Appendix

A.1  The Derivation of Ito's Lemma

A completely rigorous derivation or proof of Ito's lemma is beyond the scope of this notebook. We now present a non-rigorous definition of Ito's lemma using the Taylor series formula. For a smooth function G(x, t), the normal Taylor series expansion goes as

                     ΔG =           (A.1)       
For a non-stochastic  process, when Δx → 0, Δt → 0, the above equation becomes

      nbsp;             dG =            (A.2)                 
We now extend equation (A.2) to cover functions of variables following Ito processes. Suppose that a variable, x, follows the Ito process  

               (A.3)              
the discretised form of which is

           nbsp;                       (A.4)                           
where is a standard normal variable. From this equation,

           nbsp;                           (A.5)                 
This shows that the term involving in equation (A.1) has a component that is of order Δt and cannot be ignored.                      
The variance of a standardized normal distribution is 1. This means that

                      E() - = 1
Because E(ε) = 0, it follows that E () = 1. The expected value of Δt is of order Δand that as a result of this, we can treat as nonstochastic and equal to its expected value of Δt as Δt tends to zero. It follows from equation (A.5) that Δbecomes nonstochastic and equals to as Δt  tends to zero. Taking limits as Δx and Δt tend to zero in equation (A.1), and using this last result, we obtain   

                  dG =  dt          (A.6)  
This is  Ito's lemma. Substituting for dx from equation (A.3), equation (A.6) becomes

                        dG =  ( )dt + bdz          (A.7)

A.2  The Derivation of Black-Scholes  Formula

There are several assumptions involved in the derivation of the Black-Scholes  formula:
(1) trading takes place continuously in time. (2) the riskless interest rate r is known and constant over time. (3) the asset pays no dividend. (4) there are no transaction costs in buying or selling the asset or the option, and no taxes. (5) the assets are perfectly divisible. (6) short selling (sale of a security not own by the seller) with full use of proceeds (funds given to a borrower after all costs and fees are deducted) is possible. (7) there are no riskless arbitrage (profiting from differences in price when the same asset is traded on two or more markets) opportunities.     
One common way of pricing a derivative is to form a self-financing,  replicating hedging strategy for it. Self-financing means that the portfolio produced by the strategy must not itself take up any money apart from a possible initial investment. As we will soon see, this initial investment will be the price of the security that the strategy is replicating. The term replicating means that the strategy must replicate the payoff of the security we are trying to price. Further, the portfolio produced by the strategy should always produce the same result regardless of price changes in the underlying security. In other words, the value of the portfolio generated by the strategy should be deterministic and cannot have a stochastic component (except for the stochastic components of the underlying securities of the derivative). This explains the term 'hedging' used for the strategy.
The value of  a derivative is its expected future value discounted at the risk-free  interest rate. This is exactly the same result that we would obtain if we assumed that the world was risk-neutral.  In such a world, investors would require no compensation for risk. This means that the expected return on all securities would be the risk-free  interest rate. This is a very useful principle as it states that we can assume that the world is risk-neutral  when calculating option prices. The result would still be correct in the real world even if (as is most probably the case) it is not risk-neutral.
We derive the put-call  parity relation using this principle. To do so, use the fact that the sum of the payoff of a long call and a short put option with the same strike price and maturity (and, of course, on the same underlying securities) is given by S - K. Hence, the value of the call and put option is given by

             =           (A.8)                         
the last equality coming from risk-neutral  valuation. This gives us

                        C- P = S - K or C + K          (A.9)                  
We need one more piece of information before we can derive the Black-Scholes  formula. This is Ito's lemma: if a variable x follows a stochastic process of the form

      nbsp;           dx = a(x, t)dt + b(x, t)dz          (A.10)                              
then any smooth function G(x, t)  follows the process

                       dG =  ()dt + bdz          (A.11)                           
We are now in a position to present a derivation of the Black-Scholes  formula. Review the Black-Scholes equation derived in Chapter III:                           (A.12)                        
We have shown in Chapter III that if  , then    

         &           ln()- ln()=  σdz + (μ-)dt          (A.13)     &;
i.e,  S follows a lognormal distribution:
 &         lnS          (A.14)
The principle of risk-neutral  valuation implies that the present value of the option is the expected final value E [max ( S-K,  0 )] of the option discounted at the risk-free  interest rate. So, we have           &      c =           (A.15)                          
where g(S), the probability density function  is given by (A.14) which can be explicitly written as        g(S) =           (A.16)   & 
where μ has been replaced by r in accordance with the principle of risk-neutral   valuation. We can easily verify that this solution satisfies the principle of risk-neutral valuation by evaluating E (S) = .
The value of the integral (A.15) can be found with a bit of algebraic manipulation and the Black-Scholes  formula for standard European call option is  &            c = SN(() - K          (A.17)
       &    
          where             ,     
and is the cumulative standard normal distribution.