ÖAsk Dr. Risk! TM   Columns from 1997
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Where Do You Get Vols? (11/19/97)

Question: Dear Dr. Risk – Thank you for making Daniel Sigrist's option pricing calculator available. I have a question for you. I am new to using the Black-Scholes formula and was wondering if you could clarify for me how I can properly know what number to enter in for the field: "Volatility" (Sq. Rt. %/year) . Is there an easy way to look up this number for a particular stock? – Vic N.

Answer: Dear Mr. N. – Tricky question. I'm not aware of any web site that makes equity volatilities available for free. The usual sources for vols are market data providers, such as Reuters and Telerate, which charge for them or equivalent information (more about this, below) that they get from brokers and exchanges.

Each morning the Wall Street Journal, New York Times, and other daily financial newspapers publish prices for options on a wide range of underlyings. Those prices may meet your needs directly. Otherwise, you may want to calculate the corresponding implied volatility. That involves inputting the option's terms, the effective underlying price, perhaps a dividend yield, and an interest rate into a calculator and solving for the implied volatility that produces the observed market price.

You should be aware that implied volatilities are ordinarily not a flat function of expiration and strike. For any given expiration date the implied volatility is typically a convex (somewhat U-shaped) function of strike price. Implied volatilities for out-of-the-money put options are much higher than volatilities for at-the-money puts. This would make shorting these puts seem extremely attractive to anyone who prices them using ATM vols. By the way, a hedge fund that disintegrated recently had done this trade in size. When the market crashed on October 27, 1997, the puts jumped into the money and the vols ballooned, so the option values exploded. The next day the fund failed to meet its margin calls and had to liquidate its positions. Ironically, the market bounced back the next day.

Listed options for any given underlying cover only a limited number of expiration dates and strike prices. From these points you may want to interpolate or extrapolate to meet your needs. Interpolation via the Black-Scholes model is fairly standard, if the gap in expirations or strikes is not large. Extrapolation is a riskier proposition. – Dr. Risk

H.15 (10/28/97)

Question: Dear Dr. Risk – I've heard the term h15 thrown around lately by people who seem like they know a lot about derivatives. I think it's an interest rate of some kind, although I didn't see it in your definitions. Do you know what it is? And if so, can you send me a definition? – A. Greenspan

Answer: Dear Mr. Greenspan – Great question! The H.15 Federal Reserve Statistical Release is a weekly report of selected interest rates, including Fed Funds, Commercial Paper, Bankers Acceptances, CDs, Eurodollar Deposites, Prime Rate, Treasury Discount Rates, Treasury Bill Rates, and Constant Maturity Treasury Rates. Some derivatives contracts define their payoffs in terms of rates reported in this release, which is publicly available and – one hopes – free from corrupting influences. – Dr. Risk

How to Catch an Unfaithful Model in the Act! (10/20/97)

Question: Dear Dr. Risk – We're making a presentation to a bank next week about model risk, in hopes of winning a contract to validate their models. How can one measure and manage the Model Risk derived from using the wrong models or using models incorrectly? – Ernie Young

Answer: Dear Mr. Young – Thanks for sending in a sophisticated question about one of my favorite topics: Model Risk, the risk that you'll lose a whole lot of money, your job, even your self-respect, because some pretty model that means a lot to you, one that's been with you for a long time or one you just met, is being unfaithful – to reality. This is an equal opportunity problem. Women worry about it, as much as men. No ethnic group is free from this worry. I can't tell you how many times my secretary has interrupted my philosophical inquiries because a client with this sort of problem is on the phone. The conversation usually goes something like this:

"Hey, Frisky" – Mazie calls me 'Frisky' because, ... well, mostly it's just a pun on my nom de plume, Dr. Risk – "can you take this call? This guy's involved with a model. They had a lot of good times together. Now he's not so sure the model's faithful."

"If he wants faithful, he should get a puppy."

"The guy's at the end of his rope. Can you see him this morning?"

"He can get all the rope he needs at the hardware store. This morning, I'm tied up with Karl Popper and twentieth century positivism."

"Frisky, don't be that way. You've been down that same road. Can he see you this afternoon?"

"He can get a roadmap at the Esso station. This afternoon, I'm exploring Paul Feyerabend's Against Method. I've had it with models. They're so pretty and they never complain, even when people use them in ways that nobody should ever use a model. Pretty soon, you start depending on them, and then they let you down."

"He's hurting. You know the pain. Please, help him."

"Tell him to take two aspirin and call somebody else in the morning. End of discussion."

"Hello, sir. Sorry I took so long. Things are a little crazy around here today, but I think I can squeeze you in to see Dr. Risk this afternoon at four o'clock, if that's okay. Do you know how to get here, Mr. Jett?"

But, I digress. You wanted to know how to measure and manage model risk. That's too big a topic for this spot, today. But I can lay the philosophical foundation and talk about some practical aspects. The main thing to remember is that testing financial models is like testing scientific theories.

A scientific theory is someone’s guess about the way the world works. A good theoretical scientist exposes his theory to rejection. A pseudoscientist or quack won't do that. Other theoreticians, as well as empirical scientists and nature reject scientific theories. According to Karl Popper's school of the philosophy of science, one never validates a scientific theory – one can only hope to reject it.

Similarly, a pricing model is someone’s guess about the way a financial market works. A good model builder exposes his model to rejection. A quack will use subterfuges or political clout to avoid such a challenge. Other model builders, traders, professional testers, and the market reject pricing models. One can never validate a model – but you can always reject it, if you push it far enough. How far do you have to push? That's the question.

How does one reject a theory? In the words of that worldly philosopher, Malcolm X, "By any means necessary!" The philosopher of science, Paul Feyerabend, said basically the same thing in his book, Against Method. A tester tries to be ingenious and use any trick in the book to find out where a model fails – is unfaithful to reality. This requires an understanding of how the markets work, how the model is supposed to work, and how trader's work. One of my most useful approaches is to see if the modeler has made any of the hundreds of errors that I have made – and caught – during 25 years of building financial models. Another trick is to look for mistakes by others that I have caught in two decades of checking somebody else's financial models – as a reviewer for a scholarly journal, an employee of a derivatives dealer, or an independent consultant.

Every model has its limits. If you push the model too far, it will bite you, even if it takes five years and extreme conditions for the weakness to show itself. The trader’s job is to find the limits, not get fooled, and make a lot of money. My job – any tester’s job – is to find the limits before the trader or the market do, and help the client avoid losing a fortune to either.

