CSI 779 / STAT 789

Black-Scholes Derivative Pricing

In using the Black-Scholes formulas, it is important to remember to annualize rates, and to compute times in years.

The Black--Scholes call option formula can be written as a function in S-Plus as

bsform <- function(sig, r, t, xt, k) {
  d1  <- (log(xt/k) + (r + sig^2/2)*(t))/(sig*sqrt(t))
  d2  <- d1 - sig*sqrt(t)
  return(xt*pnorm(d1)-k*exp(-r*t)*pnorm(d2))
}

When the observed price of the option is taken as the value of the Black-Scholes formula and the volatility is taken as a variable in the resulting equation, the solution for the volatility is called the "implied volatility".

S-Plus does not have a nonlinear "solve", but Newton steps can be programmed directly to solve the nonlinear equation. The derivative is

bsformp <- function(sig, r, t, xt, k) {
  d1  <- (log(xt/k) + (r + sig^2/2)*(t))/(sig*sqrt(t))
  d2  <- d1 - sig*sqrt(t)
  d1p <- (sig^2*t - log(xt/k) - (r + sig^2/2)*(t))/(sig^2*sqrt(t))
  d2p <- d1p - sqrt(t)
  return(xt*pnorm(d1)*d1p-k*exp(-r*t)*pnorm(d2)*d2p)
}

So, after initializing sigk, for the observed price cobs, a step is

sigkp1 <- 
 sigk - (bsform(sigk, r, t, xt, k)-cobs)/bsformp(sigk, r, t, xt, k)

If the starting value is anywhere close, this is a fairly well-behaved function, so programming it directly with a convergence criterion based on the absolute difference (4 decimal places) and a limit on the number of iterations (around 50) should work reasonably well.

An alternative in packages like S-Plus that may have a nonlinear minimization function is to write an objective function as the square of the difference in the function and the specified value, that is

  (bsform(sigk, r, t, xt, k)-cobs)^2

in the expression above.

In S-Plus the minimization function requires a double to be returned by the objective function, and that function must be a function of the decision variable(s) alone. Therefore, we could write

bssig <- function(sig) {
  r    <- params[1]
  t    <- params[2]
  xt   <- params[3]
  k    <- params[4]
  cobs <- params[5]
  d1   <- (log(xt/k) + (r + sig^2/2)*(t))/(sig*sqrt(t))
  d2   <- d1 - sig*sqrt(t)
  return(as.double((xt*pnorm(d1)-k*exp(-r*t)*pnorm(d2)-cobs)^2))
}

and then

params <- c(r,t,xt,k,cobs)
assign("params", params, frame=1)

sig <- .05 ## (some reasonable guess)
nlmin(bssig,sig)

Assignment

1. Show that the Black-Scholes formula for European calls satisfies the Black-Scholes differential equation with the given boundary conditions.

2. Go to E*Trade options quotes and look at prices for Intel (INTC). Make note of the date.
Based on "last" prices for April 25, 27 1/2, 30, 32 1/2, and 35 calls, compute the implied volatility.

3. Compute (and annualize) the average volatility of INTC for the past 30 days. (Get those data from Yahoo .) How do the implied volatilities compare with each other and with the observed volatility?

4. Plot a volatility smile for the S&P 500 index. (This is the symbol SPX on E*Trade .)

5. Use Ito's formula to derive the formula for integration by parts for a stochastic integral.