Relating Call Prices to Digital and Squared-Call Payoffs

Summary

The document presents an exercise involving prices of calls across strikes and asks how to derive prices for two other terminal-payoff claims: a digital that pays when the underlying exceeds a strike, and a squared call payoff. It defines the call value as the expected positive part of terminal price minus strike, then proposes estimating the digital from the difference between call values at adjacent strikes. It next attempts a finite-difference relation for the squared payoff and encounters a boundary problem when trying to calculate values recursively.

The material is useful as a setup for thinking about how option prices across strikes encode other payoff shapes. However, it supplies no answer or derivation, so the proposed discrete identities are not verified and the missing initial value is unresolved. The notation also leaves the expectation convention and strike spacing unclear. A reader should treat the formulas as the questioner's attempts rather than established pricing results; practical numerical estimates would depend on the available strike grid and assumptions used to interpolate between strikes.

Key ideas

• A cross-section of call prices can be used to investigate prices of other terminal payoffs.
• The document proposes an adjacent-strike difference to estimate a digital payoff.
• It also proposes a finite-difference relation for a squared call payoff but cannot initialize the recursion.
• No response validates either identity or explains how to handle missing boundary values.
• Strike spacing and interpolation assumptions matter when turning discrete prices into estimates.

Tags

# Pricing Exotic options


# Pricing Exotic options












I am stuck at a assignment problem where I have to compute the price of an exotic option.

I am given the values the prices of option $C(X;k) = E[max(0,X_T - k)]$ for different strike prices $k$ and I have to compute the exotic price

$D(X;k) = E[I(X_T > k)]$ for the same set of $k$s.

I did this using difference of sum and got

$D(X;k) = C(X;k) - C(X;k+1)$

Now using these I have to compute another exotic price:

$P(X;k) = E[max(X_T - k,0)^2]$

I tried using the same method to get :

$P(X;k) - P(X;k-1) = 2C(X;k) - D(X;k+1)$

but to numerically calculate $P(X;k)$ at some k, I need the value of $P(X;k+1)$ and I have no initial value for the pricing $P$.

Any help would be great

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