# Stochastically fair price for financial options in
incomplete markets

Rustem Sadykov* and Dongming Wei

Department of Mathematics, Nazarbayev University, 53 Kabanbay
Batyr Ave, Astana, 010000, Kazakhstan.

*Corresponding author(s). E-mail(s): rustem.sadykov@nu.edu.kz;

Contributing authors: dongming.wei@nu.edu.kz;

## Abstract

In the framework of tree methods for financial options pricing, we propose to use a new notion of stochastically fair price within the fair price range for financial options in incomplete markets. This subset of fair prices is efficient in a sense that it excludes not only arbitrage, but also stochastic arbitrage based on the concept of stochastic dominance. We define stochastically fair prices and present a methodology to compute them for options by using the stochastic optimisation techniques with stochastic constraints involving doubly stochastic matrices. We use a simple trinomial model for illustration.

**Keywords:** stochastic dominance, arbitrage, stochastic arbitrage, option pricing, portfolio optimization, stochastically fair price, doubly stochastic matrices

## 1 Introduction

The concept of arbitrage is crucial for financial markets. Arbitrageurs aim to make riskless profits by exploiting mispricing of financial securities, by short-selling of overpriced assets and using these proceeds to buy underpriced assets. The activities of buying and selling assets while eliminating away the arbitrage opportunities of mispricing, the market arrives at the *fair price* for various assets, including financial options.

The idea of existence of a positive state price vector as being equivalent to a No Arbitrage (NA) condition was introduced by Ross in 1973 [1]. Risk-neutral pricing
was introduced by Cox and Ross in 1976 ([2], [3]) and formulated as a martingale by Harrison and Kreps in 1979 [4]. This collection of statements in the aforementioned papers was united in the two Fundamental Theorems of Finance. The First Theorem was formulated by Dybvig and Ross in 1987 [5]. From the 2nd Theorem, for single-period models, the market admits a unique asset price if and only if the market is complete. Otherwise, the market is incomplete, and the set of fair prices for a non-replicable option in a single stock model is a non-empty interval. As noted, for example, in [6], real-world financial markets are massively incomplete, therefore it is useful both in theory and practice to narrow down the fair price range in order to identify better deals for financial managers. In the framework of tree methods for financial options pricing, this paper proposes to define this range of fair prices in incomplete markets using the concept of stochastic dominance, and suggests a method to compute it. A price in this narrower range will be called a stochastically fair price.

Stochastic dominance (SD) is a partial order between random variables. This ordering of distribution functions does not depend on specific utility functions of economic agents and does not require restrictive parametric assumptions on distributions. [7]. It was developed in the 1970s, in prominent papers by Hadar and Russel [8] and Hanoch and Levy [9], although similar ideas were proposed as early as 1953 by Masse and Morlat [10]. Its mathematical origins go back to the classic 1934 book on inequalities by Hardy, Littlewood and Polya [11] For a history of research on stochastic orders, see a bibliography [12].

A research team at Google in 2023 tried to find a fast general algorithm of optimization with SD constraints via the Lagrangian, the paper is on ArXiv [13], but the code is yet to be posted.

As opposed to classical arbitrage, Post and Longarela [14] in 2020 introduced a weaker concept of stochastic arbitrage opportunity (SA), based on SD. They propose a linear optimization program to identify active trading strategies based on SA which provides the maximum option premium to an investor. Their approach was refined in 2022 by Beare and Seo [15] based on a 2008 fundamental result by Luedtke [16], with a considerable computational gain in efficiency.

Luedtke [16] states that this approach is based not on majorization theory, but on Strassen theorem. We propose to still use the majorization approach, and introduce a new constraint to make the corresponding matrix not only stochastic, as in [15], but doubly stochastic. We find that this approach produces the necessary dominating payoffs, allowing us to compute the narrower range of fair prices for options. Similar to how the no arbitrage (NA) condition is consistent with a fair price for an asset, we propose that the no stochastic arbitrage condition (NSA) is consistent with the new notion of a stochastically fair price.

This paper is organized as follows: in Section 2 we review concepts of stochastic dominance, majorization and arbitrage. We then extend a well-researched concept
of arbitrage into a relatively new idea of stochastic arbitrage (SA). In Section 3 we proceed to show and compare computational methods for identifying a SA portfolio, starting with [14], improving on it in [15] based on [16], and a 2023 proposal for further improvement in [13]. We propose an LP with an additional constraint. Section 4 provides an illustration of our ideas using a simple trinomial model.

## 2 Stochastic Arbitrage

First, we introduce the concept of stochastic dominance (see, for example, [9], [17]). Assume X and Y are random variables with distribution functions $F_X$ and $F_Y$, respectively. A cumulative distribution function $F_X(x)$ of a real-valued variable X is given by

$$F_X(x) = \mathbb{P}(X \leq x) \qquad (1)$$

where RHS represents the probability that random variable X takes a value less than or equal to $x \in \mathbb{R}$.

**Definition 2.1 (FSD).** X is said to dominate Y with respect to first degree stochastic dominance (FSD), written $X \gtrsim_{FSD} Y$, if

$$F_X(t) \leq F_Y(t), \quad \forall t \in \mathbb{R} \qquad (2)$$

Any rational decision maker would prefer X to Y if $X \gtrsim_{FSD} Y$ [17].

**Definition 2.2 (SSD).** X is said to dominate Y with respect to second degree stochastic dominance (SSD), written $X \gtrsim_{SSD} Y$, if

$$\mathbb{E}(t - X)^+ \leq \mathbb{E}(t - Y)^+, \quad \forall t \in \mathbb{R}, \qquad (3)$$

where $(\cdot)^+ = \max\{\cdot, 0\}$.

