1. Introduction

In this post, I work through Adding and Subtracting Black Scholes, JFE (2011) by Antonio Mele and Dennis Kristensen. This paper develops a method for approximating the price of an asset where no closed form expression exists. In the analysis below, I focus my attention on pricing a European option on a stock when the stock price displays stochastic volatility.

In Section , I describe the approximation method which hinges on the wedge between the infinitesimal generator of the asset price under the true specification (e.g., with stochastic volatility) and a simpler reference specification (e.g., with constant volatility). Then, in Section I conduct some simple numerical exercises to see how the approximation behaves in practice.

2. The Method

Consider a stock with a price in units of ‘s at time in units of ‘s whose evolution under the risk neutral measure is described by the equations below:

(1)  

Here, is the risk free rate in units of , is the instantaneous variance of the stock price in units of , is the mean reversion rate of the variance of the stock price in units of , is the mean of the variance of the stock price in units of and both and are constants parameterizing the volatility of the variance of the stock price where has units of and is unitless.

Variance and price of the underlying stock.

Let be the payout of a European option on this stock with strike price (in ). Then, define the price of this option as follows:

(2)  

As shown in Ait-Sahalia and Kimmel (2007), when there is no closed form solution for . The authors approximate by computing the difference between the infinitesimal generators of and where is the price of a European option on a stock whose volatility is constant and set to .

Definition (Infinitesimal Generator): Let be a time homogeneous Ito diffusion in . The infinitesimal generator of is defined as:

(3)  

Source: Oksendal (2003)

The infinitesimal generator characterizes how a stochastic process would look different if we peered an instant into the future. The theorem below gives a method for computing the infinitesimal generator.

Theorem (Infinitesimal Generator of an Ito Process): Let be the Ito diffusion in :

(4)  

If with a compact support then:

(5)  

Source: Oksendal (2003)

To tie this mathematical construction back into the financial application of options pricing, note that in a world with no financial frictions a la Black and Scholes (1973) the instantaneous change in the price of an asset must exactly equal the interest payment earned by depositing the current price of the asset in a bank for the next instant. Thus, in generator form the second order partial differential equation pinning down the price of a European option on a stock whose price has constant volatility can be written as follows:

(6)  

where the generator can be characterized as:

Proposition (Infinitesimal Generator given Constant Variance): The generator for the price of a European option on a stock with constant volatility is given by:

(7)  

Proof: If a stock price displays constant volatility , I can write as follows:

(8)  

where is the instantaneous volatility of the stock price in units of . Plugging this differential equation into the formula above yield:

(9)  

I can conduct a similar exercise to characterize the price of a European option on a stock with stochastic volatility. For this asset, I know that:

(10)  

What’s more, I can use the same theorem to characterize the infinitesimal generator as:

Proposition (Infinitesimal Generator given Stochastic Variance): The generator for the price of a European option on a stock with stochastic volatility is given by:

(11)  

Proof: Applying Ito’s lemma to and defining for shorthand yields an expression for as:

(12)  

Collecting terms yields expressions for and which given .

The authors’ key insight is that applying the infinitesimal generator to the difference yields the expression:

(13)  

given the boundary condition that for all and . This wedge given by has the natural interpretation of the hedging cost incurred by a trader continuously hedging the risk of the European option with stochastic volatility using the incorrect constant volatility model. Thus, we can write the true price as the Black and Scholes price plus an error correction term:

(14)  

This is a particularly nice formulation as each of the error terms is a function of only the Black and Scholes (1973) formulation with constant volatility.

3. Numerical Results

To put this approximation method to work, use a Taylor expansion around :

(15)  

Each of the expansion terms can be expressed recursively as follows:

(16)  

The Black and Scholes (1973) options price given constant volatility can be expressed as:

(17)  

Thus, expanding out the first approximating terms yields:

(18)  

In order to simplify calculations, I take to kill off the term in my numerical analysis. Below I replicate Figure from the Kristensen and Mele (2011) for and plot the mispricing as a fraction of the Black and Scholes (1973) price with for . The strike price is , the time to maturity is year, the remaining parameter values are , , , and .

