Capturing parameter risk with convex risk measures

Abstract

Adequately specifying the parameters of a financial or actuarial model is challenging. In case of historical estimation, uncertainty arises through the estimator’s volatility and possible bias. In case of market implied parameters, the solution of a calibration to market data might not be unique or the numerical routine returns a local instead of a global minimum. This paper provides a new method based on convex risk measures to quantify parameter risk and to translate it into prices, extending results in Cont (Math Finance 16(3):519–547, 2006) and Lindström (Adv Decision Sci, 2010). We introduce the notion of risk-capturing functionals and prices, provided a distribution on the parameter (or model) set is available, and present explicit examples where the Average-Value-at-Risk and the entropic risk measure are used. For some classes of risk-capturing functionals, the risk-captured price preserves weak convergence of the distributions. In particular, the risk-captured price generated by the distributions of a consistent sequence of estimators converges to the true price. For asymptotically normally distributed estimators we provide large sample approximations for risk-captured prices. Following Bion-Nadal (J Math Econ 45(11):738–750, 2009); Carr et al. (J Financ Econ 62:131–167, 2001); Cherny and Madan (Int J Theor Appl Finance 13(8):1149–1177, 2010); Xu (Ann Finance 2:51–71, 2006), we interpret the risk-captured price as an ask price, reflecting aversion towards parameter risk. To acknowledge parameter risk in case of calibration to market prices, we create a parameter distribution from the pricing error function, allowing us to compare the intrinsic parameter risk of the stochastic volatility models of Heston and Barndorff-Nielsen and Shephard as well as the Variance Gamma option pricing model by pricing different exotics.

Appendix

Proof (of Proposition 1) Let $X\in\mathcal{C}^b(\Theta)$ and $\alpha\in(0,1]$ be arbitrary, abbreviate ${f(\theta):=\mathbb{E}_\theta[X]. }$ A property of weak convergence is that it transfers to pushforward measures of continuous functions (cf. Bartoszynski [7]), especially $F_{f,R_N}(x)\to F_{f,R_0}(x), N\to\infty,$ with $F_{f,S}(x) := S(f\leq x)$ denoting the S-distribution function of f. Since the quantile function is the quasi-inverse of the distribution function, it follows $q_f^{R_N}(\beta)\to q_f^{R_0}(\beta)$ Lebesgue-a.e. on (0,1). Hence, by dominated convergence,

$$ \begin{aligned} R_N\ast\hbox{AVaR}_\alpha(X) &= \frac{1}{\alpha}\int\limits_0^\alpha \hbox{VaR}_\beta^{R_N}({\mathbb{E}}_\cdot [X]) \mathrm{d} \beta = \frac{1}{\alpha}\int\limits_0^\alpha {\it q}_{{\mathbb{E}}_\cdot[{\it X}]}^{{\it R_N}}(1-\beta) \mathrm{d} \beta \\ &\xrightarrow{N\to\infty} \frac{1}{\alpha} \int\limits_0^\alpha {\it q}_{{\mathbb{E}}_\cdot[{\it X}]}^{{\it R}_0}(1-\beta) \mathrm{d} \beta={\it R}_0\ast\hbox{AVaR}_\alpha({\it X}). \end{aligned} $$

Proof (of Corollary 1) In Acerbi [1] it is pointed out that every spectral risk measure can be represented by a Borel measure $\mu$ on [0,1] with zero mass in 0 such that

$$\rho_\phi(X) = \int\limits_0^1 \hbox{AVaR}_\alpha(X) \mu(\mathrm{d}\alpha),$$

so the AVaR w.r.t. different security level $\alpha\in (0,1)$ are the “building blocks” of spectral risk measures. Hence, it follows by Proposition 1 and dominated convergence

$$ \begin{aligned} \Gamma_\phi^{R_N}(X) &= \int\limits_0^1 R_N\ast\hbox{AVaR}_\alpha(X) \mu(\mathrm{d}\alpha) \\ &\xrightarrow{N\to\infty} \int\limits_0^1 R_0\ast\hbox{AVaR}_\alpha({\it X}) \mu(\mathrm{d}\alpha) = \Gamma_\phi^{{\it R}_0}({\it X}). \end{aligned} $$

Proof (of Proposition 2) Let $\lambda\in(0,\infty)$ be arbitrary but fix. Since ${f(\theta):=\mathbb{E}_\theta[X]}$ is assumed to be continuous and bounded, $u_\lambda \circ f$ is continuous and bounded as well for $u_\lambda(x) := \exp (\lambda x).$ Since the expectation of $u_\lambda\circ f$ w.r.t. the measure $R^N$ is exactly the $\hbox{AVaR}_1$ of $u_\lambda\circ f,$ we obtain by Proposition 1 the convergence

$$ \int\limits_\Theta {\it u}_\lambda({\mathbb{E}}_\theta[{\it X}]) {\it R_N}(\mathrm{d} \theta)\xrightarrow{N\to\infty} \int\limits_\Theta {\it u}_\lambda({\mathbb{E}}_\theta[{\it X}]) {\it R}_0(\mathrm{d} \theta) $$

and since $u_\lambda^{-1}(y) = \lambda^{-1}\log(y)$ is continuous, also

$$ u_\lambda^{-1}\left(\int\limits_\Theta u_\lambda({\mathbb{E}}_\theta[X]) R_N(\mathrm{d} \theta)\right) = \Gamma^{{\rm ent},N}_\lambda(X)\xrightarrow{N\to\infty} \Gamma^{{\rm ent},0}_\lambda(X) $$

follows.