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

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

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: 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

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 Understanding Stock Options

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

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

Trading of Stock Options & Option Strategies Example :  S t = $100  X = $95

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

Trading of Stock Options & Option Strategies Option Strategies Option Strategies a) Protective Put stock Protective put portfolio S 0 = X Put option price StSt profit -P A -S 0

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

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

Trading of Stock Options & Option Strategies d) Spreads : X1X1 X2X2 Profit STST

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.

Determinants and Restrictions of Stock Option Valuation Restrictions  C (S t,X,t,T) > 0  C > S 0 – PV (X) – PV (D)  C < S 0

Determinants and Restrictions of Stock Option Valuation Upper bound: = S 0 Lower bound = adjusted intrinsic value = S 0 – PV (X) – PV(D) C B A Value of call Share price

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.

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

Determinants and Restrictions of Stock Option Valuation If this variable increases The Value of a Call Option: Stock Price (S) Increases Exercise Price (X) Decreases Volatility ( б ) Increases Time to expiration (t) Increases Interest rate (rf) Increases Dividend payouts (D) Decreases

Determinants and Restrictions of Stock Option Valuation Put – Call Parity: + = Investor’s payoff Future stock price Investor’s payoff Buying share Future stock price 50 Buying put option 50 Protection + = Investor’s payoff Future stock price Investor’s payoff Bank deposit paying $50 Future stock price 50 Buying call option 50 Protection a) b) Value of call + present value of exercise price = value of put + share price or; C + PV (X) = P + S 0

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

Methods for Valuing Stock Options Binomial Method is based on a simple approach: 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.

Methods for Valuing Stock Options Single Period Model for Binomial Method Single Period Model for Binomial MethodEXAMPLE: For XYZ stock; S 0 = $80 at Jan 1, S 1 = $160 or S 1 = $40 for Dec 31 X = $ 120 at the end of the year r f = 6%

Methods for Valuing Stock Options 1) Buying Call Option 1) Buying Call Option 2) Buying stock with borrowing 2) Buying stock with borrowing $160 $80 $40 (stock price possibilities) $40 $80 0 (call option payoffs) Value of stock at year end $160$40 - Repayment of loan with interest - $40 -$40 Total$1200

Methods for Valuing Stock Options Solving 1 &2: Solving 1 &2: Strategy 3 ; buy stock (no loan) & write 3 call Strategy 3 ; buy stock (no loan) & write 3 call $120 $ C = C = Value of stock at year end $160$40 - Obligations from 3 calls written - $ Total$40$40 PV of Strategy 3 Portfolio = $80 – 3C = PV of Strategy 3 Portfolio = $80 – 3C = C =

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. 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 - H = C + - C - / S + - S - In the example hedge ratio equal to 0.33 In the example hedge ratio equal to 0.33 The year end value of portfolio (0.33 shares and 1 written call) equal to $ PV of $ from 6% int. rate equal to $ If we use this amount at the portfolio function we hold C = $ (Same solution) PV of $ from 6% int. rate equal to $ If we use this amount at the portfolio function we hold C = $ (Same solution)

Methods for Valuing Stock Options General Binomial Method General Binomial Method Binomial Tree

Methods for Valuing Stock Options General Binomial Method General Binomial MethodEXAMPLE:

Methods for Valuing Stock Options Black – Scholes Option Valuation Model Black – Scholes Option Valuation ModelFormula: 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

Methods for Valuing Stock Options Basic Assumptions of BS Model 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

Methods for Valuing Stock Options Single Period Model for Binomial Method Single Period Model for Binomial MethodEXAMPLE: For XYZ stock; S 0 = $80 at Jan 1, S 1 = $160 or S 1 = $40 for Dec 31 X = $ 120 at the end of the year r f = 6%

Methods for Valuing Stock Options BS Formula EXAMPLE: BS Formula EXAMPLE: For XYZ stock; S 0 = $100, X = $ 95, r f = 10%, T = 0.25 year, ∂ = 0.50  First of all we can calculate d1 and d2 values. d1 = [ln (100 / 95) + ( (0.5)2/2) 0.25] / 0.5 √0.25 = 0.43 d2 = √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) = N(0.43) = N(0.18) = N(0.18) =  Thus the value of the call option is: C = 100 * – 95 ℮ -0.10*25 * = – = $13.70 C = 100 * – 95 ℮ -0.10*25 * = – = $13.70

Methods for Valuing Stock Options Advantages of Binomial Method and BS Formula 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 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.

Methods for Valuing Stock Options Modified BS Formulas and Other Models 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

Methods for Valuing Stock Options The Greek Letters 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.

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. Delta ( ∆ ) : The sensitivity of current option value to its current underlying asset price. It is easily calculated from a binomial tree. ∆ = ∂C / ∂S t = N (d 1 ) < 1 (for call option non-dividend) ∆ = N (d 1 ) – 1 < 0 (for put option non-dividend) Option price A B Slope =  Stock price

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

Methods for Valuing Stock Options Vega ( ν ) is the rate of change of the value of a derivatives portfolio with respect to volatility Vega ( ν ) is the rate of change of the value of a derivatives portfolio with respect to volatility ν = S 0 √T N’ (d 1 ) Rho ( ρ ) is the rate of change of the value of a derivative with respect to the interest rate 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 (d 2 )

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

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