Valuing a Poisson-Arrival Machine with Geometric Brownian Motion Output

Summary

The document poses a risk-neutral valuation problem for a machine that produces an item worth a stochastic amount at random arrival times. The item’s value follows geometric Brownian motion, while arrivals follow a Poisson process with fixed intensity. Given the current item value and the risk-free rate, the task is to express the machine’s current value as a function of the model parameters.

The questioner identifies the first arrival time’s exponential density and attempts a recursive discounted-value equation that accounts for both the arriving item and the machine’s continuation value. They are unsure how to handle the dependence implied by observing a GBM at a random time, despite a hint that the process is compound Poisson. No answer or derivation is included, so the document does not supply a valuation formula or establish the conditions under which a finite value exists. It serves as a setup for analyzing discounted expected cash flows under stochastic output and random arrival timing.

Key ideas

• The machine produces an item at each arrival of a Poisson process with fixed intensity.
• The value of each produced item evolves according to geometric Brownian motion.
• Under risk neutrality, valuation involves discounting expected output and continuation value.
• The document proposes conditioning on the first arrival time but leaves the equation unsolved.
• No final formula or finiteness conditions are provided.

Tags

# How to compute this current value using no arbitrage condition?


# How to compute this current value using no arbitrage condition?












Suppose $X_t$ is a geometric Brownian motion with drift $\mu$ and volatility $\sigma$. $X_0$ is known. You have a machine that produces something worth $X_t$ at random times $t$ generated by a Poisson process with fixed intensity $\lambda$. Suppose you are risk-neutral, the risk-free rate is $r$, and markets are perfectly competitive. Compute the current value of this machine (as a function of $X_0, \mu, \sigma, \lambda$ and $r$).

The professor gives a hint that this is a compound Poisson process. However, since the jump process is a GBM, not iid, I was still confused about how to tackle this problem.

I have some starting points, but I am not sure if these make sense. Denote the first arrival as $t_1$. By the property of the Poisson process, the density of $t_1$ is $$ f_{t_1}(t)=\lambda e^{-\lambda t}, \forall t>0 $$ Let $V(x)$ denote the value of the machine conditioning on the GBM is currently at level $x$. Therefore, $$ \int_{0}^{\infty} e^{-rt} \cdot (E[X_t]+E[V(X_t)]) \cdot \lambda e^{-\lambda t} dt=V(X_0) $$ But I have no idea about how to solve this equation.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.