5.1 Option pricing: pre-analytics Lecture 5

5.2 Notation c : European call option price p :European put option price S 0 :Stock price today X :Strike price T :Life of option  :Volatility of stock price C :American Call option price P :American Put option price S T :Stock price at time T D :Present value of dividends during option’s life r :Risk-free rate for maturity T with cont comp

5.3 Calls: An Arbitrage Opportunity? Suppose that c = 3 S 0 = 20 T = 1 r = 10% X = 18 D = 0 Is there an arbitrage opportunity?

5.4 Lower Bound for European Call Option Prices; No Dividends c + Xe -rT  S 0 in fact, at maturity: if S>X, identical result if S<X, superior result consider 2 portfolios: a. buy a call and a ZCB worth X at T b. buy the stock

5.5 Lower Bound for European Call Option Prices; No Dividends c + Xe -rT  S 0 c  S 0 -Xe -rT c  * c 

5.6 arbitrage if c<3.71 TODAY sell the stock at 20, and with the proceeds: buy the call at 3 and a ZCB maturing in 1 year time for 16,28 today (i.e. 18=X) invest the diff=Y= ,28=0.72 in a ZCB maturing in 1 year AT MATURITY if S<18, you have (so better off than keeping the stock if S>18, you get: - S-X from the call (=S-18) - X from the ZCB (=18) from the residual sum up the terms and you will be happier than having kept the stock

5.7 Puts: An Arbitrage Opportunity? Suppose that p = 1 S 0 = 37 T = 0.5 r =5% X = 40 D = 0 Is there an arbitrage opportunity?

5.8 Lower Bound for European Put Prices; No Dividends p + S 0  Xe -rT p  Xe -rT - S 0 consider 2 portfolios: a. buy a put and a stock b. buy the ZCB worth X at maturity

5.9 Put-Call Parity; No Dividends Consider the following 2 portfolios: - Portfolio A: European call on a stock + PV of the strike price in cash - Portfolio B: European put on the stock + the stock Both are worth MAX( S T, X ) at the maturity of the options They must therefore be worth the same today - This means that c + Xe -rT = p + S 0

5.10 Arbitrage Opportunities Suppose that c = 3 S 0 = 31 T = 0.25 r = 10% X =30 D = 0 What are the arbitrage possibilities when p = 2.25 ? p = 1 ?

5.11 arbitrage c + Xe -rT = p + S * =p =p+31 p= if p=1, then TODAY: - sell the call at 3 and buy the put at 1. You have 2 euro remaining. - buy the stock at 31, by borrowing and using 0.77 from the options proceeds - invest 1.23 (out of the original 2) in a ZCB in 3 months time: - if S=33, for example, put=0, call=3. So you are at sell the stock at 33 and repay your debt. So you are at +2 - so you are at -1, but in reality you still have 1.23*exp(0.25*10%)

5.12 The Impact of Dividends on Lower Bounds to Option Prices

5.13 Effect of Variables on Option Pricing cpCP Variable S0S0 X T  r D ++ – – + – – –– + – + – +

5.14 American vs European Options An American option is worth at least as much as the corresponding European option C  c P  p

5.15 The Black-Scholes Model

5.16 The Stock Price Assumption Consider a stock whose price is S In a short period of time of length  t the change in then stock price is assumed to be normal with mean  Sdt and standard deviation  is expected return and  is volatility

5.17 The Lognormal Distribution

5.18 The Concepts Underlying Black-Scholes The option price & the stock price depend on the same underlying source of uncertainty We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate This leads to the Black-Scholes differential equation

of 3: The Derivation of the Black-Scholes Differential Equation

of 3: The Derivation of the Black-Scholes Differential Equation there are no stochastic terms inside

of 3: The Derivation of the Black-Scholes Differential Equation

5.22 Risk-Neutral Valuation The variable  does not appearin the Black- Scholes equation The equation is independent of all variables affected by risk preference The solution to the differential equation is therefore the same in a risk-free world as it is in the real world This leads to the principle of risk-neutral valuation

5.23 Applying Risk-Neutral Valuation 1. Assume that the expected return from the stock price is the risk-free rate 2. Calculate the expected payoff from the option 3. Discount at the risk-free rate

5.24 The Black-Scholes Formulas

5.25 Implied Volatility The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price The is a one-to-one correspondence between prices and implied volatilities Traders and brokers often quote implied volatilities rather than dollar prices

5.26 Dividends European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into Black-Scholes Only dividends with ex-dividend dates during life of option should be included The “dividend” should be the expected reduction in the stock price expected

5.27 European Options on Stocks Paying Continuous Dividends continued We can value European options by reducing the stock price to S 0 e –q T and then behaving as though there is no dividend

5.28 Formulas for European Options

5.29 The Foreign Interest Rate We denote the foreign interest rate by r f When a European company buys one unit of the foreign currency it has an investment of S 0 euro The return from investing at the foreign rate is r f S 0 euro This shows that the foreign currency provides a “dividend yield” at rate r f

5.30 Valuing European Currency Options A foreign currency is an asset that provides a continuous “dividend yield” equal to r f We can use the formula for an option on a stock paying a continuous dividend yield : Set S 0 = current exchange rate Set q = r ƒ

5.31 Formulas for European Currency Options