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The concept of the stochastic discount factor (SDF) is used in financial economics and mathematical finance. The name derives from the price of an asset being computable by "discounting" the future cash flow x ~ i {\displaystyle {\tilde {x}}_{i}} by the stochastic factor m ~ {\displaystyle {\tilde {m}}} , and then taking the expectation. [ 1 ]
Hansen–Jagannathan bound is a theorem in financial economics that says that the ratio of the standard deviation of a stochastic discount factor to its mean exceeds the Sharpe ratio attained by any portfolio. This result applies, among others, the Cauchy–Schwarz inequality. The Hansen-Jagannathan (H-J) bound is a type of mean-variance frontier.
Let [2] = (())be the bank account or money market account numeraire and = / = (())be the discount factor in the market at time 0 for maturity T.If is the risk neutral measure, then the forward measure is defined via the Radon–Nikodym derivative given by
Markov decision process (MDP), also called a stochastic dynamic program or stochastic control problem, is a model for sequential decision making when outcomes are uncertain. [ 1 ] Originating from operations research in the 1950s, [ 2 ] [ 3 ] MDPs have since gained recognition in a variety of fields, including ecology , economics , healthcare ...
Using the related stochastic discount factor - also called the pricing kernel - the asset price is computed by "discounting" the future cash flow by the stochastic factor ~, and then taking the expectation; [16] the third equation above.
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where is the discount factor corresponding to the risk-free rate to the final maturity date T years into the future. Now suppose the integral is hard to compute. We can approximate the integral by generating sample paths and then taking an average. Suppose we generate N samples then
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