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Are There Flaws In Options Pricing?
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Quote from Emilio_Lizardo:
Not true. Usual hypothesis is that mu = r + (sigma^2)/2.
The condition mu = r corresponds to the process S(t)B(t)^-1 being a martingale -- a fairly implausible assumption.
You're confusing the drift component of the log price with the average rate of return of the price. They are different things. You start with mu as a free parameter, but after solving for the call and put prices the only value of mu consistent with put-call parity is mu = r - sigma^2/2 (this is not a hypothesis, it is forced by put-call parity). When you plug that in and solve for the expected price (not the expected log price), you get E[P(T)] = P(0)exp(rT).
So I still say that the only way to make the log normal pricing assumption consistent with no arbitrage is to use a pricing model that assumes the expected rate of growth is the risk free rate, regardless of the real rate of growth of the asset.
You guys are confusing yourselves with math. Go back to the basics. The BS assumes that the underlying moves in a random walk, and that the likelihood of the underlying being at any particular price at expiration is described by a lognormal probability distribution. The shape of that probability distribution remains the same regardless of the risk-free interest rate. Doesn't matter whether it is 0%, 10%, 20% or whatever.
The risk-free rate is a cost-of-carry calculation, nothing more. In stocks the model uses it to calculate a forward price, which the model then uses as the price of the underlying in its calculations, NOT the price you entered. In both stocks and futures, it also discounts the option price by the cost of carry.
The risk-free rate is a cost-of-carry calculation, nothing more. In stocks the model uses it to calculate a forward price, which the model then uses as the price of the underlying in its calculations, NOT the price you entered. In both stocks and futures, it also discounts the option price by the cost of carry.
Quote from dmo:
You guys are confusing yourselves with math. Go back to the basics. The BS assumes that the underlying moves in a random walk, and that the likelihood of the underlying being at any particular price at expiration is described by a lognormal probability distribution. The shape of that probability distribution remains the same regardless of the risk-free interest rate. Doesn't matter whether it is 0%, 10%, 20% or whatever.
The risk-free rate is a cost-of-carry calculation, nothing more. In stocks the model uses it to calculate a forward price, which the model then uses as the price of the underlying in its calculations, NOT the price you entered. In both stocks and futures, it also discounts the option price by the cost of carry.
Go back to basics. The position of the mean of that normal curve is vitally important in determining the price of puts and calls. After all, that's what it's all about -- are you above or below the strike price at expiration, and by how much? Common sense tells you that the higher mu is the higher the mean, and indeed, increasing mu shifts the normal curve up.
The most surprising things about BS is that mu, related to the average growth of the underlying asset, doesn't appear in the final equations. That's a bit weird -- it tells us that if we have two stocks with the same current price, one growing at an average rate of 5%, and the other at an average rate of 15%, with the same volatility, and both with price behavior consistent with the lognormal hypothesis (i.e., daily returns are uncorrelated and normally distributed), their one year calls will be priced exactly the same.
That's because the no arbitrage principal forces mu to be r - sigma^2/2, and thus forces the average growth of the price to be r. That's what this pissing match is about.
I don't think my previous posts were sufficiently clear and that is probably the source of some of the contention.
There are basically two ways to price a financial instrument:
1. Consider all future prices and their probabilities. Calculate the future expected price, then discount to the present at the risk free rate to get the current expected price.
2. Find some price that is forced by arbitrage. That is, if there is some significant deviation from this price arbs will jump in and trade away the deviation, making risk free profits. This is the no arbitrage price.
It is generally agreed that if there is a no arbitrage price defined and it differs from the expected price then the no arbitrage price is what you will see in the market. After all, arbs, both human and computer, will force the price that way. But a trader can legitimately see the expected price as the "fair value". If the expected price is well above the no arbitrage price then the trader can buy at the no arbitrage price and expect to make a profit. Likewise if the expected price is below the no arbitrage price a trader can sell at the no arbitrage price and expect a profit. Unlike the arbs the trader doesn't have a guaranteed win on every trade but on average he should make money.
For options I will assume as usual that the returns are normally distributed and independent, and with constant volatility. In that case the Black Scholes equation gives the no arbitrage price, which is independent of mu. On that we have no disagreement. But what about the expected option prices? They are straightforward to compute (no stochastic calculus needed). Just do the integral and discount to the present. When you do that the equations for puts and calls have an explicit dependence on mu. As one would expect, calls become more expensive as mu goes up and puts get cheaper.
