y_{k,j}^{i}\\}_{k=1}^{12}$ $\triangleright$ Get the
market to future value.

9: for $k=1$, $k<13$,$k++$ do

10: $Y_{k,j}^{i}\leftarrow Y_{k,j}^{i}+1\text{bp}$ $\triangleright$ Apply
shock to the $k$-th yield.

11: $V_{k,j}^{i}\leftarrow\\{Y_{k,j}^{i}\\}_{k=1}^{12}$ $\triangleright$ Re-
evaluate instrument.

12: $S_{k,j}^{i}\leftarrow V_{k,j}^{i}-V_{i}^{j}$ $\triangleright$ Compute
sensitivity w.r.t. $k$-th yield.

13: $Y_{k,j}^{i}\leftarrow Y_{k,j}^{i}-1\text{bp}$ $\triangleright$ Set $k$-th
yield to its true value.

14: end for

15: $\text{IM}_{j}^{i}\leftarrow\\{S_{k,j}^{i}\\}_{k=1}^{12}$ $\triangleright$
IM is compute from the sensitivities via SIMM

16:
$\text{IM}_{j}^{i}\leftarrow\text{IM}_{j}^{i}e^{-\sum_{\hat{k}=0}^{i}r^{j}_{\hat{k}}(t_{i+1}-t_{i})}$
$\triangleright$ Apply discount to $t_{0}$.

17: end for

18: $\text{DIM}_{i}\leftarrow\dfrac{1}{M}\sum_{j=0}^{M}\text{IM}_{j}^{i}$
$\triangleright$ Approximate DIM from its MC estimator.

19:end for

###  3.3 DIM learning

The nested MC methodology presented in Section 3.2 highlights the difficulties
of being applied in realistic situations. At each time step and MC scenario,
as many portfolio re-evaluations are performed as many risk factors are
considered. Given the possibility that a portfolio may be composed of hundred
of products and complicated exotic transactions, such simulation becomes
impractical due to the computational cost and time required. Moreover, the
situation becomes even worse if one wants to compute such values for different
initial market states.

These circumstances lead to considering the use of machine learning and/or
deep learning techniques to approximate the function that defines, in this
case, the DIM values. From now on, we define the deterministic DIM function as
$\mathbf{F}:\Omega\subset\mathbb{R}^{d}\to\mathbb{R}^{N}$, the time
discretized version of the equation (2.8), where $\Omega$ is the
$d$-dimensional feature space of market state variables at time $t_{0}$, and
$N$ is the number of DIM monitoring times. This multi-output structure is one
of the design choices we made for our model approximation. It entails removing
time as an input variable (leaving only the initial market state variables as
inputs to the model) and treating the output as the set of temporal reference
points at which the DIM is intended to be known.

Such definition poses no issue due to the point-wise nature of conditional
expectations. Let be $\mathbf{X}$ the market state variable, and
$\mathbf{IM}=\left(\text{IM}_{0},\dots,\text{IM}_{N}\right)$ the vector random
variable, understood as the discretization of the IM stochastic process
defined above. From (2.8) it follows that

$\displaystyle\mathbf{DIM}=\left(\text{DIM}_{0},\dots,\text{DIM}_{N}\right)$
$\displaystyle=\mathbb{E}^{\mathbb{Q}}\left[\left(\text{IM}_{0},\dots,\text{IM}_{N}\right)\Bigl{|}\mathbf{X}\right]=\Bigl{(}\mathbb{E}^{\mathbb{Q}}\left[\text{IM}_{0}\bigl{|}\mathbf{X}\right],\dots,\mathbb{E}^{\mathbb{Q}}\left[\text{IM}_{N}\bigl{|}\mathbf{X}\right]\Bigr{)}$
(3.2)

In addition, if we assume that $\mathbf{X}$ is the complete Markov state at
time $t_{0}$, there exist a deterministic, but unknown, function
$\text{F}_{i}$ of the state. Thus, our multi-output function $\mathbf{F}$ is
defined as

$\mathbf{F}(\mathbf{X})=\left(\text{F}_{0}\left(\mathbf{X}\right),\dots,\text{F}_{N}\left(\mathbf{X}\right)\right)=\Bigl{(}\mathbb{E}^{\mathbb{Q}}\left[\text{IM}_{0}\bigl{|}\mathbf{X}\right],\dots,\mathbb{E}^{\mathbb{Q}}\left[\text{IM}_{N}\bigl{|}\mathbf{X}\right]\Bigr{)}.$
(3.3)

The idea of ML/DL applied to our context is to approximate our defined
function $\mathbf{F}$ through the mapping
$\mathbf{\hat{F}}\left(x;\boldsymbol{\omega}\right)$, with the weights
$\boldsymbol{\omega}$ that results in the best approximation. The most common
choice for $\mathbf{\hat{F}}$ in this kind of numerical approximation tasks
are feed forward networks, consisting of the composition of several layers,
where each layer is also a composition of affine maps and non-linear
transformations (see Figure 1). The coefficients of the affine maps are the
above-mentioned weights that need to be adjusted to fulfill the approximation
requirements. Such process is called training and can be performed in several
ways.

Figure 1: Schematic representation of the multi-output fully connected neural
network used as a proxy $\mathbf{\hat{F}}$ for DIM values at monitoring times.
Each connecting edge represents an updatable weight. Each vertex in the hidden
layers represents the weighted sum of its inputs (values of the previous
layer) and a non-linear function acting over such a sum.

####  3.3.1 Classical supervised learning approach

The first approach that one might think of is to train our DIM network by
supervised learning. In this training strategy, the weights are updated from a
training set consisting of samples drawn from the initial market state
variable, $\mathbf{X}^{(k)}$, and their corresponding vector DIM values,
$\mathbf{DIM}^{(k)}$. They are simulated via Algorithm 1 and can be consider
ground truth values if the number of Monte Carlo paths $M$ is large enough.

In each training iteration, the network is evaluated in a market state sample
$\mathbf{X}^{(k)}$ and the result is compared with the respective label
$\mathbf{DIM}^{(k)}$ by means of a cost function that measures how far the
prediction is from the target. Thus, if we assume that $\Psi(\cdot,\cdot)$ is
a chosen metric, we can define the cost function as

$\mathcal{J}:\boldsymbol{\omega}\in\mathbb{R}^{\text{l}(\boldsymbol{\omega})}\to\mathcal{J}(\boldsymbol{\omega})=\Psi\left(\mathbf{\hat{F}}(\,\,\cdot\,\,;\boldsymbol{\omega}),\mathbf{F}(\,\cdot\,)\right)\in\mathbb{R},$

where $\text{l}(\boldsymbol{\omega})$ is the dimension of the parameter space.
The goal of the training is to find
$\boldsymbol{\omega}^{*}\in\mathbb{R}^{\text{l}(\boldsymbol{\omega})}$ such
that

$\boldsymbol{\omega}^{*}=\arg\min_{\boldsymbol{\omega}\in\mathbb{R}^{\text{l}(\boldsymbol{\omega})}}\mathcal{J}(\boldsymbol{\omega}).$

In our case, we decide to work with the squared $L^{2}$ norm, so we get

$\displaystyle\Psi\left(\mathbf{\hat{F}}(\mathbf{X}^{(k)};\boldsymbol{\omega}),\mathbf{F}(\mathbf{X}^{(k)})\right)$
$\displaystyle=\Psi\left(\mathbf{\hat{F}}(\mathbf{X}^{(k)};\boldsymbol{\omega}),\mathbf{DIM}^{(k)}\right)$
(3.4)
$\displaystyle=\bigl{|}\bigl{|}\mathbf{\hat{F}}(\mathbf{X}^{(k)};\boldsymbol{\omega})-\mathbf{DIM}^{(k)}\bigr{|}\bigr{|}^{2}_{2}$
$\displaystyle=\sum_{i=0}^{N+1}\left(\hat{\text{F}}_{i}(\mathbf{X}^{(k)};\boldsymbol{\omega})-\text{DIM}_{i}^{(k)}\right)^{2}.$

In practice, the process of updating weights is done after evaluating the
approximator in a given number of samples (batch size), then the real loss
estimation used is the mean of the squared errors (MSE).

However, this kind of strategy is in principle not valid. The computational
cost of producing the training dataset with ground truth DIM samples is the
same as the cost of using brute force to obtain the DIM value, which does not
provide the necessary speed and flexibility to be able to include this type of
approach in a real CCR engine.

