Consider a stock whose price on 1 January is $120 and which will pay a dividend of $1 on 1 July 2000 and $2 on 1 October 2000. The interest rate is 12%. Is there an arbitrage opportunity if on 1 January 2000 the forward price for delivery of the stock on 1 November 2000 is $131? If so, compute

Question

Consider a stock whose price on 1 January is $120 and which will pay a
dividend of $1 on 1 July 2000 and $2 on 1 October 2000. The interest
rate is 12%. Is there an arbitrage opportunity if on 1 January 2000 the
forward price for delivery of the stock on 1 November 2000 is $131? If
so, compute the arbitrage profit.

Accepted Answer

The present value of the dividends is

[latex]div_0 = 1e^{-\frac{6}{12}×12\% } + 2e^{-\frac{9}{12} imes12\% }\cong 2.77 [/latex]

dollars. The right-hand side of (6.4) is equal to

[latex]F(0,T)=[[S(0)-div_0]e^{rT} [/latex] (6.4)

[latex][S(0)-div_0]e^{rT} \cong (120-2.77)e^{\frac{10}{12} imes 12\% }\cong 129.56[/latex]

dollars, which is less than the quoted forward price of $131. As a result, there will be an arbitrage opportunity, which can be realised as follows:

on 1 January 2000 enter into a short forward position and borrow $120 to buy stock;
on 1 July 2000 collect the first dividend of $1 and invest risk-free;
on 1 October 2000 collect the second dividend of $2 and invest risk-free;
on 1 November 2000 close out all positions.

You will be left with an arbitrage profit of

[latex]131 − 120e^{\frac{10}{12} imes 12\% }+1e^{\frac{4}{12} imes 12\% }+2e^{\frac{1}{12} imes 12\% }\cong 1.44[/latex]

dollars.

Question 6.3: Consider a stock whose price on 1 January is $120 and which ...

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