Replicating a Continuously Averaged-Price Derivative at Zero Interest

Summary

The document poses a portfolio-replication problem for a derivative whose terminal payoff is the time average of an underlying asset's price over the contract period, less a fixed strike. It assumes continuously sampled prices and a zero interest rate, then asks for the initial capital and a deterministic trading function specifying how many shares to hold over time so that the portfolio matches the payoff at maturity.

It also asks for the value of the replicating portfolio at intermediate times as a function of the underlying price and strike. No solution or derivation is included, so the document does not supply a hedging formula or evidence that replication is feasible under particular market assumptions. The problem is useful as an exercise in dynamic replication of an arithmetic-average claim, but solving it requires additional reasoning about the evolving average and the self-financing portfolio strategy.

Key ideas

• The derivative pays the continuous time average of the underlying price over the contract period minus a strike.
• The stated setup assumes zero interest and asks for a self-financing replication strategy.
• The requested hedge specifies a deterministic number of underlying shares to hold through time.
• The problem also asks for the replicating portfolio's value before maturity.
• No solution, derivation, or feasibility assumptions beyond those in the prompt are provided.

Tags

# Give the formula for following resulting portfolio process


# Give the formula for following resulting portfolio process












Consider the continuously sampled a derivative security with payoff function $V(T) = \frac {\int_0^TS(u)du}T -K$ but assume now that the interest rate is $r=0$. Find an initial capital $X(0)$ and a nonrandom function $\gamma(t),0\leq t \leq T$ , which will be the number of shares of risky asset held by our portfolio so that $X(T) = V(T) $ still holds. Give the formula for resulting process $X(t) $ in term of underlying asset price and K. I don't know how to start these type of questions. I started by finding $V(0)$ but couldn't do much afterwards. Can someone solve it.

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