Expressing Asian and Lookback Call Payoffs with Returns

Summary

The document asks how to rewrite path-dependent call option payoffs using returns instead of asset prices. It starts from the relation between each observation price and its cumulative return factor: the price at a given time equals the initial price multiplied by that factor. Substituting this relation into the arithmetic-average payoff gives an Asian call expressed as the average of return-scaled prices minus the strike. Applying the same substitution to the maximum observed price expresses a lookback call payoff using the maximum cumulative return factor.

These transformations are algebraic and assume the return factors are defined consistently across observation times. The initial asset price remains a multiplier in both formulas, while the strike remains in price units; it is not itself converted to a return. The source poses the derivation as a question and provides no answer, numerical example, valuation method, or discussion of sampling frequency and contract variations.

Key ideas

• Each observed price can be written as the initial price times its cumulative return factor.
• Substitution converts the Asian call’s average-price payoff into a sum of return-scaled prices.
• The lookback call’s maximum-price payoff uses the maximum cumulative return factor.
• The strike remains in price units, so the initial price multiplier must be retained.

Tags

# Prices and returns


# Prices and returns












I want to convert the payoff of an Asian and a lookback Call option with prices in their corresponding with returns. Example: for an European Call $\varphi(S_T)=(S_T-K)^+$, so knowing that $S_T=S_0(1+r_t)^T=S_0\prod_{i=1}^{T}(1+r_i)=S_0R_T$ I can write that $(S_T-K)^+=(S_0R_T-K)^+$.

If an Asian Call payoff is $\varphi(S_T)=(\frac{1}{T}\sum_{t=1}^{T}S_t-K)^+$ and a lookback Call payoff is $\varphi(S_T)=(S_{\operatorname{max}}-K)^+$, how do I obtain respectively

- $(\frac{1}{T}\sum_{t=1}^{T}S_t-K)^+=(\frac{S_0}{T}\sum_{t=1}^{T}R_t-K)^+$,

- $(S_{\operatorname{max}}-K)^+=(S_0R_{\operatorname{max}}-K)^+$ for $R_{\operatorname{max}}:=\underset{1 \leq t \leq T}{\operatorname{max}}\begin{Bmatrix} R_t \end{Bmatrix}$?

Thanks in advance.

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