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Deriving the Black-Scholes Option Pricing Formulae using the limit of a suitably constructed lattice

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Suppose we knew for certain that between time  and  the price of the underlying could move from  to either  or to , where  (as in the diagram below), that cash (or more precisely the appropriate risk-free asset) invested over that period earns an interest rate (continuously compounded) of  and that the underlying (here assumed to be an equity or an equity index) generates income, i.e. dividend yield, (continuously compounded) of .

 

 

Diagram illustrating single time-step binomial option pricing

 

Suppose that we also have a derivative (or indeed any other sort of security) which (at time ) is worth  if the share price has moved to , and worth  if it has moved to .

 

Starting at  at time , we can (in the absence of transaction costs and in an arbitrage-free world) construct a hedge portfolio at time  that is guaranteed to have the same value as the derivative at time  whichever outcome materialises.  We do this by investing (at time )  in  units of the underlying and investing  in the risk-free security, where  and  satisfy the following two simultaneous equations:

 

 

Hence:

 

 

The value of the hedge portfolio and hence, by the principle of no arbitrage, the value of the derivative at time  can thus be derived by the following backward equation:

 

 

We can rewrite this equation as follows, where  and  and hence .

 

 

Assuming that the two potential movements are chosen so that  and  are both positive, i.e. with  then   and  correspond to the relevant risk neutral probabilities for the lattice element. Getting  and  to adhere to this constraint is not normally difficult for an option like this since  is the forward price of the security and it would be an odd sort of binomial tree that did not straddle the expected movement in the underlying.

 

In the multi-period analogue, the price of the underlying is assumed to be able to move in the first period either up or down by a factor  or , and in second and subsequent periods up or down by a further  or  from where it had reached at the end of the preceding period.  or  can in principle vary depending on the time period (e.g.  might be size  in time step , etc.) but it would then be usual to require the lattice to be recombining.  In such a lattice an up movement in one time period followed by a down movement in the next leaves the price of the underlying at the same value as a down followed by an up.  It would also be common, but again not essential (and sometimes inappropriate), to have each time period of the same length, .

 

By repeated application of the backward equation referred to above, we can derive the price  periods back, i.e. at , of a derivative with an arbitrary payoff at time .  If , , , ,  and  are the same for each period then:

 

 

where:

 

 

This can be re-expressed as an expectation under a risk-neutral probability distribution, i.e. in the following form, where  means the expected value of  given the risk neutral measure, conditional on being in state  when the expectation is carried out:

 

 

Suppose we have a European-style put option with strike price  (assumed to be at a node of the lattice) maturing at time  and we want to identify its price,  prior to maturity, i.e. where . Suppose also that  and  are the same for each time period. The price of the option at maturity is given by its payoff, i.e.  where  say for some  (here  is the price ruling at time  used to construct the first node in the lattice). Applying the multi-period pricing formula set out above, we find that the price of such an option at time  in such a framework is as follows, where  is the binomial probability distribution function, i.e. , bearing in mind that :

 

 

Suppose we define the volatility of the lattice to be  and suppose too that this is constant, i.e. the same for each time period. Then if we allow  to tend to zero, keeping , ,  etc. fixed, with by, say, setting  and , we find that the above formula and hence the price of the put option tends to:

 

 

where

 

 

and  is the cumulative Normal distribution function, i.e.

 

 

The corresponding formula (in the limit) for the price,  of a European call option maturing at time  with a strike price of  can be derived in an equivalent manner as:

 

 

This formula can also be justified on the grounds that the value of a combination of a European put option and a European call option with the same strike price should satisfy so-called put-call parity, if they are to satisfy the principle of no arbitrage, i.e. (after allowing for dividends and interest):

 

 

Strictly speaking, these formulae for European put and call options are the Garman-Kohlhagen formulae for dividend bearing securities and only if  is set to zero do they become the original Black-Scholes option pricing formulae, although in practice most people would actually refer to these formulae as the Black-Scholes formulae, and call a world satisfying the assumptions underlying these formulae as a ‘Black-Scholes’ world. The volatility  used in their derivation has a natural correspondence with the volatility that the share price might be expected to exhibit in a Black-Scholes world.

 


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