Chapter 14 (Capital management) says for the rating agency approach of assessing ongoing capital: "Solvency projections can be complicated by the need for nested stochastic simulations. These calculations can be simplified using closed-form solutions". 1. What is meant by nested stochastic simulations? 2. What is meant by closed-form solutions?
1. The way I think about nested stochastic simulations is in the context of projection of the Balance Sheet (particularly as part of an ORSA exercise): There is an “outer” scenario and an “inner” scenario. The “outer” scenario can be though of as a Real-World projection in which the balance sheet assets are being projected at the real-world rate. However, each Real-World “outer” scenario will have multiple Risk-Neutral “inner” scenarios, for eg: in order to price a liability guarantee. Take for example a company wishes to project it’s balance sheet over the next year and say it models in monthly time-steps. This means that there will be 12 months of projections to considers. Now, for each month, say the company plans to do 100 real-world simulations (over which it will take the average to arrive at a point estimate). These will be the outer scenarios. For each outer scenario (corresponding to a different real-world scenario) at each projection month, the liabilities and Solvency Capital will need to be computed as well. This in turn will require risk-neutral scenarios to compute embedded guarantees (for example). Assume that the company uses 1,000 real-neutral simulations to arrive at the market consistent (risk-neutral) price of the embedded guarantees to be used in the liability computations. These will represent the inner scenarios and are the “nested” stochastic simulations since they are within each “outer” stochastic simulations. In the hypothetical example above, just to project the Balance Sheet for 12 months, the company will need to consider: 12 x 100 x 1,000 = 1.2 million simulations! 2. Continuing with the example form above, in the end the company will need to perform 1.2 million simulations just to get the asset and liability figures for the next 12 months. This might be computationally very demanding. So, an alternative might be to use a short-cut to do away with the 1,000 inner scenarios for each real-world outer scenario. This would drastically reduce the number of simulations required to 12 x 100 = 1,200. This is where the “closed-form” solutions come into the picture. Instead of using simulation technique to value the embedded guarantee, the company may use a mathematical formula such as the Black-Scholes model to estimate the same. Note that this will only be an approximate answer since the Black-Scholes model is based on certain underlying assumptions (eg: the stock price has a constant volatility) which might be hold true in practice. A similar problem also arises when computing the risk margin. The SCR needs to be projected at each future time-step, which in turn requires “nested” stochastic simulations in order to be estimated at each time-step. Hope this helps! I am also studying this subject at the moment, so please let me know if you think I have gone awry anywhere here.
Thanks for your reply and example. I found a simpler explanation on the net which resonates with what you say. "Nested stochastic simulations – where one stochastic process is affected by other variables that also have stochastic processes, requiring simulations within simulations".
