Quantile hedging (Fllmer and Leukert (1999), (S 6.2) of Mel'nikov et al. (2002)). Recall that given two

Quantile hedging (Föllmer and Leukert (1999), \(\S 6.2\) of Mel'nikov et al. (2002)). Recall that given two probability measures \(\mathbb{P}\) and \(\mathbb{Q}\), the Radon-Nikodym density \(\mathrm{d} \mathbb{P} / \mathrm{d} \mathbb{Q}\) links the expectations of random variables \(F\) under \(\mathbb{P}\) and under \(\mathbb{Q}\) via the relation

\[
\begin{aligned}
\mathbb{E}_{\mathbb{Q}}[F] & =\int_{\Omega} F(\omega) \mathrm{d} \mathbb{Q}(\omega) \\
& =\int_{\Omega} F(\omega) \frac{\mathrm{d} \mathbb{Q}}{\mathrm{d} \mathbb{P}}(\omega) \mathrm{d} \mathbb{P}(\omega) \\
& =\mathbb{E}_{\mathbb{P}}\left[F \frac{\mathrm{d} \mathbb{Q}}{\mathrm{d} \mathbb{P}}ight]
\end{aligned}
\]

a) Neyman-Pearson Lemma. Given \(\mathbb{P}\) and \(\mathbb{Q}\) two probability measures, consider the event

\[
A_{\alpha}:=\left\{\frac{\mathrm{d} \mathbb{P}}{\mathrm{d} \mathbb{Q}}>\alphaight\}, \quad \alpha \geqslant 0
\]

Show that for \(A\) any event, \(\mathbb{Q}(A) \leqslant \mathbb{Q}\left(A_{\alpha}ight)\) implies \(\mathbb{P}(A) \leqslant \mathbb{P}\left(A_{\alpha}ight)\).

Hint: Start by proving that we always have

\[
\begin{equation*}
\left(\frac{\mathrm{d} \mathbb{P}}{\mathrm{d} \mathbb{Q}}-\alphaight)\left(2 \mathbb{1}_{A_{\alpha}}-1ight) \geqslant\left(\frac{\mathrm{d} \mathbb{P}}{\mathrm{d} \mathbb{Q}}-\alphaight)\left(2 \mathbb{1}_{A}-1ight) \tag{7.54}
\end{equation*}
\]

b) Let \(C \geqslant 0\) denote a nonnegative claim payoff on a financial market with risk-neutral measure \(\mathbb{P}^{*}\). Show that the Radon-Nikodym density

\[
\begin{equation*}
\frac{\mathrm{d} \mathbb{Q}^{*}}{\mathrm{dP}^{*}}:=\frac{C}{\mathbb{E}_{\mathbb{P}^{*}}[C]} \tag{7.55}
\end{equation*}
\]

defines a probability measure \(\mathbb{Q}^{*}\).

Hint: Check first that \(\mathrm{dQ}^{*} / \mathrm{dP}^{*} \geqslant 0\), and then that \(\mathbb{Q}^{*}(\Omega)=1\). In the following questions we consider a nonnegative contingent claim with payoff \(C \geqslant 0\) and maturity \(T>0\), priced \(\mathrm{e}^{-r T} \mathbb{E}_{\mathbb{P}^{*}}[C]\) at time 0 under the risk-neutral measure \(\mathbb{P}^{*}\).

Budget constraint. In what follows we will assume that no more than a certain fraction \(\beta \in\) \((0,1]\) of the claim price \(\mathrm{e}^{-r T} \mathbb{E}_{\mathbb{P}^{*}}[C]\) is available to construct the initial hedging portfolio \(V_{0}\) at time 0 .

Since a self-financing portfolio process \(\left(V_{t}ight)_{t \in \mathbb{R}_{+}}\)started at \(V_{0}:=\beta \mathrm{e}^{-r T} \mathbb{E}_{\mathbb{P}^{*}}[C]\) may fall short of hedging the claim \(C\) when \(\beta<1\), we will attempt to maximize the probability \(\mathbb{P}\left(V_{T} \geqslant Cight)\) of successful hedging, or, equivalently, to minimize the shortfall probability \(\mathbb{P}\left(V_{T}
For this, given \(A\) an event we consider the self-financing portfolio process \(\left(V_{t}^{A}ight)_{t \in[0, T]}\) hedging the claim \(C \mathbb{1}_{A}\), priced \(V_{0}^{A}=\mathrm{e}^{-r T} \mathbb{E}_{\mathbb{P}^{*}}\left[C \mathbb{1}_{A}ight]\) at time 0 , and such that \(V_{T}^{A}=C \mathbb{1}_{A}\) at maturity \(T\).

