11.4 Econometric Models: A Brief Summary

11.4.1 Stock Prices: the CAPM

The capital asset pricing model (CAPM), developed independently by various authors in the sixties, is a classical equilibrium model for the valuation of risky securities (stocks). It is based on the following assumptions:
  1. There exists homogenous information among the market participants. This assumption can be weakened by assuming that under homogenous information a rational equilibrium is fully revealing (see the strong version of Definition 10.17).
  2. The market has no frictions, i.e., there are no transaction costs, no taxes, no restrictions on short selling or on the divisibility of stocks.
  3. There is complete competition.
  4. There are no arbitrage opportunities.
  5. There are a finite number of stocks ($ K$) and a riskless security with return $ r$.
  6. Every investor has a strictly concave utility function as a function of the risky future cash flows. This means that every investor is risk averse.

  7. Every investor maximizes his expected utility, which is dependent only on the expectation and the variance of the risky future cash flows. This is the crucial assumption of the CAPM. Sufficient conditions for this ( $ \mu-\sigma$)-criterion are either of the following:
    1. Every investor has a quadratic utility function.
    2. The stock returns are normally distributed.

In the following $ X_{i,t}$ and $ \alpha_{i,t}$ represent the price and the number of $ i$-th stock supplied in equilibrium at time $ t$. We define the market portfolio $ X_{m,t}$ as

$\displaystyle X_{m,t} = \sum_{i=1}^K \alpha_{i,t} X_{i,t}.$ (11.8)

The relative weight $ w_{i,t}$ of the $ i$-th stock in this portfolio is as follows

$\displaystyle w_{i,t} = \frac{\alpha_{i,t}X_{i,t}}{\sum_k \alpha_{k,t}X_{k,t}}.
$

Most of the well known stock indices are such value weighted indices, nevertheless often only the largest stocks in the market are included in the index (DAX, for example, contains only the 30 largest stocks). As in Definition 10.15, we define the stock return as $ R_{i,t}=\ln ( X_{i,t}/X_{i,t-1} ) $ and the market return as $ R_{m,t}= \ln (X_{m,t}/X_{m,t-1} ) $. We assume that the underlying process of the return is covariance stationary. In equilibrium according to the CAPM it holds for every stock $ i$ that

$\displaystyle {\mathop{\text{\rm\sf E}}}[R_{i,t}] = r + \beta_i({\mathop{\text{\rm\sf E}}}[R_{m,t}]-r),$ (11.9)

with the `beta' factor

$\displaystyle \beta_i = \frac{\mathop{\text{\rm Cov}}(R_{i,t},R_{m,t})}{\mathop{\text{\rm Var}}(R_{m,t})}.
$

Equation (10.9) says that in equilibrium the expected return of the $ i$-th stock is comprised of two components: the return of the riskless security and a risk premium which is specifically determined for each stock through the beta factor. Stocks that are positively correlated with the market have a positive risk premium. The larger the correlation of a stock with the market portfolio is, the larger is the premium in CAPM for portfolio risk.

Since the CAPM can be derived using theories on utilities, it is sometimes described as a demand oriented equilibrium model. In contrast to this there are other models that explain the stock returns in terms of various aggregate variables, so called factors, and are referred to as being supply oriented. In Section 12.3 we will relax the assumptions of time constant variance and covariance implicit in equation (10.9).

11.4.2 Exchange Rate: Theory of the Interest Rate Parity

For stocks one can find a large variety of econometric models and for exchange rates there are even more. There are two standard and quite simple theories. However they are not sufficient to explain the considerable price movements in currency markets, especially in the short-run. The purchasing power parity (PPP) assumes that identical goods in different countries must have the same relative price, i.e., a relative price given in units of currency. It has been empirically determined that in the long-run this theory describes reality well, but in the short-run the price movements could not be explained. The second simple theory, the theory of interest rate parity, performs better as capital flows faster than goods. The difference in interest rates can thus resemble the exchange of capital in other currencies. So does the exchange rate. The theory of the interest rate parity assumes that domestic and foreign securities are perfect substitutes with respect to duration and risk structure.