Good luck! – Dr. Risk

How to Get a High Paying Job Trading Options (9/30/97)

Question: Dear Dr. Risk – I am a 25 year old university graduate with a substantial math background, currently working on a Computer Science degree at nights. I want to get into options trading but am unsure of the method of entry due to limited contacts in the market. I have thought about picking up and going to Chicago to take some classes which the CBOT offers, possibly making some contacts in the meantime. Do you have any suggestions on how someone can do this? I feel that I have what it takes to make it. If you have any suggestions or know people I can talk to I would be more than willing to reply. – "Lucky Lucy" Arno

Answer: Dear Ms. Arno – Thanks for writing. Finding an entry level position in trading is difficult, because the competition is so fierce.

The best entry into trading is an apprenticeship. That's why so many floor traders start as runners in a trading pit and OTC traders start crunching numbers for guys running delta books. After watching and listening for months or years, plus showing sufficient ability and initiative, maybe the apprentice gets a chance to manage a little risk. Eventually, this can turn into a full-time trading job. The best things about an apprenticeship are that initially you get to see someone else's expensive mistakes and eventually someone experienced mentors you (so you don't make expensive mistakes with his money).

Finding the apprenticeship is difficult. You have to show the usual attributes of a successful job seeker – ability and desire – but at a higher quality and quantity, because trading positions are so desirable. James Cramer discusses searching for a trading job in articles you can find at Here's an url for a particularly relevant article, although its focus is finding a job trading stocks:

The more you learn, the better. Read extensively. A good business bookstore has many books about trading. If you get to NYC, try out the McGraw-Hill Bookstore on Sixth Ave. at 48th St. (McGraw-Hill Bookstore, 212-512-4100 tel, 212-512-4105 fax , 1221 Avenue of the Americas, New York, NY 10020, M-Sa, 1000-1745). Nassim Taleb's book on dynamic hedging is excellent and full of trading insights. Read the Wall Street Journal’s trading pages daily. You might look at Financial Trader.

Courses are a good idea. You mentioned the CBOT courses. I took one for educators. It was excellent. I've heard good things about Sheldon Natenberg's courses, which are widely available. Also, check out my link to Nassim Taleb's home page. He teaches a course on advanced trading. It's pricey, but its customers think it's worth it.

Talk to traders. This probably requires moving to a town with a major market and getting some sort of a job in or near the market. If you want to be a trader, take your programming skills into a trading organization, even if your entry salary is low. I did my first Derivatives consulting work on Wall Street for Fischer Black for free.

I'm pleased that you didn't mention the direct approach, which makes about as much sense as a good eater buying a restaurant without ever working in one. Anybody can buy or rent a seat on an exchange and start trading, but that would put him in the middle of a high stakes game with insufficient experience and capital. He would lose.

Good luck. – Dr. Risk

How to Get Luckier with Monte Carlo (9/30/97)

Question: Dear Dr. Risk – We use a Monte Carlo model to price some of our exotic, multivariate European payoff functions? Even with 60,000 paths, the remaining randomness is – frankly – disappointing. Do you have any ideas that might fix things up? – Al Grimaldi

Answer: Dear Mr. Grimaldi – Your question came at just the right time. My presentation in Boston at the IAFE meeting (9/24/97) focused on this topic. The control variate approach may allow you reduce the standard error around your estimate of the product's value. Here's how it works:

  1. You price the target payoff, e.g., Max(x1, ..., x6) with a MC model, and get its estimated price, MC{target}.
  2. You price a related, control payoff, e.g., Max(x1,x2) – for which you know the exact price, V{control}– and get an estimated price, MC{control}.
  3. You regress the target payoff against the control payoff, within the Monte Carlo model, and estimate the slope coefficient, b.
  4. Finally, you compute the control variate estimate of the target payoff, CV{target}, by adjusting the Monte Carlo estimate of the value of the target payoff up or down to reflect the error in the estimate of the control payoff's value and the slope coefficent.

Based on the assumption that

MC{target} - Value{target} = a + b * [MC{control} - Value{control}]

you can compute

CV{target} = MC{target} -a - b * [MC{control} - Value{control}]

You may find it advantageous to use more than one control payoff and multivariate regression. – Dr. Risk

How Do You Beat the Options Market? (9/11/97)

Question: Dear Dr. Risk – I'm an individual retail (off the floor) investor/"trader" with a full time day job looking for some help in devising a mkt neutral oex strategy (NOT DAY TRADING) that will be profitable if the mkt moves w/ in historical % ranges or if it goes to extremes.

I've done historical analysis of oex price patterns on an expiration to expiration % basis. For the past 2+ years I've experimented with both long and short OTM & ATM diagonals, butterflies, condors, winged butterflies, and horizontal time spreads (both long and short). Bottom line its been break even or a slight loss. There have been two problems for this would be premium seller:

  1. any protection that I buy is so expensive (volatility so high) that it severly curtails the profitability zone; and
  2. its also often come down to me picking the right direction. If I'm going to pick a direction then I want to be trading the underlying, which I really don't have time to do properly anymore.

All thoughts would be greatly appreciated. How to handle this high volatility and position trade in a mkt neutral way? – W.D.B. Midus

Answer: Dear Mr. Midus – I think you have collected a lot of good information, drawn some good conclusions, but not reached the final, logical – unwelcome – conclusion.

I think that expressing your views via the index (you mention OEX, but an S&P product would do as well), rather than individual stocks, makes sense, because you have a day job. Leave stock picking, short sales, and timing plays to professionals like James J. Cramer who can spend days tracking down rumors, etc. Read his informative and entertaining dispatches and enjoy the game vicariously.

I'm not so confident that a delta neutral options position is your best choice, in the following sense. If you have typical risk aversion and views on the market index and its volatility, then the market portfolio is optimal for you. Only if you differ significantly from the average investor – you're more bullish or bearish on the market or its volatility, or more or less risk averse – should your portfolio differ significantly from the average – i.e., market – portfolio. By next month in this column, I'll outline what to do in various such cases.

I can't offer you a strategy that will be profitable, whether the market moves "in historical % ranges or if it goes to extremes". Its sounds to me as though you want an option strategy that produces a profit in every case. (Am I stating your wish unfairly?) Don't we all! I wouldn't think that you would find such as system, although over a short period any given strategy might produce a profit. You seek a system that would allow you to trade against dealers and make money, consistently, despite paying them the bid-ask spread. That isn't going to happen, unless they fail to do their full-time jobs well. Your disappointing experiments with many strategies using market data from the past two years are consistent with my theory. I'm surprised they did as well as you say. Did you take brokerage commissions and bid-ask spreads into account?

