Pricing Cash Flows That End at a Random Time

# Value of a continuous cash flow until a random time


# Value of a continuous cash flow until a random time












I am trying to compute the present value of a continuous cash flow that lasts until a random time. The rate of the cash flow is denoted by $c$ and the random time is denoted by $\tau$. Then my claim is that the present value of this cash flow is given by $$V_0 = E\left[\int_0^{\tau}P(t)c\,dt\right]$$ where $P(t)$ is the discount factor for time $t$. I am pretty certain that the expectation needs to be computed under the real-world measure. But the problem is that I know the distribution of $\tau$ only under the risk-neutral measure and not under the real-world measure. I do not know the market price of risk either. Can someone confirm whether my approach is correct and if not, point out the error(s) in it? It is of course possible, though not very likely, that there is an error in this exercise.

## Answer by Antoine Conze (score 3)

https://quant.stackexchange.com/a/21955

As long as your market is complete and $\tau$ is measurable w.r.t. the filtration generated by the market the continuous cash flow paid until $\tau$ is a hedgeable contingent claim and you have to work under the risk neutral measure.

## Answer by Kiwiakos (score 2)

https://quant.stackexchange.com/a/21957

Pricing always takes place under the risk neutral probability measure. In fact, this would make the price more conservative (i.e. lower) with respect to risk; if you priced it under the true measure you would be putting a smaller hazard rate for this random time.

Completeness make the risk neutral probability measure unique. In your case you might have infinite admissible risk neutral probability measures since jumps might not be hedgeable. You need to choose one of them.

However, you say that you are given one of them for the distribution of $\tau$. Who has given it to you? Did you calibrate it on instruments that depend on $\tau$? Then these instruments might complete the market, and the jump might be hedgeable. Then this is the measure you want to use.