Finance Financial Options Professeur André Farber

June 11, 2015 DESG 09 Options |2 Résumé des épisodes précédents Objectif des décisions financières: création de valeur. Mesure: Valeur actuelle, actualisation des cash flows attendus. 1 période, certitude : 1 période, incertitude: 2 approches Calculer les prix d’état: v u, v d Calculer la rentabilité attendue:

June 11, 2015 DESG 09 Options |3 Aujourd’hui Objectives for this session: –1. Define options (calls and puts) –2. Analyze terminal payoff –3. Define basic strategies –4. Binomial option pricing model –5. Black Scholes formula

June 11, 2015 DESG 09 Options |4 Définitions Un call ou option d’achat (put ou option de vente) est un contract donne: –le droit –d’acheter (vendre) –un actif (action, obligation, portefeuille,..) –à ou avant une date future (échéance) à : option "Européenne" avant: option "Américaine" à un prix fixé d’avance (le prix d’exercice) L’acheteur de l’option paie une prime au vendeur.

June 11, 2015 DESG 09 Options |5 Valeur à l’échéance: Call européen Exercer l’option si, à l’échéance: Prix de l’action > Prix d’exercice S T > K Valeur du call à l’échéance: C T = S T - K if S T > K sinon : C T = 0 C T = MAX(0, S T - K)

June 11, 2015 DESG 09 Options |6 Valeur à l’échéance: Put européen Exercer l’option si, à l’échéance: Prix de l’action < Prix d’exercice S T < K Valeur du put à l’échéance: P T = K - S T si S T < K sinon: P T = 0 P T = MAX(0, K- S T )

June 11, 2015 DESG 09 Options |7 The Put-Call Parity relation A relationship between European put and call prices on the same stock Compare 2 strategies: Strategy 1. Buy 1 share + 1 put At maturity T:S T K Share valueS T S T Put value(K - S T )0 Total valueKS T Put = insurance contract

June 11, 2015 DESG 09 Options |8 Put-Call Parity (2) Consider an alternative strategy: Strategy 2: Buy call, invest PV(K) At maturity T:S T K Call value0S T - K InvesmtKK Total valueKS T At maturity, both strategies lead to the same terminal value Stock + Put = Call + Exercise price

June 11, 2015 DESG 09 Options |9 Put-Call Parity (3) Two equivalent strategies should have the same cost S + P = C + PV(K) where S current stock price P current put value C current call value PV(K) present value of the striking price This is the put-call parity relation Another presentation of the same relation: C = S + P - PV(K) A call is equivalent to a purchase of stock and a put financed by borrowing the PV(K)

June 11, 2015 DESG 09 Options |10 Valuing option contracts The intuition behind the option pricing formulas can be introduced in a two-state option model (binomial model). Let S be the current price of a non-dividend paying stock. Suppose that, over a period of time (say 6 months), the stock price can either increase (to uS, u>1) or decrease (to dS, d<1). Consider a K = 100 call with 1-period to maturity. uS = 125 C u = 25 S = 100 C=? dS = 80 C d = 0

June 11, 2015 DESG 09 Options |11 Key idea underlying option pricing models It is possible to create a synthetic call that replicates the future value of the call option as follow:  Buy Delta shares  Borrow B at the riskless rate r f (5% per annum) Choose Delta and B so that the future value of this portfolio is equal to the value of the call option.  Delta uS - (1+r f  t) B = C u Delta 125 – B = 25  Delta dS - (1+r f  t) B = C d Delta 80 – B = 0 (  t is the length of the time period (in years) e.g. : 6-month means  t=0.5)

June 11, 2015 DESG 09 Options |12 No arbitrage condition In a perfect capital market, the value of the call should then be equal to the value of its synthetic reproduction, otherwise arbitrage would be possible: C = Delta  S - B This is the Black Scholes formula We now have 2 equations with 2 unknowns to solve.  Eq1  -  Eq2   Delta  ( ) = 25  Delta = Replace Delta by its value in  Eq2   B = Call value: C = Delta S - B =   C = 12.20

June 11, 2015 DESG 09 Options |13 Using state price The underlying logic is the same as the logic used to calculate state prices. Here, 2 states: Up (u) or down (d). State prices are: Therefore, the value of the call option is:

June 11, 2015 DESG 09 Options |14 Risk neutral pricing Define: As: 1>p>0 and p + (1-p) = 1 p and 1-p can be interpreted as probabilities

June 11, 2015 DESG 09 Options |15 Risk neutral pricing (2) p is the probability of a stock price increase in a "risk neutral world" where the expected return is equal to the risk free rate. In a risk neutral world : p  uS + (1-p)  dS = 1+r f  t p  C u + (1-p)  C d is the expected value of the call option one period later assuming risk neutrality The current value is obtained by discounting this expected value (in a risk neutral world) at the risk-free rate.

