Monte Carlo Sensitivities Using the Absolute Measure-Valued Derivative Method

Sensitivities play a key role in the assessment and the management of risks associated with financial derivatives and must be computed along with prices as part of sound risk management practice. Given that it is not possible to compute the sensitivities let alone the prices of  many financial derivatives analytically, they are usually computed numerically, with Monte Carlo simulation being the most widely used technique. The associated sensitivities are then computed by finite difference, pathwise differentiation, or likelihood ratio methods, each with its own set of limitations. In particular, the finite difference method is biased and sensitivity estimates have relatively large variances, the pathwise derivative method cannot be applied to products with discontinuous payoffs such as digitals, and the likelihood ratio method can be subject to unbounded variance when the associated score function is not absolutely continuous with respect to the sensitivity parameter. For a detailed discussion of these methods and their limitations, refer to Glasserman (2004), chapter 7.

There has been extensive work on regularizing the pathwise method to enable it to work with discontinuities. Such work tends either to involve differentiating the discontinuity directly, which yields a delta distribution, or to utilize smoothing to avoid this differentiation. Joshi and Kainth (2004), Rott and Fries (2005), and Brace (2007) explore the explicit integration of the delta distribution, which requires at least one extra simulation that must be handcrafted to the payoff. Liu and Hong (2011) and Lyuu and Teng (2011) reduce the degree of handcrafting by employing kernel smoothing and importance sampling, respectively. Chan and Joshi (2015) modify the simulation scheme as a function of the perturbation variable so that the pathwise payoff, as a function of perturbation size, is smooth. It is then possible to apply the pathwise method at the cost of gaining a small likelihood ratio term. All these methods have the drawback that the methodology is adapted to the singularities of the payoff. This renders them much harder to implement generically than the likelihood ratio and pathwise methods.

Despite their importance, there is a relative absence of new techniques for computing Monte Carlo sensitivities, and recent advances have focused instead on improving the computational efficiency of existing techniques. For example, Capriotti and Giles (2018) apply adjoint algorithmic differentiation to speed up pathwise sensitivity calculations, while Xiang and Wang (2022) apply low discrepancy sequences to achieve variance reduction for American option sensitivities.

A notable exception is the measure-valued differentiation (MVD) method considered, for example, in Heidergott et al. (2008) and  Pflug and Thoma (2016). The MVD method does not differentiate the payoff and so discontinuities do not cause difficulties, and it has been shown to produce sensitivity estimates with lower variance than those computed using the finite difference and the likelihood ratio (LR) methods. It is also generic in the sense that it is not adapted to the payoff in any way, and so the same paths can be used to compute the sensitivities of many derivatives on the same underlying framework. In the MVD approach, the derivative of the transition density is recognized as a signed measure and decomposed as the difference of two proper measures. The simulation at each step is then branched according to the two constituent measures to produce multiple MVD paths. Sensitivity estimates from these branched paths are then recombined to produce one MVD sensitivity sample. It is shown that this approach generally results in sensitivity estimates with lower variance than alternative methods, albeit at the expense of a greater computational burden due to the branching of the paths. Further variance reduction is possible by coupling, or correlating, each pair of the branched paths.

In this paper, we introduce an alternative to the MVD method, called the absolute measure-valued differentiation (AMVD) method, which decomposes the density derivative as the product of its sign and its absolute value. This method shares many of the properties of the MVD method, such as the ability to handle discontinuous payoffs, but  instead of initiating two additional paths at each simulation step of the original path, it initiates only one additional subpath. It is found that the AMVD method gives vega estimates with lower variance than the MVD method for vanilla and digital calls in the Black–Scholes model. Moreover, the AMVD vega estimates have lower variance than the pathwise differentiation method in certain situations, such as the case of short-dated deep in-the-money vanilla calls. A consideration of vegas for the more exotic double-barrier options revealed that the relative performances of the LR, MVD, and AMVD methods depend, in general, on the positions of the barriers and the way each method distributes densities relative to the barriers.

Applying the AMVD method requires the generation of nonstandard random variates,1 and for the examples considered in this paper, we provide a highly efficient implementation of the inverse transform method for this purpose using Newton’s method with a careful selection of the initial point. The time taken to generate each vega subpath for the AMVD method is approximately equal to the time taken to generate the corresponding MVD method using the efficient generation of the double-sided Boltzmann–Maxwell distribution introduced in Heidergott et al. (2008).

