Consider the continuously sampled Asian option of Subsection...

Consider the continuously sampled Asian option of Subsection 7.5.3, but assume now that the interest rate is $r=0$. Find an initial capital $X(0)$ and a nonrandom function $\gamma(t)$ to replace (7.5.22) so that $$ X(T)=\frac{1}{c} \int_{T-c}^T S(u) d u-K $$ still holds. Give the formula for the resulting process $X(t), 0 \leq t \leq T$, to replace (7.5.24) and (7.5.26). With this function $\gamma(t)$ and process $X(t)$, Theorem 7.5.3 still holds.

Consider the continuously sampled Asian option of Subsection 7.5.3, but assume now that the interest rate is $r=0$. Find an initial capital $X(0)$ and a nonrandom function $\gamma(t)$ to replace (7.5.22) so that
$$
X(T)=\frac{1}{c} \int_{T-c}^T S(u) d u-K
$$
still holds. Give the formula for the resulting process $X(t), 0 \leq t \leq T$, to replace (7.5.24) and (7.5.26). With this function $\gamma(t)$ and process $X(t)$, Theorem 7.5.3 still holds.

Key Concepts

Asian Option

An Asian option is a type of derivative where the payoff is based on the average price of an underlying asset over a certain time period rather than its price at a single point. This averaging mechanism can reduce volatility and make the option less prone to market manipulation, which distinguishes it from standard European or American options.

Continuous Sampling

Continuous sampling refers to the method where the average underlying asset price is calculated as an integral over time. This technique captures all the price fluctuations continuously, leading to a precise mathematical description of the payoff. It contrasts with discrete sampling, where averages are computed at specific intervals.

Replicating Portfolio and Hedging

The construction of a replicating portfolio, or hedging strategy, aims to dynamically mirror the payoff of a derivative by holding a specific amount of the underlying asset and possibly cash or a risk-free asset. The use of a nonrandom (deterministic) function, such as gamma(t), in this context determines the quantity of the underlying held at each time, ensuring that the portfolio's terminal value matches the derivative’s payoff.

Zero Interest Rate Environment

When the interest rate is set to zero, the discount factor used in pricing models becomes one, meaning that future cash flows are not discounted. This simplification affects the formulation of replication strategies and the calculation of the initial capital since there is no time value adjustment to account for, focusing solely on matching the payoff through dynamic trading.

Initial Capital Determination

Determining the initial capital involves calculating the amount required to set up the replicating portfolio so that it will exactly match the derivative's payoff at maturity. In the absence of interest, this capital represents the direct cost of constructing the strategy, aligning the portfolio dynamics with the continuous sampling mechanism used in the Asian option.

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