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Suppose we have a stochastic cashflow $X_t$ from a portfolio of contracts with clients. We can simulate from $X_t$ and can calculate $E[X_t], \forall t \in [1,n]$ where $n$ represents the longest maturity of all contracts in the portfolio. The cashflow variability arises from the unexpected behavior of clients since clients have embedded options in the contract to quit the contract early et cetera. Suppose we would like to hedge the variability of such cashflows using a swap contract. We might enter into an OTC swap contract if we find a counterparty that takes on the deviation of the cashflow with the expected cashflow, let's say one leg paying $X_t-E[X_t]$ and the other leg pays $E[X_t] + \pi$, where $\pi$ is the premium paid for the contract. How would one price such a contract, i.e. determine $\pi$ ? Note that we can assume we can simulate from $X_t$.

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One example of this situation might be pricing mortgages where we know the expected prepayment rate, but it could vary either to the high side or the low side, and there is no way to hedge the residual risk using other financial instruments. Essentially we are asking how to price unhedgable risk, so I would think that you have to look at the riskiness of the flow, as measured to first order by the standard deviation $E[(X_t-E[X_t])^2]$, Denoting this by $\sigma$, one could propose that $\pi=k\sigma$ for some value of $k$, where $k$ is a measure of risk aversion.

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  • $\begingroup$ Indeed, I am thinking about bringing utility functions in the picture and trying to solve it for certain types of utility functions.. thanks.. $\endgroup$ Dec 23, 2021 at 19:00

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