####  3.3.2 Supervised learning with sampled labels

In order to improve the situation described above, what we can do is to take
advantage of one of the main ideas introduced by the authors in [22]. Thus,
they proved that it is possible to approximate functions defined in terms of
conditional expectations by means of datasets consisting of unbiased noisy
labels, which can even be generated with a single MC path per initial state.
Based on this idea, we can write

$\displaystyle\mathbf{DIM}=\mathbb{E}^{\mathbb{Q}}\left[\mathbf{IM}|\mathbf{X}\right]$
$\displaystyle=\min_{\mathbf{F}\in
L^{2}\left(\mathbf{X}\right)}\bigl{|}\bigl{|}\mathbf{F}(\mathbf{X})-\mathbf{IM}\bigr{|}\bigr{|}_{2}^{2}$
(3.5)
$\displaystyle\approx\min_{\boldsymbol{\omega}\in\mathbb{R}^{\text{l}(\boldsymbol{\omega})}}\bigl{|}\bigl{|}\hat{\mathbf{F}}(\mathbf{X},\boldsymbol{\omega})-\mathbf{IM}\bigr{|}\bigr{|}_{2}^{2}$
$\displaystyle\approx\hat{\mathbf{F}}\left(\mathbf{X},\min_{\boldsymbol{\omega}\in\mathbb{R}^{\text{l}(\boldsymbol{\omega})}}\text{MSE}\right),$

by the universal approximation theorem and the assumption of an IID dataset
sampled from the correct distribution. More details can be found in [22]
(Appendix 1).

By application of this result, we no longer have to produce a dataset with
ground truth DIM values. It will only be necessary to compute noisy
estimations via Algorithm 1. Following this philosophy, we compute each
dataset label taking a single MC path ($M=1$). This represents a major
advantage since, for the computational cost of one truly converged Monte Carlo
DIM execution, we can now produce an entire dataset to train our DIM network.
This allows us to overcome the difficulties presented before and justify its
use in real applications. This idea is schematically shown in Figure 2.

Figure 2: Schematic representation of the strategy explained in Section 3.3.2.
The graphs in faded colors represent the set of IM samples (noisy DIM
estimations), each from a different initial market state, used as training
dataset (sampled labels). The highlighted lines represent the approximations
of the DIM trajectories for two initial states obtained by the DIM network
trained with these sampled labels. As mentioned above, we work with a discrete
version of these trajectories, represented in the figure by the points at each
monitoring time.

##  4 Numerical experiments

In this section, we present numerical experiments concerning the convergence
of the algorithm as a function of the number of samples used in training, as
well as a study on the distribution of the error incurred by the trained
network in relation to its input data. Finally, we validate our approach by
computing the DIM and the MVA in a practical situation.

###  4.1 Experiment configuration

We choose to assess the experiments in two different interest rate models,
namely the Vasicek, [39], and the Hull-White, [41], models. Both follow the
general structure given in equation (2.1) and can be written as

$\displaystyle\text{d}x(t)$
$\displaystyle=-ax(t)\text{d}t+\sigma\text{d}W_{t},$ $\displaystyle r(t)$
$\displaystyle=x(t)+\theta(t),$

where $a>0,\sigma>0$ are the mean reversion speed and the volatility of the
process. $\theta(t)$ can be interpreted as the long term average rate, but it
plays a different role in the considered models. In Vasicek,
$\theta(t)=\theta$ is taken as a constant parameter and, to completely
determine the stochastic equation, we need to add the initial short-rate
$r_{0}$. Under this setting, the market state at time $t_{0}$ is completely
determined by these parameters, i.e., they are inputs in our methodology, see
Table 1. It should be noted that both the initial and the time-future yield
curve points scenarios, i.e., the market risk factor used to compute the ISDA
SIMM, are fully determined under these parameters, since the model defines the
present and the future realizations of the yield term structure, obviously
conditional on the current state.

One of the main drawbacks of the Vasicek model is that it cannot be fitted to
the yield term structure observed in the market at time $t_{0}$. Because of
that, we want to introduce a more realistic example of application with the
Hull-White model, which does allow for fitting the current yield curve. In
such a model, $\theta(t)$ is a function that depends on the $t_{0}$ market
instantaneous forward rate, i.e., which depends on the aforementioned term
structure at time $t_{0}$, see, e.g., [9], but the initial condition is always
the same, $x_{0}=0$.

In order to summarise the relevant information in the yield curve, we use a
parsimonious model for the latter. Thus, it is carried out by means of the
Nelson-Siegel parametrisation, [34], which is defined from four parameters:
$\beta_{0},\beta_{1},\beta_{2}$ and $\lambda>0$. In this work we set
$\lambda=1.37$, a common choice in the industry, [3]. Under these premises,
the market state at time $t_{0}$ is now determined by the Hull-White model
parameters $a,\sigma$, jointly with the free Nelson-Siegel parameters
$\beta_{0},\beta_{1},\beta_{2}$, see Table 1.

For testing purposes, we consider a portfolio made up with a single product, a
$1$ year forward, $5$ years expiry interest rate swap with quarterly-annual
float and semi-annual fixed leg schedules. We are interested in present
results not only for the ATM swap, but also for in-the-money (ITM) and out-
the-money (OTM) versions, so we add to the previously considered market state
variables the spread, $\delta$, between the fixed ATM rate (completely
determined by the yield term structure at time $t_{0}$) and the one chosen.

Vasicek setting
---
$\mathbb{X}$ | $a$ | $\sigma$ | $\theta$ | $r_{0}$ | $\delta$
$\text{min}(\mathbb{X})$ | $1\%$ | $0.5\%$ | $0.1$ | $-5\%$ | $-0.1\%$
$\text{max}(\mathbb{X})$ | $10\%$ | $2.5\%$ | $5\%$ | $5\%$ | $0.1\%$

Hull-White setting
---
$\mathbb{X}$ | $a$ | $\sigma$ | $\beta_{0}$ | $\beta_{1}$ | $\beta_{2}$ | $\delta$
$\text{min}(\mathbb{X})$ | $1\%$ | $0.05\%$ | $-0.5\%$ | $0\%$ | $0\%$ | $-0.1\%$
$\text{max}(\mathbb{X})$ | $5\%$ | $1.5\%$ | $5\%$ | $1\%$ | $1\%$ | $0.1\%$

Table 1: Lower and upper bounds for the market state variables in each
considered model.

To carry out the experiments, we need to generate both the training set and
the validation set for each proposed setting. We take $N=160$ equispaced
monitoring times between the settle time of the swap, $t_{0}=0$, and its
maturity, $T_{f}=6$. To generate both datasets, we follow the methodology
explained in Section 3.2. We sample $K$ vectors of initial market state values
from the given setting (Vasicek or Hull-White) via Latin Hypercube Sampling
(LHS), [12], with the upper and lower bounds specified in Table 1. The main
difference appears in the hyperparameters of the simulation procedure. On the
one hand, the training dataset is developed taking $K_{T}=2^{22}$ market state
vectors. For each of them, we get, following the Algorithm 1, a noisy vector
of DIM estimations by setting $M_{T}=1$ MC paths. On the other hand, the
validation dataset is made up of $K_{V}=2^{9}$ market state vectors and, each
corresponding DIM trajectory is again computed with the aforementioned
Algorithm 1 taking $M_{V}=2^{20}$ MC paths. We consider such DIM vectors as
the ground truth estimations for the quantity. Nevertheless, recall that, by
the nature of the MC algorithm, we have some error in our validation sets.
Furthermore, the error varies depending on both the model and the chosen
monitoring time, so we consider as a tolerance the maximum MC error deviation
estimated in each model, namely, $4.24\times 10^{-4}$ for Vasicek and
$2.12\times 10^{-4}$ for Hull-White.

