More than a year ago I posted a fairly vague description of a book project on the ways in which mathematics contributes to the success of science. I have made some progress on bringing together this material and thought it would be useful to post a summary of the chapters of the book along with an introductory chapter where I give an overview of the main conclusions of the book. Hopefully this is useful to other people working on similar projects. Critical suggestions for what is missing or who else is doing similar stuff is of course welcome!
Update (Feb. 17, 2011): The link to chapter 1 has been replaced with the final version. The link to the summary has been removed.
Monday, October 26, 2009
Thursday, October 15, 2009
The Polymath Project
Gowers and Nielsen offer in the current issue of Nature a report on the online collaboration in mathematics known as the Polymath Project. It is hard to know what to make of it all without delving into the details and trying to understand if there is anything special about this problem which lends itself to collaboration. But two passages jump out for the philosopher:
This theorem was already known to be true, but for mathematicians, proofs are more than guarantees of truth: they are valued for their explanatory power, and a new proof of a theorem can provide crucial insights.
The working record of the Polymath Project is a remarkable resource for students of mathematics and for historians and philosophers of science. For the first time one can see on full display a complete account of how a serious mathematical result was discovered. It shows vividly how ideas grow, change, improve and are discarded, and how advances in understanding may come not in a single giant leap, but through the aggregation and refinement of many smaller insights. It shows the persistence required to solve a difficult problem, often in the face of considerable uncertainty, and how even the best mathematicians can make basic mistakes and pursue many failed ideas. There are ups, downs and real tension as the participants close in on a solution. Who would have guessed that the working record of a mathematical project would read like a thriller?At over 150 000 words, these records should keep some philosopher busy for a while!
Friday, October 9, 2009
Nobel Prize for Efficient Markets Hypothesis?
One of the core ideas driving the derivation of the Black-Scholes model is the efficient markets hypothesis. Exactly what this comes to is hopefully something I'll post on next week. But for now I'll pass on this from NPR's Marketplace:
Kai Ryssdal's final note.
Not so much news as a commentary on the state of the economic profession. The Nobel Prize in economics comes out Monday morning. I obviously have no idea who's going to win, but the markets think they do. The betting line at Ladbrokes, in London, has Eugene Fama of the University of Chicago as a 2-to-1 favorite.
That's all well and good except for this: Fama's best known for something called the Efficient Markets Theory. That the markets are, in essence, always right. I dunno, I'd say that's a tough sell after the year and a half we've just had. More to come on Monday.
Tuesday, October 6, 2009
Mathematics, Financial Economics and Failure
In a recent post I noted Krugman's point about economics being seduced by attractive mathematics. Since then there have been many debates out there in the blogosphere about the failures of financial economics, but little discussion of the details of any particular case. I want to start that here with a summary of how the most famous model in financial economics is derived. This is the Black-Scholes model, given as (*) below. It expresses the correct price V for an option as a function of the current price of the underlying stock S and the time t.
My derivation follows Almgren, R. (2002). Financial derivatives and partial differential equations. American Mathematical Monthly, 109: 1-12, 2002.
In my next post I aim to discuss the idealizations deployed here and how reasonable they make it to apply (*) in actual trading strategies.
A Derivation of the Black-Scholes Model
A (call) option gives the owner the right to buy some underlying asset like a stock at a fixed price K at some time T. Clearly some of the factors relevant to the fair price of the option now are the difference between the current price of the stock S and K as well as the length of time between now and time T when the option can be exercised. Suppose, for instance, that a stock is trading at 100$ and the option gives its owner the right to buy the stock at 90$. Then if the option can be exercised at that moment, the option is worth 10$. But if it is six months or a year until the option can be exercised, what is a fair price to pay for the 90$ option? It seems like a completely intractable problem that could depend on any number of factors including features specific to that asset as well as an investor's tolerance for risk. The genius of the Black-Scholes approach is to show how certain idealizing assumptions allow the option to be priced at V given only the current stock price S, a measure of the volatility of the stock price σ , the prevailing interest rate r and the length of time between now and time T when the option can be exercised. The only unknown parameter here is σ , the volatility of the stock price, but even this can be estimated by looking at the past behavior of the stock or similar stocks. Using the value V computed using this equation a trader can execute what appears to be a completely risk-free hedge. This involves either buying the option and selling the stock or selling the option and buying the stock. This position is apparently risk-free because the direction of the stock price is not part of the model, and so the trader need not take a stand on whether the stock price will go up or down.
The basic assumption underlying the derivation of (*) is that markets are efficient so that ``successive price changes may be considered as uncorrelated random variables" (Almgren, p. 1). The time-interval between now and the time T when the option can be exercised is first divided into N -many time-steps. We can then deploy a lognormal model of the change in price δ S_j at time-step j :
δ S_j = a δ t + σ S ξ_j
The ξ_j are random variables whose mean is zero and whose variance is 1 (Almgren, p. 5). Our model reflects the assumption that the percentage size of the random changes in S remains the same as S fluctuates over time (Almgren, p. 8). The parameter a indicates the overall ``drift" in the price of the stock, but it drops out in the course of the derivation.
