Skip to content
All library documents

Risk-Neutral Discounting and Replication in Option Pricing

Article Quant Q&A · Author: Idonknow
Read the original

Summary

The document asks why an option’s initial value can be defined as the discounted expected payoff under a risk-neutral probability measure. The quoted textbook passage frames this as a pricing rule in which discounted asset values have expectations equal to their current values. The response connects the rule to replication: in standard option-pricing theory, an option is treated as redundant when its payoff can be recreated with underlying assets and cash, allowing its price to be inferred from the replicating portfolio.

This explanation sketches the no-arbitrage logic behind risk-neutral valuation, but it does not show the replication construction or derive the pricing equation. Its broad assertion that options carry no risk should be read in the idealized model setting: real trading prices can differ from theoretical values, and practical hedging may be imperfect. The document offers a conceptual pointer rather than a worked pricing example.

Key ideas

  • Risk-neutral valuation prices an option by discounting its expected payoff under the pricing measure.
  • The explanation links this method to replicating an option with underlying assets and cash.
  • Replication and no-arbitrage assumptions are the conceptual basis for the pricing argument.
  • The document does not provide a derivation or a worked example.

Tags

OptionsDerivatives pricingRisk management
Full text
# Why Joshi defined option value to be discounted payoff using risk neutral expectation?


# Why Joshi defined option value to be discounted payoff using risk neutral expectation?












Currently I am reading Mark Joshi's The Concepts and Practice of Mathematical Finance.

At page $59,$ the author mentioned the following.

> Instead of requiring that every portfolio should have expectation equal to today's value, we require that its expectation should be equal to the asset's value invested at the risk-free growth rate, or equivalently that its discounted expectation is equal to today's value. We thus want $$\mathbb{E}_{RN}\left( \frac{A_{\Delta T}}{B_{\Delta T}} \right) = \left( \frac{A_0}{B_0} \right)$$ for every asset where $$\mathbb{E}_{RN}$$ is an expectation with the risk-neutral probability $p$. This equation is trivially satisfied for the bond and we have the chosen the risk-neutral probability so that it is satisfied by construction for the stock. This leaves us with the option we wish to price. We define Opt$_0$ to satisfy equation above: $$Opt_0 = \mathbb{E}_{RN}\left( \frac{A_{\Delta T}}{B_{\Delta T}} \right) = e^{-r\Delta t} \mathbb{E}_{RN}(f(S))$$ where $f$ is the option's payoff.

My question is the following:

> Question: Why and how can Joshi define option value at time zero to be the discounted risk neutral expectation?

## Answer by Dhruv Mahajan (score 3)

https://quant.stackexchange.com/a/48998

Most of the Options pricing literature (except dynamic programming and reinforcement learning) is based on the assumption that options themselves are redundant. They can be artificially recreated with stocks and cash, hence options themselves carry no risk and every price is computed as "fair price" in a risk neutral fashion, needless to say options mostly don't trade at fair prices.

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.