142 Mathematics for Finance
multiplied by a fixed amount ($500 for futures on S&P500). If the number of
stocks included in the index is large, it is possible and convenient to assume
that the index is an asset with dividends paid continuously.
Exercise 6.10
Suppose that the value of a stock exchange index is 13,500, the futures
price for delivery in 9 months is 14,100 index points, and the interest
rate is 8%. Find the dividend yield.Our goal in this section is to study applications of index futures for hedging
based on the Capital Asset Pricing Model introduced in Chapter 5. As we know,
see (5.19), the expected return on a portfolio over a time step of lengthτis
given by
μV=rF+(μM−rF)βV,
whereβVis the beta coefficient of the portfolio,μMis the expected return on
the market portfolio andrFis the risk-free return over one time step. ByV(n)
we shall denote the value of the portfolio at thenth time step. We assume for
simplicity that the index is equal to the value of the market portfolio, so that
the futures prices are given by
f(n, T)=M(n)(1 +rF)T−n,M(n) being the value of the market portfolio at thenth time step. (Here we
use discrete time and ordinary returns together with periodic compounding in
the spirit of Portfolio Theory.)
We can form a new portfolio with valueV ̃(n) by supplementing the original
portfolio withNshort futures contracts on the index with delivery timeT.
The initial valueV ̃(0) of the new portfolio is the same as the valueV(0) of
the original portfolio, since it costs nothing to initiate a futures contract. The
value
V ̃(n)=V(n)−N(f(n, T)−f(n− 1 ,T))
of the new portfolio at thenth step includes the marking to market cash flow.
The return on the new portfolio over the first step is
KV ̃=
V ̃(1)−V ̃(0)
V ̃(0) =
V(1)−N(f(1,T)−f(0,T))−V(0)
V(0).
We shall show that the beta coefficientβV ̃of the new portfolio can be modified
arbitrarily by a suitable choice of the futures positionN.