Mispricing as a fraction of the Black and Scholes (1973) price with xi = 1/2 for N = 0, 1. The strike price is K = 100, the time to maturity is 1 year, the remaining parameter values are kappa = 2, alpha = 0.04, omega = 0.10, rho = -0.50 and r = 0.10.

The code for the simulations can be found on www.SageMath.org. The simulation represents estimates from iterations at price points between and .

1. Introduction

In this post I outline the Buckingham Theorem which shows how to use dimensional analysis to compute answers to seemingly intractable physical problems. For instance, in Geoffrey Taylor used the theorem to work out the energy payload released by the 1945 Trinity test atomic explosion in New Mexico simply by looking at slow motion video records. My main source for this post is Bluman and Kumei (1989).

Frame from slow motion footage of the Trinity nuclear test with a distance measurement showing the blast radius as well as a time measure showing seconds elapsed since denotation.

2. Basic Framework

The Buckingham Theorem concerns physical problems with the following form: There is a variable of interest, , which is some unknown function of different physical quantities .

(1)  

Each of these physical quantities is composed of measurements in only fundamental dimensions labeled . Thus, I can define a dimension operator which gives the dimensions of an arbitrary variable and write its output as:

(2)  

So, for example, if is measuring pressure on the surface of a table, I could write where , , and .

Definition (Dimensionless Quantity):
An arbitrary variable is dimensionless if:

(3)  

3. Main Result

Now, I show how to reformulate this problem and apply linear algebra to the dimensional exponents to derive a characterization of the solution to as a function of dimensionless quantities. First, I define as an matrix of dimensional exponents for and as the vector of dimensional exponents for :

(4)  

Next, we know that if has full rank, then there will solutions to the system of equations . Define as this matrix of solutions.

(5)  

Finally, define as the solution to system of equations :

(6)  

With these objects in hand, I can now state the Buckingham Theorem.

Proposition (Buckingham ):
A physical system with fundamental dimensions can be restated as:

(7)  

where , is an unknown function and are dimensionless parameters constructed from the physical parameters using equations of the form below:

(8)  

4. An Example

I now give an example of how to employ this theorem by working out the nuclear payload example from the introduction. For more information on this example, take a look at this blog post. Suppose that an atomic blast has a shock wave radius of with variables:

1. : Energery released by the explosion,
2. : Time elapsed since the explosion took place,
3. : Initial density, and
4. : Initial pressure.

For this problem and with fundamental dimensions of length , mass , and time yielding the dimensional matrix written below:

(9)  

The energy (force) released by the bomb is the amount of mass accelerating through a unit square on the surface of the blast wave. Since , we have only dimensionless constant which can be computed as the solution to the system of equations :

(10)  

Setting the scalar free parameter , I can write as:

(11)  

The dimension of the shock wave radius is in units of distance yielding the dimension matrix below:

(12)  

The system of equations has unknowns:

(13)  

Thus, the vector will thus be defined up to a single free parameter :

(14)  

Using the formula and tuning , I can compute as:

(15)  

Combining all of these results yields a formulation for in terms of a constant and an unknown function of the dimensionless quantity :

(16)  

Taylor expanding around where yields a formulation where with scaling constant given by:

(17)  

Setting , the plot of vs yielded an accurate fit where the predicted values fall on the solid line and the (declassified) empirically observed values are denoted by ‘s in the plot below:

1. Introduction

In this post, I replicate the main results from Glosten and Milgrom (1985) using the setup outlined in Back and Baruch (2003). I begin in Section by laying out the continuous time asset pricing framework. I consider the behavior of an informed trader who trades a single risky asset with a market maker that is constrained by perfect competition. Then, in Section I solve for the optimal trading strategy of the informed agent as a system of first order conditions and boundary constraints. Finally, I show how to numerically compute comparative statics for this model.

2. Asset Pricing Framework

There is a single risky asset which pays out at a random date . There is an informed trader and a stream of uninformed traders who arrive with Poisson intensity . All traders have a fixed order size of . The model end date is distributed exponentially with intensity .