Prescription: Unless you differ significantly from the average investor, keep your day job and put your money in a nice index fund! If you differ significantly, look to this column in the near future for a suitable strategy. – Dr. Risk

How to Price a Zero Coupon Swap (9/11/97)

Question: Dear Dr. Risk – How do you go about pricing a Zero Coupon Swap (i.e., a swap with one fixed future cash in-flow, and periodic floating rate cash out-flows)? – Franco Smith-Corona

Answer: Dear Mr. Smith-Corona – First, I'll assume that you know how to price a plain vanilla Interest Rate Swap (IRS). In that case, my short answer is, I would price it as a plain vanilla Swap, with the fixed coupon equal to zero. I should imagine that the main complication would be that the notional amounts for the fixed and floating legs would differ.

Second, let me elaborate, for those not so familiar with Swap pricing. Receiving fixed in an IRS is roughly the same as owning a Fixed Rate Note and financing it with a Floating Rate Note. Hence, the value of the position is the value of the Fixed Rate Note, minus the value of the Floating Rate Note. Typically, by convention the value of the Floating Rate Note is par at each reset date. Hence, the value of the Swap is the value of the Fixed Rate Note, minus par value of the Floating Rate note.

Third, in more detail, suppose today is a reset day. The Swap matures with the large cash inflow in T years. The position you want to price receives the fixed, zero-coupon leg, based on the notional amount, NFx, and pays the floating leg, based on the notional amount, NFl.Using the current LIBOR forward curve, the value of a Zero Coupon Bond ($1 principal) that matures at the time of the swap's cash inflow is Z(T). The fixed leg is worth Z(T).NFx. The floating leg is worth NFl.The swap is worth the difference, namely, Z(T).NFx- NFl. – Dr. Risk

When IRS Eyes Are Smiling (8/13/97)

Question: Dear Dr. Risk – I've got a problem. I bought 1000 shares of Fannie Mae in the 1980s at 15. It's now at about 44, after splitting three for one and four for one, which means the price would have been about 528 without the splits. That means I've got half a million at risk in that stock. I can take the heat, but the little lady doesn't like the way the market's been jumping around, lately. She wants me to sell it, so we don't end up poor if the stock market tanks. I don't want to sell it and pay the prohibitive capital gains tax. I can't sell short against the box. I tried to get a Derivatives dealer to do an Equity Swap with me, but he just laughed at me and said I was too small. Can I somehow solve my problem with Derivatives? – Fred Mack

Answer: Dear Mr. Mack – Dr. Risk sees the fix you're in, feels your pain, and wishes that he had that problem, instead of you.

When Dr. Risk provided analytic support to an Equity Derivatives sales desk, he had a hard time seeing the appeal of a "Costless Collar" to customers. The strategy can come close to taking the customers out of the underlying market, and is extremely expensive when you consider the inherent bid-ask spread. Of course, that's part of the appeal to the salesman. From the customer's POV, the name, "Worthless Collar", makes about as much sense.

However, the Wall Street Journal's August 13, page one "Tax Report" explains one reason for using this strategy: The Costless Collar is the "poor man's" (i.e., individual's) Equity Swap. If you sell at-the-money Calls against Fannie Mae and spend the premium on slightly in-the-money Puts, your Fannie Mae position will be (almost) flat, just as if you had sold short against the box (which the IRS frowns on) or equity-swapped away your stock returns (which that dealer laughed at). If the IRS smiles on this trade, you won't incur an immediate tax liability.

This solution isn't permanent, isn't cheap, and carries a risk. When those options expire, you're back in the same frying pan. Of course, you can repeat the trade until – in decreasing order of probability – you die and the basis on that stock steps up (Prob = 1), the IRS prohibits the strategy as "abusive", or a Libertarian Congress repeals the federal income tax laws (0 < Prob < epsilon: a long shot). Unfortunately, the bid-ask spreads on those options will eat you up, unless you die soon.

Many of Dr. Risk's employer's customers were tax-exempt institutions. Why did the Costless Collar appeal to some of them? Maybe they really did want some exposure, but didn't want to see the tails of the distribution. That can make sense for a professional manager who has scored big for the year, doesn't want to see that gain go away before the end of his performance period, and doesn't dare tell his customers that he's decided to take a vacation from the market.

One lingering question concerns legal risk: The Journal's piece concluded with a key question, namely, "How far apart must those strike prices be to keep IRS eyes smiling?" You'll have to ask your tax attorney or accountant about that. Dr. Risk can't say. – Dr. Risk

The Epistomology of Drop Lock Bonds (7/26/97)

Question: Dear Dr. Risk – As a complete illiterate in the area, your dictionary is something of a godsend. However, I don't seem to be able to locate "drop lock" bond. I assume it refers to a bond contract which allows the coupon to rise as a floating index (such as LIBOR) rises, but sets a floor if the index falls. Can you shed some light? – Crocodile Foster

Dear Mr. Foster – I'm glad you find it a godsend. When you're in a devilish mood, may I suggest that you check out "The Devil’s Derivatives Dictionary". But, in the words of that great critical thinker, Mort Sahl, "Back to the point."

Dr. Risk wishes he knew why you assume that a "drop lock bond" is a floating rate bond with a floor, and why you care, because context is extremely important when choosing one definition among several. One the one hand, if you are about to sign a contract that defines the bond that way, then one could hardly disagree with that definition. On the other hand, if you overheard some guy on the train mention the name or define it that way, and you’re idly curious about common usage of the term, then the meaning of that term is totally up in the air.

Dr. Risk would have thought that a drop lock bond would have a floating coupon rate that "locked in" whenever it dropped, and never rose. Thus, it would either remain constant or ratchet its way down. Without some context, Dr. Risk’s opinion is as good as yours, and your opinion as good as his. A derivative contract means exactly what it says, no more, and no less. In most cases in the OTC market, the confirm – and ISDA documents that it incorporates by reference – define the terms of the contract. The name that someone assigns to it adds no information and may confuse the issue, although it may prove convenient to use.

Rather than leave it at that, Dr. Risk searched the Internet for references to "drop lock". The first few hits referred to a sort of artificial hip joint. However, later on, Dr. Risk hit pay dirt four times. In addition, Dr. Risk found a hardbound reference.

