Pricing dynamic binary variables and their derivatives

Abstract

Many important assets or business ventures have cash flows that are not derivatives of a market security but are nevertheless dependent on some variable that is correlated with market prices. This includes many real option projects. This paper presents a methodology using a binary framework for pricing such assets by projection onto the market space. Under certain conditions, the result has the property that, given this price process, no risk-averse investor would choose to invest in this asset either long or short.

Keywords:

• Asset pricing
• Derivative pricing models
• Derivatives hedging
• Real options

Notes

†Some real options, such as those associated with gold mining or oil production, do have associated underlying securities and are amenable to the Black–Scholes approach.

†Suppose a and b are jointly distributed normal random variables. Let F be a real-valued continuously differentiable function on the reals, such that

Then cov(F(a), b) = E[F′(a)]cov(a, b). The proof follows from an integration by parts.

†There are singular cases where the Markowitz portfolio is not defined (Luenberger 2001).

†The condition M* > 0 will be satisfied if U severely penalizes final wealth close to zero. For example, U(x) = ln x and U(x) = − x ^−γ with γ > 0 are suitable.

‡The reverse may be assumed instead.

†It is implicitly assumed in the multiperiod case that the underlying variable B is the only non-marketed variable that gives information about the final payoff of the derivative.

†The law of total variance states that, in general, var(X) = E[var(X ∣ Y)] + var[E(X ∣ Y)].

†It is well-known (see, for example, Luenberger 1998) that, for a security with no holding costs, q _A  = (R − A _d )/(A _u  − A _d ). See example 3.2.