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Risk-Neutral Pricing

Absolute and Relative pricing

There are two polar approaches to asset pricing theory.
In absolute pricing, we price each asset by reference to its exposure to fundamental sources of (macroeconomic) risk. E.g. consumption-based and general equilibrium models, CAPM, and Factor models. We can use the absolute pricing to give an economic explanation for why prices are what they are, or in order to predict how prices might change if policy or economic structure changed.
In relative pricing, we ask what we can learn about an asset's value given the prices of some other assets. We do not ask where the prices of the other assets came from, and we use as little information about fundamental risk factors as possible. E.g. Black-Scholes option pricing.
We need to note that almost no problems are solved by the pure extremes. For example, the CAPM and factor models are taking the absolute approach. Yet in applications, they price assets 'relative' to the market or other risk factors, without answering what determines the market or factor risk premia and betas.

Positive and Normative statement

Asset pricing theory shares two related statements;
1) a positive statement - the way the world does work. So, why prices are what they are?
2) a normative statement - the way the world should work. So, is there any trading opportunities?

Central Question of Asset Pricing Theory

Asset pricing theory concerns about prices of uncertain payoffs. In other words, it studies an expected discounted value of 'uncertain' cash-flows. This statement has several points.

First, the 'uncertainty' entails two parts; 1) a time delay (an easy part) and 2) a risk premium for holding the asset and for correlations of risk (a hard part). Or here is the part we need a prediction or forecasting.
Second, does price equals expected discounted payoff? What does the expectation mean? And how can we discount the payoff?

Pricing Equation with Discount Factor

I follow a discount factor/GMM view of asset pricing theory (Cochran [2012]), which can be summarized by two equations: \begin{eqnarray} p_t & = &E(m_{t+1} x_{t+1}) \\ m_{t+1} &=& f(data,parameters) \end{eqnarray}

where $p_t$ = asset price, $x_{t+1}$ = asset payoff, $m_{t+1}$ = stochastic discount factor.

The $m_{t+1}$ discount factor view has several advantages:

  1. This approach separates 'the step of specifying economic assumptions of the model (eqn 2) from the step of deciding which kind of empirical representation to pursue (eqn 1). Fore a given model - (eqn 2) - we can easily see how eqn 1 can lead to predictions, stated in terms of returns, price-dividend ratios, expected return-beta representations, moment conditions, continuous vs. discrete-time implications, and so forth.
  2. Thinking in terms of discount factors often is much simpler than thinking in terms of portfolios. For example, it is easier to insist that there is a positive discount factor than to check that every possible portfolio that dominates every other portfolio has a larger price.
  3. The discount factor approach is also associated with state-space geometry in place of the usual mean-variance geometry.

Price, Payoffs, and Notation

The price $p_t$ gives rights to a payoff $x_{t+1}$.

price $p_t$ payoff $x_{t+1}$
Stock $p_t$ $p_t+d_{t+1}$
Return 1 $R_{t+1}$
Price-dividend ratio $\frac{p_t}{d_t} $ $(\frac{p_{t+1}}{d_{t+1}}+1) \frac{d_{t+1}}{d_t}$
Excess return 0 $R_{t+1}^{e} = R_{t+1}^{a} + R_{t+1}^{b} $
Portfolio (managed) $z_t$ $z_t R_{t+1}$
Moment condition $E(p_t z_t) $ $ x_{t+1} z_t$
One-period bond $ p_t $ 1
Risk-free rate 1 $R^f $
Option C $max(S_T - K, 0) $

Alternative approach to Basic pricing equation

By using a definition of a gross return $R_{t+1} := \frac{x_{t+1}}{p_t}$,
The basic pricing equation can also be written as (in terms of returns rather than price): \begin{eqnarray} 1 & = &E(m_{t+1} R_{t+1}) \\ m_{t+1} &=& f(data, parameters) \\ R_{t+1} &=& g(data, parameters) \\ \end{eqnarray}

I tries to directly model and forecast $m_{t+1}$ along with $R_{t+1}$. The first work is model the process of the stochastic discounting factor and the gross return. Next, we need to choose and use the best estimation method to pick free parameters of the model to make it fit best or minimize pricing errors in a test window sample. Then, we can evaluate the model by examining how big those pricing errors are in a real window sample, in order to decide the model is applicable in practice.

Portfolio Choice vs. Asset Pricing

Portfolio theory gives prices by finding a demand curve for assets, which intersected with a supply curve. However, current academic research and high-tech practice now go to prices directly. One can then find optimal portfolios. While portfolio theory is still interesting and useful, it has now become no longer a cornerstone of pricing.

A fundamental question in financial decision making

How should an investor allocate his or her funds among the possible investment choices?

Historical answer to the question

  1. by John Maynard Keynes (1935) - In Keynes’ economic thinking, psychological elements play a fundamental role. Keynes argued that inexplicable changes in spontaneous confidence can be responsible for economic fluctuations. “Animal spirits” is the term he coined to express the state of spontaneous confidence. Most, probably, of our decisions to do something positive, the full consequences of which will be drawn out over many days to come, can only be taken as the result of animal spirits—a spontaneous urge to action rather than inaction, and not as the outcome of a weighted average of quantitative benefits multiplied by quantitative probabilities
  2. by financial analysts - Fundamental analysis can discover the true value of an asset, and the investor should invest in those assets that offer the highest value given the price at which the asset is trading.
  3. by Markowitz - In Markowitz’s 1952 paper, he suggested that investors should consider risk as well as return specifically, they should decide the allocation of their investments on the basis of a trade-off between risk and return.
economics/asset_pricing_theory.txt · Last modified: 2016/09/21 17:09 by admin