Risk Preference Based Option Pricing in the Fractional Binomial Setting

In this chapter, we take on the setting presented in Chap. 3. There, we showed that binomial trees can be used to approximate the process of geometric fractional Brownian motion. The framework is free of arbitrage if investors are restricted to trade only on certain nodes. Concerning the valuation of derivatives, the classical binomial approach applies the idea of backward calculation. Starting from the terminal nodes, it is possible to go backwards step by step, relating two posterior nodes to one antecedent and using no arbitrage arguments. Within our modified setting, we are faced with the obvious problem that a step-by-step procedure is no longer possible as we could not adopt absence of arbitrage on this minimal time scale. On the other hand, if we leave the minimal time scale and only look at intervals where we can ensure absence of arbitrage, the way backwards is not unique: If the arbitrage-free interval is divided into n steps, we have to relate 2n subsequent nodes to only one prior node. As we will assert risk-neutrality of the market participants, one could have the idea of simply taking expectations. From this vantage point, assets should be valued by their conditional mean. However, the inconsiderate usage of this kind of valuation ignores the fact that one of the two basic assets—the risky one—is not a martingale. Hence, its discounted conditional expected value in time t generally does not equal its current value. Per se, this does not cause any problem as long as predictability cannot be exploited. For our basic setting with two assets, this was ensured by our restricted framework which we introduced in Chap. 3. However, if one coonsiders a further asset being related to the stock, it is not reasonable to apply a pricing rule that does not hold for the basic risky asset. Doing so, one obtains a disequilibrium between derivative and underlying. We give a short example to see that such a partial disequilibrium leads to an arbitrage possibility.