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Arrow Security Pricing and the Stochastic Discount Factor

Article Quant Q&A · Author: Predictor
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Summary

The document asks how to price an Arrow security that pays one unit if a specified event occurs at a future date. It frames the question through the stochastic discount factor: the current value is expressed as the expected product of that factor and the future payoff indicator. A sure payoff, such as a zero-coupon bond, is raised as a simpler case to probe why an expectation represents price, including in a one-period binomial setting.

The response gives a high-level connection to the von Neumann–Morgenstern utility theorem, which represents rational choices under uncertainty as maximizing expected utility. It does not work through the pricing equation, specify market assumptions, or derive how the stochastic discount factor is determined. Thus it offers an intuition linked to decision theory, but leaves the risk adjustment and conditions for converting expected discounted payoffs into market prices unexplained.

Key ideas

  • An Arrow security pays one unit when a specified future event occurs.
  • The document presents its price as an expectation involving a stochastic discount factor and the payoff indicator.
  • A sure future payoff is used to question why expected discounted value represents price.
  • The response invokes expected utility, but does not derive the pricing relation or its assumptions.

Tags

Derivatives pricingStatisticsFixed income
Full text
# Fundamentals of pricing theory, Arrow security pricing


# Fundamentals of pricing theory, Arrow security pricing












I aim to make the derivative's pricing theory abloslutely clear for myself, starting from the beginnigs. Before turning to risk-neutral measure and Radon-Nikodym change-of-measure, I'd like to price an Arrow security.

Suppose there is some market event $A_T$, and we bet on it: someone suggests us to pay one unit, if $A_T$ would be the case at the future time moment $T$ (e.g., some stock hits the barrier, or someone defaults etc). So, $X_T=\mathbb{I}_{A_T}$. What is the price $X_t$, at $t<T$?

Financial economics texts say that (given some probabilistic model) $X_t=\mathbb{E}[\tilde{m}_tX_T]=\mathbb{E}[\tilde{m}_t\mathbb{I}_A]$, where $\tilde{m}_t$ is a stochastic discount factor.

Why expectation? Let us even simplify it to $X_T=1$, the case of a zero-coupon bond. Why its price should be the expectation $\mathbb{E}[\tilde{m}_t]$ of something? I can even reduce it to a single-step binomial model, but cannot easily accept the expectation, which is the weighted sum.

So, why expectation is the price? And what is the price, finally?

## Answer by Daneel Olivaw (score 1, accepted)

https://quant.stackexchange.com/a/81411

Roughly speaking, because of the Von Neumann–Morgenstern utility theorem:

> the VNM utility theorem demonstrates that rational choice under uncertainty involves making decisions that take the form of maximizing the expected value of some cardinal utility function.

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.