Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities：練習問題

Which of the following describes the initial value of a forward contract at inception ( = 0) under no-arbitrage conditions?

A dividend-paying stock currently trades at \(S_0 = 80\). It will pay a single continuous dividend yield of \(q = 2\%\) per annum. The continuously compounded risk-free rate is \(r = 5\%\) per annum. What is the no-arbitrage 6-month (\(T = 0.5\)) forward price \(F_0(0.5)\)?

Consider a 1-year forward contract on an underlying asset that currently trades at \(S_0 = 1,000\). The continuously compounded annual risk-free rate is \(6\%\). Six months after inception (at \(t = 0.5\)), the spot price of the underlying asset has fallen to \(S_{0.5} = 940\). The initial forward price determined at inception was \(F_0(1) = 1,000 e^{0.06 \times 1} \approx 1,061.84\). What is the value of the long forward contract at \(t = 0.5\)?

Consider an equity index currently trading at \(S_0 = 1{,}500\). A 1-year forward contract is entered into at \(t = 0\) with a forward price \(F_0(1) = 1{,}500 e^{0.04 \times 1} \approx 1{,}561.22\), where the continuously compounded risk-free rate is \(4\%\) and the index pays no dividends. After 9 months (\(t = 0.75\)), the index is trading at \(S_{0.75} = 1{,}620\). Assuming the risk-free rate remains constant at \(4\%\), what is the value of the long forward position at \(t = 0.75\)?

Two forward contracts on the same non-dividend-paying stock are traded today (\(t = 0\)). Contract A expires at \(T_1\) with forward price \(F_0(T_1)\), and Contract B expires at \(T_2\) with forward price \(F_0(T_2)\), where \(T_2 > T_1\). Assuming a flat term structure of interest rates with a constant continuously compounded risk-free rate \(r > 0\), which of the following expressions replicates the forward price \(F_0(T_2)\) in terms of \(F_0(T_1)\)?

State whether a forward contract with an initial value of zero becomes an asset or a liability to the short position if the spot price of the underlying asset increases above the forward price prior to maturity.