Stochastic dominance, arbitrage, and risk management
- Concepts of stochastic dominance
- Relation with fundamentals of pricing and risk management of multi-asset portfolios
Hello all! Hope you are well and safe.
What I like the most on quantitative finance modeling, either on pricing or risk management, is the fact that it is all about probability. Risk management, for instance, embeds the analysis of loss probability of determined portfolio, while derivative pricing encompasses profit and loss probabilities, aiming to find the free-arbitrage value of the instrument.
This post intends to discuss the intuitive picture behind the probabilistic approach used in finance. In this sense, we will here rely on the concept of stochastic dominance, linking the discussion with the main concepts of quantitative finance.
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| Photo by Nathan Dumlao on Unsplash |
Arbitrage and first order stochastic dominance
Suppose we have to choose and bet between two different unfair dice. Moreover, suppose the payoff $Y$ of this game is given by $Y = X-4$, where $X$ is the dice outcome. The question is simple, what dice should we choose? The answer is even simpler, we choose the one that presents the highest probability of $X \geq 4$. This is the intuitive picture behind stochastic dominance. One chooses the bet that provides (almost) surely the best result.
Stochastic dominance can be defined in terms of different orders which can be directly related to the moments of the probability distributions which are being compared. Don't worry because this will become clearer throughout this post. For now, suppose we have a concave utility function $u(x)$ that describes some quantity, such as a derivative payoff or portfolio value, and that $x$ is a random variable that can be statistically described by means of two probability distributions $P_A$ and $P_B$. One says that distribution $A$ stochastically dominates $B$ in terms of first order, if one observes that:
$\int_{-\infty}^{x} u(t) [f_B(t) - f_A(t)] dt \geq 0 \quad \forall t$,
where $f_A$ and $f_B$ are the probability densities of $A$ and $B$, respectively. Definition above can also be written in terms of the cumulative distribution functions as $F_B(x) \geq F_A(x) \quad \forall x$. In our example of the unfair dice game, each dice would have a probability distribution, $A$ or $B$.
$\int_{-\infty}^{x} u(t) [f_B(t) - f_A(t)] dt \geq 0 \quad \forall t$,
where $f_A$ and $f_B$ are the probability densities of $A$ and $B$, respectively. Definition above can also be written in terms of the cumulative distribution functions as $F_B(x) \geq F_A(x) \quad \forall x$. In our example of the unfair dice game, each dice would have a probability distribution, $A$ or $B$.
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| Option A is first order stochastically dominant over option B. Image from https://www.vosesoftware.com/riskwiki/Stochasticdominancetests.php |
The intuitive picture behind the first order stochastic dominance is based on the analysis of the expected value with respect to the probability distributions being compared. Put in other words, the first order stochastic dominance is established when the mean value of a random variable is higher under a specific probability distribution.
The application of the first order stochastic dominance in finance becomes quite clear if we consider an investor that is choosing between two different portfolios. Depending on his/her modeling assumptions, she will try to identify the portfolio that stochastically dominates. This is exactly the case when an investor is checking for trends in stocks. She is looking for a situations similar to the our initial example game, that is, the highest probability of $X>4$. When talking about pricing of derivatives, the usual formalism, such as the one in Black-Scholes formulation, intends to find the expected value when no first order stochastic dominance appears.
As you may have noticed the pricing formula for derivatives clearly encompasses the idea of removing first order stochastic dominance. Let's take a closer look at the price of a security $V(t)$ with maturity at $T$:
$V(t) = \aleph_A(t) E_A[ \frac{V(T)}{\aleph_A(T)} | \digamma(t)] = \aleph_B(t) E_B[ \frac{V(T)}{\aleph_B(T)} | \digamma(t)]$
where $V(t)$ is the security price at $t$, $\aleph_A$ and $E_A[...]$ are the numeraire and expectation value operator under probability measure $A$, and $\digamma(t)$ is the filtration at $t$. Formulation above contains the fact that under a proper relation of the probability density and numeraire we obtain the same expected value under different probability measures. In other words, one calculates an unique value that guarantees arbitrage-free price no matter under which strategy an investor decides to use or under which probability measure she is relying on.