The network architecture used as a proxy follows the guidelines set out in
Section 3.3. Essentially, it is a multi-output fully connected feed forward
network with as many outputs as monitoring times. The hyperparameters
configuration is presented in Table 2. Only note that, except for the number
of inputs, we keep the same hyperparameters in both settings. A pre-processing
layer is added, which normalises the input data to values between $0$ and $1$.
Regarding the training stage, we choose to use the most common in literature
stochastic gradient-base optimizer, Adam, [27], with its default
configuration. The train is performed in batches of $4096$ samples. We apply a
learning rate policy as the training progresses. From an initial learning rate
of $10^{-3}$, we halve it if the mean square test error does not show
improvement within $50$ iterations, until it reaches the value of $10^{-6}$.
We set the number of epochs as $2000$ and an early-stopping criterion is added
to avoid wasting time once the threshold accuracy has been reached. All the
experiments are executed on a system AMD EPYC 7763 64-core processor, 32 GB of
RAM, equipped with a NVIDIA A100 GPU. The codes were implemented in Python 3.8
and Tensorflow v$2.9$.

Lastly, to check how well the proposed methodology works, we measure the
impact it has on the MVA calculation (2.9). To this end, we consider the
counterparty risk parameters given in [15] to account for the funding spread
(2.10): $\lambda_{C}=0$, $\lambda_{B}=1.67\times 10^{-2}$, $R_{B}=0.4$ and
$s_{I}=0$; and we compute the MVA for each ground truth DIM path by means of
(3.1).

| Vasicek setting | Hull-White setting
---|---|---
Num. inputs | $5$ | $6$
Num. outputs | $160$
Num. hidden layers | $3$
Num. units per layer | $256$
Activation function |  $\sigma(x)=\max\left(0,x\right)$ (ReLU)
Weight initialization | Glorot uniform, [13]
Table 2: Feed forward network hyperparameters for each setting.

###  4.2 Convergence in terms of training samples

The first test we present aims to study training behavior as a function of the
number of employed sampled labels. It is expected that, as the training set
grows in number of samples, the accuracy of the approximation will improve,
but by how much? The idyllic situation would be to get the same convergence
ratio as given by MC algorithm. This would mean that, for the price of a
single MC to generate the training set, plus the training cost, we would have
a fitted network that give us the numerical approximation of DIM at monitoring
times, with the same accuracy, for a whole space of initial market conditions.
To this end, from the total training set of sampled labels, we select shuffled
subsets with a number of samples ranging from $512$ to the total, $2^{22}$, in
powers of two. With each subset, we train $20$ neural networks with the
guidelines set out above. We compare their performance by looking at two
metrics against the validation set, the square root of the MSE (RMSE) and the
MVA computation (with the risk parameters given before).

In Figure 3, we present the mean value of the two metrics for each interest
rate setting and training subset, as well as a $95\%$ confidence interval
estimated with a $t-$distribution. The square-root convergence rate is also
plotted for comparative purposes. In practically all cases, the variability of
the training is very small, especially in terms of calculating MVA. The only
case that we can highlight in terms of variability is the Vasicek setting with
$K_{T}=2^{15}$, which was produced by the divergence of one of his trials. As
expected, as the number of samples in the training dataset increases, there is
an error reduction, but the idyllic behavior is not met, the error reduction
looses the MC convergence order. Even so, we find the result very
satisfactory. The loss of order is not very pronounced, and we should remember
that the trained networks parameterize DIM for a whole space of initial
conditions. Furthermore, no significant differences are observed between
settings.

(a) Vasicek RMSE

(b) Vasicek MVA absolute error

(c) Hull-White RMSE

(d) Hull-White MVA absolute error

Figure 3: Convergence for each dataset size and model setting. For each one,
the mean and $95\%$ confidence interval error is computed with $20$ training
trials. Square-root convergence rate is shown for comparative purposes.

###  4.3 DIM error in terms of the initial state variables

Having shown the overall error (in terms of RMSE and MVA) with respect to the
number of training samples, we are now interested in studying how the
difference between ground truth and predicted DIM is distributed across the
initial state variables. For that, we analyze the results obtained by the
networks trained with the largest dataset size in the previous test. In
particular, we focus on the differences given in the monitoring time which
presents a higher DIM variance, since larger values are expected. In our
simple portfolio case, where the only product is an interest rate swap, we
know that the instant of highest DIM variance it just before the first
cashflow exchange, so we set the monitoring time at $t_{\gamma}=1.75$ for both
settings.

In Figure 4 and Figure 5 we present the differences obtained for the Vasicek
and Hull-White, respectively. Both results are presented in terms of the mean
difference calculated from the $20$ trained networks. In the light of the
outcome, we extract the following conclusions:

  * •

The differences in both settings are centered around zero, i.e., no bias is
observed, and bounded by $0.25\%$. We see more dispersion in the Vasicek case,
which is to be expected given that its estimated variance is twice the
variance calculated in Hull-White.

  * •

In general, the largest differences are observed in the boundary values of
each market variable. This is a common pattern in supervised training
algorithms since the number of samples in these regions is less dense.

  * •

Parameter values that increase the variance generate larger differences. This
is most clearly seen in the Vasicek case, where the combination given by small
mean reversion speeds, $a$, with large model volatilities, $\sigma$, leads to
higher DIM variances and higher prediction differences. In Hull-White this
pattern is less pronounced, but the positive correlation between model
volatility and prediction differences is evident.

Figure 4: Mean differences for DIM, at monitoring time $t_{\gamma}=1.75$,
between ground truth and predictions in the Vasicek setting, in terms of the
input variables $a$ (top left), $\sigma$ (top center), $\theta$ (top right),
$r_{0}$ (bottom left) and $\delta$ (bottom right).

Figure 5: Mean differences for DIM, at monitoring time $t_{\gamma}=1.75$,
between ground truth and predictions in the Hull-White setting, in terms of
the input variables $a$ (top left), $\sigma$ (top center), $\beta_{0}$ (top
right), $\beta_{1}$ (bottom left) and $\beta_{2}$ (bottom center) and $\delta$
(bottom right).

###  4.4 DIM and MVA error in terms of the interest rate model volatility

As we have seen, the greatest source of error in the prediction comes from
extreme initial state values, especially of the parameters that control the
volatility of the interest rate model used. For this reason, we generate ad-
hoc samples of DIM trajectories to more comprehensively evaluate such cases.
In the Vasicek setting, we take $\theta=3\%$, $r_{0}=1\%$, $\delta=0\%$, and
we analyze the performance of the trained networks for combinations of
$a=1\%,\,5\%,\,10\%$ and $\sigma=0.5\%,\,1\%,\,2.5\%$. The same strategy is
followed with Hull-White, where we set $\beta_{0}=1\%$,
$\beta_{1}=\beta_{2}=0.5\%$, $\delta=0\%$ and combinations of
$a=1\%,\,2.5\%,\,5\%$, $\sigma=0.05\%,\,0.75\%,\,1.5\%$ are taken. Both cases
present the same behavior, their intrinsic volatility increase with small
values of $a$ and with large values of $\sigma$. Note that the simulation of
the DIM values in the described cases keeps all the properties specified at
the beginning of the section.

$a$ |  Target | $\sigma$
---|---|---
$0.5\%$ | $1\%$ | $2.5\%$
$1\%$ |  DIM$(t=t_{\gamma})$ | $3.67\times 10^{-4}$ | $3.58\times 10^{-4}$ | $2.95\times 10^{-3}$
DIM$(t=2t_{\gamma})$ | $1.14\times 10^{-5}$ | $1.76\times 10^{-4}$ | $3.55\times 10^{-3}$
MVA | $1.08\times 10^{-3}$ | $3.63\times 10^{-4}$ | $2.23\times 10^{-3}$
$5\%$ |  DIM$(t=t_{\gamma})$ | $1.07\times 10^{-4}$ | $1.35\times 10^{-4}$ | $4.66\times 10^{-4}$
DIM$(t=2t_{\gamma})$ | $1.80\times 10^{-4}$ | $2.55\times 10^{-4}$ | $8.27\times 10^{-4}$
MVA | $1.74\times 10^{-4}$ | $3.50\times 10^{-4}$ | $1.47\times 10^{-3}$
$10\%$ |  DIM$(t=t_{\gamma})$ | $3.57\times 10^{-4}$ | $1.81\times 10^{-4}$ | $1.80\times 10^{-4}$
DIM$(t=2t_{\gamma})$ | $8.35\times 10^{-4}$ | $3.14\times 10^{-4}$ | $4.02\times 10^{-4}$
MVA | $4.11\times 10^{-4}$ | $7.66\times 10^{-4}$ | $5.61\times 10^{-4}$
Table 3: Vasicek setting fixed $\theta=3\%,\,r_{0}=1\%,\,\delta=0\%$. Relative
errors for DIM at times $t_{\gamma}$ and $2t_{\gamma}$, and relative error in
MVA.