c) Show that if \(\alpha\) satisfies \(\mathbb{Q}^{*}\left(A_{\alpha}ight)=\beta\), the event

\[
A_{\alpha}=\left\{\frac{\mathrm{d} \mathbb{P}}{\mathrm{d} \mathbb{Q}^{*}}>\alphaight\}=\left\{\frac{\mathrm{d} \mathbb{P}}{\mathrm{d} \mathbb{P}^{*}}>\alpha \frac{\mathrm{d} \mathbb{Q}^{*}}{\mathrm{dP}^{*}}ight\}=\left\{\frac{\mathrm{d} \mathbb{P}}{\mathrm{d} \mathbb{P}^{*}}>\frac{\alpha C}{\mathbb{E}_{\mathbb{P}^{*}}[C]}ight\}
\]

maximizes \(\mathbb{P}(A)\) over all possible events \(A\), under the condition

\[
\begin{equation*}
\mathrm{e}^{-r T} \mathbb{E}_{\mathbb{P}^{*}}\left[V_{T}^{A}ight]=\mathrm{e}^{-r T} \mathbb{E}_{\mathbb{P}^{*}}\left[C \mathbb{1}_{A}ight] \leqslant \beta \mathrm{e}^{-r T} \mathbb{E}_{\mathbb{P}^{*}}[C] \tag{7.56}
\end{equation*}
\]

Hint: Rewrite Condition (7.56) using the probability measure \(\mathbb{Q}^{*}\), and apply the Neyman-Pearson Lemma of Question (a) to \(\mathbb{P}\) and \(\mathbb{Q}^{*}\).

d) Show that \(\mathbb{P}\left(A_{\alpha}ight)\) coincides with the successful hedging probability

\[
\mathbb{P}\left(V_{T}^{A_{\alpha}} \geqslant Cight)=\mathbb{P}\left(C \mathbb{1}_{A_{\alpha}} \geqslant Cight),
\]

i.e. show that

\[
\mathbb{P}\left(A_{\alpha}ight)=\mathbb{P}\left(V_{T}^{A_{\alpha}} \geqslant Cight)=\mathbb{P}\left(C \mathbb{1}_{A_{\alpha}} \geqslant Cight) .
\]

Hint: To prove an equality \(x=y\) we can show first that \(x \leqslant y\), and then that \(x \geqslant y\). One inequality is obvious, and the other one follows from Question (c).

e) Check that the self-financing portfolio process \(\left(V_{t}^{A_{\alpha}}ight)_{t \in[0, T]}\) hedging the claim with payoff \(C \mathbb{1}_{A_{\alpha}}\) uses only the initial budget \(\beta \mathrm{e}^{-r T} \mathbb{E}_{\mathbb{P}^{*}}[C]\), and that \(\mathbb{P}\left(V_{T}^{A_{\alpha}} \geqslant Cight)\) maximizes the successful hedging probability.

In the next Questions (f)-(j) we assume that \(C=\left(S_{T}-Kight)^{+}\)is the payoff of a European option in the Black-Scholes model

\[
\begin{equation*}
d S_{t}=r S_{t} d t+\sigma S_{t} d B_{t} \tag{7.57}
\end{equation*}
\]

with \(\mathbb{P}=\mathbb{P}^{*}, \mathrm{~d} \mathbb{P} / \mathrm{d} \mathbb{P}^{*}=1\), and

\[
\begin{equation*}
S_{0}:=1 \quad \text { and } \quad r=\frac{\sigma^{2}}{2}:=\frac{1}{2} \tag{7.58}
\end{equation*}
\]

f) Solve the stochastic differential equation (7.57) with the parameters (7.58).

g) Compute the successful hedging probability

\[
\mathbb{P}\left(V_{T}^{A_{\alpha}} \geqslant Cight)=\mathbb{P}\left(C \mathbb{1}_{A_{\alpha}} \geqslant Cight)=\mathbb{P}\left(A_{\alpha}ight)
\]

for the claim \(C=:\left(S_{T}-Kight)^{+}\)in terms of \(K, T, \mathbb{E}_{\mathbb{P}^{*}}[C]\) and the parameter \(\alpha>0\).

h) From the result of Question (g), express the parameter \(\alpha\) using \(K, T, \mathbb{E}_{\mathbb{P}^{*}}[C]\), and the successful hedging probability \(\mathbb{P}\left(V_{T}^{A_{\alpha}} \geqslant Cight)\) for the claim \(C=:\left(S_{T}-Kight)^{+}\).

i) Compute the minimal initial budget \(\mathrm{e}^{-r T} \mathbb{E}_{\mathbb{P}^{*}}\left[C \mathbb{1}_{A_{\alpha}}ight]\) required to hedge the claim \(C=\) \(\left(S_{T}-Kight)^{+}\)in terms of \(\alpha>0, K, T\) and \(\mathbb{E}_{\mathbb{P}^{*}}[C]\).

j) Taking \(K:=1, T:=1\) and assuming a successful hedging probability of \(90 \%\), compute numerically:

i) The European call price \(\mathrm{e}^{-r T} \mathbb{E}_{\mathbb{P} *}\left[\left(S_{T}-Kight)^{+}ight]\)from the Black-Scholes formula.

ii) The value of \(\alpha>0\) obtained from Question (h).

iii) The minimal initial budget needed to successfully hedge the European claim \(C=\) \(\left(S_{T}-Kight)^{+}\)with probability \(90 \%\) from Question (i).

iv) The value of \(\beta\), i.e. the budget reduction ratio which suffices to successfully hedge the claim \(C=:\left(S_{T}-Kight)^{+}\)with \(90 \%\) probability.