Risk management: higher order stochastic dominance
First order stochastic dominance is related to the expected value. In terms of finance, it will be related to the arbitrage-free price of a certain security (or portfolio). Nevertheless, price is not the only parameter to take into account when evaluating an investment, but also the risk involved should be taken into account. Risk management is focused on higher order moments of a probability distribution, named variance, skew, and kurtosis. This is because the focus is on the left tail of the distribution, that is, the loss probability.
In the same way that an arbitrage-free price is obtained when first order stochastic dominance is removed, the parameters for risk management mentioned above depend on the absence of higher order stochastic dominance. Let's first check the intuitive picture behind the previous statement and then we present the corresponding formulations.
Consider we have two securities of the same asset class both quoted at a fair price, but with different underlyings. These could be, for instance, two European vanilla options with two different underlyings. Without any further assumptions, an investor will choose the asset that embeds the lowest risk, which in a first order approximation will be represented by the lowest variance. In a very strict approach, one should account for higher order stochastic dominance when calculating risk. Of course, this is not usually the case. For instance, historical VaR does not account at all about stochastic dominance, simply because the assumptions behind it are different (see more in this previous post: Data-driven and market-driven models).
For second order stochastic dominance, one says that $A$ is second order stochastically dominant over $B$ if the following relation is satisfied:
$\int_{-\infty}^{x}\int_{-\infty}^{x_1} u(t) [f_B(t) - f_A(t)] dt dx_1 \geq 0 \quad \forall x$
Likewise, formulation for higher order stochastic dominance follows the pattern:
$\int_{-\infty}^{x} \int_{-\infty}^{x_i}...\int_{-\infty}^{x_1} u(t) [f_B(t) - f_A(t)] dt dx_1...dx_i \geq 0 \quad \forall x$
In the same way that an arbitrage-free price is obtained when first order stochastic dominance is removed, the parameters for risk management mentioned above depend on the absence of higher order stochastic dominance. Let's first check the intuitive picture behind the previous statement and then we present the corresponding formulations.
Consider we have two securities of the same asset class both quoted at a fair price, but with different underlyings. These could be, for instance, two European vanilla options with two different underlyings. Without any further assumptions, an investor will choose the asset that embeds the lowest risk, which in a first order approximation will be represented by the lowest variance. In a very strict approach, one should account for higher order stochastic dominance when calculating risk. Of course, this is not usually the case. For instance, historical VaR does not account at all about stochastic dominance, simply because the assumptions behind it are different (see more in this previous post: Data-driven and market-driven models).
For second order stochastic dominance, one says that $A$ is second order stochastically dominant over $B$ if the following relation is satisfied:
$\int_{-\infty}^{x}\int_{-\infty}^{x_1} u(t) [f_B(t) - f_A(t)] dt dx_1 \geq 0 \quad \forall x$
Likewise, formulation for higher order stochastic dominance follows the pattern:
$\int_{-\infty}^{x} \int_{-\infty}^{x_i}...\int_{-\infty}^{x_1} u(t) [f_B(t) - f_A(t)] dt dx_1...dx_i \geq 0 \quad \forall x$
Final comments
This post does not intend to exhaust the topic and is not interested on a strict formalism. The idea is to shed some light on the connection of an abstract statistical concept with the fundamentals of pricing and risk management. A careful reader should understand that this post is related to the details of the implicit assumptions of pricing and risk measure calculation. In fact, the assumptions for each approach needs to be understood before relying on determined approach. This is part of the saying: "Don't trust numbers without history". Simple like that.
As seen above first order stochastic dominance is all about the first moment, that is, it is related to the expected value. Second order stochastic dominance relates to variance that is the second moment of a probability distribution. As a matter of curiosity, I also presented a general formulation that can be used to retrieve higher order stochastic dominance. In fact, I have never seem this formula exposed like that (but remember that formalism is missing).
As seen above first order stochastic dominance is all about the first moment, that is, it is related to the expected value. Second order stochastic dominance relates to variance that is the second moment of a probability distribution. As a matter of curiosity, I also presented a general formulation that can be used to retrieve higher order stochastic dominance. In fact, I have never seem this formula exposed like that (but remember that formalism is missing).
As always, I hope you enjoyed the post. Leave your comments and share!
May the Force be with you and Peace Profound.
May the Force be with you and Peace Profound.
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Diogo Pedroso
LinkedIn: www.linkedin.com/in/diogomourapedroso
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