Option-pricing in incomplete markets: the hedging portfolio plus a risk premium-based recursive approachReport as inadecuate




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Issued date: 2005-01

Serie-No.: UC3M Working Papers. Bussiness Economics2005-21

Abstract: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 hedgiConsider 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, $\sum {t\leq T} Y {t}(0) e^{r(T -t)}$. 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 $\sum {t\leq T}(Y {t}(y)+y {t}\Deltat)e^{r(T-t)}$ 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 theoptions payoff which can be spanned, and $\hat{C {0}}-C {0}(0)$ isthe premium associated with the options 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.+-





Author: Ibáñez, Alfredo

Source: http://e-archivo.uc3m.es


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Universidad Carlos III de Madrid Departamento de Ingeniería Mecánica PROYECTO FIN DE CARRERA Ingeniería Técnica Industrial: Mecánica Autor: D.
Javier Moreno Fernández Tutores: Dra.
Dña.
Beatriz López Boada Dr.
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