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What is a convex risk measure?

What is a convex risk measure?

Trading and Investing

Review of Statistical Hedging: Motivating the Use of Convex Risk Measures for Hedging Portfolios of Derivatives

Hans Buehler

Let’s take a look at one of our Top 10 Quants On Wall Street 2023, Hans Buehler’s work on extending the Markowitz-type “mean-variance” portfolio optimization approach to derivative portfolios over one time step, incorporating general transaction costs.

Moreover, Buehler argues for the adoption of “cash-invariant monotone hulls” as proposed by Filipovic and Kupper to create effective convex risk measures.

This graph depicts the function ρ(X)=max⁡(0,X2)\rho(X) = \max(0, X^2)ρ(X)=max(0,X2), which shows how the risk measure increases with the portfolio value XXX.

Emphasizing the use of “Quadratic CVaR” as a superior convex risk measure compared to traditional Greek-based approaches for managing derivative portfolios.

Buehler addresses the challenge of hedging a derivative portfolio using various hedging instruments under real-world conditions, such as transaction costs and liquidity constraints, over a short period (e.g., a week). This theoretical foundation supports the more practical approach discussed in Buehler’s “Deep Hedging” work, which applies over multiple hedging steps.

Key Concepts

  1. Convex Risk Measures: Buehler advocates for convex risk measures, highlighting their numerical tractability and practical applicability. In addition, contrasts this with traditional methods that rely heavily on Greeks. Suggesting that models should become seen as tools for generating mark-to-model values. Rather than directly tied to market dynamics.
  2. Cash-Invariant Monotone Hulls: Additionally, using these hulls to construct risk measures, ensuring a more robust and realistic assessment of risk in derivative portfolios. Cash-Invariant Monotone Hulls are risk measures that remain unaffected by cash additions, ensuring that the risk assessment of a portfolio does not change with the inclusion of cash. These measures also maintain monotonicity, meaning that increasing the portfolio’s value will not decrease its assessed risk.
  3. Quadratic CVaR: Favored for its ability to handle downside risk effectively without relying on Greek calculations! Quadratic Conditional Value at Risk (CVaR) is a risk measure that extends the traditional CVaR by incorporating a quadratic penalty on negative outcomes.

Practical Implications?

Buehler’s approach departs from classic derivative literature by:

  • Avoiding the use of Greeks.
  • Treating pricing models as tools for producing model values, agnostic to their alignment with real-world market measures.
  • Allowing parts of the portfolio to be written off to manage risk effectively.

This framework aims to align more closely with real-life trading environments in investment banks, where internal models might not reflect actual market dynamics precisely but must fall within reasonable market bid-ask spreads.

Buehler concludes that the proposed method provides a more practical and theoretically sound approach to hedging derivative portfolios.

By using convex risk measures and cash-invariant monotone hulls, traders can achieve more effective risk management without the limitations of traditional Greek-based methods. Furthermore, this paper sets the groundwork for more sophisticated multi-step hedging strategies explored in subsequent work on “Deep Hedging.”

Lastly, read the full work: Statistical Hedging by Hans Buehler :: SSRN

What is a convex risk measure?