VALUING STOCK OPTIONS

1. VALUING STOCK OPTIONS HAKAN BASTURK Capital Markets Board of Turkey April 22, 2003

2. PRESENTATION PLAN • Understanding Stock Options (Definition, classifications & related terms) • Trading of Stock Options & Option Strategies • Restrictions & Determinants of Stock Option Valuation • Methods for Valuing Stock Options

3. Understanding Stock Options • A stock option is a contract which conveys to its holder the right, but not the obligation, to buy or sell shares of the underlying stock at a specified price on or before a given date. • Stock options primarily can help investors: • Protect their stock holdings against a decline in market prices • Increase their income on current or new stock holdings • Buy stock at a lower price • Benefit from a stock price’s rise or fall without owning the stock or selling it outright

4. Understanding Stock Options • An option contract is defined by the following elements: • Type (Call or Put) • Style ( American, European or capped) • Underlying Security • Unit of Trade • Strike Price • Expiration Date

5. Understanding Stock Options • Tradable Stock Options (exchanges & over the counter) • Non – tradable Stock Options (Employee Stock Option Plan) • LEAPS (Long Term Equity Anticipation Securities)

6. Trading of Stock Options & Option Strategies • The relationship between the Strike Price (X) and stock price (St) • In – the – money (For call option St > X) • At – the – money (St = X) • Out – of – the – money (St < X)

7. Trading of Stock Options & Option Strategies • Example : • St = $100 • X = $95

8. X X ST ST (a) holder (b) writer X X ST ST (c) holder (d) writer Trading of Stock Options & Option Strategies The Payoffs of Stock Options at Expiration Date Call option Put option

9. Trading of Stock Options & Option Strategies • Option Strategies a) Protective Put profit stock Protective put portfolio St A S0 = X -P Put option price -S0

10. Stock only Profit Covered call ST X Trading of Stock Options & Option Strategies b) Covered Calls Naked writing position

11. Profit X ST Trading of Stock Options & Option Strategies c) Straddle : Buying call stock Buying put

12. Trading of Stock Options & Option Strategies d) Spreads : Profit ST X1 • X2

13. Determinants and Restrictions of Stock Option Valuation Main difference of stock options from the other assets: Since the risk of stock option changes every time with changing stock prices, finding the opportunity cost of capital is impossible, so we cannot use discounted cash flow analysis to stock options.

14. Determinants and Restrictions of Stock Option Valuation • Restrictions • C (St,X,t,T) > 0 • C > S0 – PV (X) – PV (D) • C < S0

15. Upper bound: = S0 B Lower bound = adjusted intrinsic value = S0 – PV (X) – PV(D) C Determinants and Restrictions of Stock Option Valuation Value of call A Share price

16. Determinants and Restrictions of Stock Option Valuation • Difference of American & European Call • c > C ( before the maturity date) • Longer time to maturity, more valuable American call options, • Call price at least equal to exercise price before maturity date.

17. Determinants and Restrictions of Stock Option Valuation • Factors on the value of a stock option: • Stock Price • Exercise Price • Volatility of the stock price • Time to expiration • Interest rate • Dividend payouts

18. Determinants and Restrictions of Stock Option Valuation

19. Investor’s payoff Investor’s payoff Investor’s payoff Investor’s payoff Investor’s payoff Investor’s payoff Protection Protection Buying share Bank deposit paying $50 Buying put option Buying call option + + = = 50 50 Future stock price Future stock price 50 50 Future stock price Future stock price 50 50 Future stock price Future stock price Determinants and Restrictions of Stock Option Valuation • Put – Call Parity: a) b) Value of call + present value of exercise price = value of put + share price or; C + PV (X) = P + S0

20. Methods for Valuing Stock Options • Methods for Valuing Stock Options: • Binomial Method • Black – Scholes Option Valuation Model • Other Models

21. Methods for Valuing Stock Options • Binomial Method is based on a simple approach: In a single time period, underlying stock price can only move to two possible levels (up & down) • Basic Assumptions of Binomial Method • No riskless arbitrage opportunities • Perfect market conditions • Risk – neutral valuation principle.