Based on these combinations, we present in Tables 3 and 4 the relative errors
achieved for DIM at the monitoring times $t_{\gamma}$ and $2t_{\gamma}$, as
well as in the calculation of the MVA. Broadly speaking, they are presented in
such a way that we can identify the higher volatility scenarios with those
presented in the upper triangular part of the tables, and the lower ones with
those presented in the bottom triangular part. In general, the conclusions
drawn for both configurations on the previous experiment are confirmed. The
worst-case scenarios for Vasicek are those with combinations of small mean
reversion speed and large model volatility, losing about an order of magnitude
compared to the opposite cases. In some cases, the MVA values also lose
precision compared to the individual DIM predictions, which is expected due to
the accumulation of errors at each monitoring time.

$a$ |  Target | $\sigma$
---|---|---
$0.05\%$ | $0.75\%$ | $1.5\%$
$1\%$ |  DIM$(t=t_{\gamma})$ | $4.10\times 10^{-4}$ | $1.41\times 10^{-5}$ | $2.65\times 10^{-4}$
DIM$(t=2t_{\gamma})$ | $2.92\times 10^{-5}$ | $4.27\times 10^{-5}$ | $2.64\times 10^{-4}$
MVA | $2.92\times 10^{-4}$ | $1.86\times 10^{-4}$ | $7.60\times 10^{-5}$
$2.5\%$ |  DIM$(t=t_{\gamma})$ | $1.86\times 10^{-4}$ | $2.22\times 10^{-4}$ | $1.78\times 10^{-4}$
DIM$(t=2t_{\gamma})$ | $2.73\times 10^{-5}$ | $1.46\times 10^{-4}$ | $5.94\times 10^{-5}$
MVA | $9.30\times 10^{-5}$ | $1.81\times 10^{-4}$ | $2.27\times 10^{-4}$
$5\%$ |  DIM$(t=t_{\gamma})$ | $2.21\times 10^{-4}$ | $2.46\times 10^{-4}$ | $1.81\times 10^{-4}$
DIM$(t=2t_{\gamma})$ | $5.91\times 10^{-5}$ | $5.92\times 10^{-5}$ | $2.45\times 10^{-4}$
MVA | $1.68\times 10^{-4}$ | $3.08\times 10^{-4}$ | $2.06\times 10^{-5}$
Table 4: Hull-White setting fixed
$\beta_{0}=1\%,\,\beta_{1}=\beta_{2}=0.5\%,\,\delta=0\%$. Relative errors for
DIM at times $t_{\gamma}$ and $2t_{\gamma}$, and relative error in MVA.

In the case of Hull-White, however, the situation is different. The error
distribution does not show the pattern observed in Vasicek. It exhibits a more
random behavior across the combinations of $a$ and $\sigma$. This is explained
by the fact that these errors are very close to, or even below, the MC
variance threshold used for simulating the ground truth (reference) values.
All in all, the obtained estimations by the DIM learning approach are very
satisfactory since, for both models, they (almost) achieve the MC precision
level, at much less computational cost.

###  4.5 Portfolio application

This last subsection is intended to demonstrate how well the methodology
generalizes when applied to a practical example of major relevance. For this
purpose, we define a portfolio consisting of $6$ ATM interest rate swaps with
maturities $T_{\phi}=5+\phi,\,\,\phi=0,\dots,5$. Those defined by a even index
are payer swaps, while the rest are receiver swaps. In addition, the first
three have a quarterly vs. semi-annual payment schedule, and the others a
semi-annual vs. annual schedule. All of them are defined on a notional of
$100$ monetary units.

Figure 6: Samples of DIM portfolio prediction (dashed red line) versus the
ground truth trajectory (blue line). Left: Vasicek setting for the initial
market state $a=7.23\%$, $\sigma=1.1\%$, $\theta=1.2\%$, $r_{0}=1.4\%$. Right:
Hull-White setting for the initial market state $a=4.13\%$, $\sigma=1.2\%$,
$\beta_{0}=4.93\%$, $\beta_{1}=0.29\%$, $\beta_{2}=0.91\%$.

Most of the dataset simulation details are kept. The training set consists of
$K_{T}=2^{22}$ IM sampled labels from the above-mentioned market state
settings, see Table 1. The only difference is that we omit the swap spread
parameter, since now it has nonsense. In addition, we consider $N=100$ equally
spaced monitoring times in which DIM values is intended to be known. These
changes lead to a neural network architecture with one less input parameter in
each setting, and which $100$ outputs. No additional changes are made to the
other features of the network. Lastly, a test set with $K_{V}=32$ samples is
generated with the same number of MC paths as before.

In order to evaluate the stability and the performance of the methodology, we
run $10$ training trials for each setting, preserving the optimiser and the
training strategies used before. In relation to the global validation metrics,
we get RMSE $95\%$ confidence intervals of $3.439\times 10^{-3}\pm 7.19\times
10^{-4}$ and $2.373\times 10^{-3}\pm 9.71\times 10^{-4}$ for Vasicek and Hull-
White, respectively. These are positive results, in line with those shown for
the simplified case, and demonstrate stability in training convergence.

For completeness, we randomly select an initial market state from each
simulated ones (and for each setting) and study the network performance in MVA
computation for such a particular case. The $95\%$ confidence interval for the
MVA relative errors are $1.912\times 10^{-3}\pm 1.11\times 10^{-3}$ and
$1.845\times 10^{-3}\pm 7.28\times 10^{-4}$ for Vasicek and Hull-White,
respectively. The results seem to confirm the same behavior as in the previous
case study, the higher variance of the Vasicek model is manifested in higher
errors. However, making the portfolio more complex with a larger number of
assets does not hamper the performance, which is very positive. This situation
is reflected in all the others samples and trial executions, since the maximum
MVA relative error is $6.753\times 10^{-3}$ for Vasicek and $3.96\times
10^{-3}$ for Hull-White. Lastly, we show in Figure 6 the approximation of the
DIM trajectory (compared to the ground truth value), for each setting,
estimated by means of one of the trained networks. The initial market states
are also specified. As we can see, the approximation is qualitatively
indistinguishable from the reference, expected from what we have described.

##  5 Conclusion

One of the main disadvantages of training neural networks by means of
classical supervised training is that the elaboration of the training dataset
is highly expensive, specially in the kind of task where nested MC simulation
are required. The main idea of this work has been to address this problem for
the DIM computation, a CCR quantity of interest due to its role in the MVA. By
definition, DIM is calculated from the expectation of IM over the life of the
portfolio to which it is related, and the nested behaviour appears if one
wants to compute it. In this sense, training neural networks for DIM
approximation in a classical supervised fashion make their implementation in
an online CCR engine unfeasible. Since the DIM framework fits perfectly with
the sample payoffs training presented by the authors of [22], in this work we
extend their idea to the neural network computation of the quantity in
question.

First, we have condensed all the information required for its calculation into
a vector of initial market state variables at initial time. This has been
achieved by means of an interest rate evolution model, unequivocally
determined from a few parameters, together with a parsimonious
parameterisation of the current interest rate term structure. From the initial
market state vector, which is the set of input values for our neural network
model, we build a training dataset in which the labels are not ground truth
DIM values, but highly noisy and unbiased samples of DIM generated with a
single MC path, i.e., we use IM realizations as DIM approximations. Note that
the DIM for a given initial state is a function of time. We have handled this
particularity by defining a mesh of monitoring times in which training and
validation values are defined, which means that our neural network model has a
multi-output structure. This choice allows for a simple extension of the
concept of conditional expectation, as well as greatly facilitating the
training process. The proposed approach presents two important advantages.
First, for the price of a single MC execution we can build an entire dataset
ready for the network training, and second, our trained network has been
parameterised by the initial market state variables, so, once trained, we have
a neural network model that approximates DIM trajectories at monitoring times
for a whole set of initial states, largely avoiding re-training.

