LTN | Finance, Economics, and Mathematics

Someone recently asked me how to calculate the partial derivative of the Black–Scholes option price with respect to the strike. Indeed, that kind of calculation easily leads to a complete mess. You end up with page after page of hopelessly complicated expressions. You need to simplify, but how?

Fortunately, there is a trick. Express the option price as a function of d1, and write d2 as d1 minus sigma times the square root of remaining time to maturity. It turns out that the derivative of the option price with respect to d1 is zero. So when you calculate the partial derivative, for example with respect to the strike, by the chain rule, the effects that come from the dependence of d1 on the strike will disappear.

Sure enough, this trick is explained on pages 216–217 of my 1999 book, Pricing and Hedging of Derivative Securities

Low-brow, but useful.

I visited Tsinghua University in Beijing in December (2012), gave a lecture on Stochastic Methods in Finance, and served as a panelist at the finals of the Yau High School Mathematics Awards (YHMA), an event which is chaired by professor Shing-Tung Yau of Harvard University.

My colleagues on the panel were a mix of mathematicians from China, the US and Europe. There were 17 teams of students in the finals, with 1, 2, or 3 students in each team. Most of the students were from China, although there were also a few from Singapore and the US. They had been selected for the finals from a much larger pool of, I believe, 400 teams. Each team had written a research paper in either pure or applied mathematics. The teams made presentations and took questions from the panel, all in beautiful English.

Most pictures of the Forbidden City and the Temple of Heaven that you find on the net are taken in the summer. Here is what those sites look like in December.

The Forbidden City

Approaching the southern gate.

The Forbidden City

Two bronze lions stand in front of the Gate of Supreme Harmony, a male and a female. This one is the male.

The Forbidden City

Bronze tortoise in front of the Palace of Heavenly Purity.

The Forbidden City

Bronze crane in front of the Palace of Heavenly Purity.

The Forbidden City

Bronze vat.

The Forbidden City

Stone carving, covered in snow.

The Forbidden City

Two bronze Ding. According to The Palace Museum,

The Ding was a cooking vessel developed during the Bronze Age that stood on three legs. Fire could be placed under the Ding to heat the food. Gradually, Ding became ceremonial vessels and were engraved or embossed with elaborate inscriptions and designs. They became a symbol of sacrifice and power.

I have no idea why “Ding” is capitalized.

The Forbidden City

Interior of the Hall of Celestial Purity. The imperial throne.

The Forbidden City

Interior of the Hall of Celestial and Terrestrial Union. Curiously exposed to the elements. That is both snow and sunlight in the foreground.

The Forbidden City

I believe this is the last emperor’s bedroom where he would entertain his wives. Or else it was only for wedding nights. Which were numerous. Seen through a large window. I wonder whether the window was also there back when the bedroom was in use.