AAD and least-square Monte Carlo: Fast Bermudan-style options and XVA Greeks

2.1Bermudan-style options

While European-style options can be exercised only at final maturity, Bermudan-style options can be exercised on multiple dates up to trade expiry.

We denote by T₁, …, T_M the exercise dates of the option and define D(t)={Tm≥t} . We denote by η (t) the smallest integer such that T_η(t)+1 > t. An exercise policy is represented mathematically by a stopping time taking values in D(t) . We denote by T(t) the set of stopping times taking values in D(t) , see e.g., Andersen and Piterbarg (2010).

A rational investor will exercise the option that he holds in such a way as to maximise its economic value. As a result, the value of a Bermudan-style option is the supremum of the option value over all possible exercise policies. With the notation introduced above, the value of a Bermudan-style option at time t can thus be expressed by

A useful concept is the hold value of the Bermudan-style option: We denote by H (t) the value of the Bermudan-style option when the exercise dates are restricted to D(Tη(t)+1) , that is

The option holder, following an optimal exercise policy, will exercise his option if the exercise value is larger than the hold value, i.e.,

This, when combined with Equation (9), leads to the so-called dynamic programming formulation:

The dynamic programming formulation above implies that the stopping time, defining the optional exercise date as seen at time t, is given by

The optimal exercise strategy defined by Equation (12) requires the computation of the hold value H (t), m = η (t) + 1, …, M - 1. In a setting in which the underlying risk factor process {X (t) } _0≤t≤T is a generic k-dimensional Markov process, the hold value H (t) is a function of the state vector at time t. That is,

When the dimension of the Markov process k is small enough, the conditional expectation value in Equation (13) can be computed in a straightforward way by discretising the risk-factor process and performing standard backward induction on a tree or a grid, or by discretising an associated Partial Differential Equation (PDE). Here we refer to, e.g., Wilmott et al. (1995). However, the complexity of grid-based calculations is exponential in the dimension of the Markov process and numerical implementations become infeasible when k ≥ 4.

As we will review in Section 2.3, regression-based MC techniques provide an effective way of computing conditional expectation values of the form (13).

2.2XVA

We next consider the computation of the Credit Valuation Adjustment (CVA) and the Debt Valuation Adjustment (DVA) as the main measures of a dealer’s counterparty credit risk, see e.g., Cr'epey et al. (2014). For a given portfolio of trades with the same investor or institution, the CVA (resp. DVA) aims to capture the expected loss (resp. gain) associated with the counterparty (resp. dealer) defaulting in a situation in which the position, netted for any collateral posted, has a positive mark-to-market for the dealer (resp. counterparty).

This can be evaluated at time T₀ = 0 by

Equation (14) is typically computed on a discrete time grid of “horizon dates” 0 = T₀ < T₁ < … < T_M = T. For instance, we may have

In the k-dimensional Markov setting introduced above the conditional future exposure V (T_m) is a function V (X (T_m)) of the state vector at time T_m. However, only for vanilla securities and simple models for the evolution of the risk factors, such conditional future exposures can be expressed in closed form, and regression based Monte Carlo is commonly employed to produce approximate estimators see, e.g., Cesari et al. (2009).

2.3Conditional expectation values and Bermudan-style options by regression

Regression methods are based on the observation, see e.g., Friedman et al. (2001), that given a real-valued random input vector X∈ℝd and Y a real valued random output, the conditional expectation 𝔼[Y|X] is the function of X that best approximates in the least-square sense the output Y. That is,

In particular, assuming that the conditional expectation is a linear function of some unknown vector of parameters β,

Equation (16) reduces to the well-known linear regression conditions, giving for the optimal vector of parameters

In the context of the valuation of Bermudan-style options, the hold value (13) on an exercise date T_m is assumed to be of the form

These equations provide a straightforward recipe to compute the regression coefficients β_m by substituting Ψ_m and Ω_m with their sample average over N_MC MC replications, namely:

This approach was introduced by Tsitsiklis and Van Roy (2001) and they showed that the estimator V₀ converges for n→ ∞ to the true value V (0) provided that the representation (19) holds exactly.

A modification of this algorithm was proposed by Longstaff and Schwartz (2001) and it entails replacing Equation (29) in Step R3 (d) with

2.4Lower bound algorithm for Bermudan-style options

The hold value obtained by regression as described in the previous section defines an exercise policy whereby on each exercise date T_m the option is exercised if

Such policy, being an approximation of the solution of the dynamic programming Equation (11), will in general correspond to a suboptimal stopping time. As a result, when utilised in a second, independent, MC simulation, the exercise policy obtained by regression, will result in a lower-bound estimator for the Bermudan-style option value. The corresponding algorithm can be schematically described as it follows.

For each MC replication indexed by n = 1, …, N_MC perform steps (L1) to (L4) below:

2.5XVA by regression

As described in Section 2.2, the calculation of the XVA in Equation (15) requires the conditional future exposure V (t) on a set of dates determined by a discretisation time grid T₁ … T_M. The regression algorithm described in the previous section can be easily adapted to compute such quantity. Indeed, the conditional value of each of the options contained in the netting set can be obtained using the same least-square procedure.

