Bounding a Digital Call from a Traded Call Price

Summary

The document asks how to find the supremum of a digital option’s value when a call option price is known. It gives a call price of 5 at strike 100 and asks about a digital option with strike 105. The proposed setup equates the call price to the Black–Scholes call formula, then seeks the largest value of a shifted normal cumulative probability involving volatility and time to expiry.

This frames the problem as an options-pricing bound under market data, but it does not provide a derivation, an answer, or assumptions about the underlying price, interest rate, maturity, or volatility. Those missing inputs and modeling choices limit what can be concluded from the question alone. The document is useful as a prompt about relating vanilla option prices to digital option values, rather than as a complete pricing method.

Key ideas

• A call price at one strike can constrain the value of a digital option at another strike.
• The proposed approach expresses the call price using the Black–Scholes formula.
• The question seeks a supremum of a shifted normal probability term.
• No derivation or numerical bound is supplied.

Tags

# How to find the upper bound of a digital option given some market data?


# How to find the upper bound of a digital option given some market data?












Given the price of a call equals to 5 with Strike 100, please find the upper bound (sup) of the digital option with strike 105.

I am not sure about the solution, but I write the condition like this,

$S\mathcal{N}(d_1)-Ke^{-rT}\mathcal{N}(d_2) = 5$

what's the $\sup{N(d_1+\frac{\ln(\frac{100}{105})}{\sigma\sqrt{T}})}$?

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