5.  The Black-Scholes equation (for pricing European call options) and comparison with implementation of the binomial method

If C is the price of the call option, the Black-Scholes equation is obtained as follows:

First, C is a function of S and t, asset price and time: C=C(S,t).

Next, consider a portfolio similar to the one at the beginning of the talk.  Earlier, we considered a portfolio that had a mixture of stock (asset) and cash and showed how it could mimic or replicate the call option.  Here, we will create a portfolio of the call option and the stock, and show how it must mimic or replicate cash.  So let the portfolio be given as follows:
         ∏ = C - Δ * S
Then an incremental change in time causes the value of the portfolio to change by
        d∏ = dC - Δ * dS
Using the stochastic differential equation for dS, which contains a random part involving dX, and some stochastic calculus*,
         d∏  =  [ ∂C/∂t + () ] dt
*-imagine expanding C as a Taylor series: dC is thus (∂C/∂t)dt +(∂C/∂S)dS+( ) /2 + ...  Now substitute the form of dS.  But only keep those terms that have comparable order.  When dt is "small", of course is negligible.  But the case is different for dX in two ways.  First, whereas dX is stochastic, is essentially non-stochastic; and second, it's also not negligible, but rather it's essentially dt! [Why?  Return to the discussion in a random walk, and review why the variance was proportional to time (if you forgot, we did it "experimentally"; but of course it comes from the binomial distribution, whose variance is proportional to n ).  Next, how spread out is ?  Note that the variance of the variance of a binomial distribution is proportional to (exercise!  you can use Mathematica!). So the variance of is proportional to and so is neglible by comparison.  So is indeed essentially non-stochastic and also non-negligible, and can be replaced by dt.]

Whether you have read thru the above fine-print or not, you may well wonder: where did the dX parts go?  By choosing  Δ appropriately, namely  Δ = ∂C/∂S, the dX terms cancel out.  We'll do this on the board, if we have time at this point in the presentation.  But at least intuitively, this choice of how much stock to sell, Δ, is precisely the hedging: by buying a call option and selling some stock, the portfolio presumably cancels out the risky nature of the investments.  (Remember the discussion from the intermediate-algebra portfolio in section #2: if you sold someone a call option, recall that if the stock goes up and exceeds the strike price, you will owe that person money.  A way to hedge against having to pay that individual in the event of such a stock rise is to keep some of the stock on hand, so that in the event of just such a price rise, you can sell some of that stock to pay off the call.)

So by this choice of Δ, ∏ changes in a NON-STOCHASTIC way.  This means that the portfolio is risk-free.  As such, the amount of gain from it must precisely match the gain if the portfolio were invested in a risk-free bond:  namely, d∏ must equal ∏r dt.  (Here, r is the risk-free interest rate.  The section 2 discussion skipped any mention of interest rates, but they do figure in Black-Scholes...)

We therefore get the Black-Scholes equation:

         ∂C/∂t + = r ( C-  ∂C/∂S*S)
     

We note that μ is absent: whether the stock goes up or down is irrelevant to pricing the option, only the volatility σ appears.

Amazingly, the equation above is equivalent to the heat equation (by suitable change of variable).  That was not known to Black, Scholes and Merton back in the early 1970s.  But we can show that it is (if we have time).  We also need to include a suitable initial condition, and from that can easily write down an explicit solution.
  
In any case:  The solution to the Black-Scholes equation is known exactly.  We code up the exact form from the CUP reference (E stands for strike price, S for current stock price, r for interest rate, t for current time, capt for strike time, sigma for volatility):

Now compute the Black-Scholes solution for the price of two call options: Say today
the trading price is $49, the strike price is $50, the first call expires in 20 weeks, the second
call expires in one year.  Let's assume the risk-free interest rate is .05 (i.e. 5%), and the
volatility is .2 i.e. 20%.

So the fair cost of the 20-week option is $2.40, apparently (while it's $4.61 for the one-year option).

Now we will compare the Black-Scholes solution with "our" simple-minded up-down random walk version.  In what follows below, "up" refers to how large of an "up" step to take, and so on.  We have not discussed in this notebook how to compute the up/down sizes; perhaps you can tell simply from the form below.  Basically, they are obtained by a simple equating of the random-walk-with-drift mean and variance (obtained from the well-known binomial distribution) with the "given" mean and variance associated to the stock, obtained from the fact that it's prices are lognormally distributed.  (See the section above where the log of S was discretized; that shows how to obtain estimates of μ and  σ from actual stockpricing data.)

Note that we apparently must use a slightly greater "up" step than a "down" step.  Of course, this is to be expected, since in this example we chose a mu that was positive (so we need a larger 'up' step than 'down' step to mimic this positive growth).

Next, you will see two quantities, "p" and "q".  These arise simply as in our intermediate-algebra approach to pricing call options from section 2.  There, we simply solved the two-equation, two-unknown system 'by hand'; in general, we can express the fair call price in terms of a linear combination (via p and q) of the values of the call at the time of expiration.  Again, we will discuss (this as time allows) on the board...

Compare this to the Black-Scholes solution... $2.40 and $4.61.  Not bad!  The simple-minded, essentially intermediate algebra solution does quite well at mimicing the PDE solution!

Now we show how well Black-Scholes models the option pricing for
Amazon.com in spring 1999 (before the spring 2000 virtual explosion in the
NASDAQ, and the spring 2001 implosion/bear market).

If we had time, and we may, we will discuss the "volatility smile" and have you pick your own stock from the Wall Street Journal handouts and see if you can deduce the implied volatilities...