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Optimal investment and pricing in the presence of defaults
Mathematical Finance ( IF 1.6 ) Pub Date : 2019-07-12 , DOI: 10.1111/mafi.12219
Tetsuya Ishikawa 1 , Scott Robertson 2
Affiliation  

We consider the optimal investment problem with random endowment in the presence of defaults. For an investor with constant absolute risk aversion, we identify the certainty equivalent, and compute prices for defaultable bonds and dynamic protection against default. This latter price is interpreted as the premium for a contingent credit default swap, and connects our work with earlier articles, where the investor is protected upon default. We consider a multiple risky asset model with a single default time, at which point each of the assets may jump in price. Investment opportunities are driven by a diffusion X taking values in an arbitrary region urn:x-wiley:09601627:media:mafi12219:mafi12219-math-0001. We allow for stochastic volatility, correlation, and recovery; unbounded random endowments; and postdefault trading. We identify the certainty equivalent with a semilinear parabolic partial differential equation with quadratic growth in both function and gradient. Under minimal integrability assumptions, we show that the certainty equivalent is a classical solution. Numerical examples highlight the relationship between the factor process, market dynamics, utility‐based prices, and default insurance premium. In particular, we show that the holder of a defaultable bond has a strong incentive to short the underlying stock, even for very low default intensities.

中文翻译:

存在违约时的最优投资和定价

我们考虑存在违约时具有随机end赋的最优投资问题。对于具有恒定绝对风险规避的投资者,我们确定确定性等值,并计算可违约债券的价格和动态保护以防止违约。后一个价格被解释为或有信用违约掉期的溢价,并将我们的工作与之前的文章联系在一起,后者在默认情况下保护了投资者。我们考虑具有单个默认时间的多重风险资产模型,此时每个资产的价格可能会上涨。投资机会由在任意区域取值的扩散X驱动缸:x-wiley:09601627:media:mafi12219:mafi12219-math-0001。我们考虑到随机波动性,相关性和恢复性;无限制的随机end赋;和违约后交易。我们用函数和梯度均具有二次增长的半线性抛物型偏微分方程识别确定性等价物。在最小可积性假设下,我们证明确定性等价是经典解决方案。数值示例突出了要素过程,市场动态,基于公用事业的价格和违约保险费之间的关系。尤其是,我们表明,即使对于非常低的违约强度,可违约债券的持有人也有强烈的动机卖空基础股票。
更新日期:2019-07-12
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