Assume that along with forward and futures markets currency can be traded over time. The spot price is calculated by $ W_t^K$, the forward and future price by $ W_t^T$, each is given in units of the foreign currency, i.e., EUR/USD. An internationally acting investor has two choices. Either he holds a domestic capital investment with the domestic interest rate $ r_t^i$ or he chooses a foreign investment with the foreign interest rate $ r_t^a$. If he chooses the foreign investment, he must first exchange his capital into foreign currency at the spot price and at the end, exchange back again. The uncertainty about the future developments of the exchange rate can be avoided by purchasing a forward or future contract. In this case the return on the foreign investment is $ (1/W_t^K) (1+r_t^a)W_t^T -1$. If this return is not equal to the domestic interest rate, then an equilibrium has not been reached. Through immediate price adjustments the interest rate arbitrage disappeares and then equilibrium is reached. Thus in equilibrium it must hold that

$\displaystyle \frac{W_t^T}{W_t^K} = \frac{1+r_t^i}{1+r_t^a},$ (11.10)

i.e., the relationship between forward and future markets and spot markets corresponds exactly to the relationship between domestic and foreign gross interest rates. The relationship in (10.10) is also called the covered interest rate parity, since at the time of investment it deals with riskless exchange and interest rates.

In addition to the interest rate arbitrageur there are the so called forward and future speculators that compare the expected future exchange rate with the forward and future price and the corresponding risk of purchasing (selling) currency below or above the equilibrium. Consider a simple case where forward and future speculators are risk neutral. Then in equilibrium the expected exchange rate is equal to the forward and future price, i.e.,

$\displaystyle W_t^T = {\mathop{\text{\rm\sf E}}}[W_{t+1}^K \mid {\cal F}_{t}],$ (11.11)

with the information set $ {\cal F}_{t}$ which contains all relevant and available information. Here we assume that the speculators have rational expectations, i.e., the true underlying process is known and is used to build the optimal forecast by the speculators. This can also be written as the relationship

$\displaystyle W_{t+1}^K = {\mathop{\text{\rm\sf E}}}[W_{t+1}^K \mid {\cal F}_{t}] + \varepsilon_{t+1}$ (11.12)

which says that the deviations of the speculator's forecast $ {\mathop{\text{\rm\sf E}}}[W_{t+1}^K \mid {\cal F}_{t}]$ from the realized exchange rates is white noise $ \varepsilon_t$ (see Definition 10.8). The market is inefficient when the speculators actually are risk neutral and $ \varepsilon_t$ is not white noise. In this case the set $ {\cal F}_{t}$ does not reflect all of the relevant information in the expectations of the speculators - they do not have rational expectations. In order to test for market efficiency (that is, in order to test whether $ \varepsilon_t$ is white noise) we first need a model for $ {\mathop{\text{\rm\sf E}}}[W_{t+1}^K \mid {\cal F}_{t}]$. This can be formulated from (10.11) and (10.10).

Substituting (10.11) into (10.10) we obtain the so called uncovered interest rate parity,

$\displaystyle \frac{{\mathop{\text{\rm\sf E}}}[W_{t+1}^K \mid {\cal F}_{t}]}{W_t^K} = \frac{1+r_t^i}{1+r_t^a}.$ (11.13)

This interest rate parity is risky because the future exchange rates are uncertain and enter the relationship as expectations.

Together with (10.12) the risky interest rate parity (10.13) implies that the following holds

$\displaystyle W_{t+1}^K = \frac{1+r_t^i}{1+r_t^a} W_t^K + \varepsilon_{t+1}.$ (11.14)

When the difference in the long-run interest rates is zero on average, then (10.14) is a random walk (see Definition 10.9). The random walk is the first model to describe exchange rates.