Any rational and risk-averse decision maker with an increasing and concave utility function $f: \mathbb{R} \to \mathbb{R}$ would prefer X to Y if and only if $X \gtrsim_{SSD} Y$. Due to assumption of risk-aversion of a decision maker, SSD criterion is important for studying decision-making under risk [17].

Now we review the concept of majorization and its connection to SD. Majorization is defined in [18] as follows:

**Definition 2.3 (Majorization).** For $x, y \in \mathbb{R}^n$,

$$x \prec y \quad \text{if} \quad \left\{ \begin{array}{l} \displaystyle \sum_{i=1}^{k} x_{(i)} \geq \sum_{i=1}^{k} y_{(i)}, \quad k = 1, \dots, n-1 \\ \\ \displaystyle \sum_{i=1}^{n} x_{(i)} = \sum_{i=1}^{n} y_{(i)} \end{array} \right. \qquad (4)$$
where $x_{(1)} \le \dots \le x_{(n)}$ mean that entries of $x$ are rearranged in increasing order.

In this case, $x$ is said to be majorized by $y$, denoted $x \prec y$.

Hardy, Littelwood and Polya [11] showed that for some doubly stochastic matrix $\Psi$,

$$x \prec y \iff x = \Psi y \quad (5)$$

This $\Psi$ is not necessarily unique [18]. We will also need a concept of weak supermajorization.

**Definition 2.4** (Weak Supermajorization). For $x, y \in \mathbb{R}^n$,

$$x \prec y \quad \text{if} \quad \sum_{i=1}^{n} x_{(i)} \ge \sum_{i=1}^{n} y_{(i)}, \quad k = 1, \dots, n \quad (6)$$

A summary for some equivalents for weak supermajorization is provided in [18]. We focus on a connection between weak supermajorization and SSD.

**Theorem 1** (Weak supermajorization equivalents). For $x, y \in \mathbb{R}^n$, the following statements are equivalent:

(1) $x \prec y$

(2) $x \preceq_{SSD} y$

(3) $\sum_{i=1}^{n} \phi(x_i) \le \sum_{i=1}^{n} \phi(y_i)$ for all continuous decreasing convex functions $\phi$

(4) $\sum_{i=1}^{n} (z - x_i)^+ \le \sum_{i=1}^{n} (z - y_i)^+$ for all $z \in \mathbb{R}$

We will use a simplified 1-period option pricing model to introduce the concept of arbitrage [19], [20]. Due to our focus on option pricing, we formulate arbitrage in this context. The model has 2 dates: trading (buying or selling of assets) takes place at $t=0$, with asset payoffs realized at $t=1$. Consider $n$ tradable assets, with $n < \infty$. The set of scenarios, or states of nature, possible at $t=1$, contains $m$ elements, with $2 \le m < \infty$, while all other scenarios are assumed to be impossible (that is, having a probability of 0). Only one of those $m$ possible scenarios occurs at $t=1$.

Assume that an investor already has a position in an index benchmark portfolio (for example, S&P 500 market index) and aims to improve it by adding a self-financing arbitrage (“overlay”) portfolio, so that the payoff distribution of the combined portfolio’s returns dominates the payoff distribution of index portfolio alone.

Let $X$ denote the $m \times n$ matrix of payoffs. Its 1st column is a vector of bond or Treasury bill payoffs, denoted $z$. It is assumed to contain identical entries for every scenario, since T-bill is assumed to be a riskless asset. The 2nd column of $X$ is a payoff distribution of the financial index, denoted by $y' = (y_1, y_2, \dots, y_m)$, sorted $y_1 < \dots < y_m$.

Columns from 3 to $n$ consist of payoff vectors defined by $d_{i,j}$, for the $j$-th financial option and $i$-th scenario. $d_{i,j} = \max(y_i - k_j, 0)$ for call options, or $d_{i,j} = \max(k_j - y_i, 0)$ for put options, where $k_j$ denotes strike prices.

$$X = \begin{bmatrix} z & y_1 & d_{1,3} & \dots & d_{1,n} \\ \vdots & \vdots & \vdots & \dots & \vdots \\ z & y_m & d_{m,3} & \dots & d_{m,n} \end{bmatrix}$$

An investor already holds a benchmark portfolio at $t=0$ with payoff $y$ and wants to augment it with a portfolio $(\alpha, \beta)'$, with $\alpha, \beta \in \mathbb{R}_+^n$. Here $\alpha$ is a number of securities to buy from the market at ask prices $a \in \mathbb{R}_+^n$, and $\beta$ is a number of securities to sell
to the market at bid prices $b \in \mathbb{R}_+^n$, with a common sense consideration $a_i \ge b_i \ge 0$, $i = 1, ..., n$. As noted in [21], ask-bid spreads can be analyzed as taking each asset $j$ as two assets, each with its own single price (one with ask price $a$ and payoff $x_j$, another with bid price $b$ and payoff $-x_j$). Vector of asset prices $\pi$ is

$$ \pi = \begin{pmatrix} a \\ -b \end{pmatrix} \in \mathbb{R}^{2n} \qquad (7) $$

A candidate portfolio is denoted $h$, where

$$ h = \begin{pmatrix} \alpha \\ \beta \end{pmatrix} \in \mathbb{R}_+^{2n} \qquad (8) $$

Cost of a candidate portfolio at $t = 0$ (or initial income) is

$$ -\pi' h = (-a'\alpha + b'\beta) \in \mathbb{R} \qquad (9) $$

Payoff of a candidate portfolio at $t = 1$ is

$$ [X - X] h = X\alpha - X\beta = X(\alpha - \beta) \qquad (10) $$

Now we are ready to define arbitrage for this model, following [21], and introduce stochastic arbitrage, following [14].

