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A Butterfly Spread’s Price as a Discounted Expected Payoff

Article Quant Q&A · Author: Ryan J. Shrott
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Summary

The question asks why a butterfly spread’s value might be expressed using the probability that the terminal asset price falls between the outer strikes, multiplied by half the spacing between adjacent strikes. The answer frames pricing under the risk-neutral measure as the discounted expected payoff: integrate the butterfly’s payoff across terminal prices and weight it by their probability density.

The probability-times-half-spacing shortcut follows only under an additional approximation: if the terminal-price density is roughly constant between the outer strikes, the conditional average payoff over that interval is half the strike spacing, since the butterfly payoff forms a triangle. The document provides this assumption but does not establish that it holds for a particular asset or volatility model. The shortcut is therefore not a general pricing identity; accurate valuation needs the risk-neutral density and the payoff’s shape across the interval.

Key ideas

  • Risk-neutral option value is the discounted expected payoff.
  • A butterfly payoff is triangular between its outer strikes and zero outside them.
  • With equally spaced strikes, its peak payoff equals the spacing between adjacent strikes.
  • The probability-times-half-spacing shortcut assumes a constant terminal-price density across the interval.
  • Without that approximation, price by integrating payoff against the risk-neutral density.

Tags

OptionsDerivatives pricingVolatilityStatistics
Full text
# Butterfly spread model price


# Butterfly spread model price












Consider a butterfly spread with strikes $K_1, K_2, K_3$. My professor wrote the model price, $V$, was equal to the following: $$V = exp(-rT) * P(K_1<S_T<K_3) * (1/2) \Delta K$$

where $\Delta K = K_2-K_1 = K_3 - K_2$. I asked after class why this was true. He said it was obvious and that its just the probability times the area of the spread or something like that. I understand that he is discounting the expected payoff of the option. The option only has value when $S_T$ is between $K_1$ and $K_3$, but why multiply by $0.5\Delta k$. What am I not seeing? Can someone provide a rigorous proof with more steps?

## Answer by Ryan J. Shrott (score 0, accepted)

https://quant.stackexchange.com/a/25462

Under the risk neutral measure, the expected present value of the butterfly payoff is: $$V_0 = e^{-rT} * \int_{S_T=K_1}^{K_3}P(T,S_T)f_{S_T}dS_T$$

And if we assume that $f_{S_T}$ is constant from $K_1$ to $K_3$, then:

$$V_0 = e^{-rT} * \dfrac{1}{\Delta K} \int_{S_T=K_1}^{K_3}P(T,S_T)dS_T = e^{-rT} *\dfrac{\delta^2}{\Delta K} $$

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.