It should be emphasized that the derivation of this simple model occurred under the assumption of risk neutrality of the speculators. In the case of risk aversion a risk premium must be included. If, for example, we want to test the efficiency of the currency markets, we could then test the combined hypothesis of efficiency and uncovered interest rate parity using risk neutrality. A rejection of this hypothesis indicates the market inefficiency or that the interest rate parity model is a poor model for currency markets.

11.4.3 Term Structure: The Cox-Ingersoll-Ross Model

Term structure models are applied to model the chronological development of bond returns with respect to time to maturity. The classical starting point is to identify one or more factors which are believed to determine the term structure. Through specification of the dynamic structure and using specific expectation hypotheses, an explicit solution can be obtained for the returns.

As a typical example we briefly introduce the Cox, Ingersoll and Ross (CIR) model, which has been mentioned in Section 9.2.2. The price of a Zero Coupon Bond with a nominal value of 1 EUR is given by $ P_T(t)$ at time $ t$, i.e., a security with no dividend payments that pays exactly one EUR at maturity date $ T$. The log return of the zero coupon bond is given by $ Y_T(t)$. We assume that continuous compounding holds. The process $ Y_T(t)$ is frequently referred to as the yield to maturity . The relationship between the price and the return of the zero coupon bond is

$\displaystyle P_T(t) = \exp\{-Y_T(t)\tau \}
$

with the remaining time to maturity $ \tau = T - t$. This can be easily seen from the definition of a log return (Definition 10.15). For very short time intervals the short rate $ r(t)$ is defined as

$\displaystyle r(t) = \lim_{T\rightarrow t} Y_T(t).
$

In practice the short rate corresponds to the spot rate, i.e., the interest rate for the shortest possible investment (see Section 9.2.2). Consider, intuitively, the choice between an investment in a zero bond with the return $ Y_T(t)$ and repeatedly investing at a (risky) short-term interest rate in future periods. An important expectation hypothesis says that the following holds

$\displaystyle P_T(t) = {\mathop{\text{\rm\sf E}}}\left[\exp(-\int_t^T r(s) ds) \vert {\cal F}_t \right]$ (11.15)

(also see equation (9.1) for variable but deterministic interest). The short rate is frequently seen as the most important predicting factor of the term structure. As the CIR model, most one factor models use the short rate as factor. The CIR model specifies the dynamic of the short rate as a continuous stochastic process

$\displaystyle dr(t) = a\{b-r(t)\}dt + \sigma \sqrt{r(t)}dW_t$ (11.16)

with a Wiener process $ W_t$ and constant parameters $ a, b$ and $ \sigma$ - see also Section 9.2.2. The process (10.16) has a so called mean reversion behavior, i.e., once deviations from the stationary mean $ b$ occurs, the process is brought back to the mean value again through a positive $ a$. The volatility, written as $ \sigma\sqrt{r(t)}$, is larger whenever the interest level is higher, which can also be shown empirically.

Since in the equation (10.16) $ r(t)$ is specified as a Markov process, $ P_T(t)$ is, as a consequence of equation (10.15), a function of the actual short rate, i.e.,

$\displaystyle P_T(t) = V\{r(t),t\} .$

With Itô's lemma (5.10) and (9.7) we obtain from (10.16) the differential equation

$\displaystyle a(b-r)\frac{\partial V(r,t)}{\partial r} + \frac{1}{2} \sigma^2 r...
...al^2 V(r,t)}{\partial r^2}
+ \frac{\partial V(r,t)}{\partial t} - rV(r,t) = 0.
$

With the bounding constraint $ V(r,T) = P_T(T) = 1$ the following solution is obtained

