# A Guaranteed Deterministic Approach to Superhedging—The Case of Convex Payoff Functions on Options

^{1}

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## Abstract

**:**

## 1. Introduction

- there exists a strategy ${h}^{*}\in {D}_{t}(\xb7)$, such that ${h}^{*}y\ge 0$ for all $y\in {K}_{t}(\xb7)$,
- there exists a price movement ${y}^{*}\in {K}_{t}(\xb7)$, such that ${h}^{*}{y}^{*}>0$.

- no deterministic arbitrage opportunity (NDAO),
- no deterministic sure arbitrage (NDSA),
- deterministic sure arbitrage with unlimited profit (NDSAUP).

- –
- robust no deterministic arbitrage opportunity (RNDAO),
- –
- robust deterministic sure arbitrage with unlimited profit (RNDSAUP).

## 2. Convex Payoff Functions

**Proposition**

**1.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

- (1)
- Therefore, the condition (USC) from [5], i.e., the upper semicontinuity of ${v}_{t}^{*}(\xb7)$, $t=0,\dots ,N$, is fulfilled, so [5], Proposition 3.3 is applicable, and hence, there is an equilibrium with mixed extension $\mathcal{P}({K}_{t}(\xb7))$; moreover, ${\mathcal{P}}_{t}^{opt}(\xb7)\ne \varnothing $.
- (2)
- If the maximizer in expression (2.6) from [8] is unique, i.e., ${\mathcal{P}}_{t}^{opt}(\xb7)=\left\{{Q}_{t}^{*}(\xb7)\right\}$ is a one-point set, then by Lemma 1, $supp({Q}_{t}^{*}(\xb7))\subseteq ext({K}_{t}(\xb7))$. Applying a two-stage optimization defined by the relations (11) and (12) in [7] and taking into account Item 2 of Remark 3.2 in [7], we conclude that the number of points in support $|supp({Q}_{t}^{*}(\xb7))|\le n+1$. Moreover, if the conditions of Theorem 2.1 from [8] are fulfilled, the mapping $x\mapsto {Q}_{t}^{*}(x)$ is (weakly) continuous and $x\mapsto supp({Q}_{t}^{*}(\xb7))$ is a lower semicontinuous multivalued mapping.
- (3)
- In the case that ${K}_{t}(\xb7)$ are convex polyhedra, i.e., can be represented as a convex hull of a finite number of points (according to Theorem 19.1 in [29], the polyhedrality of a convex set is equivalent to its finite generation; in the case of compactness, such a set coincides with the convex hull of a finite number of points; see also [28], Definition 2.2), the set of extreme points $ext({K}_{t}(\xb7))$ is finite and $m=|ext({K}_{t}(\xb7))|\ge n+1$; so $n+1$ of these m points constitute the optimal mixed strategy support.

**Proposition**

**2.**

- (1)
- set ${K}_{t}(\xb7)$ is strictly convex $t=1,\dots ,N$;
- (2)
- ${K}_{t}(x)$ is a convex polyhedron with a constant (independent of x) number of vertices (the set of vertices of a compact convex polyhedron coincides with the set of its extreme points), $t=1,\dots ,N$.

**Proof.**

**Remark**

**2.**

**Proposition**

**3.**

- (1)
- If the condition$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 0\notin & ri(conv(A))\phantom{\rule{4.pt}{0ex}}\mathit{for}\phantom{\rule{4.pt}{0ex}}\mathit{any}\phantom{\rule{4.pt}{0ex}}A\subseteq ext({K}_{t}(\xb7)),\phantom{\rule{4.pt}{0ex}}\mathit{such}\phantom{\rule{4.pt}{0ex}}\mathit{as}\phantom{\rule{4.pt}{0ex}}\left|A\right|\le n,\phantom{\rule{4.pt}{0ex}}\hfill \end{array}\end{array}$$is fulfilled, then we have the following:
- -
- there is an optimal mixed strategy ${Q}_{t}^{*}(\xb7)$ with zero mean and $supp({Q}_{t}^{*}(\xb7))\subseteq ext({K}_{t}(\xb7))$ satisfying the condition of maximum cardinality of support, i.e., $|supp({Q}_{t}^{*}(\xb7))|\equiv n+1$;
- -
- compacts ${K}_{t}(\xb7)$ are full-dimensional, i.e., $dim{K}_{t}(\xb7)=n$; and
- -
- the robust condition of no arbitrage opportunities $RNDAO$ is fulfilled.

- (2)
- If, in addition, ${\mathcal{P}}_{t}^{opt}(\xb7)$ contains a single element, i.e., ${\mathcal{P}}_{t}^{opt}(\xb7)=\left\{{Q}_{t}^{*}(\xb7)\right\}$, the compact-valued mappings ${K}_{t}(\xb7)$ are continuous, $t=1,\dots ,N$, then multivalued mapping $x\mapsto supp({Q}_{t}^{*}(\xb7))$ can be decomposed into n non-coincident continuous everywhere branches, each of which is a vertex of one of the ${K}_{t}(\xb7)$ n-simplex (the n-simplex is a solid polyhedron in ${\mathbb{R}}^{n}$ with $n+1$ vertices (which are the extreme points of this polyhedron)). containing 0. (There can be several such n-simplexes).

**Proof.**

**Example**

**1.**

**Remark**

**3.**

**Example**

**2.**

**Remark**

**4.**

## 3. Conclusions

## Funding

## Conflicts of Interest

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Option Type | Payoff Function |
---|---|

“Best of asset or cash” | $max({S}_{1},{S}_{2},\dots ,{S}_{n},K)$ |

“Call on max” | $max(max({S}_{1},{S}_{2},\dots ,{S}_{n})-K,0)$ |

“Put on min” | $max(K-min({S}_{1},{S}_{2},\dots ,{S}_{n}),0)$ |

“Put 2 and Call 1” | $max({S}_{1}-{S}_{2},0)$ |

“Multi-strike” | $max({S}_{1}-{K}_{1},{S}_{2}-{K}_{2},\dots ,{S}_{n}-{K}_{n},0)$ |

“Pyramid” | $max(|{S}_{1}-{K}_{1}|+|{S}_{2}-{K}_{2}|+\dots +|{S}_{n}-{K}_{n}|-K,0)$ |

“Madonna” | $max(\sqrt{{({S}_{1}-{K}_{1})}^{2}+\dots +{({S}_{n}-{K}_{n})}^{2}}-K,0)$ |

Option Type | Payoff Function |
---|---|

“Call on min” | $max(min({S}_{1},{S}_{2},\dots ,{S}_{n})-K,0)$ |

“Put on max” | $max(K-max({S}_{1},{S}_{2},\dots ,{S}_{n}),0)$ |

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Smirnov, S. A Guaranteed Deterministic Approach to Superhedging—The Case of Convex Payoff Functions on Options. *Mathematics* **2019**, *7*, 1246.
https://doi.org/10.3390/math7121246

**AMA Style**

Smirnov S. A Guaranteed Deterministic Approach to Superhedging—The Case of Convex Payoff Functions on Options. *Mathematics*. 2019; 7(12):1246.
https://doi.org/10.3390/math7121246

**Chicago/Turabian Style**

Smirnov, Sergey. 2019. "A Guaranteed Deterministic Approach to Superhedging—The Case of Convex Payoff Functions on Options" *Mathematics* 7, no. 12: 1246.
https://doi.org/10.3390/math7121246