**Definition 2.5 (Arbitrage).** An arbitrage is a portfolio $h$ such that $-\pi' h \ge 0$ and $X(\alpha - \beta) \ge 0$, with either $-\pi' h \ne 0$ or $X(\alpha - \beta) \ne 0$ (or both).

**Definition 2.6 (Stochastic Arbitrage (SA)).** A stochastic arbitrage is a portfolio $h$ such that $-\pi' h \ge 0$ and $X(\alpha - \beta) + y \underset{SSD}{\gtrsim} y$.

For SA, solution $(\alpha, \beta)' = (0_n, 0_n)'$ is assumed to be feasible, as in [14]. The following theorem connects arbitrage to stochastic dominance of 1st degree (see [17]).

**Theorem 2 (Arbitrage and FSD).** *Arbitrage implies FSD.*

By analogy with Theorem 2, we propose the following simple lemma:

**Lemma 3 (Arbitrage and Stochastic Arbitrage).** *Arbitrage implies stochastic arbitrage.*

**Proof:** By arbitrage definition, $X(\alpha - \beta) \ge 0$, hence $X(\alpha - \beta) + y \ge y \implies X(\alpha - \beta) + y \underset{SSD}{\gtrsim} y$.

**Corollary 3.1.** *If the dataset admits a arbitrage opportunity, with the associated portfolio, than this portfolio is also a SA portfolio.*

We note that an arbitrage portfolio, while also being an SA portfolio, is not necessarily the optimal SA portfolio. That is, there might be a different SA portfolio which improves an objective function of the SA program more than the arbitrage portfolio does. It is well-known that for a market to be in a state of equilibrium, it is necessary that there are no arbitrage opportunities present. A price under no arbitrage (NA) condition is called a fair price. Now we define a new concept of a stochastically fair
price. First, we introduce a concept of no stochastic arbitrage (NSA) condition, by analogy to a NA condition.

**Definition 2.7** (No Stochastic Arbitrage condition (NSA)). *No stochastic arbitrage condition is satisfied if $\mathbb{h} \in \mathbb{R}^n$ such that $-\pi'\mathbb{h} \ge 0$ and $X(\alpha - \beta) + y \stackrel{\sim}{\ge} y_{SSD}$*

**Definition 2.8.** *Stochastically fair price is a price under the NSA condition.*

Stochastically fair price is a fair price which precludes a stochastic arbitrage.

For a non-replicable option with a vector of payoffs $d$, ask price $\pi_d^a$ and bid price $\pi_d^b$, the following theorem can be found in [22] (thm. 3.20).

**Theorem 4** (Non-replicable options). *The set of fair prices for a non-replicable option in a single stock model is a non-empty interval $(\pi_d^b, \pi_d^a)$.*

Despite the fact that according to Theorem 4, interval $(\pi_d^b, \pi_d^a)$ precludes arbitrage, it does allow for a stochastic arbitrage. Hence we propose the following conjectures:

**Proposition 1** (SA existence). *Within the range of fair prices $(\pi_d^b, \pi_d^a)$ there exists a set of prices that allow for SA.*

**Proposition 2** (SA initial income linearity). *For a set of fair prices that allow for SA in a sense of Proposition 1, initial income generated by SA is linear in fair prices.*

These propositions will be illustrated in Section 4. In the next section, we develop a linear programming model based on SA.

## 3 Linear Programming Models for Stochastic Arbitrage

In this section, we introduce Linear Programming (LP) models for producing SA. Post and Longarela [14] propose a linear optimization program to identify active trading strategies based on SA. Their approach was refined in 2022 by Beare and Seo [15] based on a 2008 fundamental result by Luedtke [16], with a considerable computational gain in efficiency.

Luedtke states that this approach is based not on majorization theory, but on Strassen theorem. We propose to still use the majorization approach, and introduce a new constraint to make the corresponding matrix not only stochastic, as in [15], but doubly stochastic. We find that this approach produces the necessary dominating payoffs, allowing us to compute a narrower range of fair prices for options, which we called stochastically fair prices.

Following [15], let $C_1$ and $C_2$ be $k \times n$ matrices of the LHS portfolio constraints, for some $k$, and $c = (c_\alpha, c_\beta)'$ is a vector of RHS portfolio constraints. This constraint simply indicates a market availability of assets to trade. All feasible combinations of layover portfolios is a polytope $P := \{(\alpha, \beta) \in \mathbb{R}_+^n \times \mathbb{R}_+^n : C_1\alpha + C_2\beta \le c\}$. For example, if $c_\alpha \in \mathbb{R}^n$ denotes the maximum long position in each option, and $c_\beta \in \mathbb{R}^n$ denotes the maximum short position in each option, then

$$C_1 = \begin{bmatrix} I_n \\ 0_{n \times n} \end{bmatrix}, \quad C_2 = \begin{bmatrix} 0_{n \times n} \\ I_n \end{bmatrix}, \quad c = \begin{pmatrix} c_\alpha \\ c_\beta \end{pmatrix} \quad (11)$$
The goal of the SA optimization program is to find such a SA portfolio $(\alpha, \beta)'$ that generates a positive initial income $-\pi'h > 0$ and, when added to an index payoffs $y$, will SSD-dominate it in terms of payoffs: $X\alpha - X\beta + y \underset{SSD}{\gtrsim} y$. If it is possible to find such a SA portfolio, then SA opportunity exists by definition 2.6.

