Sunday, October 28, 2012
October 22, 2012 Trades
Opened:
Pos Symbol Price Comm Net
-4 BMY 17NOV12 35.0 C 0.17 3.04 (64.96)
-5 INTC 20NOV13 25.0 C 0.43 3.80 (211.20)
-2 RIG 17NOV12 50.0 C 1.16 0.76 (231.24)
-4 STX 17NOV12 27.0 C 2.00 3.05 (796.95)
Pos Symbol Price Comm Net
-4 BMY 17NOV12 35.0 C 0.17 3.04 (64.96)
-5 INTC 20NOV13 25.0 C 0.43 3.80 (211.20)
-2 RIG 17NOV12 50.0 C 1.16 0.76 (231.24)
-4 STX 17NOV12 27.0 C 2.00 3.05 (796.95)
Probability Distribution Functions and Random Variables
A probability distribution function (p.d.f.) is the collected measure of the likelihoods of each possible outcome of a random event. So a p.d.f. for the future price of a particular stock would give a likelihood of each possible future stock price, from zero on up.
Probability distribution functions have a couple qualities:
Probability distribution functions have a couple qualities:
- they must be strictly non-negative; and
- the sum of all probabilities under a p.d.f. must be one.
If a random variable has a known p.d.f., two important values can be determined for it: its expected value, or mean, which is sometimes designated with the Greek letter μ; and variance, which is designated with σ². Variance is the expected value of the square of the difference between a random variable and its expected value.
There are a lot of things of interest about mean and variance, but for my purpose, only a couple are important.
First, the square root of variance, σ, or standard deviation, can be used as a measure of confidence intervals for a random variable: for a normally distributed variable, for example, the span within about 1.96 standard deviations of the mean of the variable forms a 95% confidence interval.
Second, both mean and variance are additive, which is to say that if X and Y are random variables, then generally the mean of X+Y is the mean of X plus the mean of Y, and the variance of X and Y is the variance of X plus the variance of Y. (The latter isn't quite true, because if X and Y are not independent -- the outcome of X affects the p.d.f. of Y -- then the variance of X+Y is the variance of X plus the variance of Y plus twice the covariance of X and Y. The covariance of two random variables is the expected value of the product of the differences between the variable and their respective means. I'll almost always be assuming independence between variables, so covariance won't matter to me much.)
In combination, these two factors create an important effect: the expected value for a sum of identically distributed independent variables grows with the number of variables included in the sum, while confidence intervals for this sum grow with the square root of the number of variables included in the sum. This gives the Law of Large Numbers: the average of a number of outcomes of independent variables from an identical distribution will approach their expected value.
Risk-Neutral Pricing
I have previously introduced the concept of the "risk premium" in investing, which holds that "risky" investments should have a higher expected rate of return than "non-risky" investments. In that earlier entry, I noted that Utility Theory provides a framework for explaining why the risk premium should exist. In this entry, I will try to explain a useful way of quantifying the disposition of risk premium: "risk-neutral pricing."
Typically, the risk premium will be spoken of in terms of "expected rate of return," as in, "the expected rate of return on common stocks is around 6%, while the expected rate of return on cash is around 1%." The higher expected rate of return stated here for stocks as opposed to cash indicates a higher risk premium for stocks.
However, when option securities are considered, this way of thinking runs aground. We might consider a call option, for example, that pays in dollars the amount (if any) by which a particular stock index exceeds a given value at close of trading on a particular day, or nothing if the stock index is less than or equal to the value. But now how do we come up with an expected rate of return for this option?
If we look at the Put-Call Parity equation,
C - P = S - X * exp(-rT),
we see that on the right hand side of the equation are terms, S and X * exp(-rT), that should have the expected returns of stocks and of cash, respectively (cash being a risk-free asset, subject to the risk-free interest rate of r).
So what if we treat the call as the difference between the index value -- discounted at the expected returns on stocks -- and a riskless payment of the strike price -- discounted at the risk-free interest rate -- on the expiration date?
