A Model for Drawdown Insurance
Many individual and institutional investors are wary of large market drawdowns as they not only lead to portfolio losses and liquidity shocks but also indicate potential imminent recessions.
As is well known, fund managers are typically compensated based on the fund’s outperformance over the last record maximum, or the high-water mark. As such, drawdown events can directly affect the manager’s income.
Also, a major drawdown may also trigger a surge in fund redemption by investors, and lead to the manager’s job termination. Hence, fund managers have a strong incentive to seek insurance against drawdowns.
These market phenomena have motivated the application of drawdowns as path-dependent risk measures.
On the other hand, existing market-traded contracts, such as vanilla and even lookback puts, are ineffective for insuring market drawdowns. These observations suggest that drawdown protection can be useful for both institutional and individual investors, and there is an interest in synthesizing drawdown insurance.
We consider an insurance contract based on a drawdown event. Specifically, the protection buyer who seeks insurance on a drawdown event of size k will pay a constant premium payment p continuously over time until the drawdown time τ. In return, the protection buyer will receive the insured amount α at time τ.
In this journal article, we have studied the practicality of insuring against market crashes and proposed a number of tractable ways to value drawdown protection.
Under the geometric Brownian motion dynamics, we have derived the formulas for the fair premium for a number of insurance contracts and examined its behavior with respect to key model parameters (see figures above).
We’ve also introduced a cancellable drawdown insurance contract, whereby the protection buyer would monitor the drawdown process and optimally stop the premium payment as the drawdown risk diminished.
The impact of other risk factors, such as default risk, is also examined and incorporated into drawdown insurance.
For future research, the valuation and optimal stopping problems herein can be studied under other price dynamics, especially when drawdown formulas are available.
Although we have focused our analysis on drawdown insurance written on a single underlying asset, it is both interesting and challenging to model drawdowns across multiple financial markets and investigate the systemic impact of drawdowns. This would involve modeling the interactions among various financial markets and developing new measures of systemic risk.
Lastly, the idea of market drawdown and the associated mathematical tools can also be useful in other areas, such as portfolio optimization problems, risk management, and signal detection.
For educational purposes only. Not investment advice.
References
H. Zhang, T. Leung, and O. Hadjiliadis, Stochastic Modeling and Fair Valuation of Drawdown Insurance [pdf], Insurance: Mathematics & Economics, 53(3), pp.840–850
T. Leung and H. Zhang, Optimal Trading with a Trailing Stop, Applied Math & Optimization [pdf]
H. Zhang (2019), Stochastic Drawdowns, Modern Trends in Financial Engineering: Volume 2, World Scientific
