Least Squares Monte Carlo applied to Dynamic Monetary Utility Functions

Dynamic monetary risk measures and utility functions, as described for instance in [1] and [4], are time consistent if and only if they satisfy a recursive relationship (see for instance [5], [19]). In the case of time-consistent valuations of cash flows, often in an insurance setting, (e.g. [12], [13], [17], [19], [20], [18], [11]), analogous recursions also appear. Recursive relationships also occur as properties of solutions to optimal stopping problems, of which valuation of American derivatives is a special case. It is well known that numerical solutions to these kinds of recursions suffer from ”the curse of dimensionality”: As the underlying stochastic process generating the flow of information grows high dimensional, direct computations of solutions of these recursions prove unfeasible.

In order to make objective of this paper more clear, consider a probability space $(\Omega,\mathcal{F},\mathbb{P})$, a $d$-dimensional Markov chain $(S_{t})_{t=0}^{T}$ in $L^{2}(\mathcal{F})$ and its natural filtration $(\mathcal{F}_{t})_{t=0}^{T}$. We are interested in computing $V_{0}$ given as the solution to the following recursion

where, for each $t$, $\varphi_{t}:L^{2}(\mathcal{F}_{t+1})\to L^{2}(\mathcal{F}_{t})$ is a given law-invariant mapping (see Section 2 and definition 3). Recursions such as (1) arise when describing time-consistent dynamic monetary risk measures/utility functions (see e.g. [4]). Alternatively, we may be interested in computing $V_{0}$ given as the solution to the following recursion

where $f:\mathbb{R}^{d}\to\mathbb{R}$ is a given function. Recursions such as (2) arise when solving discrete-time optimal stopping problems or valuing American-style derivatives (see e.g. [14] and [16]). In this article we will focus the recursive expression (1). In either case, due to the Markovian assumptions, we expect $V_{t}$ to be determined by some deterministic function of the state $S_{t}$ at time $t$. The curse of dimensionality can now be succinctly put as the statement that as the dimension $d$ grows, direct computation of $V_{t}$ often becomes unfeasible. Additionally, brute-force valuation via a nested Monte Carlo simulation, discussed in [3] and [2], is only a feasible option when $T=2$, as the number of required simulations would grow exponentially with $T$. One approach to tackle this problem is the Least Squares Monte Carlo (LSM) algorithm, notably used in [16] to value American-style derivatives, and consists of approximating $V_{t+1}$ in either (1) or (2) as a linear combination basis functions of the state $S_{t+1}$ via least-squares regression. While most often considered for optimal stopping problems ([16], [22], [7], [21], [14], [23], [24], [25]), it has also been used recently in [18] for the purpose of actuarial valuation, with respect to a recursive relationship in line with (1).

The paper is organized as follows. In Section 2 we introduce the mathematical definitions and notation that will allow us to describe the LSM algorithm in our setting mathematically, as well as to formulate theoretical results. Section 3 contains consistency results with respect to computing (1) both in a general setting, only requiring an assumption of continuity in $L^{2}$ norm, and for the special case of a Cost-of-Capital valuation, studied in [12] and [13], under the assumption that capital requirements are given by the risk measure Value-at-Risk, in line with solvency II. The lack of convenient continuity properties of Value-at-Risk pose certain challenges that are handled. Section 4 investigates the numerical performance of the LSM algorithm on valuation problems for a set of models for liability cash flows. Here some effort is also put into evaluating and validating the LSM algorithm’s performance, as this is not trivial for the considered cases.

Consider a probability space $(\Omega,\mathcal{F},\mathbb{P})$. On this space we consider two filtrations $(\mathcal{F}_{t})_{t=0}^{T}$, with $\mathcal{F}_{0}=\{\emptyset,\Omega\}$, and $(\mathcal{H}_{t})_{t=0}^{T}$. The latter filtration is an initial expansion of the former: take $\mathcal{D}\subset\mathcal{F}$ and set $\mathcal{H}_{t}:=\mathcal{F}_{t}\vee\mathcal{D}$. $\mathcal{D}$ will later correspond to the $\sigma$-field generated by initially simulated data needed for numerical approximations. Define $L^{2}(\mathcal{H}_{t})$ as the spaces of $\mathcal{H}_{t}$ measurable random variables $Z$ with $\mathbb{E}[Z^{2}]<\infty$. The subspace $L^{2}(\mathcal{F}_{t})\subset L^{2}(\mathcal{H}_{t})$ is defined analogously. All equalities and inequalities between random variables should be interpreted in the $\mathbb{P}$-almost sure sense.

We assume that the probability space supports a Markov chain $S=(S_{t})_{t=0}^{T}$ on $(\mathbb{R}^{d})^{T}$, where $S_{0}$ is constant, and an iid sequence $D=(S^{(i)})_{i\in\mathbb{N}}$, independent of $S$, where, for each $i$, $S^{(i)}=(S^{(i)}_{t})_{t=0}^{T}$ has independent components with $\mathcal{L}(S^{(i)}_{t})=\mathcal{L}(S_{t})$ (equal in distribution). $D$ will represent possible initially simulated data and we set $\mathcal{D}=\sigma(D)$. The actual simulated data will be a finite sample and we write $D_{n}:=(S^{(i)})_{i=1}^{n}$. For $Z\in L^{2}(\mathcal{F})$ we write $\|Z\|_{2}:=\mathbb{E}[|Z|^{2}\mid\mathcal{H}_{0}]^{\frac{1}{2}}$. Notice that $\|Z\|_{2}$ is a nonrandom number if $Z$ is independent of $D$.

