Many of the assumptions used by Black & Scholes (1973) and Black (1976) to derive option pricing formulae are violated. Extensive empirical evidence has shown that most financial time series reject the assumption of Geometric Brownian Motion (GBM). Volatility is not constant nor is the price path continuous, as many markets display dynamics consistent with jump processes. Furthermore, other “perfect market” assumptions such as the absence of transaction costs and perfect divisibility of the underlying assets are also violated. The introduction of unspanned sources of risk (and frictions) implies that option prices include risk premium. Prema facie evidence of such risk premium is the implied volatility smile patterns that have been extensively identified and discussed in the literature.
A non-trivial problem is isolating the risk premium from implied volatility smiles. It is clear that a logical starting point would be to consider alternative models to Black & Scholes (1973). However, there is no consensus regarding what this appropriate model is and what would be the nature of the risk premium if such a model includes non-traded sources of risk. Some authors have suggested a stochastic volatility risk premium, some have argued for jump risk and still others point to market frictions. This paper will isolate for the risk premium in options on Bond futures by first considering whether a rich class of alternative models that include jumps and stochastic volatility can account for the empirical dynamics of Bond Futures. Bond Futures and options are examined for two reasons. Firstly, these are among the most active exchange traded derivatives on interest rates and secondly, much of the work on this area has concentrated on Stock Index option markets. Thus, this work extends work by Bates (2000) and Pan (2001) for Bond Futures and Options markets.
We find that single and multi-factor stochastic volatility models with jumps can both explain the empirical regularities we observed in Bond Futures. Secondly, assuming a feasible change of measure with an assumption of no risk premium, options consistent with this model are priced. Such option prices consistent with the dynamics of the underlying markets are compared to actual options on these markets and a risk premium is isolated. It is observed that for three Bond futures and options markets: US T-Bonds, UK Gilts and German Bunds, the nature of the risk premium is extremely similar for the choice of the underlying stochastic volatility model. While this research does not attempt to assign economic significance to this risk premium, such consistency suggests that market agents are assuming a similar functional form for the risk premium and may provide insights into models of the risk premium that have recently been proposed.