Monthly Archives: April 2012

The Barter Economy and Say’s Law

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At what number of participants in an economy do the laws of economics change?  Asked another way, at what number of participants does an enlightened, benevolent, exogenous force need to step in to the system from above to use force or to manipulate the system for the participants’ own good?  This of course presupposes that the benevolent outside force possesses some knowledge of which the system’s participants are unaware and that he can use this knowledge for their good.

In a two person, cooperating economy, both actors are aware of their needs, the other’s needs, their capabilities, and the other’s capabilities.  The level of communication and the awareness of each other’s comparative advantages would be such that coordination of production would be simple.  Shortages due to natural factors or personal limitations could arise, but these would be out of the control of the actors and could be mitigated through the creativity of the two people.  A third party with more knowledge of some production technique could offer advice, but he would then be entering the economy, even if he chose to abstain from trade (he would have to be autarchic).  The same could be said of systems of three, four, or more people.  At some number of participants, however, the actions of all other participants could not be known by every other actor, so personal coordination could not take place.  On the other hand, the likelihood that someone or some combination of people will produce the goods desired by any one person increases with the number of participants in the economy.  The likelihood that some outside expert would possess knowledge unknown to any other participant decreases with the number of participants.  Does the likelihood of discoordination also increase with the number of participants?  Is each actor more secure or less secure in a two or three person economy than in an economy with more participants?  Is there an optimal number of participants?

Let’s say that for a barter economy the answer is no, that the economic laws do not break down with some high number of participants and that people in general are better off and less likely to have unfulfilled physical wants as more people, each of which is both a producer and a consumer, enter the economy.  Even if some discoordination happened, the economy could divide into smaller units, perhaps by geography, until the optimal number of actors was reached and coordination and stability returned.  What if each actor produces some specialized good which no other person makes and that person leaves the market?  A greater number of actors increases the likelihood that someone else produces something similar and can step in to fill the unmet demand.  The larger system is more robust.

Jean-Baptiste Say described what would later be called Say’s Law to explain how economic downturns are not the result of weak demand, but lack of production.  He says that the ability to demand can only come from a previous act of production, which in the barter economy described above is obvious.  But what happens when money is introduced into the economy and some people, instead of producing finished goods themselves, hire themselves out to others in exchange for wages?  Does Say’s Law still hold?

N(d1) and N(d2) in the Black-Scholes Formula

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N(d2) is the probability of exercise of an option with a lognormally distributed underlying following the standard BSM assumptions (drift is (r – .5σ2)(T-t)).  N(d2) defines the area of integration to give the probability of exercise and is

                Z > [ln(K/S) – (r – .5σ2)(T-t)]/σ*sqrt(T-t) = z0
meaning that the area of integration is z0 to ∞, or -∞ to  –z0, the part of the normal distribution where exercise takes place.  The option price formula can be viewed as a conditional expectation multiplied by a probability,
                e-r(T-t)[E[ST|ST > K]*N(d2) – K*N(d2)]
The conditional expectation E[ST|ST > K] can also be written as
                Se(r-.5σ^2)(T-t) + σ*sqrt(T-t)*Z
where the Z in the last term ensures that only values of the underlying S where exercise takes place are considered.  The greater the volatility σ, the greater the possible up move, the greater the area of integration, and hence the greater the conditional expectation of S.
The option price formula can now be written as
                e-r(T-t)[(Se(r-.5σ^2)(T-t) + σ*sqrt(T-t)*Z)*N(d2) – K*N(d2)]
The e(r-.5σ^2)(T-t) + σ*sqrt(T-t)*Z term can be combined with N(d2) since it is also a standard normal probability.  N(d1) is then just N(d2) +  σ*sqrt(T-t), which gives N(d1) a greater expectation based on σ.  The option price formula can now be written as
                e-r(T-t)[(Ser*N(d1) – K*N(d2)] or
                S*N(d1)- e-r(T-t)K*N(d2)
N(d1) simply takes into account the higher conditional expectation of the random underlying S versus the static K.

The Black-Scholes Differential Equation

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dS = μSdt + σSdW is an Ito process with drift µ and diffusion W. The instantaneous change in W, dW, is a Brownian motion scaled by a constant σ. The increments of dW are normally distributed with mean 0 and standard deviation 1.