Having described the introduced methodology, we have presented convergence and
sensitivity results of the approximations based on the initial data for a
simple example, a portfolio consisting of a single swap, and for two interest
rate models, Vasicek and Hull White. From these experiments, we can extract a
couple of main conclusions. Firstly, there is a convergence ratio depending on
the number of training samples used. This is worse than the square root order
reduction given by MC, in exchange for having parameterised a model as a
function of the initial conditions. Secondly, the obtained DIM estimations,
and they are not particularly affected by the input data near initial state
boundary domain. Finally, we have found that the methodology performs well in
a portfolio of swaps, a more realistic scenario, highlighting the general
applicability of the proposed solution.

## Acknowledgements

The authors’ research has been funded by the Spanish MINECO under research
project number PDI2019-108584RB-I00. The authors also acknowledge the support
of the CITIC research centre. CITIC is funded by the Xunta de Galicia through
the collaboration agreement between the Consellería de Cultura, Educación,
Formación Profesional e Universidades and the Galician universities for the
reinforcement of the research centres of the Galician University System
(CIGUS). Joel P. Villarino acknowledge the support received by the Xunta de
Galicia under grant ED481A/2023-202.

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$\overline{\mathrm{marg}}_{S_{-i}}\beta_{i}(s_{i})=\left(\mu_{s_{i}}^{m},...,\mu_{s_{i}}^{1}\right)\text{.}$

This shows that $(s_{i},s_{i})\in R_{i}^{1}$. Therefore
$\Delta_{S_{i}^{1}\times T_{i}}\subseteq R_{i}^{1}$, which implies
$S_{i}^{1}\subseteq\mathrm{Proj}_{S_{i}}\left(R_{i}^{1}\right)$. Conversely,
Proposition 1 yields $\mathrm{Proj}_{S_{i}}\left(R_{i}^{1}\right)\subseteq
S_{i}^{1}$.

Inductive Step ($n+1$). For each $i\in I$, we have to show that the following
properties hold:

(1) $\mathrm{Proj}_{S_{i}}\left(R_{i}^{n+1}\right)=S_{i}^{n+1}$;

(2) $\Delta_{S_{i}^{n+1}\times T_{i}}\subseteq R_{i}^{n+1}$ if $n+1\leq M$,
and $\Delta_{S_{i}^{M}\times T_{i}}\subseteq R_{i}^{n+1}$ if $n+1>M$.

Fix $i\in I$ and $s_{i}\in S_{i}^{n+1}$. Let
$k:=\min\left\\{n+1,M+1\right\\}$. Since
$(s_{i},s_{i})\in\Delta_{S_{i}^{k-1}\times T_{i}}$, by the induction
hypothesis it follows that $(s_{i},s_{i})\in R_{i}^{n}$. We show that
$(s_{i},s_{i})\in\mathbf{B}_{i}^{c}\left(R_{-i}^{n}\right)$; this will yield
$(s_{i},s_{i})\in R_{i}^{n+1}$. Write
$\beta_{i}(s_{i}):=\left(\nu_{s_{i}}^{m},...,\nu_{s_{i}}^{1}\right)$, where
$m\geq k$ because $s_{i}\in S_{i}^{k}$ and so, by construction,
$\beta_{i}(s_{i})$ must have length at least $k$. To show that $R_{-i}^{n}$ is
cautiously believed under $\beta_{i}(s_{i})$, recall that
$\mathrm{Supp}\nu_{s_{i}}^{l}=\Delta_{S_{-i}^{l-1}\times
T_{-i}}\subseteq\Delta_{S_{-i}^{k-1}\times T_{-i}}$ for each $l=k,...,m$.
Since $\Delta_{S_{-i}^{k-1}\times T_{-i}}\subseteq R_{-i}^{n}$ (induction
hypothesis), it follows that condition (i) of Definition 12 is satisfied at
level $l=m-k+1$. Recall also that
$\mathrm{Supp}\nu_{s_{i}}^{k}=\Delta_{S_{-i}^{k-1}\times T_{-i}}$. By the
induction hypothesis,
$\mathrm{Proj}_{S_{-i}}\left(R_{-i}^{n}\right)=S_{-i}^{k-1}=\mathrm{Proj}_{S_{-i}}\left(\Delta_{S_{-i}^{k-1}\times
T_{-i}}\right)$. Hence, $\beta_{i}(s_{i})$ satisfies condition (ii’) of
Proposition 2.2. Thus
$(s_{i},s_{i})\in\mathbf{B}_{i}^{c}\left(R_{-i}^{n}\right)$, as required. So,
we have shown that
$S_{i}^{n+1}\subseteq\mathrm{Proj}_{S_{i}}\left(R_{i}^{n+1}\right)$. For part
(2), note the following fact: If $n+1\leq M$, then for every
$(s_{i},s_{i})\in\Delta_{S_{i}^{n+1}\times T_{i}}$ we have $s_{i}\in
S_{i}^{n+1}$; analogously, if $n+1>M$, then for every
$(s_{i},s_{i})\in\Delta_{S_{i}^{M}\times T_{i}}$ we have $s_{i}\in
S_{i}^{n+1}$. Therefore, by proving that $(s_{i},s_{i})\in R_{i}^{n+1}$ for
each $s_{i}\in S_{i}^{n+1}$, we have proven (2).

Conversely, pick any $(s_{i},s_{i}^{\prime})\in R_{i}^{n+1}\subseteq
R_{i}^{n}$. Then, by the induction hypothesis, $s_{i}\in S_{i}^{n}$. Let
$\beta_{i}(s_{i}^{\prime}):=(\mu^{1},...,\mu^{n})$. Since $s_{i}^{\prime}$
cautiously believes $R_{-i}^{n}$ at some level $l$, it follows from
Proposition 2.2 and the induction hypothesis that

$\mathop{\textstyle\bigcup}_{k\leq
l}\mathrm{Suppmarg}_{S_{-i}}\mu^{k}=S_{-i}^{n}\text{.}$

So, by Proposition 1 in Blume et al. (1991b), there exists
$\nu\in\mathcal{M}(S_{-i})$, with $\mathrm{Supp}\nu=S_{-i}^{n}$, under which
$s_{i}$ is optimal. Therefore, by Remark 1, $s_{i}\in S_{i}^{n+1}$. This shows
that $\mathrm{Proj}_{S_{i}}\left(R_{i}^{n+1}\right)\subseteq S_{i}^{n+1}$,
establishing (1). $\blacksquare$

Proof of Lemma 3. Fix a type $t_{i}$ that cautiously believes $E_{-i}$, and
set
$\beta_{i}\left(t_{i}\right):=\left(\beta_{i}^{1}\left(t_{i}\right),...,\beta_{i}^{n}\left(t_{i}\right)\right)$.
Let $t_{i}^{\ast}:=\varphi_{i}\left(t_{i}\right)$. Note that bimeasurability
of $\left(\varphi_{i}\right)_{i\in I}$ implies that
$\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)(E_{-i})$ is an event in
$S_{-i}\times T_{-i}^{\ast}$. We show that $t_{i}^{\ast}$ cautiously believes
$\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)(E_{-i})$, that is,
$\beta_{i}^{\ast}\left(t_{i}^{\ast}\right)=\widehat{\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)}\left(\beta_{i}\left(t_{i}\right)\right)$
satisfies conditions (i) and (ii) of Definition 12.

First, note that

$E_{-i}\subseteq\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)^{-1}\left(\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)\right)\text{.}$

Hence, by definition of type morphism, it follows that, for all $l\leq n$,

$\beta_{i}^{l}\left(t_{i}\right)\left(E_{-i}\right)\leq\beta_{i}^{l}\left(t_{i}\right)\left(\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)^{-1}\left(\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)\right)\right)=\beta_{i}^{\ast,l}(t_{i}^{\ast})\left(\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)\right)\text{.}$

Since $E_{-i}$ is cautiously believed under $\beta_{i}\left(t_{i}\right)$, it
follows from condition (i) of Definition 12 that there exists $m\leq n$ such
that $\beta_{i}^{l}\left(t_{i}\right)\left(E_{-i}\right)=1$ for all $l\leq m$.
Therefore, we have that
$\beta_{i}^{\ast,l}\left(t_{i}^{\ast}\right)\left(\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)\right)=1$
for all $l\leq m$. Hence $\beta_{i}^{\ast}\left(t_{i}^{\ast}\right)$ satisfies
condition (i) of Definition 12.

