Scenario analysis for derivative portfolios via dynamic factor models

Abstract

A classic approach to financial risk management is the use of scenario analysis to stress test portfolios. In the case of an S&P 500 options portfolio, for example, a scenario analysis might report a P&L of $-\mathrm{\$}1$m in the event the S&P 500 falls 5% and its implied volatility surface increases by 3 percentage points. But how accurate is this reported value of $-\mathrm{\$}1$m? Such a number is typically computed under the (implicit) assumption that all other risk factors are set to zero. But this assumption is generally not justified as it ignores the often substantial statistical dependence among the risk factors. In particular, the expected values of the non-stressed factors conditional on the values of the stressed factors are generally non-zero. Moreover, even if the non-stressed factors were set to their conditional expected values rather than zero, the reported P&L might still be inaccurate due to convexity effects, particularly in the case of derivatives portfolios. A further weakness of this standard approach to scenario analysis is that the reported P&L numbers are generally not back-tested so their accuracy is not subjected to any statistical tests. There are many reasons for this but perhaps the main one is that scenario analysis for derivatives portfolios is typically conducted without having a probabilistic model for the underlying dynamics of the risk factors under the physical measure P. In this paper we address these weaknesses by embedding the scenario analysis within a dynamic factor model for the underlying risk factors. Such an approach typically requires multivariate state-space models that can model the real-world behavior of financial markets where risk factors are often latent, and that are sufficiently tractable so that we can compute (or simulate from) the conditional distribution of unstressed risk factors. We demonstrate how this can be done for observable as well as latent risk factors in examples drawn from options and fixed income markets. We show how the two forms of scenario analysis can lead to dramatically different results particularly in the case of portfolios that have been designed to be neutral to a subset of the risk factors.

Keywords:

• Risk management
• Filtering
• Scenario analysis
• Multi-factor models

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Martin B. Haugh http://orcid.org/0000-0002-0823-1044

Notes

1 One of the advantages of using simulation is that we can easily estimate other risk measures besides the expected P&L in a given scenario. For example we could estimate the P&L's standard deviation or VaR conditional on the scenario.

2 We are assuming that the main risks in the portfolio are underlying and volatility risks. If for example, the portfolio was exposed to substantial dividend or interest rate risk, which is quite possible in an S&P options portfolio, then additional risk factors for these risks should be included.

3 We would also have $\mathrm{V}\mathrm{a}\mathrm{r}\left({\mathit{ϵ}}_{1,t}\right)=0$ since this would be an instance where a component of ${\mathbf{\Delta }\mathbf{\text{x}}}_t$ coincides with one of the c.r.f. returns.

4 One notable exception is Rebonato (2019).

5 As an alternative to (6(6) $\underset{{\mathbf{\text{f}}}_k\in {\mathbb{R}}^m}{max}{\mathbb{P}}_{ϵ}(\mathbf{\Delta }{\mathbf{\text{x}}}_{k-1}-\mathbf{\text{B}}{\mathbf{\text{f}}}_k),$(6) ) we could obtain the point estimate ${\widehat{\mathbf{\text{f}}}}_t$ by solving a cross-sectional regression problem.

6 We also note that the conditional distribution of ${\mathit{ϵ}}_{t+1}\mid ({\mathcal{F}}_t,{\mathbf{\text{f}}}_{\mathbf{\text{s}},t+1}=\mathbf{\text{c}})$ (where ${\mathit{ϵ}}_{t+1}$ is given in (2(2) ${\mathbf{\Delta }\mathbf{\text{x}}}_t=\mathbf{\text{B}}{\mathbf{\text{f}}}_{t+1}+{\mathit{ϵ}}_{t+1},t=0,1,\dots$(2) )) is equal to its unconditional distribution since ${\mathit{ϵ}}_{t+1}$ is independent of ${\mathcal{F}}_t$ and ${\mathbf{\text{f}}}_{t+1}$ by assumption.

7 Suppose for example that $\mathrm{\Delta }{V}_t(\cdot )$ is a convex function. Then Jensen's inequality implies $\mathrm{\Delta }{V}_t^{\mathrm{a}\mathrm{l}\mathrm{t}}\left(\mathbf{\text{c}}\right)=\mathrm{\Delta }{V}_t\left(\mathbf{\text{B}}{\mathit{\mu }}_t^c\right)=\mathrm{\Delta }{V}_t\left({\mathbb{E}}_t\right[\mathbf{\text{B}}{\mathbf{\text{f}}}_{t+1}+{\mathit{ϵ}}_{t+1}\mid {\mathbf{\text{f}}}_{\mathbf{\text{s}},t+1}=\mathbf{\text{c}}\left]\right)\le {\mathbb{E}}_t\left[\mathrm{\Delta }{V}_t\right(\mathbf{\text{B}}{\mathbf{\text{f}}}_{t+1}+{\mathit{ϵ}}_{t+1})\mid {\mathbf{\text{f}}}_{\mathbf{\text{s}},t+1}=\mathbf{\text{c}}]=\mathrm{\Delta }{V}_t^{\mathrm{d}\mathrm{f}\mathrm{m}}\left(\mathbf{\text{c}}\right)$. In this case $\mathrm{\Delta }{V}_t^{\mathrm{a}\mathrm{l}\mathrm{t}}\left(\mathbf{\text{c}}\right)$ would underestimate the estimated scenario P&L when $\mathrm{\Delta }{V}_t(\cdot )$ is convex. Similarly $\mathrm{\Delta }{V}_t^{\mathrm{a}\mathrm{l}\mathrm{t}}\left(\mathbf{\text{c}}\right)$ would overestimate the estimated scenario P&L when $\mathrm{\Delta }{V}_t(\cdot )$ is concave.