Once the regression algorithm is completed, we can use the regression functions to compute the hold value of each option in the portfolio on the discretisation time grid by Hm=βmTψm . If the discretisation time T_m is not an exercise opportunity for the option under consideration, then this is also its conditional future exposure. Conversely, the conditional future exposure is obtained by comparing the hold value to the exercise value as in Equations (29) and (31). These observations translate in the following algorithm.

For each MC replication indexed by n = 1, …, N_MC perform steps (X1) to (X3) below:

3.1Function regularizations

In the context of MC methods, Capriotti and Giles (2012) show that AAD allows to calculate the sensitivities by differentiating the relevant estimator on a path by path basis. As a path-wise method, the MC estimators must satisfy specific regularity conditions, see Glasserman (2003). For instance, all the functions appearing in each step leading to the computation of the payout estimator must be Lipschitz continuous. A practical way of addressing non-Lipschitz estimators is to smoothen out the singularities they contain. This can be achieved by observing that in most cases the singularities in the payout functions, although not necessarily implemented as such, can be expressed in terms of Heaviside functions. For instance, the payoff function of a digital option (for a one-dimensional underlying asset) is,

The singularities in such payoff functions can be regularized by replacing the indicator function with one of its smoothened counterparts. A very common choice, for instance, is to approximate the step function with a “call spread” payoff functions,

In general, the payoff estimator for Bermudan-style options (34) is not differentiable with respect to the pathwise value of the approximate exercise boundary Hm(n) , and it requires the regularization described above. A common approximation among practitioners, see e.g., Leclerc et al. (2009), is to assume that that exercise boundary implied by the rule (32) is close to optimality so that the value of the contract is approximatively continuous across the exercise boundary and no regularization is required. Under this assumption, no contribution to the sensitivities is associated with the perturbations of the exercise boundary, and one can therefore keep the regression coefficients fixed while calculating the sensitivities, see Andersen and Piterbarg (2010). As discussed in Section 4, depending on the accuracy of the basis functions in representing the exercise boundary, this may or may not be an accurate approximation.

3.2AAD for the lower bound algorithm for Bermudan-style options

The AAD implementation of the lower bound algorithm for Bermudan-style options described in Section 2.4, producing the MC estimators for the sensitivities of the estimator (36) with respect to a set of model parameters θ_k, k = 1, …, N_θ, given by

3.3AAD for XVA by regression

The AAD implementation of the algorithm for the calculat ion of XVA described in Section 2.5, producing the MC estimators for the sensitivities of the estimator (45) with respect to a set of model parameters θ_k, k = 1, …, N_θ, given by

3.4AAD for the regression algorithm

In the AAD implementations presented in Sections 3.2 and 3.3 the adjoint of the regression coefficients in Equation (54) do not contribute to the calculation of sensitivities. As mentioned in Section 3.1, and illustrated with numerical examples in Section 4, this can be justified as a reasonable approximation in the case of Bermudan-style options when the exercise boundary approximated by regression is close to the optimal one. However, such arguments are generally approximations for Bermudan-style options, and cannot be invoked at all when regression is utilised for XVA. In this case, the contributions associated with the sensitivities of the regression coefficients must be taken into account in order to obtain accurate estimates of the model parameters sensitivities. In this section, we discuss how these contributions can be computed by the AAD implementation of the least-square MC algorithm described in Section 2.3.

4.1Best of two assets Bermudan-style option

As a first example, we consider the classical case of a Bermudan-style option on the maximum of two assets under a standard lognormal model for the asset price processes {X₁ (t)} and {X₂ (t)}. The payoff function at exercise time t is given by

Fig.1

Deltas (left panel) and Vegas (right panel) for the Bermudan-style option in (80) as a function of strike. The smoothening parameter in the call spread regularization (48) is δ = 0.005. The number of MC paths is 400,000. The results are obtained with the setting (81) and (82). The MC uncertainty (in parenthesis) is computed using the binning technique with 20 bins for each set of simulations.

Here the smoothening parameter for the calculation of the Greeks, discussed in Section 3.1, δ = 0.005, was chosen as a reasonable compromise between variance and bias of the estimator. This is illustrated in Fig. 2, showing how for δ = 0.005 the bias introduced by the finite δ is of the same order of magnitude of the statistical uncertainty for the chosen computational budget, and is negligible for any practical application.

Fig.2

AAD Delta (left panel) and Vega (right panel) of the Bermudan-style option in (80) for K = 1 vs the smoothening parameter δ for the call spread regularization (48). The number of simulated paths is 3,000,000 for δ = 0.01 and is increased as δ is decreased in order to keep statistical uncertainties roughly constant. The results are obtained with the setting (81) and (82). The values in the graphs are fitted based on a quadratic polynomial function (green lines).