In Ito calculus, dt and dW behave like this when multiplied:
dWdW = dt
dtdW = 0
dtdt = 0
This means that
(dS)² = σ²S²dt.

If C is an option and S is its underlying asset then, applying Ito’s lemma and collecting terms, we get:

dC = (∂C/∂t)dt + (∂C/∂S)dS + ½(∂²C/∂S²)(dS)²

dC = (∂C/∂t)dt + (∂C/∂S)( μSdt + σSdW)
+ ½(∂²C/∂S²)σ²S²dt

dC = (∂C/∂t)dt + (∂C/∂S) μSdt + (∂C/∂S)σSdW
+ ½(∂²C/∂S²)σ²S²dt

dC = [(∂C/∂t) + μS(∂C/∂S) + ½(∂²C/∂S²) σ²S²]dt
+ (∂C/∂S)σSdW,

which is the change in the option’s value over small increments of time.

The first term, called an infinitesimal operator, has a standard, Newtonian derivative,dt, and the dW term has a stochastic, or random, derivative. To create a riskless position, the random term must be hedged away.

A position of one option and Δ shares of S is created, since all changes in the option’s value depend on changes in S. The portfolio is

Π = C + ΔS

The change in the portfolio’s value is

dΠ = dC + ΔdS.

Substituting dC and dS from above gives:

dΠ = [(∂C/∂t) + μS(∂C/∂S) + ½(∂²C/∂S²) σ²S²]dt + (∂C/∂S) σSdW + Δ[μSdt + σSdW]

and

dΠ = [(∂C/∂t) + μS(∂C/∂S) + ½(∂²C/∂S²) σ²S²]dt
+ (∂C/∂S) σSdW + ΔμSdt + ΔσSdW

Set

(∂C/∂S) σSdW + ΔσSdW = 0
to get rid of the random terms. This means that (∂C/∂S) = -Δ, and the risky terms cancel each other out, leaving

dΠ = [(∂C/∂t) + ½(∂²C/∂S²) σ²S²]dt
as the portfolio’s change in value under small changes in time. The portfolio is riskless now, so it has an expected growth of r, meaning

dΠ = rΠdt = r(C + ΔS)dt = r[C – (∂C/∂S)S]dt
Setting

r[C – (∂C/∂S)S]dt = [(∂C/∂t) + ½(∂²C/∂S²) σ²S²]dt
and cancelling terms gives

rC = (∂C/∂t) + r(∂C/∂S)S + ½(∂²C/∂S²) σ²S²
which is the Black-Scholes-Merton differential equation, a 2nd order parabolic partial differential equation. Notice that μ, the drift of the underlying S, is not in the equation. The only drift term is the risk free interest rate r.

Replacing each partial differential with its corresponding Greek letter gives

rC = Θ + rSΔ + ½ σ²S² Γ

where
∂C/∂t = Θ
∂C/∂S = Δ
∂²C/∂S² = Γ

Dynamics of the Lognormal Process Used in the Black-Scholes Formula

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Start with the log return of a process, ln(ST/St)

Assume that S follows the process dSt = µdt + σdWt, where µ is drift, σ is volatility, and Wt is a random process that is N~(0,1)

Find the stochastic derivative of the process ln(S) using the Ito formula, dft = ft  + fs + 1/2  fss :

             dln(S) = 1/S(dS) – (1/2)(1/S2)(dS2)

             dln(S) = (µ – ½ σ2)dt + σdWt

 

This says that the logarithm of a random process that is normally distributed is concave and has a mean of (µ – ½ σ2) and that

             ln(ST/St) = ln(S1) – ln(S0) = (µ – ½ σ2)(T-t) + σ(W(T) – W(t))

             ln(ST) = ln(St) + (µ – ½ σ2)(T-t) + σ(W(T) – W(t))

 

Assume that St is lognormally distributed, meaning it is an exponentiated normal process.  Take the exponential of both sides of the above equation:

             ST = Ste(µ – 1/2σ^2)dt + σdWt

 

The lognormal process SeX, X ~ N(µ, σ), is convex and has a mean of (µ + ½ σ2), so the above equation has an expectation of

             ST = Steµt

 

since the- ½ σ2  concavity of the log of a normal and the + ½ σ2 implicit convexity in the exponentiated normal cancel each other and the expectation of dWt is 0.  This is necessary for the process St to be a martingale when µ = 0.