What has option pricing got to do with pizzas?

The precise connection did not dawn on us until a friend of our content editor pointed this out to us. This guy works on the Wall Street and is an FX options trader. Apparently this pizza problem is given in an excellent book called Heard On The Street: Quantitative Questions from Wall Street Job Interviews, by Timothy Falcon Crack.

A pizza is made by the baker and consumed by us according to its area. Area is what determines the size of the pizza and how much of it can be had by each of us. In other words the number of people who can consume a pizza is proportional to the area of the pizza.

So, if a pizza whose diameter is 10 inches can be consumed by three persons then what should be the diameter of the pizza that can be consumed by 7 persons?

The answer is, in fact, very straightforward once we know the relationship between the area of the pizza and its diameter, which we all must have studied in our high school math. Of course the assumption here is that the pizza is perfectly circular in shape (which in real life it is not). Anyway, if we assume it to be perfectly circular in shape then the relationship between the area of the pizza A and its diameter, D is:

From the above it can be seen that if A = 3 (three persons consuming the area of the pizza) for D = 10 inches then for A = 7, D should be equal to 15.27. In other words, the diameter of the pizza should be 15.27 inches for 7 persons to consume it.

What is D in the case of the pizzas is the price of an option (in terms of the volatility of the underlying) and what is A in the case of the pizzas is time to maturity in the case of options.

All options are priced according to the volatility of the underlying asset. This is the single most important and most influential parameter that goes into any option pricing model. Actually, volatility does not exist by itself. It exists alongside the time parameter, another input to any option pricing model. This is a curious and wonderful property of the Brownian motion. Volatility moves with the square root of time and it is never assessed independent of it. No matter when and how you look at the volatility of an asset it is always multiplied by the time to give the correct value.

Though other variables, such as interest rate, strike price, spot rate, etc. affect the option price, nothing dominates the price as much as volatility, or volatility as scaled by time.

Buyers and sellers of option are actually buying and selling volatility. That is why option traders always quote price in terms of volatility. True, you pay Dollars or Euros or Yen when you buy an option but in essence you have bought the Dollar value of the volatility that is currently influencing the underlying spot or the forward.

If we use arithmetic Brownian motion and make the assumptions of at the money (spot equal to strike) and *very low interest rate* (theoretically zero) then the Black-Scholes formula collapses to:

Using the above formula you can price an option for any maturity as a function of a certain maturity without knowing any other parameter.

In other words, if a 3 month ATM (at the money) option costs $2.00 then what will a 9 month option cost? The fact that a three month option cost $2.00 shows that volatility is priced at $2.00 and therefore the volatility as a function of the square of time, or nine months will cost: $3.464.

Reference: Conversation with an options trader, Heard on the Street: *Quantitative Questions from Wall Street Job Interviews by Timothy Falcon Crack.*

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