Valuing a Perpetual Portfolio with Poisson Arrivals of Shares

Summary

The document presents a valuation question for a perpetual portfolio that receives additional shares according to a Poisson process, while the share price follows geometric Brownian motion. The arrivals are assumed independent of the share price. A textbook solution represents portfolio value as a function of the current share price and gives a partial differential equation with a share-arrival term, yielding a value proportional to the current price when the risk-free rate exceeds the price drift.

The questioner challenges the use of the physical drift in the pricing equation, expecting the risk-free rate, and asks whether the portfolio should instead have infinite value. No answer resolving these concerns is included. The material is useful for highlighting the distinction between pricing assumptions and expected growth, as well as the need to state discounting and measure assumptions clearly. It provides a proposed equation and solution but no derivation or supporting evidence, so the valuation should not be treated as established without further analysis.

Key ideas

• The model combines Poisson arrivals of shares with a geometrically diffusing share price.
• The proposed valuation equation includes a term for the value of newly arriving shares.
• The question raises whether the price drift or risk-free rate belongs in the valuation equation.
• The stated solution depends on the drift being below the discount rate, but the document does not resolve its assumptions.

Tags

# Portfolio with Poisson Stock Units


# Portfolio with Poisson Stock Units












I am struggling with an exercise from some quantitative finance material I am studying. The question is as follows:

> What is the fair value of a portfolio that perpetually contains ${N_t}$ units of an underlying asset ${S_t}$, where ${N_t}$ is a Poisson process with intensity ${\lambda}$ and ${S_t}$ follows a Geometric Brownian Motion with drift ${\mu}$? Assume the arrival of additional shares is independent of the share price

The textbook offers the following solution in terms of solving a Partial Differential Equation:

> Since the machine is perpetual and the number of shares are independent of the share price, the value of the portfolio can be written as a function of ${S_t}$ (the expected number of shares will be captured via the mean of the Poisson process). Let ${V_t}$ be the value of the portfolio at time t. Then the Black-Scholes PDE is as follows: ${\frac{\partial V}{\partial S} \mu S_t+\frac{1}{2}\sigma^2S_t^2 \frac{\partial^2 V}{\partial S^2}+\lambda S_t-rV = 0}$ A solution to this PDE is ${V_0=\frac{\lambda}{r- \mu}S_0}$.

I am confused by the Black Scholes PDE they setup because it uses ${\frac{\partial V}{\partial S} \mu S_t}$ where I would have expected ${\frac{\partial V}{\partial S}r S_t}$, in which case the solution to the PDE they provided would not work. Has anyone seen Black Scholes used in this way before?

My other thought is why wouldn't the price of this thing be infinite? Given their solution, it seems that the drift would have to be less than the risk-free rate, but this would imply a negative sharpe ratio for the underlying stock.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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