This method falls apart pretty quickly. Suppose we look at just a piece of the call using this perspective: the portion from where the index equals the strike price to where it's the strike price plus one. Let's suppose also that the strike price is 1000 and the expiration date is one year in the future. So the value of the stock piece, using the expected rates of return above, will be between 1000 and 1001 discounted by 6%, or between about 943.40 and 944.34. But the value of the risk-free piece will be 1000 discounted by 1%, or about 990.10. This suggests that the value of this portion of the call option will be between -46.70 and -45.76: negative values for something that can only ever result in positive cash flow! No rational person will give you money to accept the possibility that he might give you more money later on.
The way around this is to discount everything at the risk-free interest rate: both the stock portion and the strike price portion. This gives the value of the portion of the call option we're looking at as being between 0 and 0.99, which is very consistent with the value at the end of the year being between 0 and 1.
But where does the risk premium appear?
The risk premium appears in adjustments to the probabilities of different outcomes. In particular, prices will be determined using probabilities that are different from actual expectations.
In looking at the portion of the call option where the final index value is between the strike price and the strike price plus one, I never considered the probability of this outcome: I essentially took each potential outcome -- from the index being equal to the strike price up to being that value plus one -- as given, and looked at what the value of the option would be under that outcome. To get an actual value of the call option, I'd have to assign a probability to each of the outcomes, and generate a weighted average for the option's value under all possible outcomes. In essence, the risk premium on stocks is created because the pricing probability assumes that high returns on stocks are less likely than actual expectations.
So, why use the risk-less interest rate rather than some other rate?
First of all, it would probably be more accurate to use actual expectations, and then have a different expected rate of return apply to each possible outcome. But this would create a mess: how do you figure out what expected rate of return to apply to a given outcome, and from where (without a LOT of work) do you get actual expectations?
Risk-neutral pricing takes advantage of the fact that prices based on uncertain events are based on the product of the probability of given outcomes and the expected rate of return that applies to that outcome. So rather than (more accurately) using actual expectations for the probabilities and some yet-to-be-determined expected rate of return for each outcome, risk-neutral pricing uses a single expected rate of return (the risk-free interest rate) and adjusted probabilities to come up with the same price that would result from the theoretically more accurate method of using actual expectations and variable expected rates of return.
So, again, why use the risk-less interest rate rather than some other rate?
The class of securities that by definition earn the risk-free interest rate -- riskless securities -- place some constraints on the adjusted pricing probabilities: the price indicated, under whatever pricing algorithm we use, for a fixed amount paid under all possible outcomes must be consistent with the risk-free interest rate. If we try to do this using some other interest rate, we will have to have our adjusted probabilities sum to less than 1 if we use an interest rate lower than the risk-free interest rate, and we will have to have our adjusted probabilities sum to more than 1 if we use an interest rate higher than the risk-free rate. The adjusted probabilities would not be proper probability distribution functions, because a proper probability distribution function will sum to exactly 1. And, if a function that did not sum to 1 was used, it would create immediate arbitrage potential in either selling a payoff under all outcomes if the function sums to less than 1, or buying a payoff under all outcomes if the function sums to more than 1
So, it is ultimately simplicity that drives us to use a single interest rate and adjusted probabilities rather than variable expected rates of returns and actual expectations. Then, having limited ourselves in this way out of concern for simplicity, we are limited to using the risk-free interest rate as our single expected rate of return for the reason given above.
Finally, the reason this method is called "risk-neutral pricing" is that, using the adjusted pricing probabilities, all securities are assumed to have an equal expected rate of return, which is consistent with a risk-neutral market.
Typically, the risk premium will be spoken of in terms of "expected rate of return," as in, "the expected rate of return on common stocks is around 6%, while the expected rate of return on cash is around 1%." The higher expected rate of return stated here for stocks as opposed to cash indicates a higher risk premium for stocks.
