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.
2 Answers
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.
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$\begingroup$ Ok I see your point but the FTAP says under the risk-neutral measure $Y_te^{-rt}$ is a martingale if $Y_t$ is the price process of a self-financing portfolio. Eventually, it is this relation that I use to price derivatives/contracts etc. But this cash flow could not possibly be a self-financing portfolio. I am having difficulties with convincing myself at this step. $\endgroup$– CalculonCommented Nov 27, 2015 at 12:49
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$\begingroup$ I think there is a confusion here. What would be $Y_t$ ? The discounted $t$ present value of a single cash flow $c 1_{\tau < T{$ is a martingale, hence it's time 0 present value is $E[e^{-rT} c 1_{\tau < T}]$, and the time 0 present value of your stream of cash flow is $\int_0^\infty E[e^{-rT} c 1_{\tau < T} dT] = E[\int_0^{\tau} e^{-rT} c dT]$ $\endgroup$ Commented Nov 30, 2015 at 12:31
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$\begingroup$ What I had in mind was that $Y_t$ would be the value of the self-financing portfolio that replicates the cash stream at time $t$. $\endgroup$– CalculonCommented Nov 30, 2015 at 13:36
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.
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$\begingroup$ "Pricing always takes place under the risk neutral probability measure". This statement sounds more like a rule of thumb than a mathematical fact.The professor who has given this assignment said the same thing by the way. Here is what I know:The market is free of arbitrage if and only if for a given numeraire (the money market account in our case) there exists an equivalent measure under which the relative prices of all self-financing assets are martingales.We can assume that the market is complete.Can you give an example of how this cash flow can be replicated using a self-financing strategy? $\endgroup$– CalculonCommented Nov 27, 2015 at 14:30
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$\begingroup$ Where in your quote does it say that the 'self financing assets' have to replicate? Existence of equivalent probability measures and completeness are two different things. There might be infinite measures that ensure arbitrage free pricing. If there exists only one then the market is complete and you can replicate. You cannot just 'assume' that the market is complete, you have to present us with the market instruments and show that it is complete. $\endgroup$– KiwiakosCommented Nov 27, 2015 at 19:31
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$\begingroup$ Under the stochastic volatility paradigm a market that consists of a risk free bank account and a stock is not complete. A market with an option added to it is. $\endgroup$– KiwiakosCommented Nov 27, 2015 at 19:32
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$\begingroup$ In the exercise we are using the Black-Scholes model in which the risky asset is paying dividends. This model is complete. If the random time in the question is a stopping time wrt the filtration wrt which the risky asset is adapted, then this random time can be written as the hitting time of the risky asset (this may not be exactly true). Would that be a way to replicate its payoff? $\endgroup$– CalculonCommented Nov 27, 2015 at 19:49
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$\begingroup$ I give up. Where in the question does it mention Black Scholes, dividends, hitting time etc? $\endgroup$– KiwiakosCommented Nov 27, 2015 at 20:22