Let $\hat{Y}$ denote a $m \times m$ strictly lower-triangular matrix with $(s, j)$ entry equal to $y_s - y_j$, for $s > j$. By construction, all entries are nonnegative since $y$ is ranked in increasing order, thus $\hat{Y}$ has the following structure:

$$ \hat{Y} = \begin{bmatrix} 0 & & & & \\ y_2 - y_1 & 0 & & & \\ y_3 - y_1 & y_3 - y_2 & 0 & & \\ \vdots & \vdots & \vdots & \ddots & \\ y_m - y_1 & y_m - y_2 & \cdots & y_m - y_{m-1} & 0 \end{bmatrix} $$

We note that $\hat{Y}$ is non-invertible. From [14], [16] and [15], the LP problem for SA is

$$ \underset{\alpha, \beta}{\text{maximize}} \quad f(\alpha, \beta) = -a'\alpha + b'\beta \quad (12a) $$

$$ \text{subject to} \quad -X(\alpha - \beta) + \Psi y \leq y, \quad (12b) $$

$$ \hat{Y}\xi \leq \hat{Y}p, \quad (12c) $$

$$ \Psi'p - \xi = 0_m, \quad (12d) $$

$$ \Psi 1_m = 1_m, \quad (12e) $$

$$ C_1\alpha + C_2\beta \leq c, \quad (12f) $$

$$ \alpha, \beta \geq 0_n, \quad (12g) $$

$$ \Psi \geq 0_{m \times m} \quad (12h) $$

where $\xi$ is some $m \times 1$ nonnegative vector and $p$ is a vector of physical probabilities for scenarios, following [15]. We propose to add one more constraint (13f), producing our LP formulation:

$$ \underset{\alpha, \beta}{\text{maximize}} \quad f(\alpha, \beta) = -a'\alpha + b'\beta \quad (13a) $$

$$ \text{subject to} \quad -X(\alpha - \beta) + \Psi y \leq y, \quad (13b) $$

$$ \hat{Y}\xi \leq \hat{Y}p, \quad (13c) $$

$$ \Psi'p - \xi = 0_m, \quad (13d) $$

$$ \Psi 1_m = 1_m, \quad (13e) $$

$$ \Psi' 1_m = 1_m, \quad (13f) $$

$$ C_1\alpha + C_2\beta \leq c, \quad (13g) $$

$$ \alpha, \beta \geq 0_n, \quad (13h) $$
$$ \Psi \geq 0_{m \times m} \qquad (13i) $$

LP (13a-13i) creates a $5m \times (2n+m^2+m)$ block matrix of constraints (as opposed to $(m^2+m) \times (2n+m^2)$ in Post, Longarela [14]), except for constraints (13g-13i) of lower and upper bounds on $\alpha, \beta, \Psi$, that can be expressed in LP solvers outside the matrix of constraints.

The new constraint (13f) together with (13e) makes $\Psi$ a doubly stochastic matrix. For references on doubly stochastic matrices and their applications, see [18]. Concerning applications for SD and asset pricing, see [23]. In the following we will solve only LP program (13a-13i), instead of (12a-12h).

We denote $1_{m \times m}^{(j)}$ a $m \times m$ matrix with ones in row $j$ and zeros everywhere else. Similarly, $p_{m \times m}^{(j)}$ is a $m \times m$ matrix with vector $p = (p_1, ..., p_m)$ in row $j$ and zeros everywhere else. Denote $\tilde{\Psi}$ as a vector of entries of $\Psi$ taken in columns. For a vector of unknowns $\eta = (\alpha, \beta, \tilde{\Psi}, \xi)' = (\alpha_1, ..., \alpha_n, \beta_1, ..., \beta_n, \Psi_{1,1}, \Psi_{2,1}, ..., \Psi_{m,m}, \xi_1, ..., \xi_m)'$, the matrix of constraints for (13a-13f) has a block structure in Fig. (1) and denoted $G$. Denoting the RHS constraints as $g = (y, \hat{Y}p, 0_n, 1_n, 1_n)'$, now LP (13a-13i) can be compactly rewritten as

$$ \underset{\alpha, \beta}{\text{maximize}} \quad f(\alpha, \beta) = -a'\alpha + b'\beta \qquad (14a) $$

$$ \text{subject to} \quad G\eta \leq g, \qquad (14b) $$

$$ C_1\alpha + C_2\beta \leq c, \qquad (14c) $$

$$ \alpha, \beta \geq 0_n, \qquad (14d) $$

$$ \Psi \geq 0_{m \times m} \qquad (14e) $$

The following proposition is a direct consequence of (5).

**Proposition 3 (SA via doubly stochastic matrix).** *Stochastic dominance condition*

$$ y + X(\alpha - \beta) \underset{\text{SSD}}{\gtrsim} y \qquad (15) $$

is equivalent to

$$ y + X(\alpha - \beta) \geq \Psi y \qquad (16) $$

where doubly stochastic matrix $\Psi$ is constrained by $\hat{Y}\Psi'p \leq \hat{Y}p$.

Eq.(16) means that the index payoff $y$ enhanced by the payoff $X(\alpha - \beta)$ of the overlay portfolio $(\alpha, \beta)'$ is always larger or equal to some averaging of entries of payoff $y$ consistent with a nonnegative vector $\xi$.