$\displaystyle P_T(t) = V\{r(t),t\} = \exp\left\{A(T-t) + B(T-t)r(t)\right\},$ (11.17)

where (see Section 9.2.4)
$\displaystyle A(\tau)$ $\displaystyle =$ $\displaystyle \frac{2a b}{\sigma^2} \ln \frac{2\psi
\exp\left\{(a+\psi)\tau/2\right\}}{g(\tau)},$  
$\displaystyle B(\tau)$ $\displaystyle =$ $\displaystyle \frac{2\left\{1-\exp(\psi \tau)\right\}}{g(\tau)}$  
$\displaystyle \psi$ $\displaystyle =$ $\displaystyle \sqrt{a^2 + 2\sigma^2}$  
$\displaystyle g(\tau)$ $\displaystyle =$ $\displaystyle 2\psi + (a+\psi)\left\{\exp(\psi \tau)-1\right\}.$  

For increasing time periods $ T-t$ the term structure curve $ Y_T(t)$ converges to the value

$\displaystyle Y_{lim} = \frac{2ab}{\psi+a}.
$

If the short-term interest lies above $ b$, then the term structure is decreasing, see Figure 10.1; if it lies below $ Y_{lim}$, then the term structure is increasing, see Figure 10.2. If the short-term interest rate lies between $ b$ and $ Y_{lim}$, then the curve could first rise and then fall.

Fig.: Term structure curve according to the Cox-Ingersoll-Ross model with a short rate of $ r_t$=0.2, $ a=b=\sigma=0.1$ and $ Y_{lim}=0.073$ (dotted line). 15404 SFEcir.xpl
\includegraphics[width=1.2\defpicwidth]{cir1.ps}

Fig.: Term structure curve according to the Cox-Ingersoll-Ross model with a short rate of $ r_t$=0.01, $ a=b=\sigma=0.1$ and $ Y_{lim}=0.073$ (dotted line). 15408 SFEcir.xpl
\includegraphics[width=1.2\defpicwidth]{cir2.ps}

11.4.4 Options: The Black-Scholes Model

Since we have thoroughly covered the Black-Scholes model on option pricing in the first part of this book, here only a brief summary of the model is given. Options are not only theoretically interesting for financial markets, but also from an empirical point of view. Just recently there have been indications of a systematic deviation of actual market prices from the Black-Scholes prices. These deviations will be discussed in more detail in later chapters, specifically in dealing with ARCH models.

As an example let's consider a European call option on a stock which receives no dividends in the considered time periods and has the spot price $ S_t$ at time $ t$. $ C(S,t)$ is the option price at time $ t$, when the actual price is $ S_t=S$. The payoff at the time to maturity $ T$ is $ C(S_T,T) = \max (0, S_T - K )$, where $ K$ is the strike price. The option price is determined from general no arbitrage conditions as

$\displaystyle C(S_t,t) = {\mathop{\text{\rm\sf E}}}[e^{-r\tau} C(S_T,T) \vert {\cal F}_{t}] ,
$

where expectations are built on an appropriate risk neutral distribution - see also (6.23). $ r$ is the fixed riskless interest rate.

Special results can only be derived when the dynamics of the stock prices are known. The assumptions made by Black and Scholes are that the stock prices $ S_t$ are geometric Brownian motion, i.e.,

$\displaystyle dS_t = \mu S_t dt + \sigma S_t dW_t .$ (11.18)

The option price $ C(S,t)$ thus satisfies the Black-Scholes differential equation (6.3) as a function of time and stock prices

$\displaystyle \frac{1}{2}\sigma^2S^2\frac{\partial^2 C}{\partial S^2} + r
S \frac{\partial C}{\partial S}+\frac{\partial C}{\partial t} = rC
$

Black and Scholes derive the following solutions (see Section 6.2):

$\displaystyle C(S,t)=S \Phi (y+\sigma\sqrt {\tau}) - e^{-r\tau} K \Phi (y) ,$ (11.19)

where $ \tau = T - t$ is the time to maturity for the option and $ y$ is an abbreviation for

$\displaystyle y=\frac {\ln\frac SK+(r-\frac {1}{2}\sigma^2)\tau}{\sigma\sqrt {\tau}}.
$

