State Variables for Pricing an Asian Lookback Option

Summary

The document poses a pricing problem for a path-dependent option on a geometric Brownian motion asset. Its European payoff depends on the running maximum and a time-averaged quantity formed from the integral of the squared asset price. The question asks how to express the option value as a deterministic function of current state variables and how to justify a proposed scaling representation.

The text does not provide a solution, derivation, numerical results, or evidence for the suggested form. It also contains a mismatch between the squared-price integral in the stated payoff and the later reference to an integral of the asset price, which would need resolving before formulating a pricing equation. The useful content is therefore the identification of the state-variable and homogeneity questions that arise when pricing this kind of path-dependent derivative, rather than a validated pricing method.

Key ideas

• The payoff depends on both the running maximum and an integral over the asset price path.
• Pricing requires a state representation that captures the relevant path history.
• The question considers whether scale invariance can reduce the number of state variables.
• The integral in the payoff differs from the integral later named as a state variable, so the specification needs clarification.

Tags

# 60466


# Pricing of strange Asian lookback option with European-style payoff $\max\{ \max_{u\in[0,T]}S_u-\frac1T\sqrt{\int_0^TS_t^2\mathrm{d}t},0\}$












I am trying to price the Asian lookback option at time $t$ with time-$T$ (European) payoff $\max\{M_T-A_T,0\}$, where $$M_t=\max_{u\in[0,t]}S_u,\quad A_t=\frac1t\sqrt{\int_0^tS_u^2\mathrm{d}u},$$ and $S_u$ is the price of an asset following GBM.

How does one price such a formula? What can we do to show that $V_t(t, s, m, i)$ is a deterministic function of $M_t$ and $I_t=\int_0^tS_u\mathrm{d}u$? Is it because we expect $V_t$ to follow some form of Black-Scholes-like equation due to the European-style payoff, and how can I prove it? Also, I am told that $V_t(t, s, m, i)=sf(t,\frac{m}s,\frac{s}i)$, where should I begin to show this?

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