A code to solve LP (14a)-(14e) was developed in MATLAB employing the MOSEK solver.
<table><caption>Figure 1: Matrix G of constraints for LP (13a - 13f)</caption><tr><td>-X<sub>m×n</sub></td><td>X<sub>m×n</sub></td><td>I<sub>m</sub>y<sub>1</sub></td><td>I<sub>m</sub>y<sub>2</sub></td><td>...</td><td>I<sub>m</sub>y<sub>m</sub></td><td>0<sub>m×m</sub></td></tr><tr><td>0<sub>m×n</sub></td><td>0<sub>m×n</sub></td><td>0<sub>m×m</sub></td><td>0<sub>m×m</sub></td><td>...</td><td>0<sub>m×m</sub></td><td>Ŷ<sub>m×m</sub></td></tr><tr><td>0<sub>m×n</sub></td><td>0<sub>m×n</sub></td><td>-p<sup>(1)</sup><sub>m×m</sub></td><td>-p<sup>(2)</sup><sub>m×m</sub></td><td>...</td><td>-p<sup>(m)</sup><sub>m×m</sub></td><td>I<sub>m</sub></td></tr><tr><td>0<sub>m×n</sub></td><td>0<sub>m×n</sub></td><td>I<sub>m</sub></td><td>I<sub>m</sub></td><td>...</td><td>I<sub>m</sub></td><td>0<sub>m×m</sub></td></tr><tr><td>0<sub>m×n</sub></td><td>0<sub>m×n</sub></td><td>1<sup>(1)</sup><sub>m×m</sub></td><td>1<sup>(2)</sup><sub>m×m</sub></td><td>...</td><td>1<sup>(m)</sup><sub>m×m</sub></td><td>0<sub>m×m</sub></td></tr></table>

## 4 Trinomial model: Stochastic Arbitrage bounds

Consider a 1-period trinomial model 1 (Fig.2), with $m = 3$ scenarios and $n = 2$ base assets (bond and stock), which we borrow from [22], (referred there as model 3.1). We shall use this example to demonstrate that employing the framework of the previous Section, we find that SA exists and the corresponding optimal portfolios can be calculated using LP (13a-13i).

$$ 
\begin{array}{l}
\begin{array}{c}
S_u = 230 \\
\hline
S_0 = 220 \frac{q_2}{q_3} S_m = 210 \\
\hline
S_d = 190
\end{array}
\\
B_0 = 1 \quad B_1 = 1
\end{array}
$$

<caption>Figure 2: 1-period trinomial model 1, for stock S and bond B</caption>

Prices at $t = 0$ for bond B and stock S are $B_0 = 1$, $S_0 = 220$. Payoffs of the stock at $t = 1$ can be represented as $y = (y_1, y_2, y_3)' = (S_d, S_m, S_u)'$, and payoff matrix as

$$X = \begin{bmatrix} z & y_1 \\ z & y_2 \\ z & y_3 \end{bmatrix}$$
which for model 1 in Fig. 2 is

$$X = \begin{bmatrix} 1 & 190 \\ 1 & 210 \\ 1 & 230 \end{bmatrix}$$

Riskless rate of return is 0%. The market is incomplete, hence there is no unique risk-neutral probability vector and no unique vector of fair prices. Employing the pricing equation $\pi = \mathbb{E}_Q[\bar{X}]$ for discounted payoffs $\bar{X}$, using $\sum q_i = 1$ and setting a free variable $q_1 = \lambda$, for some $0 < \lambda < 1$, obtain

$$S_O = \bar{y}_1\lambda + \bar{y}_2(1 - q_2 - \lambda) + \bar{y}_3q_3 = \lambda(\bar{y}_1 - \bar{y}_2) + q_3(\bar{y}_3 - \bar{y}_2) + \bar{y}_2$$

$$0 < q_3 = \frac{S_0 - \lambda(\bar{y}_1 - \bar{y}_2) - \bar{y}_2}{\bar{y}_3 - \bar{y}_2} < 1$$

which gives bounds

$$\frac{\bar{y}_2 - S_0}{\bar{y}_3 - \bar{y}_1} < \lambda < \frac{\bar{y}_3 - S_0}{\bar{y}_3 - \bar{y}_1} \qquad (17)$$

Similarly, obtain bounds

$$\frac{\bar{y}_2 - S_0}{\bar{y}_2 - \bar{y}_1} < \lambda < \frac{\bar{y}_3 - S_0}{\bar{y}_2 - \bar{y}_1} \qquad (18)$$

on top of

$$0 < \lambda < 1 \qquad (19)$$

Combining (17) - (19), obtain bounds

$$\max\left\{\frac{\bar{y}_2 - S_0}{\bar{y}_3 - \bar{y}_1}, \frac{\bar{y}_2 - S_0}{\bar{y}_2 - \bar{y}_1}, 0\right\} < \lambda < \min\left\{\frac{\bar{y}_3 - S_0}{\bar{y}_3 - \bar{y}_1}, \frac{\bar{y}_3 - S_0}{\bar{y}_2 - \bar{y}_1}, 1\right\} \quad (20)$$

Expression (20) provides bounds on $q_1$ risk-neutral probability for smallest scenario payoff of index $y$ at $t = 1$, with the remaining risk-neutral probabilities $q_2, q_3$ obtained from

$$\begin{pmatrix} S_0 \\ 1 \end{pmatrix} = \begin{bmatrix} \bar{y}_1 & \bar{y}_2 & \bar{y}_3 \\ 1 & 1 & 1 \end{bmatrix} \begin{pmatrix} \lambda \\ q_2 \\ q_3 \end{pmatrix}$$

