Description: This paper presents a new methodology for obtaining fast algorithms and closed form solutions for pricing callable bonds in continuous time models. Results are directly applicable to regulatory scenario testing, immunization and market value accounting.

Description: Insurance companies utilize derivatives in a variety of ways to manage and mitigate risks that are inherent in their investment or liability portfolios. They can be characterized by three main features: medium long-term duration; large volumes; and significant market risk exposure. Given the persistent low interest rate environment across the curve since the 2008 financial crisis, large companies need to hedge their liabilities, as illustrated by the significant recent increase from $786 billion as of fiscal year 2010 to $1,885 billion as of FY 2014 by insurance companies.As the Variable Annuities (VA) guarantees embedded within those liabilities hold a strong convex risk profile with respect to the underlying stock, VA carriers need to buy some convex hedge assets such as options to match the liability risk profile to improve hedge effectiveness, implying the growing cost of hedging. Such market impact of the quantity on the cost of hedging is further magnified in the recent period with the short-lived, but sharp transitions from low volatility to high volatility, when market volatility suddenly increases, even when there were no well-known fundamental catalysts. This partially stems from the growing managed volatility funds embedded within VA ($bn 400).VA carriers willing to trade a large quantity of options will see such market impact as an important dilemma, as the cost of placing one large order to close position will be far greater than the sum of infinitely small orders differed in time. In that context, managing market impact requires introducing an explicit market impact function, the influence of which the agent will try to minimize. The effect on hedge asset prices significantly depends on whether there is or not significant delta hedging activity. In case of no significant delta hedging activity, the risk appetite has a significant influence on the optimal execution strategy. If there is significant delta hedging activity, the optimal trading involve