The mappings $\rho_{t}$ and $\varphi_{t}$ appearing in Definitions 1 and 2 below can be defined analogously as mappings from $L^{p}(\mathcal{H}_{t+1})$ to $L^{p}(\mathcal{H}_{t})$ for $p\neq 2$. However, $p=2$ will be the relevant choice for the applications treated subsequently.

The elements $\rho_{t}$ of the dynamic monetary risk measure $(\rho_{t})_{t=0}^{T-1}$ are called conditional monetary risk measures.

Note that if $(\rho_{t})_{t=0}^{T-1}$ is a dynamic monetary risk measure, $(\rho_{t}(-\cdot))_{t=0}^{T-1}$ is a dynamic monetary utility function. In what follows we will focus on dynamic monetary utility function of the form

where $(\rho_{t})_{t=0}^{T}$ is a dynamic monetary risk measures in the sense of Definition 1 and $(\eta_{t})_{t=0}^{T-1}$ is a sequence of nonrandom numbers in $(0,1)$. We may consider a more general version of this dynamic monetary utility function by allowing $(\eta_{t})_{t=0}^{T-1}$ to be an $(\mathcal{F}_{t})_{t=0}^{T-1}$ adapted sequence, however we choose the simpler version here. That $(\varphi_{t})_{t=0}^{T}$ is indeed a dynamic monetary utility function is shown in [12].

We will later consider conditional monetary risk measures that are conditionally law invariant in the sense of Definition 3 below. Conditional law invariance will then be inherited by $\varphi_{t}$ in (9).

We now define the value process corresponding to a dynamic monetary utility function $(\varphi_{t})_{t=0}^{T-1}$ in the sense of Definition 2 with respect to the filtration $(\mathcal{F}_{t})_{t=0}^{T-1}$ instead of $(\mathcal{H}_{t})_{t=0}^{T-1}$. The use of the smaller filtration is due to that a value process of the sort appearing in Definition 4 is the theoretical object that we aim to approximate well by the methods considered in this paper.

Whenever it will cause no confusion, we will suppress the argument $\varphi$ in $V_{t}(L,\varphi)$ in order to make the expressions less notationally heavy.

In the following section we prove what essentially are two consistency results for the LSM estimator $\widehat{V}^{(M)}_{N,t}(L)$ along with providing conditions for these to hold. These consistency results are analogous to Theorems 3.1 and 3.2 in [7]. The first and simplest result, Lemma 1, is that if we have a flexible enough class of basis functions, $\widehat{V}_{N,t}(L)$ will asymptotically approach the true value $V_{t}(L)$. The second consistency result, Theorem 1, is that when $N$ is kept fixed, then $\widehat{V}^{(M)}_{N,t}(L)$ will approach the least-squares optimal $\widehat{V}_{N,t}(L)$ for each $t$ as $M$ grows to infinity. Hence, we show that the LSM estimator for a fixed number of basis functions is consistent in the sense that the simulation-based projection operator $P^{(M)}_{\mathcal{B}_{t,N}}$ will approach $P_{\mathcal{B}_{t,N}}$ even in the presence of errors in a multiperiod setting. Lemma 7 and Theorem 3 furthermore extends these results to the case of a Cost-of-Capital valuation, studied in [12] and [13], which here is dependent on the non-continuous risk measure Value-at-Risk. Note from Section 2 that these results presume simulated data not to be path dependent, in contrast to the results in [25].

We should note that these results do not give a rate of convergence, which is done in the optimal stopping setting in for instance [14] and [23]. Especially, these papers provide a joint convergence rate in which $M$ and $N$ simultaneously go to infinity, something which is not done here. There are three main reasons for this. First of all, in this paper the purpose is to investigate LSM methods given by standard OLS regression, i.e. we do not want to involve a truncation operator as we believe this would not likely be implemented in practice. The use of truncation operators is necessary for the results in [14] and [23], although one can handle the case of unbounded cash flows by letting the bound based on the truncation operator suitably go to infinity along with $N$ and $M$. Secondly, we believe that the bounds involved in the rates of convergence would be quite large in our case if we applied repeatedly the procedure in [14] or [23] (see remark 3). Thirdly, we want to consider mappings which are $L^{2}$-continuous (Definition 5) but not necessarily Lipschitz. In this case it is not clear how convergence can be established other than at some unspecified rate.

In this section we will test the LSM algorithm empirically for the special case of the mappings $\varphi$ being given by $(\varphi_{t,\alpha})_{t=0}^{T-1}$ in (21). The LSM algorithm described below, Algorithm 1, will differ slightly from the one previously, in the sense that it will contain the small inefficiency of having two regression steps: One for the $\operatorname{VaR}$ term of the mapping, one for the expected value term. The reason for introducing this split is that it will significantly simplify the validation procedures of the algorithm. Heuristically, we will be able to run a forward simulation where we may test the accuracy of both the $\operatorname{VaR}$ term and the expected value term.