Solving

$$\begin{bmatrix} \bar{y}_2 & \bar{y}_3 \\ 1 & 1 \end{bmatrix} \begin{pmatrix} q_2 \\ q_3 \end{pmatrix} = \begin{pmatrix} S_0 - \bar{y}_1 \lambda \\ 1 - \lambda \end{pmatrix}$$

for $q_2, q_3$, obtain

$$\begin{pmatrix} q_2 \\ q_3 \end{pmatrix} = \frac{1}{\bar{y}_2 - \bar{y}_3} \begin{pmatrix} S_0 - \bar{y}_1 \lambda - \bar{y}_3 + \bar{y}_3 \lambda \\ -S_0 + \bar{y}_1 \lambda + \bar{y}_2 - \bar{y}_2 \lambda \end{pmatrix} \qquad (21)$$

Bounds on $q$ provide bounds on corresponding fair prices for options on index $y$, since violations of these bounds lead to arbitrage. Alternatively, following [24], [25], as explained in [26], denoting

$$f_q := \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad V := \begin{bmatrix} \bar{y}_1 & \bar{y}_2 & \bar{y}_3 \\ 1 & 1 & 1 \end{bmatrix}, \quad v := \begin{pmatrix} S_0 \\ 1 \end{pmatrix}$$
we can solve LP

$$ \text{minimize} \quad f'_{q}q \qquad (22a) $$

$$ \text{subject to} \quad Vq = v, \qquad (22b) $$

$$ q_i \ge 0, \quad i = 1, \dots, m \qquad (22c) $$

to find the lower bound on $q_1$, and solve LP

$$ \text{maximize} \quad f'_{q}q \qquad (23a) $$

$$ \text{subject to} \quad Vq = v, \qquad (23b) $$

$$ q_i \ge 0, \quad i = 1, \dots, m \qquad (23c) $$

to find the upper bound on $q_1$. For model 1, a closed-form (20) or LPs (22-23) give bounds $0 < \lambda < 0.25$, consistent with [22].

Adding a Put option on index $y$ with strike $K = 220$, the payoff matrix $X$ with additional vector $d$ of option payoffs becomes

$$ X = \begin{bmatrix} z & y_1 & d_1 \\ z & y_2 & d_2 \\ z & y_3 & d_3 \end{bmatrix} = \begin{bmatrix} z & y_1 (K - y_1)^+ \\ z & y_2 (K - y_2)^+ \\ z & y_3 (K - y_3)^+ \end{bmatrix} $$

which for model 1 is

$$ X = \begin{bmatrix} 1 & 190 & 30 \\ 1 & 210 & 10 \\ 1 & 230 & 0 \end{bmatrix} $$

Trying to solve using the pricing equation $\pi = X'\pi_w$ with $\pi_w > 0$, we see that no such state price vector $\pi_w$ exists, confirming that there is no unique risk-neutral probability vector and no unique vector of fair prices. As per thm. 4, the non-attainable option in this model admits a range of fair prices $(\pi_d^b, \pi_d^a)$.

Having obtained bounds $0 < \lambda < 0.25$ on risk-neutral probability $q_1$, bounds on the option price via $\pi = E_Q[X]$ are $E_{Q_\lambda}[\bar{d}] \in (5, 7.5)$ excluding the endpoints.

For lower bound on option price $\pi_d^b = E_{Q_{\lambda=0}}[\bar{d}] = 5$, corresponding (unattainable) risk-neutral probability vector $q$ is $(\lambda, q_2, q_3)' = (0, 0.5, 0.5)'$. For upper bound on option price $\pi_d^a = E_{Q_{\lambda=0.25}}[\bar{d}] = 7.5$, corresponding (unattainable) risk-neutral probability vector $q$ is $(\lambda, q_2, q_3)' = (0.25, 0, 0.75)'$.

The set $Q$ of risk-neutral probabilities is convex and shown on the $(q_2, q_3)$-plane on Fig. 3 (blue segment). For a recent reference on optimization on a set of risk-neutral probabilities, see [27].

Now we apply LP (13a-13i). We assume every asset has 100 units available for buying and selling (or selling short). We model vector of physical probabilities $p$ as distributed normally, with an equally-distributed case added. We assume no ask-bid
<figure><img src="image_1.png" /><figcaption>Figure 3: set Q of risk-neutral probabilities for model 1</figcaption></figure>

<table><caption>Figure 4: matrix of constraints G for model 1</caption><tbody><tr><td>-1 -190 -30</td><td>1 190 30</td><td>190 0 0</td><td>210 0 0</td><td>230 0 0</td><td>0 0 0</td></tr><tr><td>-1 -210 -10</td><td>1 210 10</td><td>0 190 0</td><td>0 210 0</td><td>0 230 0</td><td>0 0 0</td></tr><tr><td>-1 -230 0</td><td>1 230 0</td><td>0 0 190</td><td>0 0 210</td><td>0 0 230</td><td>0 0 0</td></tr><tr><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td></tr><tr><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>20 0 0</td></tr><tr><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>40 20 0</td></tr><tr><td>0 0 0</td><td>0 0 0</td><td>-p<sub>1</sub> -p<sub>2</sub> -p<sub>3</sub></td><td>0 0 0</td><td>0 0 0</td><td>1 0 0</td></tr><tr><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>-p<sub>1</sub> -p<sub>2</sub> -p<sub>3</sub></td><td>0 0 0</td><td>0 1 0</td></tr><tr><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>-p<sub>1</sub> -p<sub>2</sub> -p<sub>3</sub></td><td>0 0 1</td></tr><tr><td>0 0 0</td><td>0 0 0</td><td>1 0 0</td><td>1 0 0</td><td>1 0 0</td><td>0 0 0</td></tr><tr><td>0 0 0</td><td>0 0 0</td><td>0 1 0</td><td>0 1 0</td><td>0 1 0</td><td>0 0 0</td></tr><tr><td>0 0 0</td><td>0 0 0</td><td>0 0 1</td><td>0 0 1</td><td>0 0 1</td><td>0 0 0</td></tr><tr><td>0 0 0</td><td>0 0 0</td><td>1 1 1</td><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td></tr><tr><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>1 1 1</td><td>0 0 0</td><td>0 0 0</td></tr><tr><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>0 0 0</td><td>1 1 1</td><td>0 0 0</td></tr></tbody></table>

