This paper studies modeling and existence issues for market models of stochastic implied volatility in a continuous-time framework with one stock, one bank account, and a family of European options for all maturities with a fixed payoff function h. We first characterize absence of arbitrage in terms of drift conditions for the forward implied volatilities corresponding to a general convex h. For the resulting infinite system of SDEs for the stock and all the forward implied volatilities, we then study the question of solvability and provide sufficient conditions for existence and uniqueness of a solution. We do this for two examples of h, namely, calls with a fixed strike and a fixed power of the terminal stock price, and we give explicit examples of volatility coefficients satisfying the required assumptions.
In this work, we investigate SDEs whose coefficients may depend on the entire past of the solution process. We introduce different Lipschitz-type conditions on the coefficients. It turns out that for existence and uniqueness of a strong solution it suffices to have Lipschitz continuity in mean, in a sense to be made precise. We then investigate when it suffices to have local Lipschitz conditions. Furthermore we consider the case of drift coefficients which are locally Lipschitz in mean. Finally we show how these results can be applied to prove existence and uniqueness of solutions in interest rate term structure models.The well-known existence and uniqueness result for strong solutions of SDEs with Lipschitztype coefficients can be obtained in several settings of varying generality. The setting chosen in this work is motivated by and in fact tailor-made for applications to term structure models arising in mathematical finance, which are typically of the following form. Let I ⊂ [0, ∞) be an interval, and X an infinite-dimensional process (t, ω 1 ) → X (t, T, ω 1 ) T ∈I (describing a collection of market observables) on a space Ω 1 , which satisfies an SDE of the form X (0, T ) = X 0 (T ), dX (t, T ) = α(t, T, X )dt + σ (t, T, X )dW 1 t (0.1) * Tel.
We analyze mean-variance-optimal dynamic hedging strategies in oil futures for oil producers and consumers. In a model for the oil spot and futures market with Gaussian convenience yield curves and a stochastic market price of risk, we find analytical solutions for the optimal trading strategies. An implementation of our strategies in an out-of-sample test on market data shows that the hedging strategies improve long-term return-risk profiles of both the producer and the consumer.
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