However, when option securities are considered, this way of thinking runs aground. We might consider a call option, for example, that pays in dollars the amount (if any) by which a particular stock index exceeds a given value at close of trading on a particular day, or nothing if the stock index is less than or equal to the value. But now how do we come up with an expected rate of return for this option?
If we look at the Put-Call Parity equation,
C - P = S - X * exp(-rT),
we see that on the right hand side of the equation are terms, S and X * exp(-rT), that should have the expected returns of stocks and of cash, respectively (cash being a risk-free asset, subject to the risk-free interest rate of r).
So what if we treat the call as the difference between the index value -- discounted at the expected returns on stocks -- and a riskless payment of the strike price -- discounted at the risk-free interest rate -- on the expiration date?
This method falls apart pretty quickly. Suppose we look at just a piece of the call using this perspective: the portion from where the index equals the strike price to where it's the strike price plus one. Let's suppose also that the strike price is 1000 and the expiration date is one year in the future. So the value of the stock piece, using the expected rates of return above, will be between 1000 and 1001 discounted by 6%, or between about 943.40 and 944.34. But the value of the risk-free piece will be 1000 discounted by 1%, or about 990.10. This suggests that the value of this portion of the call option will be between -46.70 and -45.76: negative values for something that can only ever result in positive cash flow! No rational person will give you money to accept the possibility that he might give you more money later on.
The way around this is to discount everything at the risk-free interest rate: both the stock portion and the strike price portion. This gives the value of the portion of the call option we're looking at as being between 0 and 0.99, which is very consistent with the value at the end of the year being between 0 and 1.
But where does the risk premium appear?
The risk premium appears in adjustments to the probabilities of different outcomes. In particular, prices will be determined using probabilities that are different from actual expectations.
In looking at the portion of the call option where the final index value is between the strike price and the strike price plus one, I never considered the probability of this outcome: I essentially took each potential outcome -- from the index being equal to the strike price up to being that value plus one -- as given, and looked at what the value of the option would be under that outcome. To get an actual value of the call option, I'd have to assign a probability to each of the outcomes, and generate a weighted average for the option's value under all possible outcomes. In essence, the risk premium on stocks is created because the pricing probability assumes that high returns on stocks are less likely than actual expectations.
So, why use the risk-less interest rate rather than some other rate?
First of all, it would probably be more accurate to use actual expectations, and then have a different expected rate of return apply to each possible outcome. But this would create a mess: how do you figure out what expected rate of return to apply to a given outcome, and from where (without a LOT of work) do you get actual expectations?
Risk-neutral pricing takes advantage of the fact that prices based on uncertain events are based on the product of the probability of given outcomes and the expected rate of return that applies to that outcome. So rather than (more accurately) using actual expectations for the probabilities and some yet-to-be-determined expected rate of return for each outcome, risk-neutral pricing uses a single expected rate of return (the risk-free interest rate) and adjusted probabilities to come up with the same price that would result from the theoretically more accurate method of using actual expectations and variable expected rates of return.
So, again, why use the risk-less interest rate rather than some other rate?
The class of securities that by definition earn the risk-free interest rate -- riskless securities -- place some constraints on the adjusted pricing probabilities: the price indicated, under whatever pricing algorithm we use, for a fixed amount paid under all possible outcomes must be consistent with the risk-free interest rate. If we try to do this using some other interest rate, we will have to have our adjusted probabilities sum to less than 1 if we use an interest rate lower than the risk-free interest rate, and we will have to have our adjusted probabilities sum to more than 1 if we use an interest rate higher than the risk-free rate. The adjusted probabilities would not be proper probability distribution functions, because a proper probability distribution function will sum to exactly 1. And, if a function that did not sum to 1 was used, it would create immediate arbitrage potential in either selling a payoff under all outcomes if the function sums to less than 1, or buying a payoff under all outcomes if the function sums to more than 1
So, it is ultimately simplicity that drives us to use a single interest rate and adjusted probabilities rather than variable expected rates of returns and actual expectations. Then, having limited ourselves in this way out of concern for simplicity, we are limited to using the risk-free interest rate as our single expected rate of return for the reason given above.