spreads $a = b$ and no transaction costs, but note that this formulation allows for their easy introduction. Matrix G is shown in Fig. (4).

LP produces SA overlay portfolios that, augmenting an index portfolio $y$ above, provide a strictly positive initial income between 20 and 15.75 dollars for the range of fair prices ($\pi_d^b, \pi_d^a$) = (5, 7.5), as shown in Fig. 5.

For example, for risk-neutral probability vector $q = (0.001, 0.498, 0.501)'$ and associated fair option price $\pi_d = \mathbb{E}_{Q_{\lambda=0.001}}[\bar{d}] = 5.01$, LP (14a-14e) has $a = (-1, -220, -5.01)'$, $b = (1, 220, 5.01)'$, $g =$
<figure><img src="image_2.png" /><figcaption>Figure 5: Initial income for trinomial model 1</figcaption></figure>

$(190, 210, 230, 0, 16, 32.4, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1)'$, $c = (100, 100, 100, 100, 100, 100)'$, and

$$C_1 = \begin{bmatrix} I_3 \\ 0_{3 \times 3} \end{bmatrix}, \quad C_2 = \begin{bmatrix} 0_{3 \times 3} \\ I_3 \end{bmatrix}$$

LP outputs a SA overlay portfolio $\alpha = (100, 0, 1.699)'$, $\beta = (0, 0.584, 0)'$ for an initial income $-a'\alpha + b'\beta = 19.98$ dollars at $t = 0$, and SA enhanced portfolio has the payoff

$$y + X(\alpha - \beta) = \begin{pmatrix} 190 \\ 210 \\ 230 \end{pmatrix} + \begin{pmatrix} 40.00 \\ -5.66 \\ -34.34 \end{pmatrix} = \begin{pmatrix} 230 \\ 204.34 \\ 195.66 \end{pmatrix} \underset{\text{SSD}}{\gtrsim} \begin{pmatrix} 190 \\ 210 \\ 230 \end{pmatrix} = y$$

The dominance relation holds since combined portfolio payoff after ranking is $(195.66, 204.34, 230)'$, which dominates index $y$ distribution-wise.

For model 1, corresponding to the optimal solution of LP (13a-13i), the following doubly stochastic matrix is obtained

$$\Psi = \begin{bmatrix} 0 & 0 & 1 \\ 0.28 & 0.72 & 0 \\ 0.72 & 0.28 & 0 \end{bmatrix}$$

This numerical result is consistent with the form of the doubly stochastic matrices found in literature, (e.g. [28]), for $m = 3$,

$$\Psi = \begin{bmatrix} \rho_1 & \rho_2 & 1 - \rho_1 - \rho_2 \\ \rho_3 & \rho_4 & 1 - \rho_3 - \rho_4 \\ 1 - \rho_1 - \rho_3 & 1 - \rho_2 - \rho_4 & \rho_1 + \rho_2 + \rho_3 + \rho_4 - 1 \end{bmatrix}$$

In this model, we have $\rho_1 = \rho_2 = 0$, $\rho_3 = 0.28$, $\rho_4 = 0.72$.

Recalling our Proposition 3 that $y + X(\alpha - \beta) \underset{\text{SSD}}{\gtrsim} y \iff y + X(\alpha - \beta) \ge \Psi y$, we
note that

$$y + X(\alpha - \beta) = \begin{pmatrix} 230 \\ 204.34 \\ 195.66 \end{pmatrix} \ge \begin{pmatrix} 230 \\ 204.34 \\ 195.66 \end{pmatrix} = \Psi y$$

We observe that the SA overlay portfolio and doubly stochastic matrix $\Psi$ obtained for fair option price $\pi_D = 5.01$ remain the same throughout the entire range of fair prices $(\pi_D^b, \pi_D^a) = (5, 7.5)$, always producing a stochastic arbitrage. Model 1 appears to be unbalanced in a sense that trading is stochastically beneficial at any fair price for the option. Their is no stochastically fair price in this model.

Now consider a trinomial model 2 (Fig.6), which we borrow from [22], (referred there as model 3.2).

$$ \begin{array}{c} S_u = 110 \\ S_0 = 80 \quad \begin{array}{c} q_1 \\ q_2 \\ q_3 \end{array} \quad S_m = 90 \\ S_d = 75 \\ B_0 = 10 \quad \text{---} \quad B_1 = 11 \end{array} $$

<caption>Figure 6: 1-period trinomial model 2, for stock S and bond B</caption>

Prices at $t=0$ for bond and stock are $B_0 = 10, S_0 = 80$. Payoffs at $t=1$ can be represented as

$$X = \begin{bmatrix} 11 & 75 \\ 11 & 90 \\ 11 & 110 \end{bmatrix}$$

Riskless rate of return is 10%. Like in model 1, the market is incomplete, hence there is no unique risk-neutral probability vector and no unique vector of fair prices. Employing (20), obtain bounds $0.1333 < \lambda < 0.6286$ on $q_1$, with corresponding $q_2, q_3$ from (21).