The remainder of the paper is structured as follows: the measure-valued differentiation method is briefly reviewed in Section 2, and the absolute measure-valued differentiation method is presented in Section 3. This is followed by Section 4 in which the notation and key identities for the remainder of the paper are presented. Sensitivities for vanilla calls and digital calls are computed analytically in Section 5 and Section 6, respectively. A more complex example of the double knock-out barrier option is investigated numerically in Section 7, and the paper concludes with Section 8.

In this section, we provide a heuristic overview of the MVD method and refer the interested reader to Heidergott et al. (2008) and Pflug and Thoma (2016) for the details.

Let X be an $\mathbb{R}$-valued random variable, and let $\varphi (x;\theta )$ be the probability density function of X with $\theta$ as one of its parameters. Then, given a payoff, $f:\mathbb{R}\to \mathbb{R}$, associated with a financial derivative on X, the likelihood ratio (LR) sensitivity with respect to the parameter $\theta$ is given by ignoring discounting. At this point, the likelihood ratio method rewrites the above expression as and computes the sensitivity as the expectation of the modified payoff obtained by scaling $f\left(x\right)$ by the score function $\partial ln\varphi (x;\theta )/\partial \theta$.

Now, suppose $\phi \left(x\right)=\partial \varphi (x;\theta )/\partial \theta$ is a signed measure on $\mathbb{R}$, and that it decomposes as the difference where ${\phi }_1$ and ${\phi }_2$ are proper measures on $\mathbb{R}$. Then, under the MVD method, the sensitivity with respect to $\theta$ is computed as the difference where $\mu \left({\phi }_i\right)={\int }_{\mathbb{R}}{\phi }_i\left(x\right)dx>0$. The two summands are computed separately, although they may be coupled to reduce the total variance. It should be noted that ${\overline{\phi }}_i\left(x\right)={\phi }_i\left(x\right)/\mu \left({\phi }_i\right)$, for $1\le i\le 2$, are probability measures on $\mathbb{R}$, and so the decomposition in (4) provides the basis for a numerical approximation of the sensitivity using, for example, a Monte Carlo simulation in which the integrals in (4) are computed numerically using random draws from the distributions determined by ${\overline{\phi }}_i\left(x\right)$.

The above description of the MVD method applies to path-independent European derivatives. For the more general case, suppose that a Monte Carlo simulation, with simulation times $0=t_0<t_1<\cdots <t_n=T$, is used to value a given derivative with expiry T. If the underlying process is assumed to be Markovian, then the probability density of a simulated path is given by the product where ${\varphi }_i(x_i\mid x_{i-1};\theta )$ is the transition density from the $t_{i-1}$ to $t_i$ conditional starting from $x_{i-1}$ at $t_{i-1}$. For notational convenience, write $\varphi (\mathit{x};\theta )=\varphi (x_0,x_1,\dots ,x_n;\theta )$, ${\varphi }_i(x_i;\theta )={\varphi }_i(x_i\mid x_{i-1};\theta )$, and define The likelihood ratio sensitivity of a general payoff, $f\left(\mathit{x}\right)=f(x_0,x_1,x_2,\dots ,x_n)$, with  respect to the parameter, $\theta$, is then given by Once again, if it is assumed that the derivatives are signed measures on $\mathbb{R}$, and decompose as differences where ${\phi }_{i,j}\left(x\right)$ are proper measures, then the MVD estimate of the sensitivity is given by where ${\partial }_{i,\theta }^{\mathrm{MVD}}\varphi (\mathit{x};\theta )$ is obtained from ${\partial }_{i,\theta }(\mathit{x};\theta )$ by substituting the derivative term, $\partial {\varphi }_i(x_i;\theta )/\partial \theta$, in  (6) with its decomposition where $\mu \left({\phi }_{i,j}\right)={\int }_{\mathbb{R}}{\phi }_{i,j}\left(x_i\right)d{x}_i$.

