January 26, 2025
As a fledgling quant at Bankers Trust in the 90’s, a trader came to me with an idea for a new product. He wanted to value a barrier option that knocked in/out the second time the underlying hit the barrier instead of the first time. Robert Merton showed how to price barrier options in … and I was familiar with his result. It used the reflection principle of Brownian motion. I learned about Brownian motion from Ito and McKean… Infinite oscillatory behavior. Fefferman’s proof the dual of H^1 is BMO. H^1_0(D) = BMOA(D).
Modeling stock price as geometric Brownian motion and assuming continuous time trading stunted growth.
It can’t answer questions a junior trader has the first day on the job. What vol do I use?
It can’t answer questions a junior trader has the second day on the job. When should I rehedge?
Market instruments have prices and cash flows.
Trades happen at discrete times.
How well does a trading strategy replicate derivative payoffs?