Transformed Tail Quantiles

• Transformed tail quantiles are a framework that defines and applies local tail monotonic transformations to preserve quantile-based risk measures.
• The method extends classical quantile transformation theorems by incorporating threshold conditions to accurately compute risk measures in non-globally monotonic settings.
• Its applications in option pricing and insurance simplify extreme risk assessment and support effective strategies for regulatory capital reduction.

Transformed tail quantiles formalize and generalize the impact of applying (possibly only locally monotonic) transformations to the tails of a random variable, with particular emphasis on the preservation and computation of quantile-based risk measures in the presence of such transformations. This concept is fundamental in actuarial science and mathematical finance, where nonlinear transformations frequently arise in payoff functions, insurance contract valuations, and dependence modeling, particularly within the context of risk mitigation and regulatory capital determination.

1. Monotone Tail Functions: Definitions and Categorization

The framework begins by extending classical monotonicity to define monotone tail functions. A function $h$ possesses a non-decreasing (resp. non-increasing) upper tail if there exists a threshold $c$ such that $h(x)$ is bounded (e.g., by $h^* = \sup_{x < c} h(x)$) for $x < c$ and is non-decreasing (resp. non-increasing) for $x \ge c$. Formally, for a non-decreasing upper tail (Definition 2.1):

• For $x < c$, $h(x) \le h^*$.
• For $x_1 \le x_2$, $x_1, x_2 \ge c$, $h(x_1) \le h(x_2)$.

Analogous definitions characterize non-increasing upper tails, and similarly for lower tails by redefining $c$ as a supremum with reversed inequalities. This approach captures relevant structural properties in functions that are not globally monotone but exhibit tail monotonicity—a situation prevalent in payoff mappings arising in option strategies and insurance products.

2. Transformation Rule for Tail Quantiles

A principal contribution is the extension of the quantile transformation theorem to monotone tail functions. For a random variable $X$ with cumulative distribution function $F_X$ and a transformation $h$ (monotone on its upper tail beyond threshold $c$ with $h^* = \sup_{x < c} h(x)$), let $\pi_c = P[h(X) \le h^*]$. Then, for $p > \pi_c$,

$F_{h(X)}^{-1}(p) = h(F_X^{-1}(p)),$

where $F_{h(X)}^{-1}(p)$ denotes the $p$-th quantile of the transformed variable.

This result (Theorem 3.4) generalizes the classical case—in which $h$ is globally non-decreasing and continuous—by focusing the transformation to regions of the probability distribution beyond a specific threshold. The analogous formulations cover non-increasing tails and lower tails, with the left or right quantile function and the probability parameter adjusted accordingly (e.g., $1-p$ for lower tails).

3. Applications: Option Payoff Mappings and Insurance

Monotone tail functions naturally arise in the design and risk analysis of synthetic financial and insurance products:

• Option strategies: Payoff functions of options (straddle, strangle, backspread, etc.) are typically not monotone everywhere, but become monotone in the tail(s). For instance, the straddle payoff $h(x) = \max\{K - x, 0\} + \max\{x - K, 0\}$ is non-decreasing in the upper tail. Accordingly, VaR or TVaR of the strategy’s terminal value can be computed by applying the transformation rule to the tail quantile of the underlying.
• Insurance contracts: Endowment assurance contracts combine survival and death benefits, with their present value function formed by distinct non-decreasing and non-increasing segments. The tail quantiles of the present value can be directly computed using the transformed quantile result, which simplifies the evaluation of extreme risk measures for mixed cash flow streams.

The key advantage is a simplification in the risk assessment workflow: tail-based risk measures (VaR, TVaR) for such products can be deduced from the quantiles of the underlying stochastic variable by simple function application, provided the tail monotonicity and threshold probability conditions are met.

4. Relationship to Dependence Structures: Quadrant Perfect Dependence

Transformed tail quantiles interconnect with dependence modeling via the notion of quadrant perfect dependence, an extension of comonotonicity. For two random variables $X_1$ and $X_2$, the difference of their (possibly generalized) quantile functions $$h_a(p) = F_{X_1}^{-1,\alpha}(p) - F_{X_2}^{-1,\alpha}(p)$$ is central.

Upper–upper comonotonicity is shown to be equivalent to the function $h_a$ possessing a non-decreasing upper tail; specifically, the vector $(X_2, X_1 - X_2)$ is upper–upper comonotonic with threshold $a = (F_{X_2}^{-1,\alpha}(\pi), h_a(\pi))$ if and only if $h_a$ is non-decreasing beyond $\pi$. This equivalence provides a precise structural condition under which the joint distribution in one quadrant exhibits maximal co-movement in the tails, a property exploited in capital allocation and hedging.

5. Implications for Risk Reduction Strategies

A core motivation for transformed tail quantiles is risk mitigation. Consider a liability $R_1$ and a candidate hedging asset $R_2$, with the total position $Z=R_1-R_2$. When $R_1$ and $R_2$ are comonotonic, their difference’s distribution can be represented as $Z = h(U)$, where $U \sim \text{Uniform}(0,1)$ and $h(p) = F_{R_1}^{-1}(p) - F_{R_2}^{-1}(p)$.

Under the condition that $h$ has a non-decreasing upper tail (with appropriate threshold), regulatory VaR capital admits a direct decomposition: for $p$ in the relevant tail region,

$\text{VaR}_p(Z) = h(F_X^{-1}(p)).$

A sufficient condition for the risk reducer to admit regulatory capital reduction is

$\rho[R_2] \leq \text{VaR}_p[R_2],$

ensuring the "premium" for the counter-balancing asset does not exceed its own tail risk at the critical quantile. Similar (but sign-reversed) conditions govern the case where $h$ has a non-increasing upper tail.

6. Generalizations and Technical Framework

The formalism accommodates both left- and right-continuity in the monotone tail function, as well as discontinuity at the threshold. The technical apparatus is compatible with both classical and generalized quantile functions (left, right, or median), and links directly to the risk theory literature—particularly extending the results of Dhaene et al. (2002a) to the setting of tail monotonicity.

Summary of Monotone Tail Quantile Transformation:

Type	Quantile Formula	Tail Region
Non-decreasing	$F_{h(X)}^{-1}(p) = h(F_X^{-1}(p))$	$p > \pi_c$
Non-increasing	$F_{h(X)}^{-1}(p) = h(F_X^{-1}(p))$	$p > \pi_c$
Lower tail types	Formulas with $F_X^{-1,+}(p)$, $1-p$	$p < \pi$

Threshold $\pi$ is determined by the probability mass below the tail monotonicity threshold in $h$.

7. Significance and Prospects

This theory clarifies when and how quantile-based risk measures for complex cash flows and payoffs can be efficiently calculated without recourse to global monotonicity. By extending quantile transformation theorems to the context of monotone tail functions, the results facilitate accurate and computationally efficient evaluation of extreme risk—critical for regulatory compliance (e.g., Solvency II, Basel III) and for quantitative risk management in high-stakes financial and actuarial domains. Additionally, the connection to quadrant perfect dependence strengthens the theoretical underpinnings of multivariate risk reduction and capital allocation. The conditions derived provide practitioners with explicit, practically checkable criteria for verifying risk mitigation effectiveness through hedging or portfolio construction in settings characterized by non-globally monotonic but tail-monotonic payoff structures (Hanbali et al., 18 Aug 2025).