Adding a Call option on index $y$ with strike $K = 80$, the payoff matrix $X$ with additional vector $d$ of option payoffs becomes

$$X = \begin{bmatrix} z & y_1 & d_1 \\ z & y_2 & d_2 \\ z & y_3 & d_3 \end{bmatrix} = \begin{bmatrix} z & y_1 & (y_1 - K)^+ \\ z & y_2 & (y_2 - K)^+ \\ z & y_3 & (y_3 - K)^+ \end{bmatrix}$$

which for model 2 is

$$X = \begin{bmatrix} 11 & 75 & 0 \\ 11 & 90 & 10 \\ 11 & 110 & 30 \end{bmatrix}$$
Bounds on the fair price for the Call are $E_{Q_{\lambda}}[d] \in (7.8788, 10.1299)$. excluding the endpoints, for any $0 < \lambda < 1$. For lower bound on option price $\pi_d^b = E_{Q_{\lambda=0.1333}}[d] = 7.8788$, corresponding (unattainable) risk-neutral probability vector $q$ is $(\lambda, q_2, q_3)' = (0.1333, 0.8666, 0)'$. For upper bound on option price $\pi_d^a = E_{Q_{\lambda=0.6286}}[d] = 10.1299$, corresponding (unattainable) risk-neutral probability vector $q$ is $(\lambda, q_2, q_3)' = (0.6286, 0, 0.3714)'$.

Solving LP (13a-13i) for model 2 produces SA overlay portfolios that, augmenting an index portfolio $y$, provide a strictly positive initial income between 10 and 6.75 dollars for the range of fair prices $(\pi_d^b, \pi_d^a) = (7.8788, 10.1299)$. Results are shown in Fig.7.

<figure><img src="image_3.png" /><figcaption>Figure 7: Initial income for trinomial model 2</figcaption></figure>

In accordance with Proposition 2, for a set of prices that allow SA, initial income is linear in fair prices. For a range of fair prices (9.106, 9.4477) which do not allow for SA, initial income is 0, hence this is a range of stochastically fair prices.

For the range of fair prices (7.8788, 9.105) “on the left” of the stochastically fair price range (9.106, 9.4477), the doubly stochastic matrix and SA overlay portfolio are

$$ \Psi = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad \alpha = \begin{pmatrix} 52.5 \\ 0 \\ 8.25 \end{pmatrix}, \quad \beta = \begin{pmatrix} 0 \\ 7.5 \\ 0 \end{pmatrix} $$

We note that doubly stochastic matrix $\Psi$ is binary (becomes a permutation matrix) in this model for any fair price of the option, meaning that the model admits not just SSD, but in fact, FSD, as noted by Luedtke [16].

For example, for a fair option price $\pi_d = 7.8794$, the SA enhanced portfolio payoff is

$$ y + X(\alpha - \beta) = \begin{pmatrix} 75 \\ 90 \\ 110 \end{pmatrix} + \begin{pmatrix} 15 \\ -15 \\ 0 \end{pmatrix} = \begin{pmatrix} 90 \\ 75 \\ 110 \end{pmatrix} \underset{\text{FSD}}{\gtrsim} \begin{pmatrix} 75 \\ 90 \\ 110 \end{pmatrix} = y $$

which is not only SSD but also FSD, keeping in mind the symmetry of the SD relation. This strategy brings in 9.99 dollars in initial income at $t = 0$.
For the range of fair prices (9.4478, 10.1299) “on the right” of the stochastically fair price (9.106, 9.4477), the doubly stochastic matrix and SA overlay portfolio (which we leave as MATLAB outputs it without simplification $\alpha - \beta$) are

$$ \Psi = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} \quad \alpha = \begin{pmatrix} 45.45 \\ 100 \\ 90 \end{pmatrix}, \quad \beta = \begin{pmatrix} 100 \\ 92 \\ 100 \end{pmatrix} $$

For example, for a fair option price $\pi_d = 10.1292$, the enhanced portfolio payoff is

$$ y + X(\alpha - \beta) = \begin{pmatrix} 75 \\ 90 \\ 110 \end{pmatrix} + \begin{pmatrix} 0 \\ 20 \\ -20 \end{pmatrix} = \begin{pmatrix} 75 \\ 110 \\ 90 \end{pmatrix} \underset{FSD}{\gtrsim} \begin{pmatrix} 75 \\ 90 \\ 110 \end{pmatrix} = y $$

which is not only SSD but also FSD, keeping in mind the symmetry of the SD relation. This strategy brings in 6.7 dollars in initial income at $t = 0$.

For future studies, adding ask-bid spreads and transaction costs is a natural extension of the model, and the proposed LP makes it straightforward. Another extension is to expand the set of benchmark portfolios (as opposed to containing only the index), such that the goal becomes to find a combined portfolio that dominates all of them, not just the index. Also multi-time period models can be considered.

# 5 Conclusion

In this paper, a new notion of stochastically fair price within the fair price range is proposed, which is employed for pricing financial options in incomplete markets. We present a methodology to compute them for options by using the stochastic optimization techniques with stochastic constraints involving doubly stochastic matrices. We use a simple trinomial model for illustration.

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