December 21, 2012 Options Expirations and Update

Closed:
 Pos    Symbol                   Price     Comm       Net
 -2 RIG 22DEC12 47.0 C            0.00     0.00        0.00
     Basis 11/19/2012:                              (199.22)
     Profit/(Loss):                                  199.22

 -5 CREE 22DEC12 15.0 P           0.00     0.00        0.00
     Basis 05/21/2012:                              (473.57)
     Profit/(Loss):                                  473.57

     Value 08/31/2012:                               (68.97)

     Post-August Profit/(Loss):                       68.97

 -4 DOW 22DEC12 25.0 P            0.00     0.00        0.00
     Basis 04/26/2012:                              (338.85)
     Profit/(Loss):                                  338.85

     Value 08/31/2012:                              (248.12)

     Post-August Profit/(Loss):                      248.12

 -3 NE 22DEC12 30.0 P             0.00     0.00        0.00
     Basis 04/23/2012:                              (496.72)
     Profit/(Loss):                                  496.72

     Value 08/31/2012:                              (199.08)

     Post-August Profit/(Loss):                      199.08

 -3 POT 22DEC12 32.5 P            0.00     0.00        0.00
     Basis 04/23/2012:                              (387.82)
     Profit/(Loss):                                  387.82

     Value 08/31/2012:                              (150.20)

     Post-August Profit/(Loss):                      150.20


 -5 TIE 22DEC12 10.0 P            0.00     0.00        0.00
     Basis 04/23/2012:                              (222.37)
     Profit/(Loss):                                  222.37

     Value 08/31/2012:                              (142.82)

     Post-August Profit/(Loss):                      142.82

Also since the last update, $7,500.00 was withdrawn from the account on December 4, $112.50 in dividends was paid on December 1 for 500 shares INTC, $82.50 was paid December 12 on 300 shares WAG, $142.00 was paid December 14 on 400 shares WM, $172.00 in dividends became payable December 12 on 400 shares ARCC, $102.00 became payable December 18 on 200 shares CSE, $10.00 became payable December 5 on 100 shares FTR, $190.00 became payable December 20 on 1,000 shares GE, $6.56 interest accrued and $8.33 was paid.

The Black-Scholes Risk-Neutral Pricing Probability Distribution Function

I wrote previously that the Black-Scholes options pricing model replaces the equity drift μ with r - ½σ²,  where r is the riskless interest rate and σ is the stochastic element of the equity's price movement. This substitution has the effect of eliminating the equity drift coefficient from the model, and obviating the need to decide what the drift coefficient actually is.

The reasoning given by Black and Scholes in their paper that introduced their model (as well as one of the very first uses of risk-neutral pricing generally) were (1) that, through continuous rebalancing, the second-order (dz²) stochastic terms in the return on a portfolio became riskless (this argument implicitly relies upon Itō's Lemma, although the paper did not explicitly refer to it), and (2) that even if they were not riskless due to discontinuity, the second-order terms should be subject to the riskless interest rate because they are uncorrelated with the market rate of return. Thus, they argued, second-order terms must be priced with the riskless interest rate directly, and any risk premium must apply only to any first-order stochastic terms.

This can only be achieved by adapting only the drift coefficient -- and not the stochastic coefficient -- in constructing a risk-neutral probability distribution. This means replacing the equity drift μ with r - ½σ².

And this is all roughly consistent with traditional utility theory.

Delta Hedge

A delta hedge is a portfolio made from two or more distinct securities that are related to the same underlying security. The delta hedge is designed to cancel out, at least instantaneously, the first order relationship (delta) of any constituent securities to the underlying security.

Itō's Lemma

Itō's Lemma is a theorem of stochastic calculus that holds that within a closed integral, dz² can be replaced by dt, where dz is a stochastic variable with order of magnitude equal to the square root of dt. The Lemma is sometimes erroneously stated as "dz² equals dt," which is not generally true. Integration, with continuity, invokes the Law of Large Numbers. In the absence of continuity the variance of dz² is proportional to Δt, length of measurement intervals taken over the range.

Itō's Lemma is significant in finance because it provides the basis according to which a delta hedge is assumed to be riskless, an assumption that is essential to the Black-Scholes Equation.

The Lognormal Distribution and Risk-Neutral Pricing

I wrote previously that the ergodic property of the normal distribution is so useful that it often makes sense to assume a normal distribution even in the face of evidence to the contrary.

Before going to that extreme, however, it is sometimes possible to arrive at a normal distribution by looking at a function of an original variable rather than the variable itself. One example of this is the lognormal distribution, which is a distribution whose log is a normal distribution. In finance, the future price of a stock is often considered to have a lognormal distribution, which gives the rate of return on the stock a normal distribution.

In considering a lognormal distribution, we generally refer to aspects of its log. In particular, we usually define a particular lognormal distribution based upon the mean, μ, and variance, σ², of the log.

