Risk and Foreign Direct Investment - Index of

Since the portfolio must duplicate the payoffs to the option
ux + Rb = Cu
and dx + Rb = Cd
which could be written, if the stress is on the hedging of the option:
Rb = ux – Cu
Rb = dx – Cd
There are two equations and two unknowns, so it is easy to solve for
both x, the hedging ratio, and for b, the borrowings.
x = Cu – Cd/ u – d
b = Cu – ux/R = uCd – dCu/R(u – d)
Therefore, the value of the portfolio and of the call option is:
x + b = Cu – Cd/u – d + uCd – dCu/ R(u – d)
or rearranging terms:
= 1/R{(R – d)/(u –d). Cu + (u – R)/(u – d).Cd
Simplifying by letting (R – d)/(u – d) = q
and therefore 1 – q = (u – R)/(u – d) {since the last term = 1 – (R – d)/(u – d)}
C = 1/R(qCu + (1 – q)Cd)
Responses to Risk 201
q and 1 – q can be regarded as risk-free probabilities since the option
and the replicating portfolio offer the same payoff; they are not the
subjective probabilities held by managers concerning the likelihood of
future outcomes. 1 The analysis is a risk-neutral approach to valuation,
based on risk-neutral probabilities which ensure the outcome is a riskfree
return. This approach breaks away from direct dependence on the
actual probabilities of outcomes and avoids the use of expected returns
and evaluation of the degree of risk aversion, and therefore the need to
estimate a risk premium.
The analysis can be extended to more than two periods. Note that
the hedge ratio and the borrowing will be different in each period.
The procedure is to move backwards, from later valuations to earlier
values, moving from the final outcomes to successively value the