We design three continuous-time models in finite horizon of a commodity price, whose dynamics can be affected by the actions of a representative risk-neutral producer and a representative risk-neutral trader. Depending on the model, the producer can control the drift and/or the volatility of the price whereas the trader can at most affect the volatility. The producer can affect the volatility in two ways: either by randomizing her production rate or, as the trader, using other means such as spreading false information. Moreover, the producer contracts at time zero a fixed position in a European convex derivative with the trader. The trader can be price-taker, as in the first two models, or she can also affect the volatility of the commodity price, as in the third model. We solve all three models semi-explicitly and give closed-form expressions of the derivative price over a small time horizon, preventing arbitrage opportunities to arise. We find that when the trader is price-taker, the producer can always compensate the loss in expected production profit generated by an increase of volatility by a gain in the derivative position by driving the price at maturity to a suitable level. Finally, in case the trader is active, the model takes the form of a nonzero-sum linear-quadratic stochastic differential game and we find that when the production rate is already at its optimal stationary level, there is an amount of derivative position that makes both players better off when entering the game.

中文翻译：

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无套利商品期权定价与市场操纵

我们在商品价格的有限范围内设计了三个连续时间模型，该模型的动态性可能会受到代表风险中立的生产者和代表风险中立的交易者的行为的影响。根据模型，生产者可以控制价格的波动和/或波动性，而交易者最多可以影响价格的波动性。生产者可以通过两种方式影响波动率：通过随机化其生产率或作为交易者使用其他手段（例如散布虚假信息）。此外，生产者在零时与交易者签订欧洲凸型衍生工具的固定头寸合约。交易者可以像前两个模型一样接受价格，或者像第三个模型一样可以影响商品价格的波动。我们半隐式地求解所有三个模型，并在较短的时间范围内给出衍生产品价格的封闭形式，以防止出现套利机会。我们发现，当交易者接受价格时，通过将到期日价格推高至适当水平，生产商总是可以通过衍生产品头寸的增加来补偿由于波动性增加而产生的预期生产利润的损失。最后，在交易者活跃的情况下，该模型采用非零和线性二次方随机微分博弈的形式，我们发现当生产率已经处于其最佳固定水平时，存在一定数量的导数头寸双方玩家进入游戏时都会变得更好。我们发现，当交易者接受价格时，通过将到期价格推高至适当水平，生产商总是可以通过衍生产品头寸的增加来补偿由于波动性增加而产生的预期生产利润的损失。最后，在交易者活跃的情况下，该模型采用非零和线性二次方随机微分博弈的形式，我们发现当生产率已经处于其最佳固定水平时，存在一定数量的导数头寸双方玩家进入游戏时都会变得更好。我们发现，当交易者接受价格时，通过将到期价格推高至适当水平，生产商总是可以通过衍生产品头寸的增加来补偿由于波动性增加而产生的预期生产利润的损失。最后，在交易者活跃的情况下，该模型采用非零和线性二次方随机微分博弈的形式，我们发现当生产率已经处于其最佳固定水平时，存在一定数量的导数头寸双方玩家进入游戏时都会变得更好。