Let denote the net position of the noise traders up to but not including date and let denote the net position of the informed up to but not including date . The market maker sees an anonymous order flow at each time of so that generates a -field which represents the market maker’s information set.

Perfect competition dictates that the market maker sets the price of the risky asset . Let and denote the bid and ask prices at time .

(1)  

Let be the left limit of the price at time . Given that , we can interpret as the probability of the event at time given the information set . The informed trader chooses a trading strategy in order to maximize his end of game wealth at random date with discount rate. Let and let .

(2)  

In order to guarantee a solution to the optimization problem posed above, I restrict the domain of potential trading strategies to those that generate finite end of game wealth.

(3)  

I then look for probabilistic trading intensities which make the net position of the informed trader a martingale.

Definition: At each time , an equilibrium consists of a pair of bid and ask prices,

(4)  

as well as a vector of trading intensities,

(5)  

such that the prices equal the conditional expectation of the asset value relative to given a sell or buy order and the trading intensities solve the informed agent’s objective function, satisfy the finiteness conditions and are martingales relative to the informed trader’s information set :

(6)  

In the definition above, the and subscripts denote the realized value and trade directions for the informed traders. So, for example, denotes the trading intensity at some time in the buy direction of an informed trader who knows that the value of the asset is .

3. Optimal Trading Strategies

I now characterize the equilibrium trading intensities of the informed traders. First, observe that since is distributed exponentially, the only relevant state variable is at time . Thus, in the equations below, I drop the time dependence wherever it causes no confusion.

Since the bid and ask prices are conditional expectations, we can compute their values using Bayes’ rule.

(7)  

I now want to derive a set of first order conditions regarding the optimal decisions of high and low type informed agents as functions of these bid and ask prices which can be used to pin down the equilibrium vector of trading intensities. Let and denote the value functions of the high and low type informed traders respectively.

Condition 1: No arbitrage implies that for all with and since:

(8)  

Thus, for all it must be that and . What’s more, and are absorbing points for meaning that while .

Condition 2: The around a buy or sell order, the price moves by jumping from or from so we can think about the stochastic process as composed of a deterministic drift component and jump components with magnitudes and .

(9)  

The probabilities and can be computed using Bayes’ rule.

(10)  

Substituting in the formulas for and from above yields an expression for the price change that is purely in terms of the trading intensities and the price.

(11)  

However, via the conditional expectation price setting rule, must be a martingale meaning that .

(12)  

Condition 3: At the time of a buy or sell order, smooth pasting implies that the informed trader was indifferent between placing the order or not. For instance, if he strictly preferred to place the order, he would have done so earlier via the continuity of the price process.

(13)  

Condition 4: It is not optimal for the informed traders to bluff. i.e., a high type informed trader can never increase his value function by selling at time and vice versa for a low type informed trader.

(14)  

Condition 5: In all time periods in which the informed trader does not trade, smooth pasting implies that he must be indifferent between trading and delaying an instant . There are forces at work here. In each instant , there is a probability that will arrive and the informed trader’s value function will plummet to . This cost has to be offset by the value delaying. For the high type informed trader, this value includes the value change due to the price drift , the value change due to an uninformed trader placing a buy order with probability and the value change due to an uninformed trader placing a sell order with probability . Similar reasoning yields a symmetric condition for low type informed traders.

(15)  

This combination of conditions pins down the equilibrium.

Proposition: If the trading strategies are admissible, is a non-increasing function of , is a non-decreasing function of , both value functions satisfy the conditions above, and the trading strategies are continuously differentiable on the interval , then the trading strategies are optimal for all .

In the section below, I solve for the equilibrium trading intensities and prices numerically.

4. Numerical Solution

In the results below, I set and for simplicity. I compute the value functions and as well as the optimal trading strategies on a grid over the unit interval with nodes. Let denote the vector of prices.

