Last time, we journeyed through the realm of the Capital Asset Pricing Model (CAPM), but stumbled upon its limitation as a static model. It couldn’t quite capture the dynamics of human behavior over time. To sculpt a more nuanced understanding, we embarked on a quest fueled by curiosity by asking some basic questions: how do individuals make choices when outcomes are uncertain? How can preferences be represented and measured? How to model risk aversion level? These musings guided us to the shores of Expected Utility Theory.
Crafted by the brilliant minds of John von Neumann and Oskar Morgenstern in their landmark 1944 tome, “Theory of Games and Economic Behavior,” this theory laid the mathematical keel for sailing through decisions under uncertainty. They introduced us to utility functions that map outcomes to a numerical value, representing the satisfaction or preference level of the decision-maker, and by proposing that decisions are made so as to maximize expected utility, rather than expected monetary value.
Moreover, the contour of the utility function reveals much about a person’s stance towards risk, categorizing them into three adventurers: the risk-averse, who tread cautiously; the risk-neutral, who stride indifferently; and the risk-seeking, who chase the thrill.
We now have the equipment of expected utility to dive into the intriguing world of equilibrium pricing. You might wonder, “How do we figure out the ‘right’ price for things in a market where everyone has their own utility functions?” This is where the concept of equilibrium pricing comes into play, acting as the superhero that balances everyone’s desires and resources. Think of it as the party planner who ensures everyone has a good time without running out of snacks. In financial terms, it’s the price at which supply meets demand, and everyone is as happy as they can be without making someone else worse off.
Now, imagine a world where every conceivable outcome in the future could be bought and sold today. This magical marketplace is known as the Arrow-Debreu market, named after two economists who were like the financial world’s version of Batman and Robin. They showed that if we could trade ‘state-contingent’ claims (essentially bets on every possible future event), we could achieve an incredibly efficient allocation of resources. It’s as if you could buy insurance for every minor inconvenience in life, from spilling coffee on your shirt to rain on your wedding day. The Arrow-Debreu model demonstrates how markets can theoretically lead to optimal use of resources, under the assumption of complete markets and perfect competition.
But what does “optimal” really mean? Enter Pareto Optimality, a fancy term for a situation where no one can be made better off without making someone else worse off. It’s like a perfectly balanced see-saw. The Arrow-Debreu model assures us that the equilibrium prices will lead to a Pareto Optimal allocation of resources, ensuring that the economic pie is sliced in the most efficient way possible, even if it doesn’t address how big a slice everyone gets.
This brings us to the superhero duo of economic theory: the First and Second Theorems of Welfare Economics. The first theorem is like the optimistic friend who believes everything will work out for the best, stating that any competitive equilibrium (like our equilibrium pricing) naturally leads to a Pareto Optimal allocation of resources. The second theorem is the resourceful friend who knows how to work the system, claiming that with the right initial distribution of wealth, we can achieve any Pareto Optimal outcome through competitive markets. Essentially, if we can just figure out how to divide the initial slice of the economic pie, the market will take care of the rest.
But real life is messier than these elegant theories suggest, which is where the Consumption-Based Capital Asset Pricing Model (C-CAPM) comes in. Building on the CAPM, which tells us how to price assets in a static world, C-CAPM incorporates the idea of expected utility and personal consumption over time. It connects the dots between how we value future consumption, our aversion to risk, and the prices of assets today. Imagine deciding between spending all your money on a lavish party today or investing in a startup that makes flying cars. C-CAPM helps explain why and how you make such decisions, by considering your habits, how much you value future consumption, and your fear of missing out on future parties or successful investments.
To sum up, our journey through financial economics shows us how individual preferences and risk attitudes shape decisions (Expected Utility Theory), how these decisions lead to market prices that balance supply and demand (Equilibrium Pricing), and how, in an ideal world, these prices lead to the most efficient allocation of resources (Arrow-Debreu Market and Pareto Optimality). The Welfare Theorems give us hope that, under the right conditions, markets can lead to outcomes that are good for society, while C-CAPM bridges the gap between our personal consumption choices and the broader financial markets. Together, these concepts form a tapestry that illustrates the intricate dance between individual choices and market dynamics.
In the next note, we’ll dive into the world of relative pricing and its foundational role in understanding financial derivatives. This journey will bridge the abstract with the practical, showcasing how the theories we’ve discussed not only illuminate the underpinnings of financial markets but also directly influence the valuation and trading of derivatives like options.
