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### The Theory of Speculation

No, I am not going to write on this year’s Nobel laureates in economics, but this is the title of Louis Bachelier’s Ph.D. thesis, published in 1900 (in French), in which he developed the stochastic process of Brownian motion. He is considered to be the father of modern financial mathematics.

His approach for valuating stock options modelled the movement of the underlying equity as an arithmetic Brownian motion. Equivalently, the changes in stock price are normally distributed. For longer time horizons, this leads to some problems, the positive probability that the stock price may become negative is the most important one.

The Black-Scholes-Merton model, published in 1973, assumes that not the changes in stock price are normally distributed but the return rates on small time scales. For the model problem (no dividends, constant volatility), the Black-Scholes-Merton model  yields lognormally distributed stock prices.

For small volatilities (appropriately scaled), there is not too much difference between the realisation of a Bachelier path (red) and a Black Scholes path (black) as the following figure indicates. We have used the same random numbers for both models: Every up movement in one model is mirrored by an up movement in the other model.

However, with large volatilities, the Bachelier paths may become negative.

For interest rates, the Black (Black 76) model is widely used for quoting implied volatilites of caps or swaptions, and the Vasicek or the Hull-White model were often criticised for allowing negative interest rates. However, in reality it was observed that negative forward rates were quoted for some currencies (CHF and JPY are popular examples). On July 1, 2013, the German Basiszinssatz (forming the baseline of interest in many contracts when one party is late with their payment obligations) was quoted as -0.38% (minus 38 basis points).

If the underlying quantity (here: a forward interest rate) becomes negative, then you get in trouble when you try to apply the logarithm in the Black76 framework. A Bachelier model may help.

More on this topic on next mathematics Wednesday.