In general put-call parity does not apply to the expected option prices. (Put-call parity is of course a particular instance of the no arbitrage rule.) It is natural to ask whether there is a value of mu that makes put-call parity work for the expected prices. There is -- set mu = r - sigma^2/2. When you do that put-call parity works at all strikes. Furthermore, the expected price equations reduce to the BS equations. If you compute the expected future value of the underlying asset with mu = r - sigma^2/2 it turns out to be E[P(T)] = P(0)*exp(rT), that is, the expected growth rate is the risk free rate.
The bottom line is this: The expected option prices match the no arbitrage prices only when the underlying has an expected growth rate equal to the risk free rate. For all other growth rates (i.e., mu != r - sigma^2/2) the expected prices are different from the no arbitrage prices.
So... when Warren Buffet complained that BS doesn't work, what he was really saying was that he expects the stock market overall to grow significantly above the risk free rate, with the result that the expected price of long term puts is well below the no arbitrage (BS) price. So he was happy to sell puts in large size at the BS price. Indeed, he sold at such large size it is clear he considers the expected price of the puts to be very close to 0.
There are basically two ways to price a financial instrument:
1. Consider all future prices and their probabilities. Calculate the future expected price, then discount to the present at the risk free rate to get the current expected price.
2. Find some price that is forced by arbitrage. That is, if there is some significant deviation from this price arbs will jump in and trade away the deviation, making risk free profits. This is the no arbitrage price.
It is generally agreed that if there is a no arbitrage price defined and it differs from the expected price then the no arbitrage price is what you will see in the market. After all, arbs, both human and computer, will force the price that way. But a trader can legitimately see the expected price as the "fair value". If the expected price is well above the no arbitrage price then the trader can buy at the no arbitrage price and expect to make a profit. Likewise if the expected price is below the no arbitrage price a trader can sell at the no arbitrage price and expect a profit. Unlike the arbs the trader doesn't have a guaranteed win on every trade but on average he should make money.
For options I will assume as usual that the returns are normally distributed and independent, and with constant volatility. In that case the Black Scholes equation gives the no arbitrage price, which is independent of mu. On that we have no disagreement. But what about the expected option prices? They are straightforward to compute (no stochastic calculus needed). Just do the integral and discount to the present. When you do that the equations for puts and calls have an explicit dependence on mu. As one would expect, calls become more expensive as mu goes up and puts get cheaper.
In general put-call parity does not apply to the expected option prices. (Put-call parity is of course a particular instance of the no arbitrage rule.) It is natural to ask whether there is a value of mu that makes put-call parity work for the expected prices. There is -- set mu = r - sigma^2/2. When you do that put-call parity works at all strikes. Furthermore, the expected price equations reduce to the BS equations. If you compute the expected future value of the underlying asset with mu = r - sigma^2/2 it turns out to be E[P(T)] = P(0)*exp(rT), that is, the expected growth rate is the risk free rate.
The bottom line is this: The expected option prices match the no arbitrage prices only when the underlying has an expected growth rate equal to the risk free rate. For all other growth rates (i.e., mu != r - sigma^2/2) the expected prices are different from the no arbitrage prices.
So... when Warren Buffet complained that BS doesn't work, what he was really saying was that he expects the stock market overall to grow significantly above the risk free rate, with the result that the expected price of long term puts is well below the no arbitrage (BS) price. So he was happy to sell puts in large size at the BS price. Indeed, he sold at such large size it is clear he considers the expected price of the puts to be very close to 0.
I have to look at this in more detail... I have never seen put-call parity argued this way.
In every fields there is always flaws that we can encounter and even the ââ¬Årisk-free rateââ¬Â also have an up downs. The flaws that we encounter are part of our daily life and serve as our experience.
Here's a toy example that vividly demonstrates the difference between the no arbitrage and the expected prices of an option.
To keep the math simple, I will assume a risk free rate of 0.
<a href="http://www.yoyodyne.com/">Yoyodyne Corporation</a> (ticker YOYO) has a remarkable pricing history. It has grown at 3% a month, every month, for the past 2 years. What's more, the price curve is a perfect exponential, with no down days. Its price on January 20, 2011 is 9.71.
