Life in the Hotel Okura

Introduction to the Japanese Edition of "My Life as a Quant"

I first visited Tokyo in 1990, when I worked at Goldman, Sachs & Co. as head of the Quantitative Strategies group. Our job was to write software and models for the equity derivatives traders at Goldman, and we had just created a risk management system for managing the “Nikkei book,” the portfolio of derivative contracts we owned that had in common their dependence on the value of the Nikkei index. Our equity derivatives traders were a global team, with members in London, Tokyo and New York, and the management of the book was passed around the world from team to team like a 400 meter relay runners’ baton as one market closed and the next one opened. I traveled to Tokyo to introduce the new system to our traders there.

POSTED — September 9, 2005 — Archives

Letter to the NY Times (October 18 2008): Who’s the Boss: You or Your Computer?

The truth is that computers are created by humans. They are not superintelligent. Even on Wall Street, they merely count very fast at the behest of their human masters, and only fools or idolators imagine they are wise.

You don’t have to be a rocket scientist to look at the world around you and conclude that natural stupidity trumps artificial intelligence any time.

POSTED — October 10, 2008 — Archives

Contribution by Forex Traders: The Search for the Holy Grail Continues and “StatArb” Forex May Be It

“Incentives work” is a truism that most of us discover early on in our commercial careers, but it is no more obvious in any sector of our economy than within our trading markets.  The amount of intellectual energy and resources devoted to finding the next “secret” correlation in stocks, commodities, or currencies is ongoing evidence and confirmation of this statement.  Complex mathematical models continually search for imperfect pricing in our markets because the arbitrage opportunity, though fleeting, can generate huge rewards in the bat of an eyelash.

POSTED — September 9, 2013 — Archives

Options on Periodically-Settled Stocks

In some countries, for example France, stocks bought or sold during an account period have their settlement deferred to a designated settlement date. We explain how to value and hedge cash-settled European- or American-style options on stocks whose settlement is deferred. To do this we introduce the notion of “bare” (immediately settled) and “dressed” (deferred-settled) stock prices. It is the volatility of bare prices that is fundamental. 

POSTED — November 11, 1992 — Archives

The Volatility Smile and Its Implied Tree

The market implied volatilities of stock index options often have a skewed structure, commonly called “the volatility smile.” One of the long-standing problems in options pricing has been how to reconcile this structure with the Black-Scholes model usually used by options traders. In this paper we show how to extend the Black-Scholes model so as to make it consistent with the smile.


The Black-Scholes model assumes that the index level executes a random walk with a constant volatility. If the Black-Scholes model is correct, then the index distribution at any options expiration is log- normal, and all options on the index must have the same implied vol- atility. But, ever since the ‘87 crash, the market’s implied Black- Scholes volatilities for index options have shown a negative relation- ship between implied volatilities and strike prices – out-of-the-money puts trade at higher implied volatilities than out-of-the-money calls. The graph above illustrates this behavior for 47-day European-style March options on the S&P 500, as of January 31, 1994. The data for strikes above (below) spot comes from call (put) prices.

POSTED — January 1, 1994 — Archives

Static Options Replication (The Journal of Derivatives)

This paper presents a method for replicating or hedging a target stock option with a portfolio of other options. It shows how to con- struct a replicating portfolio of standard options with varying strikes and maturities and fixed portfolio weights. Once constructed, this portfolio will replicate the value of the target option for a wide range of stock prices and times before expiration, without requiring further weight adjustments. We call this method static replication. It makes no assumptions beyond those of standard options theory.

You can use the technique to construct static hedges for exotic options, thereby minimizing dynamic hedging risk and costs. You can use it to structure exotic payoffs from standard options. Finally, you can use it as an aid in valuing exotic options, since it lets you approx- imately decompose the exotic option into a portfolio of standard options whose market prices and bid-ask spreads may be better known.

POSTED — April 4, 1994 — Archives

Enhanced Numerical Methods for Options with Barriers (Financial Analysts Journal)

Most real-world barrier options have no analytic solutions, either because the barrier structure is complex or because of volatility skews in the market. Numerical solutions are a necessity. But options with barriers are notoriously difficult to value numerically on binomial or multinomial trees, or on finite-difference lattices. Their values converge very slowly as the number of tree or lattice levels increase, often requiring unattainably large comput- ing times for even a modest accuracy.

In this paper we analyze the biases implicit in valu- ing options with barriers on a lattice. We then sug- gest a method for enhancing the numerical solution of boundary value problems on a lattice that helps to correct these biases. It seems to work well in practice. 

POSTED — October 10, 1995 — Archives

Implied Trinomial Trees of the Volatility Smile (The Journal of Derivatives)

In options markets where there is a significant or persis- tent volatility smile, implied tree models can ensure the consistency of exotic options prices with the market prices of liquid standard options.

Implied trees can be constructed in a variety of ways. Implied binomial trees are minimal: they have just enough parameters – node prices and transition probabil- ities – to fit the smile. In this paper we show how to build implied trinomial tree models of the volatility smile. Tri- nomial trees have inherently more parameters than bino- mial trees. We can use these additional parameters to conveniently choose the “state space” of all node prices in the trinomial tree, and let only the transition probabili- ties be constrained by market options prices. This free- dom of state space provides a flexibility that is sometimes advantageous in matching trees to smiles.

A judicially chosen state space is needed to obtain a rea- sonable fit to the smile. We discuss a simple method for building “skewed” state spaces which fit typical index option smiles rather well. 

POSTED — January 1, 1996 — Archives