11.4.5 The Market Price of Risk

In a risk neutral world the market price of risk, see Section 9.2.3, is equal to zero. In the following section, we will consider the market price of risk and derive once again the Black-Scholes formula. To do this we will consider derivatives of financial instruments that are determined by a single random process $ \theta_t.$ We will assume that the process $ \theta_t$ is geometric Brownian motion :

$\displaystyle d \theta_t =m \theta_t dt + s \theta_t dW_t \; .$ (11.20)

The variable $ \theta_t$ does not necessarily represent a financial value. It could be the state of the market, a measure for the popularity of a politician or the frequency of an ad-hoc announcement at time $ t$. Assume that $ V_{1t}$ and $ V_{2t}$ are the prices for two derivatives of financial instruments that depend only on $ \theta_t$ and $ t$. As a simplification, no payments are allowed during the observed time period. This process $ V_{jt} = V_j(\theta, t)$, $ j= 1,2$ also follows the schema (10.20) with the same Wiener process $ W_t$

$\displaystyle d V_{jt} = \mu_{jt} V_{jt} dt + \sigma_{jt} V_{jt} dW_t , \qquad j=1,2$ (11.21)

where $ \mu_{jt}, \sigma_{jt}$ could be functions of $ \theta_t$ and $ t$. The random process $ W_t$ in (10.20) and (10.21) is always the same since we assume that this is the only source that creates uncertainty.

The observation of (10.21) in discrete time leads to:
$\displaystyle \Delta V_{1t}$ $\displaystyle =$ $\displaystyle \mu_{1t} V_{1t} \Delta t + \sigma_{1t} V_{1t} \Delta W_t$ (11.22)
$\displaystyle \Delta V_{2t}$ $\displaystyle =$ $\displaystyle \mu_{2t} V_{2t} \Delta t + \sigma_{2t} V_{2t} \Delta W_t$ (11.23)

We could ``eliminate the random variable $ \Delta W_t$'' by constructing a riskless portfolio that continually changes. To do this we take $ \sigma_{2t} V_{2t}$ units of the first instrument and $ - \sigma_{1t} V_{1t}$ of the second instrument, i.e., we short sell the second instrument. Letting $ \Pi_t$ be the total value of the portfolio at time $ t$ we have:

$\displaystyle \Pi_t = (\sigma_{2t} V_{2t})V_{1t} - (\sigma_{1t} V_{1t})V_{2t}$ (11.24)

and

$\displaystyle \Delta \Pi_t = (\sigma_{2t} V_{2t})\Delta V_{1t} - (\sigma_{1t} V_{1t})\Delta V_{2t}$ (11.25)

Substituting in (10.22) and (10.23) we have:

$\displaystyle \Delta \Pi_t = (\mu_{1t} \sigma_{2t} V_{1t} V_{2t} - \mu_{2t} \sigma_{1t} V_{1t} V_{2t})\Delta t.$ (11.26)

This portfolio should be riskless, thus in time period $ \Delta t$ it must produce the riskless profit $ r \Delta t$:

$\displaystyle \frac{\Delta \Pi_t}{\Pi_t}= r \Delta t.$ (11.27)

Substituting (10.24) and(10.26) into this equation produces:
$\displaystyle (\mu_{1t} \sigma_{2t} V_{1t} V_{2t} - \mu_{2t} \sigma_{1t} V_{1t} V_{2t})\Delta t$ $\displaystyle =$ $\displaystyle (\sigma_{2t} V_{1t} V_{2t} - \sigma_{1t} V_{1t} V_{2t})r \Delta t$  
$\displaystyle \mu_{1t} \sigma_{2t} - \mu_{2t} \sigma_{1t}$ $\displaystyle =$ $\displaystyle r\sigma_{2t} - r\sigma_{1t}$  
$\displaystyle \frac{\mu_{1t} -r}{\sigma_{1t}}$ $\displaystyle =$ $\displaystyle \frac{\mu_{2t} -r}{\sigma_{2t}}$  