Given that V is a function of both S and t we can approximate a change in V for a small time-step &delta t using a series expansion known as a Taylor series
δ V = V_t δ t + V_s δ S + 1/2 V_{SS} δ S^2
where additional higher-order terms are dropped. Given an interest rate of r for the assets held as cash, the corresponding change in the value of the replicating portfolio Π = DS+C of D stocks and C in cash is
δ Π = Dδ S + r C δ t
The last two equations allow us to easily represent the change in the value of a difference portfolio which buys the option and offers the replicating portfolio for sale. The change in value is
δ(V-Π)=(V_t - rC)δ t + (V_S - D)δ S + 1/2 V_{SS} δ S^2
The δ S term reflects the random fluctuations of the stock price and if it could not be dealt with we could not derive a useful equation for V . But fortunately the δ S term can be eliminated if we assume that at each time-step the investor can adjust the number of shares held so that
D=V_S
Then we get
δ(V-Π)=(V_t - rC)δ t + 1/2 V_{SS} δ S^2
The δ S^2 remains problematic for a given time-step, but we can find it for the sum of all the time-steps using our lognormal model. This permits us to simplify the equation so that, over the whole time interval Δ t ,
Δ(V-&Pi) = (V_t - rC + 1/2 σ^2 S^2 V_{SS})Δ t
Strictly speaking, we are here applying a result known as Ito's Lemma.
What is somewhat surprising is that we have found the net change in the value of the difference portfolio in a way that has dropped any reference to the random fluctuations of the stock price S . This allows us to deploy the efficient market hypothesis again and assume that Δ(V-Π) is identical to the result of investing V-Π in a risk-free bank account with interest rate r . That is,
Δ(V-Π) = r (V-Π)Δ t
But given that V-Π = V - DS - C and D = V_S , we can simplify the right-hand side of this equation to
(rV - rV_S S - rC)Δ t
Given our previous equation for the left-hand side, we get
(*) V_t + 1/2 σ^2 S^2 V_{SS} + rSV_S - rV = 0
after all terms are brought to the left-hand side.
My derivation follows Almgren, R. (2002). Financial derivatives and partial differential equations. American Mathematical Monthly, 109: 1-12, 2002.
In my next post I aim to discuss the idealizations deployed here and how reasonable they make it to apply (*) in actual trading strategies.
A Derivation of the Black-Scholes Model
A (call) option gives the owner the right to buy some underlying asset like a stock at a fixed price K at some time T. Clearly some of the factors relevant to the fair price of the option now are the difference between the current price of the stock S and K as well as the length of time between now and time T when the option can be exercised. Suppose, for instance, that a stock is trading at 100$ and the option gives its owner the right to buy the stock at 90$. Then if the option can be exercised at that moment, the option is worth 10$. But if it is six months or a year until the option can be exercised, what is a fair price to pay for the 90$ option? It seems like a completely intractable problem that could depend on any number of factors including features specific to that asset as well as an investor's tolerance for risk. The genius of the Black-Scholes approach is to show how certain idealizing assumptions allow the option to be priced at V given only the current stock price S, a measure of the volatility of the stock price σ , the prevailing interest rate r and the length of time between now and time T when the option can be exercised. The only unknown parameter here is σ , the volatility of the stock price, but even this can be estimated by looking at the past behavior of the stock or similar stocks. Using the value V computed using this equation a trader can execute what appears to be a completely risk-free hedge. This involves either buying the option and selling the stock or selling the option and buying the stock. This position is apparently risk-free because the direction of the stock price is not part of the model, and so the trader need not take a stand on whether the stock price will go up or down.
The basic assumption underlying the derivation of (*) is that markets are efficient so that ``successive price changes may be considered as uncorrelated random variables" (Almgren, p. 1). The time-interval between now and the time T when the option can be exercised is first divided into N -many time-steps. We can then deploy a lognormal model of the change in price δ S_j at time-step j :
δ S_j = a δ t + σ S ξ_j
The ξ_j are random variables whose mean is zero and whose variance is 1 (Almgren, p. 5). Our model reflects the assumption that the percentage size of the random changes in S remains the same as S fluctuates over time (Almgren, p. 8). The parameter a indicates the overall ``drift" in the price of the stock, but it drops out in the course of the derivation.
Given that V is a function of both S and t we can approximate a change in V for a small time-step &delta t using a series expansion known as a Taylor series
δ V = V_t δ t + V_s δ S + 1/2 V_{SS} δ S^2
where additional higher-order terms are dropped. Given an interest rate of r for the assets held as cash, the corresponding change in the value of the replicating portfolio Π = DS+C of D stocks and C in cash is
δ Π = Dδ S + r C δ t
The last two equations allow us to easily represent the change in the value of a difference portfolio which buys the option and offers the replicating portfolio for sale. The change in value is
δ(V-Π)=(V_t - rC)δ t + (V_S - D)δ S + 1/2 V_{SS} δ S^2
The δ S term reflects the random fluctuations of the stock price and if it could not be dealt with we could not derive a useful equation for V . But fortunately the δ S term can be eliminated if we assume that at each time-step the investor can adjust the number of shares held so that
D=V_S
Then we get
δ(V-Π)=(V_t - rC)δ t + 1/2 V_{SS} δ S^2
The δ S^2 remains problematic for a given time-step, but we can find it for the sum of all the time-steps using our lognormal model. This permits us to simplify the equation so that, over the whole time interval Δ t ,
Δ(V-&Pi) = (V_t - rC + 1/2 σ^2 S^2 V_{SS})Δ t
Strictly speaking, we are here applying a result known as Ito's Lemma.
What is somewhat surprising is that we have found the net change in the value of the difference portfolio in a way that has dropped any reference to the random fluctuations of the stock price S . This allows us to deploy the efficient market hypothesis again and assume that Δ(V-Π) is identical to the result of investing V-Π in a risk-free bank account with interest rate r . That is,
Δ(V-Π) = r (V-Π)Δ t
But given that V-Π = V - DS - C and D = V_S , we can simplify the right-hand side of this equation to
(rV - rV_S S - rC)Δ t
Given our previous equation for the left-hand side, we get
(*) V_t + 1/2 σ^2 S^2 V_{SS} + rSV_S - rV = 0
after all terms are brought to the left-hand side.
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