## Fractional Brownian motion and managing risk with short-term futures contracts

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In this dissertation, we consolidate previous research done by [3], [5], [13], [14] and [22] on an optimal strategy to reduce the running risk in hedging a long-term supply commitment with short-term futures contracts. Under the assumption that the market price of the commodity is modeled by the fractional Brownian motion (fBm), we study the following optimization problem: Under the constraint $$\ \int_{0}^{1}\int_{0}^{1}(1-g(u))(1-g(v))|u-v|^{2H-2}dudv\leq\theta,$$ Which measurable function (g: [0,1]\rightarrow R[0,1]) will minimize the value of $$\ \sup_{t\in[0,1]}\int_{0}^{t}\int_{0}^{t}(t-g(u))(t-g(v))|u-v|^{2H-2}dudv?$$ where (\theta\in[0,\frac{H^{2H}}{(H+1)^{2H+2}}]) and (H\in(\frac{1}{2},1)). Under the fractional market model, we gave the spot risk function by using the hedging strategies provided by Glasserman [5] and found that when the hurst index (H) is equal to (0.5) the maximal spot risk is the same as the result given in [5]. The main work in this dissertation is to show that a unique solution to this optimization problem always exists.