Equating this as in (9.2.3) to $ \lambda_t$ we see that the price $ V_t$ of a derivative instrument, an instrument that depends only on $ \theta_t$ and $ t$, follows the dynamics

$\displaystyle d V_t = \mu_t V_t dt + \sigma_t V_t dW_t,$ (11.28)

the value

$\displaystyle \lambda_t = \frac{\mu_t - r}{\sigma_t} = \frac{\mu(\theta_t,t) - r}{\sigma(\theta_t,t)}$ (11.29)

represents the market price of risk . This market price of risk can depend on $ \theta_t$ (using $ \mu_t, \sigma_t$), but not on the actual price of the instrument $ V_t$! We can rewrite equation (10.29) as:

$\displaystyle \mu_t - r = \lambda_t \sigma_t$ (11.30)

Furthermore we can interpret $ \sigma_t$, which in this interpretation can also be negative, as the level of the $ \theta_t$-risk in $ V_t$. Equation (10.30) has strong ties to the CAPM model, which we discussed in Section (10.4.1) - for further details see also Hafner and Herwartz (1998).

Example 11.2   Assume that there are two objects, both are dependent on a 90 day interest rate. The first instrument has an expected return of $ 6
\%$ per year and a volatility of $ 20 \%$ per year. For the second instrument a volatility of $ 30 \%$ per year is assumed. Furthermore, $ r=3 \%$ per year. The market price of risk for the first instrument according to (10.29) is:

$\displaystyle \frac{0.06 - 0.03}{0.2} = 0.15$ (11.31)

By substituting into equation (10.30) for the second object we obtain:

$\displaystyle 0.03 + 0.15 \cdot 0.3 = 0.075$ (11.32)

or $ 7.5 \%$ expected value.

Since $ V_t$ is a function of $ \theta_t$ and $ t$, we can determine the dependence on $ \theta_t$ using Itô's lemma. The direct application of Itô's lemma (5.10) on $ V(\theta,t)$ gives, in comparison to (10.28), the parameters in this equation

$\displaystyle \mu_t V_t$ $\displaystyle =$ $\displaystyle m \theta_t \frac{\partial V_t}{\partial \theta_t} +
\frac{\partia...
...partial t} + \frac{1}{2}s^2 \theta_t^2
\frac{\partial^2 V_t}{\partial \theta^2}$  
$\displaystyle \sigma_t V_t$ $\displaystyle =$ $\displaystyle s \theta_t \frac{\partial V_t}{\partial \theta}.$  

Due to equation (10.30) we have $ \mu_t V_t - \lambda_t
\sigma_t V_t = r V_t$, so that we obtain the following differential equation for $ V_t$:

$\displaystyle \frac{\partial V_t}{\partial t} + (m - \lambda_t s) \theta_t \fra...
...rac{1}{2} s^2 \theta_t^2 \frac{\partial^2 V_t}{\partial \theta^2} = r \cdot V_t$ (11.33)

This equation (10.33) is very similar to the Black-Scholes differential equation and is in fact identical to (6.3) for $ \theta_t = S_t$, where $ S_t$ denotes the stock price with no dividends. In this case $ \theta_t$ itself is the price of the risk bearing instrument and must therefore satisfy (10.30), like the price $ V_t$ of any derivative based on the stock. Thus we obtain

$\displaystyle m-r = \lambda_t s,$ (11.34)

so that the second term in (10.33) is equal to

$\displaystyle r \theta_t \frac{\partial V_t}{\partial \theta}.$ (11.35)

Thus we have a differential equation:

$\displaystyle \frac{1}{2} s^2 \theta_t^2 \frac{\partial^2 V_t}{\partial \theta^...
...ac{\partial V_t}{\partial \theta} - r V_t + \frac{\partial V_t}{\partial t} = 0$ (11.36)

which is identical to (6.3) after renaming the variables. More explicitly, let $ S_t = \theta_t, b = r$ (since there are no dividends) and let $ \sigma = s$ using the notation in Section 6.1.