The implementation of the MVD method begins with the simulation, in the usual manner, of the underlying asset price path. Then, for each $1\le i\le n$, the i-th step is branched into two by drawing from distributions corresponding to the probability densities, ${\overline{\phi }}_{i,j}\left(x\right)={\phi }_{i,j}\left(x\right)/\mu \left({\phi }_{i,j}\right)$, for $1\le j\le 2$. The remaining steps in the two branched paths are then resimulated by drawing from the original distributions, ${\varphi }_j(x_j;\theta )$ for $i+1\le j\le n$. Thus, the full MVD method generates a set of $2n$ paths for each regular path and is consequently more computationally intensive than, for  example, the likelihood ratio method.2 In Pflug and Thoma (2016), less computationally demanding approximations to the full MVD method are proposed in which branching is performed only in a small subset of the simulation steps.

An important factor taken into consideration when deciding on the form of the decomposition of the transition density in (9) is the additional computation burden associated with the need to generate random variates from the component distributions. The decomposition chosen in Heidergott et al. (2008) and Pflug and Thoma (2016) is due, at least in part, to ensuring that one of the components coincides with the distribution already used in generating the underlying process. The remaining factor, however, will generally be from a different distribution.

The main motivation behind the absolute measure-valued differentiation (AMVD) method is improve the computational efficiency of the MVD method by requiring the introduction of only one additional distribution and branching off only one path at each simulation step, while retaining the key features of the MVD method, viz., not requiring the payoff derivative, and producing sensitivity estimates with low variances. Using the notation from the previous section, the AMVD estimate for the sensitivity of a payoff f with respect to the parameter $\theta$ is defined by    where $\epsilon \left({\phi }_i\left(x_i\right)\right)={\mathbf{1}}_{\{{\phi }_i\left(x_i\right)\ge 0\}}-{\mathbf{1}}_{\{{\phi }_i\left(x_i\right)<0\}}$, $\mu \left(|{\phi }_i|\right)={\int }_{\mathbb{R}}\left|{\phi }_i\left(x_i\right)\right|d{x}_i$, and  It follows from $\epsilon \left({\phi }_i\left(x_i\right)\right)\left|{\phi }_i\left(x_i\right)\right|={\phi }_i\left(x_i\right)$ that ${\Delta }_{\theta }^{\mathrm{AMVD}}\left(f\right)$ is an unbiased estimator for ${\Delta }_{\theta }\left(f\right)$. Moreover, although the AMVD method introduces what appears to be a problematic term, viz., $\epsilon \left({\phi }_i\left(x_i\right)\right)$, since $\epsilon \left({\phi }_i\left(x_i\right)\right)=\pm 1$, and these terms are squared when computing the associated variance, they do not adversely affect the AMVD variance.

In situations where it applies, the pathwise differentiation (PW) method usually gives sensitivity estimates with lower variances than other methods, and it was shown in Pflug and Thoma (2016) that the MVD sensitivities have lower variance than the corresponding finite difference estimates. In subsequent sections, we will compare the variances of sensitivity estimates computed using the LR, MVD, AMVD, and  PW methods. Although the analysis could be performed for any sensitivity and with any underlying model, we will focus on delta and vega under the Black and Scholes (1973) model. This allows us to obtain, analytically, the variances of deltas and vegas for vanillas and digitals, and avoid the consideration of more complex models that do not contribute, in any substantial way, to the analysis of the relative performance of the various approaches for computing sensitivities.

For vanilla options, it is possible to derive analytic expressions for the variances of deltas and vegas for the LR, MVD, AMVD, and PW methods. We only provide the details for European calls, since the corresponding calculations for puts are entirely similar. Throughout this section, let K be the strike for a European call with expiry T, and let $\varsigma =\sigma \sqrt{T}$.

In this section, we compute the LR, MVD, and AMVD variances for the digital call deltas and vegas. Once again, we only provide details for digital calls; let K be the strike for a digital call with expiry T.

In this section, we consider the delta and vega for a double-barrier option that knocks out if the closing price of the underlying asset lies outside the range, $]B_L,B_U[$, on at least three days prior to expiry, where $B_L<B_U$, and pays the constant rebate $R=100$ otherwise. We take $S_0=100$, $T=0.25$, $\sigma =0.20$, and subdivide the interval $[0,25]$ into 90 equal subintervals. This is the example considered in Chan and Joshi (2015) Subsection 5.3.