(16)  

Let and denote the vector of value function levels over each point in the price grid after iteration . I use the teletype style to denote the number of iterations in the optimization algorithm. denotes the level of the value function at price point after iterations.

(17)  

The algorithm below computes , , and . The equilibrium trading intensities can be derived from these values analytically. I seed initial guesses at the values of and .

(18)  

Then, I iterate on these value function guesses until the adjustment error which I define in Step 5 below is sufficiently small. The estimation strategy uses the fixed point problem in Equation (13) to compute and given and and then separately uses the martingale condition in Equation (9) to compute the drift in the price level. The algorithm updates the value function in each step by first computing how badly the no trade indifference condition in Equation (15) is violated, and then lowering the values of for near when the high type informed trader is too eager to trade and raising them when he is too apathetic about trading and vice versa for the low type trader. Along the way, the algorithm checks that neither informed trader type has an incentive to bluff.

x-axis: Price of risky asset. Panel (a): Value function for the high (red) and low (blue) type informed trader. Panel (b): Bid (red) and ask (blue) prices for the risky asset. Panel (c): Between trade price drift.

Below I outline the estimation procedure in complete detail. Code the for the simulation can be found on my GitHub site.

Step 1. Numerically compute and  at each point.

(19)  

I fill in each of the boundary derivatives manually.

(20)  

Step 2. Solve for bid and ask prices using Equation (13).

(21)  

I interpolate the value function levels at and linearly. Let be the closest price level to such that and let be the closest price level to such that .

(22)  

Step 3. Compute using Equation (9).

(23)  

I then plug in Equation (10) to compute and .

(24)  

I then use Equation (7) to solve for and  in terms of only prices.

(25)  

Combining these equations leaves a formulation for which contains only prices.

(26)  

Step 4. At each for , set and ensure that Equation (14) is satisfied. If the high type informed traders want to sell at price , increase their value function at price by .

(27)  

If the low type informed traders want to buy at price , decrease their value function at price by .

(28)  

Step 5. Update and by adding times the between trade indifference error from Equation (15).

(29)  

Step 6. Evaluate update error.

(30)  

1. Introduction

In this post, I compute industry level summary statistics the CRSP monthly file using different industry classification schemes:

1. Fama and French (1988)
2. Moskowitz and Grinblatt (1999)

All of the code for the results below as well as a JSON file containing the industry classification schemes can be found at my GitHub page. I use the Zoom.it API to make it convenient to scroll around and inspect the large summary statistic plots I create. Each of these plots can be expanded to full screen mode using the controls at the lower right hand corner of the figure.

2. Data

In this section, I describe my data sources for the plots below.

CRSP Monthly File

I gather my stock data from the CRSP monthly file via the WRDS database. Thus, the unit of observation is a firm month pair. I restrict my attention to the time period from January 1988 to December 2010 to focus on the period of time over which the Fama and French (1988) industry classification scheme would have been widely known. I keep only actively traded firms listed on the NYSE, NASDAQ and AMEX exchanges. I require that the firm reports a non-missing price, return, share count and SIC code for a given month. I also remove any observations which lack valid data in the previous month. This leaves me with total firm month observations covering firms. The figure below plots the total number of firms in the dataset each month.

Number of firms in the monthly CRSP database from January 1988 to December 2010.

Industry Classifications

I created a JSON file to house CRSP-COMPUSTAT industry classification data. The data can be found in various places throughout the web; e.g., see Ken French’s website used in Fama and French (1988). However, everywhere I looked, the data came as a txt file with quirky formatting. For example, below is the first industry coding from the file on Ken French’s site:

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 1 Agric  Agriculture
          0100-0199 Agric production - crops
          0200-0299 Agric production - livestock
          0700-0799 Agricultural services
          0910-0919 Commercial fishing
          2048-2048 Prepared feeds for animals
 ...

