Skip to content
All library documents

Risk-Neutral Probabilities in Option Hedging

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

Summary

The document uses a one-period stock example to explain why an option hedge cannot be evaluated by treating physical outcome probabilities as pricing probabilities. A call is paired with a short stock position, and the stated up and down probabilities make the proposed hedge appear to have an unexpected loss.

The answer identifies the mismatch: the quoted option value is inconsistent with the assumed market setup. Under the example’s no-discounting assumptions, the risk-neutral probabilities are equal across the two outcomes, giving a call value of five rather than eight. The short stock position is therefore not a riskless offset to a call priced using the physical 80/20 probabilities. This is a compact illustration of risk-neutral valuation, but the answer does not develop a general hedging derivation or discuss transaction costs, dividends, or incomplete markets.

Key ideas

  • Physical probabilities describe expected outcomes but do not directly determine arbitrage-free option prices.
  • Option valuation uses risk-neutral probabilities consistent with the traded underlying and financing assumptions.
  • A stock hedge that appears to have a residual loss may reflect an incorrectly valued option rather than a failed hedge.
  • In the example, equal risk-neutral probabilities imply a call value of five under no discounting.

Tags

OptionsEquitiesDerivatives pricingRisk management
Full text
# Hedging Options


# Hedging Options












Scenario: stock trading at 100 today, 80% chance it will trade at 110 tomorrow, 20% chance it will trade at 90 tomorrow

A new 100 strike call option on this stock is worth 8 today (assuming no discounting).

Assume you buy this call option. Now, you can hedge this option by selling 50 shares today, giving a payoff of -3 tomorrow. This is clearly a poor trade.

Logically, shouldn’t the payoff of the hedged position be zero in this scenario? What am I missing here?

## Answer by dm63 (score 2)

https://quant.stackexchange.com/a/36138

You are missing the fact that the option is worth 5 not 8. You cannot use the 80/20 probabilities to value the option, you have to use the risk/neutral probabilities , which are 50/50.

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.