Lookback Option Boundary Conditions at the Running Maximum

Summary

The document introduces a lookback option whose value depends on the underlying asset price and its running maximum. It defines the underlying as a diffusion process and presents the running maximum as the limit of an integral-based approximation using increasingly high powers of the asset price. The option value is written as a function of the current price, time, and running maximum.

It then states a boundary condition for the pricing partial differential equation: at the point where the asset price equals its running maximum, the derivative of option value with respect to that maximum is set to zero. The question asks whether a textbook explanation of this condition is reasonable, but the document supplies no explanation or response. Thus it identifies a mathematical boundary condition and the proposed rationale of eliminating the running-maximum increment, without deriving the PDE, specifying the option payoff, or resolving the conceptual question. It is useful as a starting point for studying path-dependent option pricing, but is not a complete derivation.

Key ideas

• A lookback option’s value depends on the asset price and its running maximum.
• The running maximum is represented as a limit of an integral-based power approximation.
• The stated PDE boundary condition sets sensitivity to the running maximum to zero when price equals that maximum.
• The document links the condition to removing the running-maximum increment but gives no derivation.
• The question about the reasonableness of a textbook explanation remains unanswered in the source.

Tags

# Boundary condition of lookback option


# Boundary condition of lookback option












This is a well know conclusion of the `boundary condition` of `lookback option`. Here $$\dfrac{d S_t}{S_t} = (\mu - D)dt + \sigma dW_t$$ is `underlying asset`. $$M_t^{(n)} =\left[\int^t_0S^n_u d u\right]^{\dfrac{1}{n}}$$ $$M_t = \max\limits_{t_0\leq u \leq t}S_u$$ is `maximal process` of $S_t,$ and we have the conclusion: $$\lim\limits_{n\rightarrow\infty}M_t^{(n)} = M_t$$ $V(S_t,t,M_t)$ is the price of `lookback option` at time $t.$

One of the `boundary condition` of `PDE` for $V$ is $$\dfrac{\partial V}{\partial M_t}\big|_{S_t = M_t} = 0$$ this is to make the $d M_t$ term zero.

But, some books explain this condition as following way, do you think it is reasonable?

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