  1. According to Singapore Technologies Capital Limited (SCTL), the "DROP-LOCK INTEREST SCHEME (Auto-DLis)" is a car loan that carries a rate of interest that drops whenever the lender starts making loans at a lower rate of interest. That sounds a lot like Dr. Risk’s idea of a drop lock bond. However, since the rate of interest is something the lender can control directly, rather than a market risk factor, it is not exactly Dr. Risk’s definition.
  2. According to Western Connecticut State University’s Financial Dictionary, it is "[a]n arrangement whereby the interest rate on a floating-rate note or preferred becomes fixed if it falls to a specified level." That sounds like your definition.
  3. According to Alessia Ambrosini in "FINANZAfacile", a definition of "Obbligazioni drop lock" that will have to speak for itself for now is, "Le obbligazioni drop lock proteggono il sottoscrittore da un eccessivo ribasso del tasso di interesse." Dr. Risk has requested a translation of that definition into a language he understands.
  4. According to the online Dictionary of Financial Risk Management, Revised Edition, a Drop-Lock Floating Rate Note is "[a] floating rate instrument that converts into a fixed rate note when the reference index rate drops below a pre-set trigger rate."
  5. Finally, it is a Floating Rate Bond (q.v.) that turns into a fixed-rate bond with a coupon equal to its Floor (q.v.) if the bond’s Index Rate ever hits the Trigger Level. (CSFB, 1988.).

Dr. Risk thinks these several definitions make his main point: the contract defines the product. If you take a definition that conflicts with #1 into the Singapore courts against STCL, don’t expect to win. – Dr. Risk

The P's and Q's of Managing Risk for Power Companies (7/26/97)

Question: Dear Dr. Risk – I work for a consulting company. My boss wants me to set up a risk management function for clients. I studied the Black-Scholes model in business school. Will this be all I need for pricing electricity options and managing risk for a power company? – Reddy Killawaddy

Answer: Dear Mr. Killawaddy – Thanks for your challenging question. The Black-Scholes model is suitable for some problems in pricing power options and managing their risk. It won’t work in others. Sometimes, the sort of delta hedging that is commonplace in currency, equity, fixed income, and other commodity derivatives isn't practical with any pricing model. Unfortunately, knowing when a model is appropriate and when it isn’t requires knowledge of the specific case. We can say that two key issues are (1) the underlying price process and (2) quantity risk.

The Price Process. The price of electrical power (power price) has some unique statistical properties that complicate stochastic modeling, which complicates option pricing and risk management. Dr. Risk has enlisted David Bucknall (mailto: of KW International (, energy consultants, to identify some of the key properties and outline an approach for option pricing and risk management. According to Mr. Bucknall, power prices are

  • Highly seasonal. Think of the high prices associated with the peak demand for electricity to run air conditioners in Texas in the summer.
  • Somewhat local. The ability to move power from regional grid to regional grid – say from Tennessee to Florida – is not free. Moving power from Scandinavia to Florida is extremely difficult, although it may be possible to do so, indirectly, at times, to a small degree. Dr. Risk thinks of towing a power plant on a barge.
  • Sometimes linked to a fuel cost, such as the price of natural gas, coal, or fuel oil. Thus, when fuel costs rise or fall, so do power costs. However, sometimes the price of power may fluctuate largely independent of the price of one fuel or all fuels. This depends on what the source of fuel is for the marginal power plant. If the marginal plant is hydroelectric or nuclear, then the price of fuel oil is largely irrelevant.
  • Leptokurtic. Movements in power prices tend to have fat tails. They fluctuate around one level for a while. Then something drastic happens, such as the arrival of a heat wave or a drought. Power prices move to a new level, where the vibrate for a while. And so on.

Quantity Risk. The quantity of power exposed to price risk is variable. This is analogous to the situation of the farmer. Assuming that the right futures market exists, the farmer and the power company can both hedge the price risk of their output. However, ordinarily, neither one can be sure of the quantity.

In the U.S. market, government regulates the retail price of power. When the regulated price exceeds the marginal cost of producing the power, an increase in demand increases power company profit. However, after production exceeds the level for which the marginal cost of power equals the regulated price, the power company loses money on each additional kilowatt hour it sells. A large jump in demand can cost a power company a great deal of money. That's why some power companies pay a bonus to users who are willing to have their air conditioners shut off during peak usage times.

Implications for Option Pricing and Risk Management. These stylized facts about the energy market have some implications for pricing and managing the risk of energy derivatives.

  • Quantity risk throws a monkey wrench into the entire risk management process for power companies. If you don't know how much power you will produce, and the quantity depends on a factor other than the price of power or an input, then you don't know how much price risk to hedge. Thus, traditional delta hedging doesn't work so well in the power industry. One is likely to need to combine it with portfolio analysis. Compare this to hedging an equity call option, where the quantity risk is random (the delta), but depends only on the underlying price. Delta hedging can still work. The rest of the comments pertain to price risk, only.
  • Seasonality means that spot energy prices are not as useful for option pricing and risk management as spot share prices are for equity option pricing and risk management. However, futures prices refer to conditions at a specific delivery time, reducing the problem of seasonality. Black’s futures option pricing model may work better than the Black-Scholes model.
  • Local markets mean that basis risk can be a serious problem when hedging power price risk in one location with derivatives based on power prices in another location.
  • The link of power and fuel prices offers potential for cross hedging power price risk with more developed fuel futures markets, under certain circumstances.
  • Leptokurtosis is an empirical question, and the jury is still debating its origin and significance. If the underlying distribution simply has fat tails, then the Black-Scholes model may not be a satisfactory approximation of reality – particularly for ITM and OTM options. If it stems from a "switch in regimes", then one might be able to use a variation of Black’s model to price options

To sum up, I don’t think anybody’s published the "cookbook" for power options, yet. Cooking up models for option pricing and risk management still requires a "master chef". – Dr. Risk

Cyberspace: Hype or Cause for Heavy Breathing? (7/23/97)

Question: Dear Dr. Risk: I've read a lot of hype about the Internet, recently. Do you think it's as important as people say? – Webster (Web) Masters

Answer: Dear Mr. Masters: Dr. Risk aspires to a blasé attitude that the late George Sanders (e.g., "All About Eve") would have envied. Maintaining this dégagé Weltanshauung has been exceedingly easy, given that Dr. Risk has seen over the years more than he cares to or can remember. However, when contemplating the Internet Dr. Risk finds it difficult to avoid hyperventilating.