8 The assumption that some of the c.r.f. returns might display moderate dependence is not a strong assumption since even uncorrelated c.r.f. returns can display moderate dependence. Suppose for example that the c.r.f. returns have a joint multivariate t distribution with ν degrees-of-freedom. These factor returns can be uncorrelated and yet still have extreme tail dependence (McNeil et al. 2015). As a result the distribution of these factors conditional on an extreme scenario can display strong dependence.

9 We assume the risk-free rate of interest and dividend yield remain constant throughout and therefore do not model risk factors associated with them. This is typical for an equity options setting unless the p.m. wishes to trade with a specific view on dividends. We also acknowledge that in practice one trades futures on the S&P 500 index rather than the index itself. Given the assumption of a constant risk-free rate and dividend yield, there is essentially no difference in assuming we can trade the index itself, however, and so we will make that assumption here.

10 GARCH models are a well-established class of time-series models that can model empirically observed behavior such as volatility clustering in equity markets. McNeil et al. (2015) and Tsay (2010) can be consulted for a more detailed exposition as well as related references.

11 We acknowledge that the absence of arbitrage imposes restrictions on the magnitude of permissible c.r.f. stresses. For example, Rogers and Tehranchi (2010) have shown that the implied volatility surface cannot move in parallel without introducing arbitrage opportunities. Indeed it is well known that moves in the implied volatilities are more likely to follow a ‘square-root-of-time’ rule and we will model this below with our first latent c.r.f. For another example, it is also well-known that the volatility skew at any fixed maturity cannot become too steep without introducing arbitrage. We don't explicitly rule out scenarios that allow for arbitrage but note that such scenarios would have to be very extreme indeed. Moreover, it is easy to check a given scenario for arbitrage (by solving a small LP) and so ruling out such scenarios would be very straightforward. This would be more expensive computationally if we were to test every Monte-Carlo sample for arbitrage but if this was considered desirable then variance reduction techniques could be employed to speed up the calculations.

12 An implied volatility skew is the cross-section of the implied volatility surface that we obtain when we hold the time-to-maturity fixed. There is therefore a different skew for each time-to-maturity. There are various skew models in the literature and we refer the interested reader to the work of Derman and Miller (2016) who describe some of these models.

13 Roughly speaking, they build an implied volatility surface based on each day's closing prices (of the S&P 500 and its traded options) and then use this surface to read off volatilities for the various delta-maturity combinations.

14 Standard approaches to fitting GARCH models are Maximum Likelihood Estimation (MLE) and Quasi-MLE methods. For details of these approaches, refer to McNeil et al. (2015).

15 Recall that $z_t$ has unit standard deviation and so a fall of $-2$ in $z_t$ implies via (17(17) $r_t={\sigma }_t{z}_t,t=1,2,\dots$(17) ) an S&P 500 log-return of $-2{\sigma }_t$.

16 Indirect evidence is obtained via the various statistical tests that any multivariate dynamic model can be subjected to. A sample of such tests is described in section 7.

17 In Appendix 3 we discuss the issue of inferring the realization of latent c.r.f. returns in the context of a historical back-test and why that results in a bias that overstates the accuracy of DFMSA. Our results in Appendix 3 suggest the bias may be quite small, however.

18 We can back-out the realization of the market c.r.f. return, defined as the innovation term in the GARCH model (17(17) $r_t={\sigma }_t{z}_t,t=1,2,\dots$(17) ), by taking the realized S&P 500 log-return and dividing it by the ${\sigma }_t$ computed from the estimated GARCH model via (18(18) ${\sigma }_t^2={\alpha }_0+{\alpha }_1{r}_{t-1}^2+{\text{β}}_1{\sigma }_{t-1}^2,t=1,2,\dots$(18) ).

19 For a proof of this statement see Lemma 9.5 in McNeil et al. (2015), for example.

20 This is because if scenarios include the stressing of latent c.r.f. returns, then it will be necessary to estimate the realized values of these c.r.f. returns which could potentially introduce significant bias into the statistical testing of the scenario VaR exceptions. This is discussed further in Appendix 3.

21 We have in mind here the distinction between statistical and practical significance. Given that all models are ‘wrong’, it's inevitable that even good models will eventually fail tests of statistical significance given sufficient data. It is important then to consider issues of practical significance when accepting or rejecting models.

22 The complete model parameters are available upon request.

23 Diebold and Li (2006) chose a value of $\lambda =0.7308$ for the U.S. Treasury yield curve.

24 It's worth emphasizing that our back-tests are not at all concerned with why the p.m. has this particular view or whether or not it is ever justified. The view is simply used to construct a portfolio to which we then apply SSA and DFMSA.