Question

Problom 2.5.4. (Price of continuous random dividends) The current price of a stock is 100. The stock pays dividends continuously at a rate proportional to its price. The dividend yields is 396. The continuously compounded risk-free interest rate is 7%. Calculate the price of the stream of dividends to be paid in the next 5 years. (Note: Because the dividends are stochastic, their present value is also stochastic and hence cannot be their price.)

Answer 1

Answers

Say that an investor longs one share of stock. Then, with continuous reinvestment of dividends the investor’s profit can be expressed, in our usual notation, as S(T)e ^δT − S(0)e ^rT .

A rational investor who demands to be compensated for risk would only invest if the expected profit above were positive.

So, X = E[S(T)] > S(0)e ^(r−δ)T = 100e ^(0.07-0.03)5=100e^(0.04)5=100e^(0.2)= 100(1.2214)=122.14(Answer)

r= risk free rate=7%1

^{δ=dividend yield =3%}

^{S= spot price}

^{T= time period = 5 years}

^{x= price}