Consider a non-spanned security $C_{T}$ in an incomplete market. Westudy the risk/return tradeoffs generated if this security is soldfor an arbitrage-free price $hat{C_{0}}$ and then hedged. Weconsider recursive "one-period optimal" self-financing hedgingstrategies, a simple but tractable criterion. For continuoustrading, diffusion processes, the one-period minimum varianceportfolio is optimal. Let $C_{0}(0)$ be its price. Self-financingimplies that the residual risk is equal to the sum of the one-periodorthogonal hedging errors, $sum_{tleq T} Y_{t}(0) e^{r(T -t)}$. Tocompensate the residual risk, a risk premium $y_{t}Delta t$ isassociated with every $Y_{t}$. Now let $C_{0}(y)$ be the price ofthe hedging portfolio, and $sum_{tleq T}(Y_{t}(y)+y_{t}Deltat)e^{r(T-t)}$ is the total residual risk. Although not the same, theone-period hedging errors $Y_{t}(0) and Y_{t}(y)$ are orthogonal tothe trading assets, and are perfectly correlated. This implies thatthe spanned option payoff does not depend on y. Let$hat{C_{0}}-C_{0}(y)$. A main result follows. Any arbitrage-freeprice, $hat{C_{0}}$, is just the price of a hedging portfolio (suchas in a complete market), $C_{0}(0)$, plus a premium,$hat{C_{0}}-C_{0}(0)$. That is, $C_{0}(0)$ is the price of theoption's payoff which can be spanned, and $hat{C_{0}}-C_{0}(0)$ isthe premium associated with the option's payoff which cannot bespanned (and yields a contingent risk premium of sum $y_{t}Delta$t$e^{r(T-t)}$ at maturity). We study other applications of option-pricing theory as well.
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机译：考虑不完整市场中的非跨期证券$ C_ {T} $。如果该证券以无套利价格$ hat {C_ {0}} $出售然后进行套期，则对产生的风险/收益权衡进行评估。我们考虑递归的“一期最优”自筹资金对冲策略，这是一个简单但易于处理的标准。对于连续交易，扩散过程，单周期最小方差组合是最佳的。令$ C_ {0}（0）$为价格。自筹资金意味着剩余风险等于单周向对冲误差之和$ sum_ {t leq T} Y_ {t}（0）e ^ {r（T -t）} $。为了补偿剩余风险，将风险溢价$ y_ {t} Delta t $与每个$ Y_ {t} $相关联。现在，将$ C_ {0}（y）$作为对冲投资组合的价格，并将$ sum_ {t leq T}（Y_ {t}（y）+ y_ {t} Deltat）e ^ {r（Tt ）} $是总剩余风险。尽管不尽相同，但单期对冲误差$ Y_ {t}（0）和Y_ {t}（y）$与交易资产正交，并且完全相关。这意味着跨期期权收益不取决于y。设$ hat {C_ {0}}-C_ {0}（y）$。主要结果如下。任何无套利价格$ hat {C_ {0}} $都是对冲投资组合的价格（例如，在一个完整的市场中）$ C_ {0}（0）$，再加上溢价$ hat { C_ {0}}-C_ {0}（0）$。也就是说，$ C_ {0}（0）$是可以跨越的期权收益价格，$ hat {C_ {0}}-C_ {0}（0）$是与期权收益相关的溢价，无法跨越（到期时产生总和$ y_ {t} Delta $ t $ e ^ {r（Tt）} $的或有风险溢价）。我们也研究期权定价理论的其他应用。
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