Interfaces with Other DisciplinesModel risk in the over-the-counter market
Introduction
The use of models in financial markets is accompanied by their own risk, which leads to various attempts to identify and measure model risk. The academic research on model risk in continuous-time finance dates back to Derman (1996). Measuring model risk is an essential concern in the industry as well (Basel Committee on Banking Supervision, 2009, European Banking Authority, Federal Reserve Board of Governors).
The focus of this paper is the measurement of model risk for pricing models in the over-the-counter (OTC) market, where market prices are not available. Specifically, we consider exotic option pricing models, under both the physical and risk-neutral probability measures. Based on Bayesian methods, we investigate the parameter estimation risk (PER) and model specification risk (MSpR) of models used to price exotic options; and, by taking PER into consideration, we study the model selection risk (MSeR) of a model class for both long and short positions in the OTC market; in an empirical analysis, we apply our proposed methodology for variance swap (VS) rates and forward starting (FS) option prices to estimate the model risk of affine jump-diffusion (AJD) models and Lévy jump models.
Many previous studies differentiate between PER and MSpR as essential components of model risk. For example, Green & Figlewski (1999) find that option writers are exposed to considerable model risk due to imperfect model specification and inaccurate parameter estimation. However, only a handful of studies attempt to measure PER and MSpR. Kerkhof, Melenberg, & Schumacher (2010) propose a worst-case type risk measure over a set of models. They compute PER based on the confidence interval of parameters and quantify MSpR as compared to a reference model in the model set. In addition, Lazar, Qi, & Tunaru (2020) evaluate PER and MSpR of continuous-time finance models with expected shortfall (ES) type risk measures based on Bayesian estimators.
Moreover, the majority of studies use point-wise estimation methods. However, such an approach would ignore PER and underestimate model risk. Compared with the asymptotic distribution based measure of PER in Kerkhof et al. (2010), a Bayesian estimation approach is a more reliable way to measure PER. Jacquier & Jarrow (2000) study PER based on Bayesian estimation methods for the Black-Scholes (BS) model. Jacquier, Polson, & Rossi (2002) further apply Bayesian estimators for stochastic volatility models and find that the Bayesian approach outperforms the moments and likelihood estimators for pricing. Johannes & Polson (2010) point out that the marginal posterior distribution obtained via the Bayesian approach can quantify PER, also see Chung, Hui, & Li (2013) and Tunaru (2015). Tunaru & Zheng (2017) apply Bayesian methods to study the PER of the BS and Merton jump-diffusion (MJD) models. Most recently, Leung, Li, Pantelous, & Vigne (2021) find that it is essential to consider parameter uncertainty for risk management; and they suggest using a Bayesian estimation method for this. We follow Lazar et al. (2020) and compute PER in a Bayesian framework, using an ES-type measure based on the estimated price distribution; such a measure would highlight the risk that market players are exposed to due to the risk of inaccurate parameter estimation.
“All models are wrong, but some are useful” (Box, Draper et al., 1987), and no models will fit market prices perfectly. Chen & Hong (2011) emphasize the importance of model specification. MSpR is regarded as the main source of model risk (Derman, 1996). Lazar et al. (2020) propose a measure of MSpR for liquid products; however, this approach requires market prices, which are generally not available in the OTC market. As such, the framework put forward in this paper relaxes the need for option prices because it only requires payoffs which are easily calculated, so it works for thinly traded exotic options in the OTC market. OTC options are the result of a private transaction between the buyer and the seller and deals are not always done at the best price. Moreoever, the OTC options have no secondary market where investors buy and sell products they already own. Typically option traders in the OTC market hold options until maturity, and holders have to enter into separate transactions to offset losses or leverage gains. Compared with exchange-traded options traders, OTC options investors are typically “buy and hold investing” (Deuskar, Gupta, & Subrahmanyam, 2011), since holders realize their profits or losses at expiry. Fabozzi (1997) also state that OTC options investors intend to hold options to expiration because of the illiquidity of the OTC market. Given the reasons above, it is valid to use realized payoffs to compute MSpR in the OTC market.
Our first contribution is that we propose an approach to assess the PER and MSpR of models used to price exotic options in the OTC market in a Bayesian framework. To the best of our knowledge, this is the first study to study MSpR in the OTC market, and we approximate MSpR of models using realized payoffs of derivatives. The measurement also enables us to investigate the PER for long and short positions. Additionally, the time evolution of PER and MSpR of models can be obtained; which can reflect the models’ ability to capture the market dynamics.
Our second contribution concerns MSeR of a model class. The MSeR has been often defined in terms of price differences within a set of models and measured using the worst-case approach (Barrieu, Scandolo, 2015, Cont, 2006, Coqueret, Tavin, 2016). However, Turner (2015) states that model uncertainty should also be considered when selecting models. In line with this, we take PER into consideration when assessing MSeR. As such, we propose a framework that computes time-varying MSeR using information on PER, which allows for differentiation between MSeR for long and short positions. A detailed discussion of MSeR can be found in Section 2.
Our third contribution is that we investigate the model risk in estimating fair variance strikes of variance swaps and forward starting option prices. These two types of exotic options have the advantage that closed-form pricing formulae are available for the continuous-time finance models we investigate in this paper. We find that modeling jumps, especially for variance jumps, reduces the MSpR in estimating fair variance strikes of variance swaps significantly. On the contrary, the distribution of jumps affects the performance of models in estimating forward starting option prices differently; and the log-stable jumps are preferred. Moreover, we show that the size of the MSeR is considerable.
The paper is organized as follows: Section 2 proposes the model risk measurement framework; Section 3 introduces the models and the related derivative pricing methods; the empirical study is presented in Section 4; and the last section concludes.
Section snippets
Model risk measures
This section introduces the model risk measurement framework for products in the OTC market. We consider model risk from two aspects: (1) individual model risk of a single pricing model, which can be decomposed into PER and MSpR, and here our methodology extends the framework of Lazar et al. (2020) using Bayesian inference; (2) the model risk associated with a class of models, named MSeR, building on the works of Cont (2006), Barrieu & Scandolo (2015) and Coqueret & Tavin (2016).
Before
Models and derivatives pricing
This section describes the set of models used in this paper under both probability measures, the physical measure and the risk-neutral measure with the detailed estimation methods of the models considered described in the Supplementary Appendix. The change of measure is also discussed in this section. Moreover, the closed-form pricing formulae for variance swaps and forward starting options are presented in Sections 3.3 and 3.4, respectively.
Empirical analysis
We explore the main aspects of model risk for VS’s and FS options. For VS’s, we study the model risk in computing fair variance strikes associated with various pricing model specifications, and find that the model risk shows large variations depending on market conditions. Given the estimated fair variance strikes, one important extension we consider is the investigation of how model risk affects the significance of VRP. We then turn our focus to the model risk in pricing FS options.
Conclusions
In this paper, we propose a model risk measure based on realized payoffs that is able to quantify PER and MSpR of pricing models, respectively, for options in the OTC market, where the market prices of products are not available, and option holders tend to follow the “buy and hold” strategy. Our model risk measurement also considers MSeR of a model class for long and short positions. We then apply this measurement to AJD models and Lévy jump models to investigate the model risk when estimating
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