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Local Time

The most obscure of the derivations is the one involving the concept from stochastic calculus known as 'local time.' Local time is a very technical idea involving the time a random walk spends in the vicinity of a point.

The derivation is based on the analysis of a stop-loss strategy in which one attempts to hedge a call by selling one share short if the stock is above the present value of the strike, and holding nothing if the stock is below the present value of the strike. Although at expiration the call payoff and the stock position will cancel each other exactly, this is not a strategy that eliminates risk. Naively you might think that this strategy would work, after all when you sell short one of the stock as it passes through the present value of the strike you will neither make nor lose money (assuming there are no transaction costs). But if that were the case then an option initially with strike above the forward stock price should have zero value. So clearly something is wrong here.

To see what goes wrong you have to look more closely at what happens as the stock goes through the present value of the strike. In particular, look at discrete moves in the stock price.

As the forward stock price goes from K to K + e sell one share and buy K bonds. And then every time the stock falls below the present value of the strike you reverse this. Even in the absence of transaction costs, there will be a slippage in this process. And the total slippage will depend on how often the stock crosses this point. Herein lies the rub. This happens an infinite number of times in continuous Brownian motion.

If U(e) is the number of times the forward price moves from K to K + e, which will be finite since e is finite, then the financing cost of this strategy is

Now take the limit as e 0 and this becomes the quantity known as local time. This local-time term is what explains the apparent paradox with the above example of the call with zero value.

Now we go over to the risk-neutral world to value the local-time term, ending up, eventually, with the Black-Scholes formula.

It is well worth simulating this strategy on a spreadsheet, using a finite time step and let this time step get smaller and smaller.

Parameters as Variables

The next derivation is rather novel in that it involves differentiating the option value with respect to the parameters strike, K, and expiration, T , instead of the more usual differentiation with respect to the variables S and t. This will lead to a partial differential equation that can be solved for the Black-Scholes formula. But more importantly, this technique can be used to deduce the dependence of volatility on stock price and time, given the market prices of options as functions of strike and expiration. This is an idea due to Dupire (1994) (also see Derman & Kani, 1994, and Rubinstein, 1994, for related work done in a discrete setting) and is the basis for deterministic volatility models and calibration.

We begin with the call option result from above

that the option value is the present value of the risk-neutral expected payoff. This can be written as

where p(S*, t*; S, T) is the transition probability density function for the risk-neutral random walk with S* being today's asset price and t* today's date. Note that here the arguments of V are the 'variables' strike, K, and expiration, T.

If we differentiate this with respect to K we get

After another differentiation, we arrive at this equation for the probability density function in terms of the option prices

We also know that the forward equation for the transition probability density function, the Fokker-Planck equation, is

Here a (S, t) is evaluated at t = T. We also have

This can be written as

using the forward equation. Integrating this by parts twice we get

In this expression a (S, t)has S = K and t = T .After some simple manipulations we get

This partial differential equation can now be solved for the Black-Scholes formula.

This method is not used in practice for finding these formula, but rather, knowing the traded prices of vanillas as a function of K and T we can turn this equation around to find a ,since the above analysis is still valid even if volatility is stock and time dependent.

 
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