June 11, 2015 DESG 09 Options |16 Risk neutral pricing illustrated In our example,the possible returns are: + 25% if stock up - 20% if stock down In a risk-neutral world, the expected return for 6-month is 5%  0.5= 2.5% The risk-neutral probability should satisfy the equation: p  (+0.25%) + (1-p)  (-0.20%) = 2.5%  p = 0.50 The call value is then: C = 0.50  25 / = 12.20

June 11, 2015 DESG 09 Options |17 Making Investment Decisions: NPV rule NPV : a measure of value creation NPV = V-I with V=PV(Additional free cash flows) NPV rule: invest whenever NPV>0 Underlying assumptions: –one time choice that cannot be delayed –single roll of the dices on cash flows But: –delaying the investment might be an option –what about flexibility ?

June 11, 2015 DESG 09 Options |18 Real option (based on McDonald, R. Derivative Markets, 2ed. Pearson 2006 – Chap 17) Risk-free interest rate 6% Market risk premium 4% Project Investment 10 at time 0, 95 at time 1 Payoff time 1: C u = 120 Proba = 0.60 C d = 80 Proba = 0.40 Beta payoff = 1.25

June 11, 2015 DESG 09 Options |19 Valuing flexibility Suppose that the investment in year 1 is optional. Decision to invest is like a call option: invest if C 1 > I 1 The investment in year 0 is the price we pay to have the option to invest in year Invest  V 1 = 120 – 95 = Do not invest  V 1 = 0  NPV = – 10 = 1.389

June 11, 2015 DESG 09 Options |20 Volatility In the binomial model, u and d capture the volatility (the standard deviation of the return) of the underlying stock u and d are given by the following formulas: The value a call option, is a function of the following variables: 1. The current stock price S 2. The exercise price K 3. The time to expiration date T 4. The risk-free interest rate r 5. The volatility of the underlying asset σ

June 11, 2015 DESG 09 Options |21 Multi-period model: European option For European option, follow same procedure (1) Calculate, at maturity, - the different possible stock prices; - the corresponding values of the call option - the risk neutral probabilities (2) Calculate the expected call value in a neutral world (3) Discount at the risk-free rate

June 11, 2015 DESG 09 Options |22 An example: valuing a 1-year call option Same data as before: S=100, K=100, r f =5%, u =1.25, d=0.80 Call maturity = 1 year (2-period) Stock price evolution Risk-neutral proba.Call value t=0 t=1t= p² = p(1-p) = (1-p)² = Current call value : C = 0.25  56.25/ (1.025)² = 13.38

June 11, 2015 DESG 09 Options |23 Option values are increasing functions of volatility The value of a call or of a put option is in increasing function of volatility (for all other variable unchanged) Intuition: a larger volatility increases possibles gains without affecting loss (since the value of an option is never negative) Check: previous 1-period binomial example for different volatilities VolatilityudCP (S=100, K=100, r=5%,  t=0.5)

June 11, 2015 DESG 09 Options |24 Black-Scholes formula For European call on non dividend paying stocks The limiting case of the binomial model for  t very small C = S N(d 1 ) - PV(K) N(d 2 )  Delta B In BS: PV(K) present value of K (discounted at the risk-free rate) Delta = N(d 1 ) N(): cumulative probability of the standardized normal distribution B = PV(K) N(d 2 )

June 11, 2015 DESG 09 Options |25 Black-Scholes : numerical example 2 determinants of call value: “Moneyness” : S/PV(K) “Cumulative volatility” : Example: S = 100, K = 100, Maturity T = 4, Volatility σ = 30% r = 6% “Moneyness”= 100/(100/ ) = 100/79.2= Cumulative volatility = 30% x  4 = 60% d 1 = ln(1.2625)/0.6 + (0.5)(0.60) =0.688  N(d 1 ) = d 2 = ln(1.2625)/0.6 - (0.5)(0.60) =0.089  N(d 2 ) = C = (100) (0.754) – (79.20) (0.535) = 33.05

June 11, 2015 DESG 09 Options |26 Cumulative normal distribution This table shows values for N(x) for x  0. For x<0, N(x) = 1 – N(-x) Examples: N(1.22) = 0.889, N(-0.60) = 1 – N(0.60) = 1 – = In Excell, use Normsdist() function to obtain N(x)

June 11, 2015 DESG 09 Options |27 Black-Scholes illustrated