I am psyched! After months of trying to understand the logic, derivations and theorems that lead to the result today we finally derived the famous Black-Scholes option pricing formula. Weeks ago, I already introduced a one-period derivative pricing model. Well, since then the topic has become a lot more complex and in this post, I will tell about the logic of anther magic formula.
As we did already in the one period case we will start this analysis by assuming that there is no arbitrage in the market. Thus, every contingent claim X(t) with a specific payoff can be replicated. Our goal is to price this contingent claim.
In this case the contingent claim is a European call option with strike price K. The payoff of this option at time is equal to X(T) = max(S(T) – K, 0) as we have seen before already. Additionally, we already know that there are always two ways to price a contingent claim when markets are complete:
- Find a strategy that replicates the payoff. The price of the option then equals the cost of setting up the following strategy, where Sx(t) is the price of the contingent claim at time t, θ0(T)B(t) is the position in the risk-free security and θ1(t)S(t) represents the position in the risky security.
- The expectation at time t under the risk neutral probability of the discounted price at time T equals equals the price at time t as can be seen below, where EQ means expectation under the risk-neutral probability Q, eδ(T-t) is the discount factor between T and t in the continuous time and the information that is known is the information at time t, ft.
With the help of advanced integrals, we can model the development of the stock price as
This formula tells us that the stock price at time t equals the stock price at time 0 times an exponential. The exponential depends on: the instantaneous risk-free rate of interest δ, the volatility σ and the standard Brownian motion W*(t) which corresponds to the part of the return that is totally random and unpredictable. However, W*(t) is distributed according to a normal distribution with mean 0 and variance t (N(0,t)).
So, what do we have so far? We have the equation for the payoff of our call option, we have the condition for the price under the risk neutral probability (we choose this approach to value the option) and we have the development of the stock price. Let’s put all this together. The price of the call option at time 0 (c(0)) can be calculated as follows:
For convenience, we start our valuation at time 0 which makes the equation easier to work with. Next we standardize the standard Brownian motion so that W*(t) becomes sqrt(T)Z where Z is normally distributed with zero mean and variance of 1 (N(0,1)).
Thanks to various Russian and Japanese mathematicians the expression above can be expressed as an integral. Thus, we get
where the last part is the density function of the normal distribution.
Now we are at a point where we should ask us which development actually is of relevance for the price of the call option. Obviously, we would only pay for the option if we think it to be possible that the option will be in the money. Consequently, we only consider the case where S(t) => K. What does this depend on? On Z! The stock price develops exponentially according to different values that Z takes on. However, K is fixed and thus there is a unique value of Z that equalizes K and the stock price as can be seen below.
Thus, the system below can be solved for a unique value of Z.
This is a big step since it enables us to get rid of the max function in the integral. We can re-write the equation
This consists of two parts:
- There average price of the stock if we exercise the option
- The average cost of exercising the option (the present value of the strike price times the probability of being in the money)
The difference between these two is the cost of the option. I will not bother you anymore with further derivations but finally the option can be priced as follows:
c(0) = S(0)N(d1) – e-δTKN(d2)
d1 = (ln(S(0)/K) + (δ+0.5σ2)T)/(σT)
d1 = (ln(S(0)/K) + (δ-0.5σ2)T)/(σT)
This is it. This is the famous (magic) formula that allows us to price options.
One side note. I did some research about the normal distribution assumption implied by the formula and found out that indeed it is possible to introduce bimodal models or jumps in the distribution. Thus, what I have writing in my entry about Deutsche Bank regarding possible option pricing inefficiencies might not be true after all.
Wow, we got a lot done here. Is your head exploding as well? But how cool that we just derived one of the most famous and fundamental formulas in Finance!