Consider now an elementary cylinder
$\hat{C}_{s_{-i}}:=\left\\{s_{-i}\right\\}\times T_{-i}^{\ast}$ satisfying
$\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)\cap\hat{C}_{s_{-i}}\neq\emptyset$.
First, note that

$\displaystyle\left(\left\\{s_{-i}\right\\}\times T_{-i}\right)\cap E_{-i}$
$\displaystyle\subseteq$ $\displaystyle\left(\left\\{s_{-i}\right\\}\times
T_{-i}\right)\cap\left(\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)^{-1}\left(\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)\right)\right)$
$\displaystyle=$
$\displaystyle\left(\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)^{-1}\left(\hat{C}_{s_{-i}}\right)\right)\cap\left(\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)^{-1}\left(\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)\right)\right)$
$\displaystyle=$
$\displaystyle\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)^{-1}\left(\hat{C}_{s_{-i}}\cap\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)\right)\text{.}$

Hence, by definition of type morphism, it follows that, for all $l\leq n$,

$\displaystyle\beta_{i}^{l}\left(t_{i}\right)\left(\left(\left\\{s_{-i}\right\\}\times
T_{-i}\right)\cap E_{-i}\right)$ $\displaystyle\leq$
$\displaystyle\beta_{i}^{l}\left(t_{i}\right)\left(\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)^{-1}\left(\hat{C}_{s_{-i}}\cap\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)\right)\right)$
$\displaystyle=$
$\displaystyle\beta_{i}^{\ast,l}\left(t_{i}^{\ast}\right)\left(\hat{C}_{s_{-i}}\cap\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)\right)\text{.}$

Since $E_{-i}$ is cautiously believed under $\beta_{i}\left(t_{i}\right)$ at
level $m\leq n$, and since
$\hat{C}_{s_{-i}}\cap\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)\neq\emptyset$
implies $\left(\left\\{s_{-i}\right\\}\times T_{-i}\right)\cap
E_{-i}\neq\emptyset$, by condition (ii) of Definition 12 there exists $k\leq
m$ such that
$\beta_{i}^{k}\left(t_{i}\right)\left(\left(\left\\{s_{-i}\right\\}\times
T_{-i}\right)\cap E_{-i}\right)>0$. Therefore, we obtain

$\beta_{i}^{\ast,k}\left(t_{i}^{\ast}\right)\left(\hat{C}_{s_{-i}}\cap\left(\mathrm{Id}_{S_{-i}},\varphi_{-i}\right)\left(E_{-i}\right)\right)>0\text{.}$

Thus, $\beta_{i}^{\ast}\left(t_{i}^{\ast}\right)$ satisfies condition (ii) of
Definition 12. $\blacksquare$

Proof of Theorem 2. Part (i): Fix a type structure $\mathcal{T}:=\langle
S_{i},T_{i},\beta_{i}\rangle_{i\in I}$. If
$\mathop{\textstyle\prod}\nolimits_{i\in
I}\mathrm{Proj}_{S_{i}}\left(R_{i}^{\infty}\right)=\emptyset$, then the result
is immediate. So in what follows we will assume that this set is non-empty.
For each $i\in I$ and
$s_{i}\in\mathrm{Proj}_{S_{i}}\left(R_{i}^{\infty}\right)$, there exists
$t_{i}\in T_{i}$ such that $(s_{i},t_{i})\in R_{i}^{\infty}$. Since
$(s_{i},t_{i})\in R_{i}^{1}$, it follows that $s_{i}$ is a lexicographic best
reply to
$\overline{\mathrm{marg}}_{S_{-i}}\beta_{i}(t_{i})\in\mathcal{N}^{+}(S_{-i})$.
Therefore, by Proposition 1, $s_{i}$ is admissible, hence condition (a) of
Definition 3 is satisfied.

Next note that, for each $k\geq 1$, type $t_{i}$ cautiously believes
$R_{-i}^{k}$. So, it follows from Proposition 2.1 that $R_{-i}^{\infty}$ is
cautiously believed under $\beta_{i}(t_{i}):=(\mu^{1},...,\mu^{n})$ at some
level $m$. Moreover, Proposition 2.2 entails $\cup_{l\leq
m}\mathrm{Suppmarg}_{S_{-i}}\mu^{l}=\mathrm{Proj}_{S_{-i}}\left(R_{-i}^{\infty}\right)$.
Since $s_{i}$ is a lexicographic best reply to
$\overline{\mathrm{marg}}_{S_{-i}}\beta_{i}(t_{i})$, Proposition 1 in Blume et
al. (1991b) yields the existence of some $\nu\in\mathcal{M}(S_{-i})$ under
which $s_{i}$ is optimal and such that
$\mathrm{Supp}\nu=\mathrm{Proj}_{S_{-i}}\left(R_{-i}^{\infty}\right)$. Remark
1 entails that $s_{i}$ is admissible with respect to
$S_{i}\times\mathrm{Proj}_{S_{-i}}\left(R_{-i}^{\infty}\right)$, establishing
condition (b) of Definition 3.

Finally, by Corollary A1 in Brandenburger and Friedenberg (2010), every
$s_{i}^{\prime}$ that supports $s_{i}$ is a lexicographic best reply to
$\overline{\mathrm{marg}}_{S_{-i}}\beta_{i}(t_{i})$ as well. It follows that
$(s_{i}^{\prime},t_{i})\in R_{i}^{\infty}$, and this in turn implies that
$s_{i}^{\prime}\in\mathrm{Proj}_{S_{i}}\left(R_{i}^{\infty}\right)$,
establishing condition (c) of Definition 3.

Part (ii): Let $Q\in\mathcal{Q}$ be a non-empty SAS. Fix $i\in I$ and
$s_{i}\in Q_{i}$. By conditions (a) and (b) of Definition 3, and by Remark 1,
there exist $\nu_{s_{i}}^{2},\nu_{s_{i}}^{1}\in\mathcal{M}(S_{-i})$ such that
$\mathrm{Supp}\nu_{s_{i}}^{2}=S_{-i}$ and
$\mathrm{Supp}\nu_{s_{i}}^{1}=Q_{-i}$, and such that $s_{i}$ is optimal under
$\nu_{s_{i}}^{2}$ and $\nu_{s_{i}}^{1}$. Hence, $s_{i}$ is a lexicographic
best reply to $(\nu_{s_{i}}^{1},\nu_{s_{i}}^{2})\in\mathcal{N}^{+}(S_{-i})$.
Moreover, as in BFK (p. 328), we can choose $\nu_{s_{i}}^{2}$ and
$\nu_{s_{i}}^{1}$ in such a way that every strategy $s_{i}^{\prime}$ is
optimal under $\nu_{s_{i}}^{2}$ and $\nu_{s_{i}}^{1}$ if and only if
$s_{i}^{\prime}$ is supported by $s_{i}$. Now we construct a finite type
structure $\mathcal{T}:=\langle S_{i},T_{i},\beta_{i}\rangle_{i\in I}$ as
follows.

For each $i\in I$, let $T_{i}:=Q_{i}$. For every $s_{i}\in T_{i}$, define
$\mu_{s_{i}}^{1},\mu_{s_{i}}^{2}\in\mathcal{M}(S_{-i}\times T_{-i})$ as

$\displaystyle\mu_{s_{i}}^{1}\left(\left\\{\left(s_{-i},s_{-i}\right)\right\\}\right)$
$\displaystyle:$
$\displaystyle=\nu_{s_{i}}^{1}(\left\\{s_{-i}\right\\}),\forall s_{-i}\in
Q_{-i}\text{,}$
$\displaystyle\mu_{s_{i}}^{2}\left(\left\\{\left(s_{-i},\bar{s}_{-i}\right)\right\\}\right)$
$\displaystyle:$
$\displaystyle=\nu_{s_{i}}^{2}(\left\\{s_{-i}\right\\}),\forall s_{-i}\in
S_{-i}\text{,}$

where $\bar{s}_{-i}\in T_{-i}$ is arbitrarily chosen. Let
$\beta_{i}(s_{i}):=(\mu_{s_{i}}^{1},\mu_{s_{i}}^{2})$. Finiteness of each type
set guarantees that each belief map is measurable (in fact, continuous). This
completes the definition of the type structure $\mathcal{T}$.

