6.4 APPENDIX

Applying It 's lemma to the value of the option, V( x,t), implies:

or

This means that the instantaneous rate of return on the option will be:

where:

Now, suppose we invest W ₁ in the underlying capital project, W ₂ in options and W ₃ in the risk-free asset, but in such a way that the investment is self-financing, or W ₁ + W ₂ + W ₃ = 0. Then differentiation shows:

Substituting the stochastic differential equations for d x and d V into the right-hand side of this expression will then give:

Now, suppose we pursue an investment policy which eliminates all uncertainty, so that:

or

Since all uncertainty has been eliminated, arbitrage will dictate that the nonstochastic component of the investment policy will also have to be zero, or:

Simplifying the right-hand side of this equation gives:

which is the fundamental valuation equation contained in the text.

In the text we develop a pricing formula for a call option written on the net present value variable, with an exercise price of zero. Here, we solve the fundamental valuation equation under the more general boundary condition:

where E is a (non-trivial) exercise price. We again make the substitution V( x,t) = exp[ ? r( T ? t)] F( ?, ?), based on the co-ordinate system and ? = ( T ? t)/2. The fundamental valuation...