Forward Price with Continuous Dividend Yield and Storage Cost

Scenario

Consider an underlying asset trading at spot price $S_0$. The market provides a continuous risk-free borrowing and lending rate $r$. The asset pays a continuous dividend yield $q$ and incurs a continuous proportional storage (or holding) cost $u$.

Your Task

1. By constructing two replicating portfolios at $t=0$, prove via no-arbitrage that the theoretical forward price for delivery at time $T$ must be: $F_0 = S_0 e^{(r - q + u)T}$
2. Describe the cash-and-carry arbitrage executed if the forward market price satisfies $F_{\text{market}} > S_0 e^{(r - q + u)T}$.
3. Compute the forward price and arbitrage profit per unit of asset when S_0 = \50.00$,$r = 4%$,$q = 1%$,$u = 2%$,$T = 0.5$years, and the traded forward is$F_{\text{market}} = $52.00$.

Constraints

• Continuous compounding of rate, dividend yield, and storage cost.
• Frictionless short sales.

Examples With net cost of carry $c = r - q + u = 0.04 - 0.01 + 0.02 = 0.05$. For $T = 0.5$, $F_0 = 50 e^{0.025} \approx 51.2657$.