We now show that $\mathcal{T}$ satisfies the required properties. Note that
each type $s_{i}\in T_{i}$ is cautious because
$\mathrm{Supp}\nu_{s_{i}}^{2}=S_{-i}$; hence, $\mathcal{T}$ is a cautious type
structure. For every $i\in I$ and $s_{i}\in Q_{i}$, strategy-type pair
$(s_{i},s_{i})$ is cautiously rational by construction; for every
$s_{i}^{\prime}\notin Q_{i}$, condition (c) of Definition 3 implies that
$s_{i}^{\prime}$ does not support $s_{i}$, so by construction the pair
$(s_{i}^{\prime},s_{i})$ is not rational. Hence,
$\mathrm{Proj}_{S_{i}}\left(R_{i}^{1}\right)=Q_{i}$. Now, suppose by way of
induction that $(s_{i},s_{i})\in R_{i}^{m}$ for each $i\in I$ and each
$s_{i}\in Q_{i}$. We show that type $s_{i}$ cautiously believes $R_{-i}^{m}$,
establishing that $(s_{i},s_{i})\in R_{i}^{m+1}$; this will yield
$(s_{i},s_{i})\in R_{i}^{\infty}$. Note that
$\mathrm{Supp}\mu_{s_{i}}^{1}=\left\\{(s_{-i},s_{-i}):s_{-i}\in
Q_{-i}\right\\}\subseteq R_{-i}^{m}$, where the inclusion follows from the
induction hypothesis. Moreover, since
$\mathrm{Suppmarg}_{S_{-i}}\mu_{s_{i}}^{1}=\mathrm{Supp}\nu_{s_{i}}^{1}=Q_{-i}$,
Proposition 2.2 entails that $R_{-i}^{m}$ is cautiously believed under
$\beta_{i}(s_{i})$ at level $1$. Therefore, we conclude that
$\mathrm{Proj}_{S_{i}}\left(R_{i}^{\infty}\right)=Q_{i}$. $\blacksquare$

For the proof of Theorem 5, we need an auxiliary technical fact.

Lemma C.3 Fix two sequences of pairwise disjoint topological spaces
$\left(X_{n}\right)_{n\in\mathbb{N}}$ and
$\left(Y_{n}\right)_{n\in\mathbb{N}}$. Let $X:=\cup_{n\in\mathbb{N}}X_{n}$ and
$Y:=\cup_{n\in\mathbb{N}}Y_{n}$. Suppose that, for each $n\in\mathbb{N}$,
there is a map $f_{n}:X_{n}\rightarrow Y_{n}$. If each map $f_{n}$ is
continuous (resp. surjective), then the union map
$\cup_{n\in\mathbb{N}}f_{n}:X\rightarrow Y$ is continuous (resp. surjective).

Proof. Let $O$ be open in $Y$. By definition of direct sum topology, the set
$O$ can be written as $O=\cup_{n\in\mathbb{N}}O_{n}$, where each $O_{n}:=O\cap
Y_{n}$ is open in $Y_{n}$ (see Engelking 1989, p. 74). Thus,

$\left(\mathop{\textstyle\bigcup}_{n\in\mathbb{N}}f_{n}\right)^{-1}\left(O\right)=\mathop{\textstyle\bigcup}_{n\in\mathbb{N}}f_{n}^{-1}\left(O_{n}\right)\text{.}$

So, if each $f_{n}$ is continuous, then each $f_{n}^{-1}\left(O_{n}\right)$ is
open, and this in turn implies that
$\left(\cup_{n\in\mathbb{N}}f_{n}\right)^{-1}\left(O\right)$ is open. The
conclusion that $\cup_{n\in\mathbb{N}}f_{n}$ is surjective if each $f_{n}$ is
surjective is immediate by inspection of the definitions. $\blacksquare$

Proof of Theorem 5. The desired type structure $\mathcal{T}:=\langle
S_{i},T_{i},\beta_{i}\rangle_{i\in I}$ is constructed as follows. For each
$i\in I$, let $T_{i}$ be the Baire space
$\mathbb{N}_{0}^{\mathbb{N}_{0}}$,292929Here $\mathbb{N}_{0}$ denotes the set
$\left\\{0,1,2,...\right\\}$, i.e.,
$\mathbb{N}_{0}:=\mathbb{N}\cup\left\\{0\right\\}$. The Baire space is
sometimes defined as the set $\mathbb{N}^{\mathbb{N}}$ of all infinite
sequences of natural numbers. This difference is immaterial for all the
relevant topological properties we are going to use in this proof. so that
each $t_{i}\in T_{i}$ is an infinite sequence of non-negative integers. The
set $\mathbb{N}_{0}$ is endowed with the discrete topology, and
$\mathbb{N}_{0}^{\mathbb{N}_{0}}$ is endowed with the product topology. The
basic open sets of $\mathbb{N}_{0}^{\mathbb{N}_{0}}$ are sets of the form

$O_{k}:=\left\\{\left(n_{1},n_{2},...\right)\in\mathbb{N}_{0}^{\mathbb{N}_{0}}:(n_{1},...,n_{k})=\left(o_{1},...,o_{k}\right)\right\\}$

for each $k\in\mathbb{N}_{0}$ and
$(o_{1},...,o_{k})\in\left(\mathbb{N}_{0}\right)^{k}$. With this topology, a
basic open set is also closed, so sets of the form $O_{k}$ constitute a clopen
basis. The space $\mathbb{N}_{0}^{\mathbb{N}_{0}}$ is Polish and uncountable,
but not compact.

For each $i\in I$, we partition $T_{i}$ into a countable family of non-empty
Borel subsets. For each $k\geq 0$, let

$\mathrm{T}_{i}^{k}:=\left\\{\left(n_{1},n_{2},...\right)\in\mathbb{N}_{0}^{\mathbb{N}_{0}}:n_{1}=k\right\\}\text{.}$

Each $\mathrm{T}_{i}^{k}$ is a subbasic clopen subset of $T_{i}$; moreover,
each $\mathrm{T}_{i}^{k}$ is homeomorphic to the Baire space. It is clear that
$T_{i}=\cup_{k\geq 0}\mathrm{T}_{i}^{k}$, and all the $\mathrm{T}_{i}^{k}$’s
are pairwise disjoint.

The next step is to construct the belief maps in such a way that, for all
$k\geq 0$, $t_{i}\in\mathrm{T}_{i}^{k}$ and $s_{i}\in S_{i}$, the pair
$(s_{i},t_{i})$ _does not_ belong to $R_{i}^{k+1}$. For each $i\in I$, we
construct a countable partition of $\mathcal{N}(S_{-i}\times T_{-i})$ that
“mirrors” the above partition of $T_{i}$. This is done as follows: For each
$i\in I$, let

$\Lambda_{i}^{0}:=\mathcal{N}(S_{-i}\times
T_{-i})\backslash\mathcal{C}_{i}^{0}\text{,}$

where $\mathcal{C}_{i}^{0}$ is the set of all LPS’s
$\bar{\mu}_{i}\in\mathcal{N}(S_{-i}\times T_{-i})$ such that
$\overline{\mathrm{marg}}_{S_{-i}}\bar{\mu}_{i}\in\mathcal{N}^{+}\left(S_{-i}\right)$.
Since $\left|S_{i}\right|\geq 2$ for each $i\in I$, it follows that
$\Lambda_{i}^{0}\neq\emptyset$.

Next, let

$\Lambda_{i}^{1}:=\left\\{\bar{\mu}_{i}\in\mathcal{C}_{i}^{0}:\mu_{i}^{1}(S_{-i}\times\mathrm{T}_{-i}^{0})>0\right\\}\text{,}$

and, for each $k\geq 2$,

$\Lambda_{i}^{k}:=\mathop{\textstyle\bigcap}_{m\in\left\\{1,...,k-1\right\\}}\left\\{\bar{\mu}_{i}\in\mathcal{C}_{i}^{0}:\mu_{i}^{1}(S_{-i}\times\mathrm{T}_{-i}^{m-1})=0\right\\}\cap\left\\{\bar{\mu}_{i}\in\mathcal{C}_{i}^{0}:\mu_{i}^{1}(S_{-i}\times\mathrm{T}_{-i}^{k-1})>0\right\\}\text{.}$

In words: $\Lambda_{i}^{1}$ is the set of all LPS’s on $S_{-i}\times T_{-i}$
such that the marginal on $S_{-i}$ has full support and the first component
measure assigns strictly positive probability to
$S_{-i}\times\mathrm{T}_{-i}^{0}$; $\Lambda_{i}^{2}$ is the set of all LPS’s
on $S_{-i}\times T_{-i}$ such that the marginal on $S_{-i}$ has full support
and the first component measure assigns probability $0$ to
$S_{-i}\times\mathrm{T}_{-i}^{0}$, and strictly positive probability to
$S_{-i}\times\mathrm{T}_{-i}^{1}$; and so on.

