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\}.

Advertisements

One comment

  1. Pingback: Neyman-Pearson Lemma « 01law's Blog


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s