On January 20, 2011 Marvin Marketmaker prices the $10 strike one year YOYO call option. The log price chart of YOYO is a straight line with slope mu = 12*ln 1.03 = 0.3547. The daily returns are constant, i.e., their distribution is a Dirac delta function. But that's just the degenerate case of a normal curve with standard deviation 0. With constant returns the volatility is, of course, 0. So the price behavior is an edge case, but it fits the BS model. With both the volatility and the risk free rates at 0 the Black Scholes price for the $10 strike one year YOYO calls is 0.
Marvin wants to make money so he sets the ask price for the calls at 0.05, enough above the BS price to cover hedging costs and leave a profit.
On that same day Suzy decides to calculate the expected price of the Jan '12 $10 YOYO calls. She looks at the markets and operations of Yoyodyne and concludes that the stock price will continue to grow at 3% a month. In other words, the log price will continue to be a straight line, with mu = 0.3547. With the volatility at 0 there is no random component so the future expected price is trivial to compute: 9.71 * (1.03)^12 - 10 = 3.84. The risk free rate is 0 so that's also the current expected price. Suzy sees that the calls are priced at 0.05, well below 3.84, so she happily pays $5,000 for 1,000 calls. Note that that high value of mu was vitally important to Suzy but ignored by Marvin.
Marvin's balance sheet after the sale to Suzy looks like this:
Short 1,000 YOYO Jan '12 $10 calls
Cash: $5,000
Marvin does not buy any YOYO shares because the delta on the calls is 0.
On February 20, 2011 YOYO stock hits 10.00. At that point the delta instantly changes from 0 to 1. Marvin hedges by buying 100,000 shares of YOYO. Due to slippage and commissions he gets them at an average price of 10.01. Marvin's new balance sheet looks like this:
Short 1,000 YOYO Jan '12 $10 calls
Long 100,000 YOYO shares, bought at 10.01
Cash -996,000
Fortunately Marvin is a good friend of Ben Bernanke and can borrow at 0%.
Not much happens for the next 11 months, the delta stays at 1 so there's nothing for Marvin to do.
On January 20, 2012 (expiration Friday) Suzy sells her calls for their intrinsic value. YOYO is at 13.84, so Suzy gets 3.84 for the calls she bought for 0.05. Her profit is 384,000 - 5,000 = $379,000.
Marvin buys back the calls from Suzy for 3.84, and sells his YOYO shares. Because of slippage and commissions he averages 13.83 per share.
Marvin is left with -996,000 + 1,383,000 - 384,000 = $3,000.
So, BS theory worked. Marvin made a profit by selling above the BS price and delta hedging.
But Suzy didn't do at all badly, she made 126 times as much as Marvin.
Marvin used the no arbitrage price, appropriate for a delta hedger. Suzy used the expected price, appropriate for a speculator. To make the math trivial I assumed a magical chart with 0 volatility. But had I made the example a bit more realistic, with, say 10% annual volatility, the disparity between the expected and no arbitrage prices would still be stark. The BS price would be about 0.27, and the expected price would be about the same as before. Suzy would still get better than a 10 bagger.
To keep the math simple, I will assume a risk free rate of 0.
<a href="http://www.yoyodyne.com/">Yoyodyne Corporation</a> (ticker YOYO) has a remarkable pricing history. It has grown at 3% a month, every month, for the past 2 years. What's more, the price curve is a perfect exponential, with no down days. Its price on January 20, 2011 is 9.71.
On January 20, 2011 Marvin Marketmaker prices the $10 strike one year YOYO call option. The log price chart of YOYO is a straight line with slope mu = 12*ln 1.03 = 0.3547. The daily returns are constant, i.e., their distribution is a Dirac delta function. But that's just the degenerate case of a normal curve with standard deviation 0. With constant returns the volatility is, of course, 0. So the price behavior is an edge case, but it fits the BS model. With both the volatility and the risk free rates at 0 the Black Scholes price for the $10 strike one year YOYO calls is 0.
Marvin wants to make money so he sets the ask price for the calls at 0.05, enough above the BS price to cover hedging costs and leave a profit.