It is immediate to check that $\mathcal{N}(S_{-i}\times T_{-i})=\cup_{k\geq
0}\Lambda_{i}^{k}$ and all the $\Lambda_{i}^{k}$’s are non-empty, pairwise
disjoint sets; so the countable family of all $\Lambda_{i}^{k}$’s is a
partition of $\mathcal{N}(S_{-i}\times T_{-i})$.

Claim C.1 For each $k\geq 0$, $\Lambda_{i}^{k}$ is a Borel subset of
$\mathcal{N}(S_{-i}\times T_{-i})$.

Proof. Since $\mathcal{C}_{i}^{0}$ is Borel (see Lemma D.2 in Catonini and De
Vito, 2020), so is $\Lambda_{i}^{0}$. For each $k\geq 1$, let

$\mathrm{P}_{i}^{k}:=\left\\{\bar{\mu}_{i}\in\mathcal{C}_{i}^{0}:\mu_{i}^{1}(S_{-i}\times\mathrm{T}_{-i}^{k-1})>0\right\\}\text{.}$

Note that $\Lambda_{i}^{1}=\mathrm{P}_{i}^{1}$, and for each $k\geq 2$,
$\Lambda_{i}^{k}$ is the intersection of $\mathrm{P}_{i}^{k}$ with the
complements of $\mathrm{P}_{i}^{1},...,\mathrm{P}_{i}^{k-1}$. Thus, in order
to show that each $\Lambda_{i}^{k}$ is Borel in $\mathcal{N}(S_{-i}\times
T_{-i})$, it is sufficient to show that each $\mathrm{P}_{i}^{k}$ is Borel in
$\mathcal{N}(S_{-i}\times T_{-i})$. Let

$\mathrm{M}_{i}^{k}:=\left\\{\mu\in\mathcal{M}\left(S_{-i}\times\mathrm{T}_{-i}\right):\mu(S_{-i}\times\mathrm{T}_{-i}^{k-1})>0\right\\}\text{.}$

By Theorem 17.24 in Kechris (1995), if $X$ is a Polish space, then the Borel
$\sigma$-field on $\mathcal{M}\left(X\right)$ is generated by sets of the form
$\left\\{\mu\in\mathcal{M}\left(X\right):\mu\left(E\right)\geq p\right\\}$,
where $E\in\Sigma_{X}$ and $p\in\mathbb{Q\cap}\left[0,1\right]$. Hence, for
every $E\in\Sigma_{X}$, the set
$\left\\{\mu\in\mathcal{M}\left(X\right):\mu\left(E\right)>0\right\\}$ is
Borel, since it can be written as
$\cap_{n\in\mathbb{N}}\left\\{\mu\in\mathcal{M}\left(X\right):\mu\left(E\right)\geq\frac{1}{n}\right\\}$.
This implies that $\mathrm{M}_{i}^{k}$ is Borel in
$\mathcal{M}\left(S_{-i}\times\mathrm{T}_{-i}\right)$. Moreover, for each
$n\in\mathbb{N}$, the canonical projection map

$\begin{array}[]{cccc}\mathrm{Proj}_{1,n}:&\mathcal{N}_{n}(S_{-i}\times
T_{-i})&\rightarrow&\mathcal{M}\left(S_{-i}\times\mathrm{T}_{-i}\right)\text{,}\\\
&\left(\mu_{i}^{1},...,\mu_{i}^{n}\right)&\mapsto&\mu_{i}^{1}\text{,}\end{array}$

is continuous, hence the set
$\mathrm{Proj}_{1,n}^{-1}\left(\mathrm{M}_{i}^{k}\right)$ is Borel in
$\mathcal{N}_{n}(S_{-i}\times T_{-i})$. With this, the conclusion follows from
the observation that $\mathrm{P}_{i}^{k}$ can be written as

$\mathrm{P}_{i}^{k}=\left(\mathop{\textstyle\bigcup}_{n\in\mathbb{N}}\mathrm{Proj}_{1,n}^{-1}\left(\mathrm{M}_{i}^{k}\right)\right)\cap\mathcal{C}_{i}^{0}\text{.}$

$\square$

Recall that every Borel subset of a Polish space is a Lusin space, when
endowed with the relative topology. Moreover, every Lusin space is also
analytic (see Cohn 2003, Proposition 8.6.13). Thus, by Claim C.1, each
$\Lambda_{i}^{k}$ is analytic. Since each $\mathrm{T}_{i}^{k}$ is homeomorphic
to the Baire space, it follows from Corollary 8.2.8 in Cohn (2003; see also
Kechris 1995, p. 85) that, for every $k\geq 0$, there exists a surjective
continuous map
$\beta_{i}^{\left[k\right]}:\mathrm{T}_{i}^{k}\rightarrow\Lambda_{i}^{k}$. For
each $i\in I$, let $\beta_{i}$ be the union of the
$\beta_{i}^{\left[k\right]}$’s, i.e., $\beta_{i}:=\cup_{k\geq
0}\beta_{i}^{\left[k\right]}:T_{i}\rightarrow\mathcal{N}(S_{-i}\times
T_{-i})$. The map is well defined because the $\mathrm{T}_{i}^{k}$’s are
pairwise disjoint. By Lemma C.3, $\beta_{i}$ is a continuous (and so Borel)
surjective map. This completes the definition of the type structure
$\mathcal{T}$.

We now show that $\mathcal{T}$ satisfies the required properties.

Claim C.2 For each $i\in I$ and for each $k\geq 0$,

$(S_{i}\times\mathrm{T}_{i}^{k})\cap R_{i}^{k+1}=\emptyset\text{.}$

Proof. By induction on $k\geq 0$.

(Basis step: $k=0$) Fix $i\in I$ and $(s_{i},t_{i})\in S_{i}\times T_{i}$ with
$t_{i}\in\mathrm{T}_{i}^{0}$. We clearly have $(s_{i},t_{i})\not\in R_{i}^{1}$
because $\beta_{i}(t_{i})\in\Lambda_{i}^{0}$, hence $t_{i}$ is not cautious.
Therefore $(S_{i}\times\mathrm{T}_{i}^{0})\cap R_{i}^{1}=\emptyset$.

(Inductive step: $k\geq 1$) Suppose we have already shown that
$(S_{i}\times\mathrm{T}_{i}^{k-1})\cap R_{i}^{k}=\emptyset$ for each $i\in I$.
Fix $i\in I$ and $(s_{i},t_{i})\in S_{i}\times T_{i}$ with
$t_{i}\in\mathrm{T}_{i}^{k}$. Thus
$\beta_{i}(t_{i}):=(\mu_{i}^{1},...,\mu_{i}^{n})\in\Lambda_{i}^{k}$, hence
$\mu_{i}^{1}(S_{-i}\times\mathrm{T}_{-i}^{k-1})>0$. Since, by the induction
hypothesis, $(S_{-i}\times\mathrm{T}_{-i}^{k-1})\cap R_{-i}^{k}=\emptyset$, it
must be the case that $\mu_{i}^{1}(R_{-i}^{k})<1$. Therefore $R_{-i}^{k}$ is
not cautiously believed under $\beta_{i}(t_{i})$; this implies
$(s_{i},t_{i})\not\in\mathbf{B}_{i}^{c}(R_{-i}^{k})$. Hence
$(s_{i},t_{i})\not\in R_{i}^{k+1}$. $\square$

To conclude the proof, pick any $(s_{i},t_{i})\in S_{i}\times T_{i}$. Then
there exists $k\geq 0$ such that $t_{i}\in\mathrm{T}_{i}^{k}$. By Claim C.2,
it follows that $(s_{i},t_{i})\not\in R_{i}^{k+1}$. Since
$R_{i}^{\infty}:=\cap_{m\geq 1}R_{i}^{m}$, this shows that
$R_{i}^{\infty}=\emptyset$, as required. $\blacksquare$

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