Finally, the reason this method is called "risk-neutral pricing" is that, using the adjusted pricing probabilities, all securities are assumed to have an equal expected rate of return, which is consistent with a risk-neutral market.
Friday, October 26, 2012
Put-Call Parity
The "Put-Call Parity" equation is an instance of the Law of One Price. Its mathematical form is
C - P = S - X * exp(-rT),
where S is the price of a stock at time 0, C is the price at time 0 of a European call option on S with strike price X and expiration time T, P is the price at time 0 of a European put option on S with strike price X and expiration time T, and r is the risk-free interest rate (assumed to be constant).
The term X * exp(-rT)in the equation is constant, and it represents the value at time 0 of the amount X paid with certainty at time T. Figuring out the value of a call and a put with the given strike for any given price of the stock on the expiration date will show that being long such a call and short such a put will always have the save payoff at time T as the stock itself less a payment of X at the same instant, which invokes the Law of One Price.
Put-Call Parity is model independent, meaning that it will hold regardless of under what model prices are determined. Considered differently, any model that violates Put-Call Parity cannot be realistic.
C - P = S - X * exp(-rT),
where S is the price of a stock at time 0, C is the price at time 0 of a European call option on S with strike price X and expiration time T, P is the price at time 0 of a European put option on S with strike price X and expiration time T, and r is the risk-free interest rate (assumed to be constant).
The term X * exp(-rT)in the equation is constant, and it represents the value at time 0 of the amount X paid with certainty at time T. Figuring out the value of a call and a put with the given strike for any given price of the stock on the expiration date will show that being long such a call and short such a put will always have the save payoff at time T as the stock itself less a payment of X at the same instant, which invokes the Law of One Price.
Put-Call Parity is model independent, meaning that it will hold regardless of under what model prices are determined. Considered differently, any model that violates Put-Call Parity cannot be realistic.
Laws of Price: The Law of One Price
I think it's a useful exercise to try clearly to state as many requirements as possible that apply to efficient free market prices. So from time to time I'll devote a post to one or more of these, which I will call "Laws of Price."
The first that I'll look at is a very widely known law with many implementations: the Law of One Price.
Very simply, the Law of One Price states that if two different portfolios of securities have exactly the same payoff at a future date under all possible outcomes, then they must have the same price. Stated differently, there can be only one price for a given pattern of future cash flows, regardless of how the pattern is achieved.
So, a bet that pays $1 if the Steelers beat the Redskins this coming Sunday, plus a bet that pays $1 if the Redskins beat the Steelers in the same game, plus a bet that pays $1 in the event of a tie in the game, should have the same price as $1 paid after the game Sunday with certainty.* If they don't, then a guaranteed profit can be made by selling the more expensive position and buying the less expensive one: simple arbitrage. (The Law of One Price is sometimes referred to with the qualification "no arbitrage," or "arbitrage free.")
* -- These bets will not in fact add up to the value of $1 paid with certainty, but this is because the sports betting market is not efficient: the bookies and the odds-makers arrange to take a percentage of each bet as their compensation, so making all of these bets will cost more than $1 paid with certainty. Also, the "frictional costs" -- costs associated with transaction fees or expenses from acquiring or disposing of positions -- of attempting arbitrage will allow some deviation of prices away from a singular price. But the prices cannot stray too far apart.
The first that I'll look at is a very widely known law with many implementations: the Law of One Price.
Very simply, the Law of One Price states that if two different portfolios of securities have exactly the same payoff at a future date under all possible outcomes, then they must have the same price. Stated differently, there can be only one price for a given pattern of future cash flows, regardless of how the pattern is achieved.