22. Methods for Valuing Stock Options • Single Period Model for Binomial Method EXAMPLE: For XYZ stock; S0 = $80 at Jan 1, S1 = $160 or S1 = $40 for Dec 31 X = $ 120 at the end of the year rf = 6%

23. Methods for Valuing Stock Options • 1) Buying Call Option $160 $80 $40 (stock price possibilities) $40 $80 0 (call option payoffs) • 2) Buying stock with borrowing

24. Methods for Valuing Stock Options • Solving 1 &2: $120 $42.26 0 3C = 42.26 C = 14.09 • Strategy 3 ; buy stock (no loan) & write 3 call • PV of Strategy 3 Portfolio = $80 – 3C = 37.74 C = 14.09

25. Methods for Valuing Stock Options • The hedge ratio equals the ratio of ranges because the option and the stock are perfectly correlated, a perfect hedge requires that the option and stock be held in a fraction determined only by relative volatility. H = C+ - C- / S+ - S- • In the example hedge ratio equal to 0.33 • The year end value of portfolio (0.33 shares and 1 written call) equal to $13.33 • PV of $13.33 from 6% int. rate equal to $12.57 If we use this amount at the portfolio function we hold C = $14.09 (Same solution)

26. Methods for Valuing Stock Options • General Binomial Method • Binomial Tree

27. Methods for Valuing Stock Options • General Binomial Method EXAMPLE:

28. Methods for Valuing Stock Options • Black – Scholes Option Valuation Model • Formula: The variables are: S = stock price X = strike price t = time remaining until expiration, expressed as a percent of a year r = current continuously compounded risk-free interest rate v = annual volatility of stock price (the standard deviation of the short-term returns over one year). ln = natural logarithm N(x) = standard normal cumulative distribution function e = the exponential function

29. Methods for Valuing Stock Options • Basic Assumptions of BS Model • The stock pays no dividends during the option's life • European exercise terms are used • Markets are efficient • No commissions are charged • Interest rates remain constant and known • Returns are lognormally distributed

30. Methods for Valuing Stock Options • Single Period Model for Binomial Method EXAMPLE: For XYZ stock; S0 = $80 at Jan 1, S1 = $160 or S1 = $40 for Dec 31 X = $ 120 at the end of the year rf = 6%

31. Methods for Valuing Stock Options • BS Formula EXAMPLE: For XYZ stock; S0 = $100 , X = $ 95, rf = 10%, T = 0.25 year, ∂ = 0.50 • First of all we can calculate d1 and d2 values. • d1 = [ln (100 / 95) + (0.10 + (0.5)2/2) 0.25] / 0.5 √0.25 = 0.43 • d2 = 0.43 - 0.5 √0.25 = 0.18 • Next we will find N (d1) and N (d2) from the tables about the values of the normal distribution. We can from the table that: • N(0.43) = 0.6664 • N(0.18) = 0.5714 • Thus the value of the call option is: • C = 100 * 0.6664 – 95 ℮-0.10*25 * 0.5714 = 66.64 – 52.94 = $13.70

32. Methods for Valuing Stock Options • Advantages of Binomial Method and BS Formula • Binomial Model can be used to American options • Binomial Model isn’t related with the returns of stocks. • BS Formula is easy to use • BS Formula gives a rapid solution • Disadvantages of Binomial Method and BS Formula • Binomial Method relatively has a slow speed. Not easy to calculate with lots of sub period. • BS Formula is only related with European type options. • Two models have lots of assumptions.

33. Methods for Valuing Stock Options • Modified BS Formulas and Other Models • Modified BS European Model : equities with dividend and for other securities, • Modified BS American Model : minimum value for option at intrinsic value • Modified BS French Model : trading days instead of calendar year, • Whaley (Quadratic Approximation) Method: It gives a value for American options which equals to the value of European option plus American options’ early exercise option • Pseudo – American Method : cash flows such as cash dividends

34. Methods for Valuing Stock Options • The Greek Letters These are related with the degrees of change in option price according to the change in the variables. They give the sensitivity of option price to the change in variables.

35. Option price Slope = D B Stock price A Methods for Valuing Stock Options • Delta (∆): The sensitivity of current option value to its current underlying asset price. It is easily calculated from a binomial tree. ∆ = ∂C / ∂St = N (d1) < 1 (for call option non-dividend) ∆ = N (d1) – 1 < 0 (for put option non-dividend)

36. Methods for Valuing Stock Options • Gamma (Γ) measures the rate at which the delta changes as the underlying asset price changes. Γ = ∂2C / ∂St2 = N’ (d1) > 1 (for call option non-dividend) • Theta (Θ) measures the sensitivity of the current option value to a reduction in time-to-expiration. Θ = - ∂C / ∂(T-t) < 0

37. Methods for Valuing Stock Options • Vega (ν) is the rate of change of the value of a derivatives portfolio with respect to volatility ν = S0√T N’ (d1) • Rho (ρ) is the rate of change of the value of a derivative with respect to the interest rate ρ = X T e –rT N (d2)

38. Methods for Valuing Stock Options Using of Option Pricing Models • Delta hedging • Portfolio Insurance (dynamic hedging)

39. Methods for Valuing Stock Options Rules & Comments of Regulatory Authorities • Financial Accounting Standards Board (FASB) Statement No. 123 • SEC, NASD & stock exchanges