Before we get to the Internet, think back to the Eisenhower years, when the federal government began constructing all those interstate highways. At huge expense to the taxpayers, they made it possible to moves goods between places faster and cheaper. Land near them became much more valuable. In particular, land in the suburbs near the cities grew in value, because it was more accessible to other places in the urban area than it had been. Land in the central cities became less valuable, because it wasn't so important to be so close to other places. This showed up as urban blight, because investment in the cities made less sense. In a nutshell – and slightly oversimplifying things – the Interstate Highway System drained central cities and nourished suburbs.

Now, consider the Internet. At a relatively trivial expense to the taxpayers, the Internet makes it possible to move information between computers faster and cheaper. Computers connected to the Internet have become much more valuable. In particular, computers in small businesses operating in the same field as large businesses have grown in value, because they are more accessible to other computers in the business world than before. Computers in large businesses with internal computer networks have become relatively less valuable, because it isn't so important to be in a large firm with vast resources. Now, small firms can be in Cyberspace with greater resources. For example, today Dr. Risk pays well under $100 per month for global Internet resources. Six years ago such resources would have required a corporate investment of millions of dollars. This change in relative prices will lead to a "substitution effect" – a relative decline in large businesses, as investment in them makes less sense. Small businesses will flower, relatively. People who connect meaningfully to the Internet will increase their wealth, while those married to the old ways of moving information will suffer. We have yet to see if the old institutions will decay the way the cities decayed. Clearly, the Internet has a generally positive impact on human welfare ("wealth effect"). However, in a nutshell, relatively speaking, the Internet is stifling large corporations and encouraging small businesses.

The crucial breakdown in the analogy between (a) land and Interstates and (b) computers and Internet is that the supply of land is relatively fixed, while the supply of computers is nearly totally elastic. Thus, the Interstate System didn't change the quantity of land, but increased the value of suburban land, relative to inner city land. The Internet won't affect the cost of computers – which is fixed at the competititve level – but the substitution effect will be to reduce (increase) the number of computers in larger (smaller) businesses. The "income" effect will be to increase the number of computers in both kinds of businesses. – Dr. Risk

How to Pitch to a Heavy Hitter (7/9/97)

Question: Dear Dr. Risk: Recently, an executive from a brand name brokerage firm asked me if he could send me some literature about some of his firm's money management programs. After I said yes, I asked him how he got my name. He said, "You're on the HH List." When I asked him what that was, he replied, almost reverentially, "That's the Heavy Hitter List." It sounded distinguished, and I was too embarrased to tell him I still didn't know what he was talking about. What is it? How did I get on it? – Carl Marks

Answer: Dear Mr. Marks: In your context, the "Heavy Hitter List" certainly contains names of qualified, prospective brokerage clients with lots of money, almost certainly "accredited investors" in the meaning of SEC Regulation D, Rule 505. Your caller has just told you in code words that he believes you have a large sum of liquid, investable money. No doubt, he hopes that you have even enough loose cash kicking around to qualify you for partnerships and other less regulated investments, which tend to have bigger profit margins and smaller fiduciary responsibilities for the brokers. Your reaction to the literature that he sends you and his follow up questions will help him tailor his sales pitch to your "needs".

You qualified for the HH list by acting or speaking in a way that distinguishes you as a person of much more than average means. Maybe you did something like buying one of the more expensive BMW's or any estate in Greenwich, or perhaps speaking lightly of large sums of money with an investment professional. In contrast, merely answering an ad in Barron's or Financial Trader could get you on a basic prospect list. You can get on the list of candidates for jury duty by registering to vote or listing a telephone in your name.

I wouldn't be too happy about being on that list if I were you, unless your self-esteem needs a boost or you like sparring with professionals in the sweet science of separating you from your money. The person who put you on that list is Wall Street's equivalent of the Depression-era hobo who marked my Grandmother's curb after she gave him a ham sandwich. The result: a parade of bums at the back door, looking for food. Some of them would actually work for it! Others knew that all they had to do is say they were from Tennessee. Similarly, you haven't had your last call as a result of being on this list. It's too valuable, and its compiler will sell it and sell it again. I know one heavy hitter who lost a small fortune in penny stocks. He died in 1992, brokers tried to call him for years after that, and mailed him solicitations as recently as June of 1997.

Dealing with these callers will be a challenge. The salespersons who work off these lists know a great deal about their products and getting people to say yes. They know less about the general topics of investment, trading, hedging, and speculation. You won't satisfy the callers with a ham sandwich, but when you've tired of speaking with these people, express polite interest, moan about financial setbacks, and ask them if they can lend you the money to invest. – Dr. Risk

Convertible Paper (6/24/97)

Question: Dear Dr. Risk – I'm a student in europe writing a paper on how to best price our convertible bonds. My professor sends you his regards. Using the Margrabe model for exchange options has worked quite well. Could you please share your experience and knowledge on this matter with me? If would be very grateful if you could teach me a lesson or two on this. Thanks and regards – Toni Edelweiss

Answer: Dear Mr Edelweiss – Thanks for writing. Please give my regards to your professor.

Pricing models for convertible bonds can be intricate. I'm not sure what ground you've covered, already, so I'll cover a broad range of models. I'm sure you prefer to keep the challenge of pricing the bonds, yourself, so I won't delve too deeply into any of them.

  1. I'm glad to hear that you have had success applying the Margrabe model to convertible bonds. If the conversion period is short, compared to the underlying bond's maturity, it's reasonable to model the convertible bond as an ordinary bond plus an option to exchange the bond for shares. Thus, you're using a two-factor model, which I would think would be a minimum requirement for a satisfactory convertible bond model.
  2. I can't say I'm surprised that you've had good success. What surprises me is how many people - and which people - have modeled and even continue to model convertible bonds with a single risk factor, as a bond plus a call option on shares. Some dealers have used a one-factor model for marking their positions surprisingly recently, and I would not find it surprising to hear that some do even today.
  3. Convertible bonds come in many varieties. Probably the simplest version is a European convertible, with a short period until a single possible moment of conversion, and a long-term underlying bond. The Margrabe model fits this product relatively well.
  4. For an American, Bermudan, or similar convertibility option, one might want to use a binomial variant on the Margrabe model to price the American option to exchange the bond for the shares.
  5. If the option is American and the underlying bond's maturity is relatively short, then one may be better off combining a binomial bond option model and a binomial stock option model into some sort of multinomial model with American exercise.
  6. The application of bond option models with more than two risk factors (e.g., one for the equity and two for the term structure) is possible, but may be more time-consuming than the problem merits. It all depends on the character of the underlying bond market, the terms of the bond, and the "juice" in the convertible bond market.

