Lecture 2: Option Theory

How To Price Options u The critical factor when trading in options, is determining a fair price for the option.

u Computes the value of a call option u The value can change only at the end of the period (t+1) and the possible maximal and minimal values are currently (t) known. The binomial model for option pricing

The Binomial Option Pricing Formula

Explanations of the parameters = Call option C = Call option = Call value at t+1, when stock price goes to max. = Call value at t+1, when stock price goes to max. = Call value at t+1, when stock price goes to min. = Call value at t+1, when stock price goes to min. = Riskless interest rate = Riskless interest rate = Multiplicative upward movement in stock price (S) u = Multiplicative upward movement in stock price (S) = Multiplicative downward movement in stock price (S) d = Multiplicative downward movement in stock price (S)

Features of the binomial model u The formula doesn’t depend on investors’ attitudes towards risk u Investors can agree on the relationship between and even if they have different expectations about the upward or downward movement of u Investors can agree on the relationship between C, S and r even if they have different expectations about the upward or downward movement of C

u The only random variable on which the call value depends, is the stock price itself u can take values between 0 and 1 (0 < < 1). If investors were risk neutral then. u p can take values between 0 and 1 (0 < p < 1). If investors were risk neutral then p=q.

A Short Example = 10 monetary units = 10 monetary units = 0 mu = 0 mu = 1,06 ( 6 %) r = 1,06 ( 6 %) u = 1,25 d = 0,75 = 10 monetary units = 10 monetary units = 0 mu = 0 mu = 1,06 ( 6 %) r = 1,06 ( 6 %) u = 1,25 d = 0,75 K= 15 mu (striking price) S = 20 mu = 25 mu = 15 mu

References u Cox & Rubinstein pp u Copeland & Weston p 259.

The Black & Scholes formula (A continuous time formula) u “It is possible to create a risk free portfolio” by owning 1 stock and writing h call options on it.” u The most frequently used option pricing formula u Originally a heat transfer equation in physics

Assumptions behind the formula: 1. The stock price follows a continuous Wiener- process and the future stock prices are lognormally distributed. 2. There exist no transaction costs or taxes. 3. No dividends are paid during the lifetime of the option. 4. The capital market is perfect: there exist no arbitrage-possibilities. 5. The composition of the portfolio can be continuously adjusted. 6. The risk free interest rate is constant during the lifetime of the option.

Black & Scholes Option Pricing Formula where Copeland & Weston p 276.

Parameters C = The price of the call option S = Stock price N = The standard normaldistribution = The continuous risk free rate of return [=ln(1+ )] = The continuous risk free rate of return [=ln(1+ )] t = The time to expiration (if 63 days, then t=63/365) K = Striking price   = The variance of the stock return

N(d 1 ) = the inverse hedge ratio e.g. for each stock that is owned, 1/N(d ) options has to be written for the portfolio to be risk free. e.g. for each stock that is owned, 1/N(d ) options has to be written for the portfolio to be risk free. = the discounted value of the striking price. = the discounted value of the striking price. N(d 2 ) = the probability for the option to be “in the money” on due date (e.g. the option will be exercised).

The price of an option is dependent on the following parameters: PC PC  Current stock price (S)   Time to expiration (t)   Striking price (K)   Stock volatility   Interest rates  (Cash dividends) 

References u Cox & Rubinstein p 37.

The Put-Call Parity: u There is a connection between the price of a call and a put option u If the prices differ from this equation, there exist arbitrage opportunities

References u Cox John C.&Rubinstein Mark: OPTIONS MARKETS. (USA 1985). Pp

An intuitive example u Buy a share: S = 19 mu u Buy a put option: P = 1 mu (striking price K = 20 mu) l Buy a call option with the same striking price and maturity: C = 1,50 mu l Deposit 18,50 mu at the risk free rate r = 10,95% = MAX(19-20,0) + 20 = 20

Outcome 9 months later on expiration date Put + Share = Call + Deposition Put + Share = Call + Deposition = = out of the money in the money Input vaules: r=10,95%;t=0,75; K=18,50;C=1,50;P=1;S=19 Discounted to t(0) Prolonged to t

u If we take cash dividends (D) into consideration the formula is slightly modified u Further modifications required if applied on american options Dividends

Arbitrage restrictions on call values (1) The value of a call is never less than the larger of: u zero u S - K (2) The value of a call is never greater than: u the price of its underlying stock In other words: The amount of cash deposited (equals to K)

References u Cox & Rubinstein p 129.

Call price Stockprice Cox & Rubinstein p 131. C = S

Cox & Rubinstein p 136. Striking price price Call price Call price C = S - K S S C C

Arbitrage restrictions on put values The value of a put is never less than the larger of u zero u K-S The value of a put is never greater than u its striking price In other words: The amount of cash deposited (equals K)

References u Cox & Rubinstein p 145.

Put price Stockprice K - S Cox & Rubinstein p 147. K K

Put price Strikingprice Cox & Rubinstein p 148. P = K P = K - S P

Popular research subjects u Forecasting the future volatility (Engle Robert F.: “Statistical Models for Financial Volatility”, Financial Analysts’ Journal Jan/Feb 1993) (Engle Robert F.: “Statistical Models for Financial Volatility”, Financial Analysts’ Journal Jan/Feb 1993) u Comparing the theoretical and the actual pricing of options (Kahra Hannu: “Pricing FOX Options Under Conditional Heteroscedasticity In Returns”, Tampere Economic Studies 1/1992) (Kahra Hannu: “Pricing FOX Options Under Conditional Heteroscedasticity In Returns”, Tampere Economic Studies 1/1992)

u Option Valuation under Stochastic Volatility. With Mathematica Code. (Alan L. Lewis, Finance Press, NewPort California, 2000) (Alan L. Lewis, Finance Press, NewPort California, 2000) u Option pricing, using distributions other than the standard normal distribution (Cox John C & Steven A. Ross: “The Valuation of Options for Alternative Stochastic Processes”, Journal of Financial Economics Jan-Mar 1976)

Findings: u Call options tend to be overpriced u Put options tend to be underpriced u Volatility seems to be forecastable in the short run u Only the broker makes profits in the long run