So, a bet that pays $1 if the Steelers beat the Redskins this coming Sunday, plus a bet that pays $1 if the Redskins beat the Steelers in the same game, plus a bet that pays $1 in the event of a tie in the game, should have the same price as $1 paid after the game Sunday with certainty.* If they don't, then a guaranteed profit can be made by selling the more expensive position and buying the less expensive one: simple arbitrage. (The Law of One Price is sometimes referred to with the qualification "no arbitrage," or "arbitrage free.")
* -- These bets will not in fact add up to the value of $1 paid with certainty, but this is because the sports betting market is not efficient: the bookies and the odds-makers arrange to take a percentage of each bet as their compensation, so making all of these bets will cost more than $1 paid with certainty. Also, the "frictional costs" -- costs associated with transaction fees or expenses from acquiring or disposing of positions -- of attempting arbitrage will allow some deviation of prices away from a singular price. But the prices cannot stray too far apart.
Monday, October 22, 2012
October 19, 2012 Options Expirations, Trades, and Update
Closed:
Pos Symbol Price Comm Net
-4 BMY 20OCT12 34.0 C 0.00 0.00 0.00
Basis 09/24/2012: (102.48)
Profit/(Loss): 102.48
-2 RIG 20OCT12 50.0 C 0.00 0.00 0.00
Basis 09/24/2012: (150.48)
Profit/(Loss): 150.48
-3 WAG 20OCT12 33.0 C 2.84 2.28 854.28
Basis 05/21/2012: (457.72)
Profit/(Loss): (396.56)
Value 08/31/2012: (940.89)
Post-August Profit/(Loss): 86.61
-3 DD 20OCT12 40.0 P 0.00 0.00 0.00
Basis 02/29/2012: (288.55)
Profit/(Loss): 288.55
Value 08/31/2012: (20.29)
Post-August Profit/(Loss): 20.29
-5 INTC 20OCT12 25.0 P 3.735 0.00 1,867.50
Basis 08/17/2012: (242.70)
Profit/(Loss): (1,624.80)
Value 08/31/2012: (494.90)
Post-August Profit/(Loss): (494.90)
-3 SNDK 20OCT12 35.0 P 0.00 0.00 0.00
Basis 02/21/2012: (554.08)
Profit/(Loss): 554.08
Value 08/31/2012: (164.04)
Post-August Profit/(Loss): 164.04
-4 STX 20OCT12 28.0 P 0.10 0.00 40.00
Basis 08/21/2012: (208.96)
Profit/(Loss): 168.96
-2 TM 20OCT12 55.0 P 0.00 0.00 0.00
Basis 02/29/2012: (109.39)
Profit/(Loss): 109.39
Value 08/31/2012: (2.47)
Post-August Profit/(Loss): 2.47
-3 UPS 20OCT12 33.0 P 0.00 0.00 0.00
Basis 02/23/2012: (249.39)
Profit/(Loss): 249.39
Value 08/31/2012: (12.47)
Post-August Profit/(Loss): 12.47
Opened:
Pos Symbol Price Comm Net
500 INTC INTEL CORPORATION 21.265 0.00 10,632.50
400 STX SEAGATE TECH PLC 27.90 0.00 11,160.00
-4 BBBY 20OCT12 40.0 P 0.50 1.14 (148.86)
Also, on October 3 I withdrew $2,500.00 from the account and a dividend of $136.00 became payable November 1 on 400 shares of BMY; on October 5 a dividend of $206.00 became payable November 1 on 400 shares of VZ; and on October 17 a dividend of $112.40 became payable November 15 on 200 shares of PG.
Opened:
Pos Symbol Price Comm Net
500 INTC INTEL CORPORATION 21.265 0.00 10,632.50
400 STX SEAGATE TECH PLC 27.90 0.00 11,160.00
-4 BBBY 20OCT12 40.0 P 0.50 1.14 (148.86)
Also, on October 3 I withdrew $2,500.00 from the account and a dividend of $136.00 became payable November 1 on 400 shares of BMY; on October 5 a dividend of $206.00 became payable November 1 on 400 shares of VZ; and on October 17 a dividend of $112.40 became payable November 15 on 200 shares of PG.
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