Digital Option Prices Under Normal and Lognormal Spot Models

Summary

The document raises an interview question about how a digital, or binary, option is priced when the spot price is modeled with a normal distribution versus a lognormal distribution. It suggests that the lognormal model’s heavier right tail might make the digital option cheaper, reasoning that a binary payoff does not grow with very large spot prices. The post does not provide a derivation, numerical comparison, or answer, so this proposed explanation remains unverified.

For a cash-or-nothing call with the same strike and cash payoff, the price depends on the risk-neutral probability that spot finishes above the strike, adjusted for discounting. Comparing the models therefore requires specifying their parameters and matching conventions; a heavier right tail alone does not determine which model assigns a higher probability above a particular strike. The document is a useful prompt about distributional assumptions and digital payoffs, but it does not establish a pricing conclusion.

Key ideas

• A digital option pays a fixed amount when its terminal condition is met.
• Its price is related to the risk-neutral probability of finishing in the money.
• A heavier right tail alone does not determine the comparison at a given strike.
• The post poses a pricing question but supplies no derivation or evidence.

Tags

# Difference of digital (binary) option price under normal and log-normal assumption


# Difference of digital (binary) option price under normal and log-normal assumption












Here is the question asked in interview：

> What's is the difference of digital (binary) option price under normal and log-normal assumptions of spot?

I think the log normal has fatter right tail than normal. However digital option is not sensitive to the large spot, therefore the price of log normal will be smaller than normal.

I don't know if I got to the point, but I hope you can add something.

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