Could you perhaps send me a description of the terms of the convertible bond that you want to price? I'd like to see which of the above models – if any – might apply best? Good luck! – Dr. Risk

If the Cap Gives You Fits ... Look on the Internet (6/5/97)

Question 1: Dear Dr. Risk – What can you tell me about Sticky Caps and Periodic Caps? – Chuck Woods

Answer 1: Dear Mr. Woods – The term, Cap, usually refers to an Interest Rate Cap, but may refer to a Commodity Price Cap or other, similar instruments.

  • A Periodic Cap is a Cap for which the strike can change from period to period, usually as a function of recent LIBOR or historical LIBOR rates.
  • A Ladder Periodic Cap is a Periodic Cap with a strike that depends on the previous LIBOR reset but can take only values on a discrete "ladder".
  • A Lookback Periodic Cap has a strike that equals the highest or lowest observed LIBOR over some window of time.

See our "Derivatives Dictionary" for more complete definitions of Periodic Cap, Ladder Periodic Cap, and Lookback Periodic Cap.

Sticky Cap is a new one on me. Can you tell me where you saw it, or give me a lead? I'd like to find out and report more. – Dr. Risk

Question 2: Dear Dr. Risk – I finally found a reference to a Sticky Floater (also called a One Way Collared Note or Ratchet Floater) in "The Structured Note Market" by Peng and Dattatreya. The coupon cap is the previous coupon + 0.25% (say) and the coupon floor is the previous coupon. – Chuck Woods

Answer 2: Dear Mr. Woods – Thanks for the information and the lead. Dattatreya works for Sumitomo Bank Capital Markets, which has a Web site ( I revisited the site and found a page ( where Peter Fink discusses Sticky Floaters in the context of Monte Carlo models. Peter tells me that they put the page up within the last week or so.

One can presume that a Sticky Cap would work the same way, but one doesn't know, without looking at the Confirm. – Dr. Risk

Impossible Dream? (5/14/97)

Question: Dear Dr. Risk – We make a market in a wide range of OTC derivatives, and want to price a product that depends on a risk factor that is not an asset, futures, or forward price or interest rate. Is that possible? Can you give us some guidance for pricing it? –

Answer: Dear – I wish I had some idea what underlying risk factor you had in mind. Anyway, thank you for your general question, because that makes it easy for me to give you a general answer. In general, you can go any of several ways:

  • One of the best ways to go is always to find a customer to take both sides of the transaction at prices that leave you the bid-ask spread. The word on the street is that a number of hedge funds are stepping up and providing liquidity in markets for complicated derivatives, where it allows them to express their complicated views. Network with them.
  • The other great technique is to overhedge, guaranteeing that you break even, with a possibility of a large score, and charging your customer for the excessive hedge. You have to be both lucky or wise in the structure and smarter than the client to do this.
  • In principle, you can solve this problem mathematically. In equilibrium in a perfect market, "Arrow-Debreu" (A-D) prices, everyone's subjective probabilities, and everyone's utility functions must be consistent. (A-D prices are proportional to risk neutral probabilities in world of known interest rates.) Thus, if you have a subjective probability distribution for the risk factor and know your utility function, you can compute consistent A-D prices. You can then use these prices to put a value on the troublesome derivative product. Warning: I've never known anyone to actually do this.
  • A final approach is extremely naughty. You design the product to mature or expire after bonus day. You develop a model to price the product and convince your boss, controllers, and internal and external auditors that it is right. Then you sell the product for what the customer will pay and mark your short position for as little as your controllers will allow. The difference is your P&L. When bonus day comes – if not before – you'll get your reward. – Dr. Risk

  • Management Theory and Derivatives (4/5/97)

    Question: Dear Dr. Risk – Michael S. Malone summarized twelve recent, important major theories of management science in "A Way Too Short History of Fads" (Forbes ASAP, 4/7/97). However, he didn't apply them directly to Derivatives. How would you do that, and do you think they hold water? – Mel Michaels

    Answer: Dear Mr. Michaels – Malone's article was splendidly comprehensive, and the way he stretched out some of the discussions was perfectly appropriate for a general management audience – even if it would put the average Derivatives trader or bookrunner to sleep. (His bloated article filled an entire page of the magazine, and he covered only twelve theories! In a proper Derivatives periodical this would crowd out a great deal of gossip or P.R.)

    From the superior vantage point of someone who has spent years studying the Derivatives industry, I would say that he should have called it "A Way Too Long History of Fads". We all know that somebody has written at least one book about each of these theories, and sometimes the number of books runs into the dozens, but that doesn't mean that any of the theories holds water, or that it takes more than a sentence or two to communicate the kernel of sense in any theory that actually contains one. In fact, a lot of people in the Derivatives industry have done pretty darn well, thank you, without any of these management theories, without any other published management theories, indeed without even any traditional "management", whatsoever. Others have applied the essential parts of these theories without reading any of the books, because if you need a book to figure out the key points in any of these theories, you probably aren't smart enough to work in Derivatives, anyway. In most cases, a couple of sentences should be more than enough to explain the ideas, and a few seconds should be more than enough time to understand the explanations.

    Derivatives people need to capture their information in quicker gulps than Mr. Malone could deliver, so we're going to try to abbreviate his summary to match the average attention span in our industry, give each explanation a "Derivatives Twist", make it clear why some of the theories he mentioned are totally irrelevant for the Derivatives industry and the rest don't require books, and summarize a theory (#13) – which he omits – that describes a lot of Derivatives reality.