中文版
上次，我们讲了资本资产定价模型（CAPM），但作为一个静态模型，它无法描述价格和人们的行为随时间的动态变化。为了对这个局限性有更为深入的理解，我们自然可以提出一些基本问题，例如，当结果不确定时，个人如何做出选择？个人的偏好如何被量化？如何建立风险厌恶水平的数学模型？解答了这些问题的理论就是期望效用理论。
这个理论由冯·诺伊曼和摩根斯坦在他们1944年的著作《博弈论与经济行为》中首次建立，为在不确定性下做出决策提供了数学基础，其中最重要的是效用函数的提出。效用代表了决策者的满意度或偏好水平，并提出应通过最大化期望效用而非期望货币价值来做出决策。
此外，效用函数曲线深刻揭示了个人对风险的态度，大致可以将所有人分为三类：谨慎行事的风险厌恶者；漫不经心的风险中性者；以及追求刺激的风险寻求者。
有了期望效用的工具，就可以深入探索平衡定价的世界了。你可能会好奇：“在一个每个人都有自己的效用函数的市场中，我们如何确定物品的‘正确’价格？”平衡定价概念在这里发挥作用，它像是平衡每个人欲望和资源的超级英雄。简单来说，就是供给与需求相遇的价格点，在这个点上，没有人因别人的损失而受益，从而达到大家都相对满意的状态。
现在，想象一个你今天就能买卖任何可想象的未来结果的世界。这个神奇的市场被称为Arrow-Debreu市场，以两位就像金融世界中的蝙蝠侠和罗宾的经济学家命名。他们展示了，如果我们可以交易“状态条件”的索赔（本质上是对每个可能的未来事件的赌注），我们就能实现资源的极度有效分配。就好像你能为生活中的每一个小不便买保险，从衬衫上洒咖啡到婚礼那天下雨。Arrow-Debreu模型演示了在完全市场和完美竞争的假设下，市场如何理论上导致资源的最优使用。
但是，“最优”真正意味着什么呢？这就要提到帕累托最优，用于描述一个在不损害他人利益的情况下，所有人都已经达到最优利益的情况。这就像一个完美平衡的跷跷板。Arrow-Debreu模型向我们保证，平衡价格将导致资源的帕累托最优分配，确保以最有效的方式切分经济蛋糕，即便它没有解决每个人得到的份额大小问题。
这引出了经济理论中的超级英雄组合：福利经济学的第一和第二定理。第一个定理就像是那个乐观的朋友，相信一切都会迎刃而解，ta声明任何竞争性平衡（如我们的平衡定价）自然会导致资源的帕累托最优分配。第二定理则是那个知道如何利用体系的机智朋友，ta声称通过正确的初始财富分配，我们可以通过竞争市场实现任何帕累托最优结果。本质上，如果我们能够弄清楚如何分配经济馅饼的初始片段，市场会处理好剩下的一切。
但现实生活比这些优雅的理论要复杂得多，这就是消费资本资产定价模型（C-CAPM）发挥作用的地方。在CAPM的基础上，C-CAPM融入了期望效用和随时间变化的个人消费的概念。它连接了我们如何评估未来消费、我们对风险的厌恶，以及今天资产价格之间的点点滴滴。想象一下，在今天花光所有的钱举办一场奢华的派对，或者投资于一家制造飞行汽车的创业公司之间做出决定。C-CAPM帮助解释了你为什么以及如何做出这样的决策，通过考虑你的习惯、你对未来消费的重视程度，以及你对错过派对或者投资的恐惧。
总而言之，我们通过金融经济学的旅程看到了个人偏好和风险态度如何塑造决策（期望效用理论），这些决策如何导致平衡供需的市场价格（平衡定价），以及在理想世界中，这些价格如何导致资源的最有效分配（Arrow-Debreu市场和帕累托最优性）。福利定理给了我们希望，在正确的条件下，市场可以导致对社会有益的结果，而C-CAPM则在我们的个人消费选择和更广泛的金融市场之间架起了一座桥梁。这些概念共同构成了一幅图案，展示了个人选择与市场动态之间错综复杂的联系。
在下一篇笔记中，我们将探讨相对定价的世界及其在金融衍生品中的基础性作用，展示我们讨论的理论不仅仅停留在纸面上，而且还直接影响了期权等衍生品的估值和交易。