As discussed in Section 3.1, neglecting to smoothen out the exercise boundary, although common in the financial practice, introduces a bias in the computation of sensitivities because the exercise boundary is in general not optimal. This is illustrated in Fig. 3, in which we compare the Delta, with and without smoothening, for different choices of the basis functions. These results are obtained with the inputs (81) and (82). Here for Delta, the smoothened estimator turns out to be more accurate, especially when the exercise boundary obtained by regression is a less accurate approximation of the real one. However, it is difficult to establish a priori whether the unsmoothened estimator provides a smaller or a larger bias than the smoothened one. This is because the bias introduced by the lack of smoothening may be offset by the bias arising from the sub-optimality of the exercise boundary. This is illustrated in the right panel of Fig. 3 showing that for Vega the smoothened and unsmoothened estimators have a comparable accuracy. In any case, a consideration to keep in mind is that smoothening the exercise boundary is generally required to obtain stable second-order risk values.

Fig.3

AAD Delta (left panel) and Vega (right panel) of the Bermudan-style option in (80) for K = 1 as obtained with the unsmoothened and the smoothened estimators with the call-spread regularization (48) for five choices of the regression basis functions. The number of simulated paths is 1,000,000 with 20 bins and the smoothening parameter is δ = 0.005.

The quasi-optimality of the exercise boundary is also generally invoked among industry practitioners as a justification for neglecting the contributions to the sensitivities arising from the exercise boundary. Clearly, the quality of this approximation is dependent on the accuracy of the regression functions in reproducing the actual exercise boundary. This is illustrated in Fig. 4 where we plot Delta and Vega of the Bermudan-style option (80) for different choices of the regression basis functions. Here we compare the results obtained by the AAD algorithm, as described in Section 3.4, in the case where a) the exercise boundary is kept fixed, and b) when accounting instead for its contributions to the sensitivities. As expected, the difference between the two approaches vanishes as the regression functions become more accurate. However, as it is shown for the Vega, it can lead to a significant bias if a simple (e.g., linear or quadratic) representation of the exercise boundary is adopted.

Fig.4

AAD Delta (left panel) and Vega (right panel) of the Bermudan-style option in (80) for K = 1 as obtained by keeping the exercise boundary fixed (Fixed) and accounting instead for its contributions to the sensitivities (Flexible). The number of simulated paths is 1,000,000 with 20 bins and the call-spread smoothening parameter is δ = 0.005.

4.2XVA sensitivities

As another example, we compute the sensitivities of XVA (14) for the same option defined in Equation (80). Here, for simplicity, we assume that the hazard rate and the volatility are piecewise constants. The following results are obtained with the specifications (81) and (82). The XVA sensitivities with respect to some of the model parameters, namely the term structure of hazard rates of the counterparty and of volatilities of the underlying, obtained with the AAD algorithm described in Section 2.5 are compared with the ones obtained by standard bumping in Tables 2 and 3. As expected, the AAD sensitivities are in excellent agreement with those obtained by bumping with any discrepancies attributable to the bias of the finite-difference approach completely masked in this example by the MC uncertainties.

Similar to the case of Bermudan-style option Greeks, keeping the regression boundary fixed while computing the sensitivities, introduces a bias. However, for XVA, this issue is much more severe than in the case of the Bermudan-style option Greeks because no quasi-optimality argument can be invoked. As shown in Fig. 5, the volatilities sensitivities obtained by bumping and AAD keeping into account of the regression contributions are almost identical. Instead, the results of AAD keeping the regression coefficients fixed are remarkably different and, if utilised for risk management, would lead to significant mis-hedging. On the other hand, as expected, the XVA sensitivities with respect to the hazard rates are not affected by the contribution arising from the regression functions. This is because the hazard rates do not enter the computation of the portfolio value conditional on default, and hence do not appear in the regression parameters.

Fig.5

XVA sensitivities with respect to the piecewise volatility (left panel) and hazard rate (right panel), computed by AAD keeping into account of the contributions of the regression coefficients (Flexible), by AAD keeping the regression coefficients fixed (Fixed), and by finite-differences (Bumping). The number of MC paths is 1,000,000 and the number of bins is 25. The results are obtained with the setting (81) and (82).

Finally, the remarkable computational efficiency of the AAD approach is illustrated in Fig. 6. Here we plot the cost of computing the XVA sensitivities with respect to the term structure of the counterparty hazard rate and the underlying volatility, relative to the cost of performing a single valuation. The calculation of the sensitivities by means of AAD can be performed in about three times the cost of computing the XVA value, that is, well within the theoretical bound (7). In contrast, the cost of bumping, for one-sided finite-difference estimators, is in general (1 + N) times the cost of as single valuation, where N is the number of model parameters, i.e., in this case over 20 times the cost of computing the value of the XVA.

Fig.6

Ratio of the CPU time required for the calculation of the XVA, and its sensitivities, and the CPU time spent for the computation of the XVA alone.