    1. Total Quality Management. Delivering customized Swaps and Options at razor-thin spreads that leave customers swooning with satisfaction and dealers gasping at bottom lines bleeding red ink. An obvious non-starter, because it can't justify million dollar bonuses.
    2. Computer-Integrated Manufacturing. Turning the market-making / risk management process over to computers that run artificial intelligence algorithms without a trace of common sense – for example, giving a VAR system some teeth. Only for the suicidal.
    3. Management by Objective / Theory Z. In theory, this means sailing smoothly between the Scylla of abdicating all responsibility to faceless minions who must carry out vague, meaningless corporate objectives, and the Charybdis of micromanaging global operations from world headquarters. This excellent theory amounts, in practice, to telling the troops by winks and nods to make money any way they can, putting apparently strong, but easily ignored corporate controls down on paper, and nailing any violator to the wall – unless he can implicate higher-ups.
    4. The Learning Organization. This is nothing more than the old "Just in Time" inventory theory, applied to employee skills and knowledge. This fine theory amounts, in practice, to hiring only people with I.Q.'s in excess of 130, preferably those who have graduated from Ivy League or equivalent institutions, who can learn anything they need to know instantly – or overnight, at worst.
    5. Reengineering. Surgically trimming an organization's fat, without touching the crucial muscle and sinew. As an ideal, this is a no-brainer. It's analogous to what you want your surgeon to do when he does your prostatectomy take out the bad boy, but please leave the nerve intact, so I can still have erections. In practice, it's hard to implement.
    6. Virtualization . If you truly love your employees, set them free, let them fly away and do what they must to maximize profits through (in Malone's words) a "more integrated relationship with suppliers, distributors, retailers, customers, and even competitors." In practice, you must clip employee wings with golden handcuffs, and if they don't freely come back with enough profits, email pink slips to them in Cyberspace.
    7. Decentralization. Let every desk have its own front office systems, back office systems, controllers, legal department, research department, etc. This theory sounds crazy, but corporate bureaucracy is even crazier. In practice, to get anything done, most sales and trading operations have taken big steps in this direction.
    8. Flat Organization. All 200 people on six desks report directly to the big boss, who knows absolutely nothing about products on five of the desks. A fine prescription for abdicating management responsibility, leading to page one embarrassment. "He who controls everything controls nothing."
    9. Critical Path Analysis. Paying consultants to streamline the introduction of new products. Only an idiot would do that in the Derivatives world. Instead, snag a copy of the competition's literature from greedy customers who want a better price, turn the documents over to "research", and let them retype the termsheets on your stationery for redistribution. Just to be safe, change some of the numbers by the way, it usually doesn't matter whether you get the numbers right or not.
    10. Sales Force Automation. Integrating the sales force into the MIS structure, so management can figure out what the hell the salesmen are doing to create sales, turn those tasks over to poorly paid clerks, and reduce bonuses paid to salesmen. Management hasn't been able to pull off this trick, yet. Given the current state of the art, the most likely result of attempting this in the short term is immediate passive aggressive behavior from the salesmen (e.g., loss of or damage to expensive equipment meant to monitor their activities), in the intermediate term is loss of the sales force after bonus checks clear the issuing bank. In the long term, the most likely result is the growth of powerful new competition.
    11. Chaordic Organizations. Maintaining a rigid overall structure, while all hell breaks loose on every desk. This captures the flavor of many Derivatives departments.
    12. Post-Capitalism / Co-Opetition. Recognizing that the pie will be bigger if we can all just work as a team. In theory, this is difficult to dispute. In practice, particularly in the Derivatives world, everybody cares exclusively about the size of his own piece of the pie!
    13. The Shogun Theory. Sometimes, particularly in the early year of Derivatives at a commercial or investment bank, an ambitious Managing Director leads his department in a ruthless, (figuratively) bloody war for control of floor space, head count, and the P&L they produce. If his department defeats the enemy (who is within his own corporation, of course), heads roll (again, figuratively). The losing MD goes on display, unless he decides to absent himself (pursuing other opportunities, spending more time with his family, involving himself more actively in his church), and he and his key loyalists head out to pasture, rather than to join their ancestors. Some will end up in risk management, some will remain as senior sales honchos, trying to keep their loyal customers from deserting. Sounds brutal and destructive, but it also reduces the cost of certain internal frictions. Soon, the winning MD explains to his assembled army (including conquered "Samurais" – salespersons and traders – and "vassals" – everyone else) that "we are now all one big happy family, trying to maximize our departmental P&L" – and prepares for the next war – at the next higher level – in his quest to become "Shogun".

    Background to Shogun Theory: For anyone interested – probably not a Derivatives trader, because he grasped the main idea, long ago, and is on to his next trade – Japan was in near anarchy around 1550, when hundreds of provincial daimyos held independent power. From 1568 to 1615, Oda Nobunaga, Toyotomi Hideyoshi, and Tokugawa Ieyasu unified Japan through bloody warfare. Nobunaga controlled, then eliminated the last Ahikaga shogun. His lieutenant, Hideyoshi, conquered most of the daimyos, made deals with the rest, and pretty much unified Japan. Ieyasu, Nobunaga's vassal and Hideyoshi's ally, succeeded Hideyoshi, destroyed Hideyoshi's family to gain total control, and had the emperor grant him the hereditary title of shogun.

    James Clavell sets his novel about feudal Japan, Shogun (New York: Atheneum, 1975), in this period. In the novel, Lord Toranaga directs his army of Samurais in a ruthless, bloody war to control land, peasants, and the agricultural products they produce. As Clavel describes the outcome on the last page:

    "THAT YEAR, at dawn on the twenty-first day of the tenth month, ... the main armies clashed. ... By late afternoon Toranaga had won the battle and the slaughter began. Forty thousand heads were taken.

    "Three days later Ishido was captured alive and Toranaga genially reminded him of the prophecy and sent him in chains to Osaka for public viewing, ordering the eta to plant the General Lord Ishido's feet firm in the earth, with only his head outside the earth, and to invite passersby to saw at the most famous neck in the realm with a bamboo saw. Ishido lingered three days and died very old."

    Sounds awful, but the Tokugawa shogunate led to a period of peace, relative prosperity – and isolationism aimed at maintaining family control of Japan. – Dr. Risk

    Vol Greeks (3/21/97)

    Question: Dear Dr. Risk: I have often heard risk managers refer to the dVega/dVol of an option's position. What is it and what is its significance in portfolio management. – "Andrew"

    Answer: Dear "Andrew": In a nutshell, the dVega/dVol is analogous to Gamma for options and Convexity for bonds. It sheds light on how Vega changes as volatility changes, just as Gamma tells how Delta changes as the underlying price changes and Convexity tells how Duration or DV01 changes as yield changes. A trader uses it the same way he might use Gamma or Convexity – to compute quickly and approximately the required change in a hedge position for a given change in the underlying risk factor.

    Let's start with definitions and a light discussion.

