Quantile hedging problems

We review several problems on the quantile hedging given by [FL99].

We assume that the discounted price process of stock is given by a semimartingale ${X}$ on a probability space ${(\Omega, \mathcal{F}, \mathbb{P}, \mathcal{F}_t)}$. Under self-financing, the wealth process ${V_t}$ with inital capital ${v}$ and a predictable strategy ${\xi}$ follows

$\displaystyle V_t^{v,\xi} = v + \int_0^t \xi_s d X_s, \ \forall t \in [0,T], \ \mathbb{P}-a.s.$

${\xi}$ belongs to a space of adimissible strategy given by ${\mathcal{A}(v)}$,

$\displaystyle \mathcal{A}(v) = \{\xi: V_t^{v,\xi} \ge 0, \ \forall t\in [0,T], \ \mathbb{P}-a.s.\}$

Let ${H\in L^0(\mathcal{F}_T)}$ be a given claim.

Problem [SP]. Given a fixed constant ${\hat v>0}$, maximize success probability, i.e. find

$\displaystyle \sup_{v\le \hat v} \sup_{\xi\in \mathcal{A}(v)} \mathbb{P}\{ V_T^{v, \xi} \ge H\}.$

Problem [SR] Given a fixed constant ${\hat v>0}$, maximize success ratio, i.e. find

$\displaystyle \sup_{v\le \hat v} \sup_{\xi\in \mathcal{A}(v)} \mathbb{E} \Big[ I_{\{H\le V_T^{v, \xi}\}} + \frac{V_T^{v,\xi}}{H} I_{\{H > V_T^{v, \xi}\}} \Big].$

Problem [CP] Given a constant ${\varepsilon \in (0,1)}$, minimize the cost ${v}$ to beat ${H}$ with probability ${1-\varepsilon}$, i.e. find

$\displaystyle \inf_v\Big\{v: \mathbb{P}\{V_T^{v,\xi} \ge H\} \ge 1-\varepsilon \hbox{ for some } \xi\in \mathcal{A}(v) \Big\}.$

Problem [CR] Given a constant ${\varepsilon \in (0,1)}$, minimize the cost ${v}$ to beat ${H}$ with ratio ${1-\varepsilon}$, i.e. find

$\displaystyle \inf_v\Big\{v: \mathbb{E} \Big[ I_{\{H\le V_T^{v, \xi}\}} + \frac{V_T^{v,\xi}}{H} I_{\{H > V_T^{v, \xi}\}} \Big] \ge 1-\varepsilon \hbox{ for some } \xi\in \mathcal{A}(v) \Big\}.$

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