This format is particularly difficult to read as it is irregularly spaced and little mark-up around the data. In response to this problem, I used Emacs Regexp to convert the file on Ken French’s website into a JSON format. I also coded up the firm industry classification used by Moskowitz and Grinblatt (1999). The JSON file contains major directories, one for the Fama and French (1988) industry classification scheme using different clusters and one for the Moskowitz and Grinblatt (1999) scheme with different clusters. The industry groupings are based on the SIC codes. Below I post a sample entry for the industry from the Fama and French (1988) scheme:

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{"Fama and French (1988)": {
    "Agriculture": {
	"Agric production - crops": {"start":100, "end":199},
	"Agric production - livestock": {"start":200, "end":299},
	"Agricultural services": {"start":700, "end":799},
	"Commercial fishing": {"start":910, "end":919},
	"Prepared feeds for animals": {"start":2048, "end":2048}
    },
    ...
}

Note that under the main heading there are several subindustry headings. The and  tags denote the initial and ending SIC codes for each subindustry. The Moskowitz and Grinblatt (1999) scheme is less complex. There is a simple start and stop date for each of the broad industry groupings:

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 "Moskowitz and Grinblatt (1999)": {
     "Mining": {"start":1000, "end":1499},
     "Food": {"start":2000, "end":2099},
     "Apparel": {"start":2200, "end":2399},
     "Paper": {"start":2600, "end":2699},
     "Chemical": {"start":2800, "end":2899},
     "Petroleum": {"start":2900, "end":2999},
     "Construction": {"start":3200, "end":3299},
     "Prim. Metals": {"start":3300, "end":3399},
     "Fab. Metals": {"start":3400, "end":3499},
     "Machinery": {"start":3500, "end":3599},
     "Electrical Eq.": {"start":3600, "end":3699},
     "Transportation Eq.": {"start":3700, "end":3799},
     "Manufacturing": {"start":3800, "end":3999},
     "Railroads": {"start":4000, "end":4099},
     "Other Transport.": {"start":4100, "end":4799},
     "Utilities": {"start":4900, "end":4999},
     "Retail": {"start":5000, "end":5299},
     "Dept. Stores": {"start":5300, "end":5399},
     "Retail": {"start":5400, "end":5999},
     "Financial": {"start":6000, "end":6999}
 }

3. Fama and French (1988) Classification

In this section, I plot different summary plots of the CRSP data split by the Fama and French (1988) industry classification. First, I plot the number of firms in each industry. In all of the plots, I omit the “Other” industry containing firms with no clear industry classification. All of the industries except for Candy and Soda, Coal, Non-Metalic and Industrial Mining, Pharmaceutical Products, Precious Metals and Trading display a single peaked pattern indicating that the number of firms in each industry dramatically expanded around 2000.

Next, I break down this firm count by industry plot even further into sub-industries in the figure below. This plot reveals that there is wide variation in the number of subindustries. What’s more, this single peaked pattern does not persist as strongly at the sub-industry level.

I then turn to market capitalization by industry rather than firm counts. In the figure below, we see that while the number of firms in most industries has been shrinking since 2000, each industry’s market capitalization has been rising steadily. Thus, the combination of the first figure with the figure below reveals that industries have been consolidating.

Finally, I look at the distribution of monthly excess returns defined as where is the month T-Bill by industry. Due to space constraints, it was not possible to plot box plot for each month of observations, so instead I first computed the mean monthly excess return for each firm in each year and then computed yearly box plots. Thus, a data point in the plot below is a mean monthly return for a particular firm over the whole year. This figure reveals that there are large outliers in the return distribution that need to be addressed before any further data work can be done. For instance, in 1992 a firm in the entertainment industry earned an average monthly excess return of over .

4. Moskowitz and Grinblatt (1999) Classification

I also create similar plots for the industry classification system used in Moskowitz and Grinblatt (1999) which contains only industries rather than the in the Fama and French (1988) system. These charts generally mirror the insights from above—just at a much more granular level. First, I plot the number of firms in each of the industries. These industry grouping were chosen in part to balance out the partition of firms across industries and, as a results, display a much more even cross-sectional distribution.

Again, the market capitalization plot reveals that firms have been consolidating within each industry since .

Finally, I plot the distribution of excess returns for each firm within industry as above.