There are any number of things to be known about lognormal distributions, but the most important fact about them for my purposes is that the expected value of a lognormal distribution is exp(μ + ½σ²).

Getting back to finance, the future price of a stock at time t can be considered to have a lognormal distribution with log-mean lnS + μt and log-variance σ²t, where S is the price of the stock at time 0. (here I have effectively used μ and σ² as the mean and variance of the instantaneous rate of return of the stock, so that at any given point in time in the future t the rate of return on the stock will have mean μt and variance σ²t). This gives the expected value of the stock at time t as exp(lnS + μt + ½σ²t).

Under a risk-neutral pricing regime, however, the expected future price of the stock should be exp(lnS + rt), where r is the riskless interest rate. So if we want to preserve the lognormal distribution of the stock price, we somehow have to adapt our real world expectations into risk-neutral probabilities so that exp(lnS + μt + ½σ²t) equals exp(lnS + rt), or more simply so that μ + ½σ² equals r.

One method to accomplish this is simply to replace μ with r - ½σ². And this is precisely what the Black-Scholes options pricing model does, about which more later.

Risk Arbitrage

Conventional arbitrage is most succinctly defined as creating a portfolio with zero cost that has no possibility of a negative future value, and at least some possibility of a positive future value. There would be no reason that an investor would not create as much of this portfolio as possible -- it costs zero -- and maximize his possible future value.

Conventional arbitrage is riskless: there is no possibility of any loss. There are sometimes references to "risk arbitrage," which is (vaguely) taking a position that has a better return than its risk profile would suggest.

I have developed a more precise definition of risk arbitrage, based on the ability to short a portfolio with very little likelihood of loss. In particular, if for any positive x, positive or negative y, and positive z, a portfolio can be created to short for which the probability of any future positive value at all is less than x, the expected rate of return is less than y, and the maximum future value is less than z, then an opportunity for risk arbitrage exists.

This definition relies on the fact that prices are set at the margin: if any investor is willing to entertain the possibility of additional risk, then regardless of how small z is set (which will tend to diminish the size of the short position taken), the price paradigm will be broken.

Basically, the choice of z sets the maximum loss from shorting the portfolio, and then x and y can be selected to make arbitrarily remote the probability of any loss and arbitrarily small the expected loss: if no one is interested in selling short a portfolio that has 1% chance of having any positive future value at all at a given future point in time and that has -50% expected rate of return, how about a portfolio with 0.5% chance of having any positive future value and that has -75% expected rate of return? And the probability of future value, and the expected future value, can be continually reduced until a reasonable investor would not be able to resist at least a taste.

Laws of Price: Prices Are Set At The Margin

In a free market, a prevailing price means just one thing: that the demand and the supply are (for a moment) equal at that price.

In a grand sense, this is not a very powerful meaning: we conventionally think of the value of portfolios as the sum of the products of the positions in every investment and the last price of the particular investment. So with 1,000,000 shares of IBM trading at $200.00 a share, this would have a value of $200,000,000.00. But if someone holding this position were to try to convert it to cash all at once, he would almost certainly net less than $200,000,000.00: his position is equivalent to about 25% of the average daily volume of the stock, and it is unlikely that buyers would arrive conveniently to absorb the additional sales that he has introduced, and leave the stock price unaffected.

But in a microscopic sense, it is quite a powerful meaning. In particular, if there is reason to believe that a realistic investor would want to buy rather than sell (or vice versa) a particular position at a given price, then it does not matter that this tendency would be exhausted with just a tiny position: the price cannot be expected to hold, other things being equal.

The latter fact is central to my understanding of the pricing of financial risk.

November 30, 2012 Update

    Cash:                             $     15.48
    Accrued Dividends:                     337.00
    Interest Accruals:                      (7.50)
    Stocks:                            120,665.58
    Options:                            (8,294.33)

  Account Value:                      $112,716.23

  Equity:                             $120,912.18
  Margin Requirement:                   73,669.98
  Available Funds:                      47,242.20

  Regulation T Margin Requirement:     102,157.88
  Special Memorandum Acct:              37,567.91

The account value of $112,716.23 compares with the value a month earlier of  $114,712.06, a return of -1.74%. However, this does not take into account $2,500.00 withdrawn from the account during the month. Adding this in to the final value gives a return for the month of 0.44%.

This compares with the change in the S&P 500 Index from 1,412.16 to 1,416.18. Adding an estimate of a quarter percent dividend payment (one twelfth of 3%), this gives a return of 0.53% for the S&P 500, so my portfolio underperformed the market in the month of November.

November 30, 2012 Trade

Opened:
 Pos    Symbol                   Price     Comm       Net
 -2    HES 18MAY13 37.5 P         0.85     1.52     (168.48)

November 29, 2012 Trade

Opened:
 Pos    Symbol                   Price     Comm       Net
 -4    MDT 18MAY13 30.0 P         0.20     1.52      (78.48)