On that same day Suzy decides to calculate the expected price of the Jan '12 $10 YOYO calls. She looks at the markets and operations of Yoyodyne and concludes that the stock price will continue to grow at 3% a month. In other words, the log price will continue to be a straight line, with mu = 0.3547. With the volatility at 0 there is no random component so the future expected price is trivial to compute: 9.71 * (1.03)^12 - 10 = 3.84. The risk free rate is 0 so that's also the current expected price. Suzy sees that the calls are priced at 0.05, well below 3.84, so she happily pays $5,000 for 1,000 calls. Note that that high value of mu was vitally important to Suzy but ignored by Marvin.
Marvin's balance sheet after the sale to Suzy looks like this:
Short 1,000 YOYO Jan '12 $10 calls
Cash: $5,000
Marvin does not buy any YOYO shares because the delta on the calls is 0.
On February 20, 2011 YOYO stock hits 10.00. At that point the delta instantly changes from 0 to 1. Marvin hedges by buying 100,000 shares of YOYO. Due to slippage and commissions he gets them at an average price of 10.01. Marvin's new balance sheet looks like this:
Short 1,000 YOYO Jan '12 $10 calls
Long 100,000 YOYO shares, bought at 10.01
Cash -996,000
Fortunately Marvin is a good friend of Ben Bernanke and can borrow at 0%.
Not much happens for the next 11 months, the delta stays at 1 so there's nothing for Marvin to do.
On January 20, 2012 (expiration Friday) Suzy sells her calls for their intrinsic value. YOYO is at 13.84, so Suzy gets 3.84 for the calls she bought for 0.05. Her profit is 384,000 - 5,000 = $379,000.
Marvin buys back the calls from Suzy for 3.84, and sells his YOYO shares. Because of slippage and commissions he averages 13.83 per share.
Marvin is left with -996,000 + 1,383,000 - 384,000 = $3,000.
So, BS theory worked. Marvin made a profit by selling above the BS price and delta hedging.
But Suzy didn't do at all badly, she made 126 times as much as Marvin.
Marvin used the no arbitrage price, appropriate for a delta hedger. Suzy used the expected price, appropriate for a speculator. To make the math trivial I assumed a magical chart with 0 volatility. But had I made the example a bit more realistic, with, say 10% annual volatility, the disparity between the expected and no arbitrage prices would still be stark. The BS price would be about 0.27, and the expected price would be about the same as before. Suzy would still get better than a 10 bagger.
The difference between "expected value of the option" and arbitrage free price is quite simple. This is because the arbitrage free price IS the "expected value of the option", just with different probabilities, the risk neutral ones.
I didn't read your last post(Rex) because it looks excessively long, but the conclusion of your second to last post was absolutely correct. I'm quite confused as to what all the fuss is still about...
What I will say though, is that the "expected value of the option" as you are calling it, is quite subjective. Everyone has a different mu in mind, and each mu defines a new "expected value of the option". For this reason, I don't find it at all troubling that it doesn't show up in the BS formula.
I didn't read your last post(Rex) because it looks excessively long, but the conclusion of your second to last post was absolutely correct. I'm quite confused as to what all the fuss is still about...
What I will say though, is that the "expected value of the option" as you are calling it, is quite subjective. Everyone has a different mu in mind, and each mu defines a new "expected value of the option". For this reason, I don't find it at all troubling that it doesn't show up in the BS formula.
Quote from VGSSD:
What I will say though, is that the "expected value of the option" as you are calling it, is quite subjective. Everyone has a different mu in mind, and each mu defines a new "expected value of the option". For this reason, I don't find it at all troubling that it doesn't show up in the BS formula.
Well, each sigma also defines a new expected value for the options, and guessing what the future volatility will be is not much easier than guessing mu. My point is this: People would not buy stocks unless they believed they will, over the long run, grow at faster than the risk free rate. If long term calls are priced according to Black Scholes then they are priced at what the expected price would be *if* the stock market grew at the risk free rate. (If you are going to argue against that then you didn't understand my previous post at all.) If in fact the average growth rate of the stock market is higher that then risk free rate (i.e., if stock buyers are rational) then that means that calls are on average priced below their expected price. Likewise the long term puts are priced over their expected price.
Of course if we look at a ten year stock chart we might well question whether stock prices really do grow faster than the risk free rate. So maybe stock buyers are just irrational.