1. Option pricing models

2. Objective Learn to estimate the market value of option contracts.

3. Outline The Binomial ModelThe Binomial Model The Black-Scholes pricing modelThe Black-Scholes pricing model

4. The Binomial Model A very simple to use and understand lattice model that provides the theoretical foundation for other, more advanced models

5. The one-period binomial model Exemplification Assume the stock of ABC currently sells for $100/share. A call option on the ABC is struck at $120/share and expires in one year. In one year, the stock can either go up by 50% or down by 25%. How much should you pay for the call today if the risk free rate is 10%?

6. The one-period binomial model: A hedged portfolio Question: Can we combine the call option and the stock in a risk- free portfolio? Yes. Buy 4 shares of stock and sell 10 call options.

7. A hedged portfolio S 0 = $100 S d = $75 S u = $150 1+ u = 1.5 1+ d = 0.75 Buy 4 shares and sell 10 calls Cash flow = 10C 0 - $400 Portfolio value: $150(4) - $30(10) = $300 Portfolio value: $75(4) = $300

8. A hedged portfolio: Summary Our portfolio is risk-free. We invested 10C 0 - $400 and we obtained $300 over one year Our portfolio must earn the risk-free rate. -(10C 0 - $400)(1.1) = $300 C 0 = $12.73 The call option on stock ABC, struck at $120 and expiring in one year, should sell today for $12.73

9. How did we know to buy 4 shares and write 10 calls? Calculating the hedge ratio, h We required that the value of the portfolio stays the same regardless of the stock price: -C u + h(1+u)S 0 = -C d + h(1+d)S 0 h = the ratio of number of shares to number of calls h = (C u - C d )/(S u - S d )‏ In our case: h =($30 - $0)/$100(1.5 - 0.75) =0.4

10. The one-period binomial model It follows that: (hS 0 - C 0 )(1+r) = h(1+d)S 0 - C d C 0 = [h(1+r)S 0 + C d - h(1+d)S 0 ]/(1+r)‏ Remember, h = (C u - C d )/(S u - S d )‏ C 0 = [ (C u - C d )/(S u -S d ) (1+r)S 0 + C d - (C u - C d )/(S u -S d ) (1+d)S 0 ]/(1+r)‏

11. The one-period binomial model C 0 = [pC u + (1-p)C d ]/(1+r) where p = (r-d)/(u-d)‏ 1 - p = (u-r)/(u-d)‏

12. The two-period binomial model Exemplification Assume the stock of ABC currently sells for $100/share. A call option on the ABC is struck at $120/share and expires in two years. Each year, the stock can either go up by 50% or down by 25%. How much should you pay for the call today if the risk free rate is 10%?

13. The two-period binomial model S 0 = $100 S d = $75 S u = $150 1+ u = 1.5 1+ d = 0.75 1+ u = 1.5 S uu = $225 S dd = $56.25 1+ u = 1.5 1+ d = 0.75 S ud = $112.50

14. The two-period binomial model C 0 = [p 2 C uu +2p(1-p)C ud + (1-p) 2 C dd ]/(1+r) 2

15. What if we have n-periods? Assume Returns are log-normal distributed The risk-free rate and the returns volatility are constant Each period becomes extremely short Interest is compounded continuously No transaction costs The options are European and/or American but no dividend is paid on the stock until after expiration

16. The Black-Scholes formula C = SN(d 1 ) - Ee -rT N(d 2 )‏ S = current stock price E = exercise price d 1 = [ln(S/E) + (r + σ 2 /2)T]/ σ T 1/2 d 2 = d 1 - σ T 1/2 N(d 1 ), N(d 2 ) = cumulative normal probabilities σ = annualized standard deviation of stock’s return r = continuously compounded risk-free rate T = time to maturity

17. The Black-Scholes formula: A numerical example Digital Equipment July 165 call has 35 days to maturity (assume 365 days in a year)‏ The current stock price is $164 The risk-free rate is 5.35% The standard deviation of the return is 29% Calculate the theoretical value of the call option

18. Step 1 Express the risk-free rate as a continuously compounded yield (find out that continuously compounded rate that would yield 5.35%): (1 + r*/n) n = 1.0535 e r* = 1.0535 r* = ln(1.0535) = 0.0521

19. Step 2 Calculate d 1 and d 2 : d 1 = [ln(164/165) + (0.0521 + 0.0841/2)0.0959]/(0.29)(0.309677)‏ d 1 = [ - 0.006079 + 0.009029 ]/(0.0898063) = 0.0328485 d 2 = 0.0328485 - 0.0898063 = - 0.0569578

20. Normal distribution table: p(x < z-statistic)‏

21. Step 3 N(0.03) = 0.5 + 0.012 = 0.512 N(-0.06) = 0.5 - 0.0239 = 0.4761

22. Step 4 C = 164(0.512) - 165e -(0.521)(0.0959) (0.4761)‏ C = 83.968 - 165(0.995)(0.4761) = $5.803

23. Discussion Right now, the stock is at $164 The call is out-of-the-money (intrinsic value is zero)‏ The market value should be $5.803, reflecting the probability that, by expiration, the stock price might increase This probability calculation is based on observing the volatility of return

24. Relationship between several variables and the market value of options