    1. I'll assume that your Vega is what some people call Kappa, and both equal the derivative of option value with respect to a change in volatility. You asked the question, so we'll use the term, Vega. If you graphed option value versus volatility, the slope of the graph at any point would be its Vega. Vega tells you the rate at which value changes, locally, for an infinitessimal change in volatility. Note that the rate at which value changes for a finite change in volatility may differ from Vega, and may be more useful to a risk manager, because an infinitessimal change in volatility isn't worth worrying about, while a larger change is. Vega, the option value's sensitivity to a change in volatility, is analogous to Delta, the option value's sensitivity to a change in the underlying price. Volatility and the underlying price are two risk factors, and Vega and Delta measure the sensitivity of option value to these risk factors.
    2. Hence, dVega/dVol is the second derivative of option premium with respect to a change in volatility. If you graphed Vega versus volatility, the slope of that graph would be dVega/dVol. The dVega/dVol tells you the rate at which Vega changes, locally, for an infinitessimal change in volatility. As with Vega, the change in Vega for a finite change in volatility may be more useful to a risk manager than dVega/dVol. The dVega/dVol, Vega's sensitivity to a change in volatility, is analogous to Gamma, Delta's sensitivity to a change in the underlying price.

    Now that we know the definitions we can discuss why the risk manager might care about dVega/dVol. In a sense, he cares about it for the same reason he care about Gamma – he can use it to figure out how much he will have to change his hedge for a given change in the underlying risk factor. Suppose the trader starts out flat – with no exposure to a small change in the underlying price or volatility. Multiply the Gamma by the change in underlying price and you know approximately the resulting change in Delta, which indicates the required hedge trade in the underlying instrument to remain Delta neutral. Multiply dVega/dVol by the change in volatility and you know approximately the change in Vega, which indicates the required hedge trade in the appropriate option to remain Vega neutral.

    The following examples give us something concrete to discuss about option value, Vega, and dVega/dVol as functions of volatility. Suppose that the dollar sells for 100 yen, and that the yen and dollar rates of interest are five percent. A picture's worth a thousand words. Unfortunately, including graphics at this site is not yet convenient, so I'll look at the graphs and describe them to you.

    First, consider the dollar value of a one-year Call Option on one yen, struck ATM forward at a penny per yen.

    1. For zero volatility this option is worthless, because it will expire at the money. For levels of volatility between zero and 100%, value appears to be a roughly linear function of volatility. As volatility increases without limit, option value asymptotically approaches a limit, which is the underlying spot price, discounted at the underlying yen's interest rate. This is counterintuitive to me, but if you look at the Black-Scholes-Merton equation, it is obvious.
    2. As vol increases from zero to 100%, Vega declines from about 0.38 to 0.335. As vol continues to increase, Vega approaches zero.
    3. As vol increases from zero to 100%, dVega/dVol declines from about zero to -8. (I used a finite difference approximation to compute the second derivative.) It reaches a minimum at a volatility of about 200%, then rises toward a limiting value of zero for large volatility.

    For at-the-money (ATM) forward, ordinary European options, option value is a good approximation of a linear function of volatility for low levels of volatility (below about fifty percent) and expiration within about one year. You would be better off in most cases, using that approximation, rather than using a Binomial model, which has random "binomial error".

    Second, consider an out-of-the-money (OTM) forward option.

    1. For zero vol the option is worthless. At a sufficiently high level of volatility an increase in vol increases option value. For a sufficiently high level of volatility, option value peaks, and an increase in volatility no longer matters.
    2. Vega measures the slope of the option value function – from zero for zero vol it rises sharply to a high level, then falls relatively gradually and asymptotically back down to zero for enormous vol.
    3. For zero vol, dVega/dVol is zero. As vol increases, dVega/dVol rises sharply, peaks, falls sharply to a negative number, then increases asymptotically to zero.

    Third, consider an in-the-money (ITM) forward option.

    1. For zero vol the option is valuable, because it will end up ITM. For low vol the option has value and is nearly insensitive to an increase in vol. At a sufficiently high level of volatility an increase in vol increases option value. For a sufficiently high level of volatility, option value approaches its asymptotic value, and an increase in volatility no longer matters.
    2. Vega as a function of vol looks much as it does for the OTM call.
    3. The dVega/dVol resembles its counterpart for the OTM call. – Dr. Risk

    Potpourri from Italy (3/14/97)

    Question: Dear Dr. Risk – Thanks for your immediate answer. Could I ask you something?

    1a) What is the difference between Interest Rate Options (IROs) and Debt Options?
    1b) Why do many papers start explaining the IRO's pricing model, but then speak about bond options?

    2) Do you know where could I find the articles of Hull and White ?

    3) Could you give me some information about the CIR model?

    – Carlo di Roma

    Answers: Dear Mr. di Roma – You're welcome and I hope my answer is immediate enough. Yes, ask away.

    1a) The difference between Interest Rate Options and Debt Options is the risk factor in the payoff function. An IRO's payoff function depends on one or more rates of interest. A Cap or Floor is an example. A Debt Option's payoff function depends on the value of the debt. A Bond Option is an example.

    1b) If you have a general understanding of Interest Rate Options, then you can price an arbitrary payoff function of interest rates. If you realize that a bond's value is a function of interest rates, then you can compute the value of a Bond Option from a general IRO model.

    2) The Hull and White articles appeared originally in a variety of places, including Risk and academic journals. You could look them up in the Journal of Economic Literature or a computerized bibliography. Some large financial bibliographies are on the WWW, and might include the articles you seek. I have not yet included links from my site to them, but that will happen before long. Risk has published a collection of the Hull and White papers, and would be glad to sell you a copy.

    3) CIR published a pair of papers, and you might be referring to either of them.
    3a) Cox, Ingersoll, Ross, "An Intertemporal General Equilibrium Model of Asset Prices," Econometrica 53 (March, 1985), pp. 363-384, develops a general equilibrium model of an entire economy. A key result is a general partial differential equation that describes the motion of an arbitrary asset price.
    3b) Cox, Ingersoll, Ross, "A Theory of the Term Structure of Interest Rates," Econometrica 53 (March, 1985), pp. 385-407, develops a variation on Vasicek's model. The Vasicek and CIR models have the same mean reversion, but Vasicek has a normal disturbance, while the CIR disturbance is proportional to the square root of the rate of interest. As a result, in Vasicek's model the rate of interest can go negative, but in the CIR model it cannot. The final forms of the two models bear striking similarities, but key factors have different functional forms. – Dr. Risk

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