Suppose a stock priced at $100 will either go up by 10% or down by 10%.

Call Option

• Consider a call option with a strike of $105.
  
• It ends with a value of $5 if the stock goes to $110 and a value of $0 if the stock goes to $90.
• We want to find its value at the beginning.

Delta

• Define \(\Delta\) as the difference in the option values divided by the difference in the stock values.
• This is \((5-0)/(110-90) = 1/4\).
  
• Here is the value of 1/4 share of the stock.

Debt

• Consider borrowing the PV of the bottom value from the previous figure.
• Suppose the interest rate is 5%. The PV of $22.50 is $21.43. Here is how the debt evolves.

Buy \(\Delta\) shares on margin

Conclusion

• A call option in this simple model is equivalent to buying \(\Delta\) shares with leverage.
• The value of the call must be the equity needed in the levered portfolio.
• This is the cost of \(\Delta\) shares minus the amount borrowed.

Calculation in code

S = 100     # initial stock price
K = 105     # strike
u = 0.10    # up return
d = -0.10   # down return
r = 0.05    # interest rate

Su = S * (1+u)
Sd = S * (1+d)
Cu = max(0, Su-K)
Cd = max(0, Sd-K)
delta = (Cu-Cd) / (Su-Sd)
lev = (delta*Sd - Cd) / (1+r)
call = delta*S - lev
call

3.57142857142858

• Suppose a $100 stock goes up by 5% or down by (1/1.05-1) = -4.76% in each of two periods.

• A call option with a strike of $105 expires at the end of the 2nd period.
• The call evolves as

Recursion

• At each of the two nodes at the middle date, repeat the one-period analysis to get call values \(c_u\) and \(c_d\).
• Then repeat the one-period analysis at the initial date using \(c_u\) and \(c_d\).

Recursion in code

def one_period_call(S, Su, Sd, Cu, Cd, r):
    delta = (Cu-Cd) / (Su-Sd)
    lev = (delta*Sd - Cd) / (1+r)
    return delta*S - lev

S = 100
K = 105
u = 0.05  # up return each period
r = 0.03  # interest rate each period