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Call Option Example • Suppose that at the February expiry date we hold one April $6 ABC Inc. call (the exercise price is $6). If the price of a share in ABC Inc. is S = $7, would the call be exercised? A: Yes (buy @ $6 and sell @ $7) Would it be exercised if S = $5.50? A: No (buy @ $6 ?? No way) E.5 Basic Call Option Pricing Relationships • At expiry, an American call option is worth the same as a European option with the same characteristics. • If the call is in-the-money, it is worth ST - X. • If the call is out-of-the-money, it is worthless. For a call, the intrinsic value is given by: CaT = CeT = Max[ST - X, 0] • Where ST is the value of the stock at expiry (time T) X is the exercise price. CaT is the value of an American call at expiry CeT is the value of a European call at expiry E.6 Call Option Payoffs 6 4 Buy a call Option payoffs ($) 2 0 0 1 2 3 4 5 6 7 8 9 10 -2 Stock price ($) Write a call -4 -6 Exercise price = $6 E.7 Call Option Profits 6 Breakeven = Exercise + Premium 4 Buy a call Option profits ($) 2 0 0 1 2 3 4 5 6 7 8 9 10 -2 Stock price ($) Write a call -4 -6 Exercise price = $6; option premium = $1 E.8 Basic Put Option Pricing Relationships • At expiry, an American put option is worth the same as a European option with the same characteristics. • If the put is in-the-money, it is worth X - ST. • If the put is out-of-the-money, it is worthless. For a put, the intrinsic value is given by: PaT = PeT = Max[X - ST, 0] E.9 Put Option Payoffs 6 Buy a put 4 Option payoffs ($) 2 0 0 1 2 3 4 5 6 7 8 9 10 -2 Stock price ($) -4 Write a put -6 Exercise price = $6 E.10 Option profits ($) Put Option Profits 6 Breakeven = Exercise + Premium 4 2 Write a put 1 0 0 1 2 3 4 5 6 7 8 9 10 -1 Buy a put -2 Stock price ($) -4 -6 Exercise price = $6; option premium = $1 E.11 More Options Relationships & Definitions • Option premium: this is the value of the option • Speculative Value OR Time Value – The difference between the option premium and the intrinsic value of the option. This is the part of the option’s value that derives the possibility of future favorable movements in the stock price. Option Intrinsic Time Value Premium = Value + E.12 Put-Call Parity c0 p0 S 0 PV ( X ) where c0 = value of call p0 = value of put S0 = value of share PV(X) = exercise price discounted at risk-free rate • Note This relationship ONLY holds for European options but American options it becomes as follows: S 0 X c 0 p 0 S 0 PV ( X ) E.13 Valuing Options • The last section concerned itself with the value of an option at expiry. • This section considers the value of an option prior to the expiration date. • A much more interesting question. E.14 Option Value Determinants Call Put 1. Stock price + – 2. Exercise price – + 3. Interest rate + – 4. Volatility in the stock price + + 5. Expiration date + + The value of a call option C0 must fall within max (S0 – E, 0) < C0 < S0. The precise position will depend on these factors. E.15 Binomial Pricing Models • Dynamic tracking strategy: – used for derivatives whose future payoff is not a linear function of payoff of the underlying asset • Two types: – one-period binomial trees • tracking-portfolio method • risk-neutral valuation method – multiperiod binomial trees E.16 Binomial Process (One Period) • At each point (node), there are only two possible values – up – down £620 (up) £570 £480 (down) • Good approximation E.17 The Tracking-Portfolio Method • Identify the tracking portfolio – units (shares) of the underlying asset – £’s in the risk-free bond • Find the current value of the option E.18 Identifying the Tracking Portfolio Su S = value of underlying today S Su = value of underlying in up state Sd Sd = value of underlying in down state (1+rf) rf = risk-free rate (1+rf) E.19 Identifying the Tracking Portfolio t=1 t=0 Up Down Tracking S+ Su+(1+rf) Sd+(1+rf) portfolio = = Derivative ? Vu Vd System of two equations with two unknowns E.20 Identifying the Tracking Portfolio Su (1 rf ) Vu (1) S d (1 rf ) Vd (2) Subtracting (2) from (1) Su S d Vu Vd Vu Vd spread option prices (3) Su S d spread share prices Vu S u (4) 1 rf E.21 Finding the Current Value of the Option • If arbitrage doesn’t exist: V S (5) Example: – what is the current value of a call option with a strike price (X) of 90? E.22 Valuing a Call Option I Su=120 Vu=30 = 120 - 90 S=100 rf=0.10 V=? Sd=80 Vd=0 t=1 t=0 Up Down Tracking 100+ 120+1.1 80 +1.1 portfolio Derivative V? 30 0 E.23 Valuing a Call Option I Vu Vd 30 0 3 Su S d 120 80 4 3 30 120 Vu Su 4 54 .55 1 rf 1.1 Under no arbitrage: 3 V S 100 (54.55) 20.45 4 E.24 Two Remarks • The valuation process does not refer to the probabilities of the up and down states – this information is already reflected in the current price of the underlying • The process does not refer to the risk preferences of the investor E.25 The Risk-Neutral Valuation Method Two-step procedure for the valuation of derivatives: 1. determine the risk neutral probabilities given that investors are risk neutral 2. multiply each future value of the option by the corresponding risk-neutral probability and discount the sum of these values by the risk-free rate E.26 Valuing a Call Option II Su=120 Vu=30 S=100 rf=0.10 V=? Sd=80 Vd=0 1. Determine the risk neutral probabilities 120 80 1.1 (1 ) 100 100 0.75 E.27 Valuing a Call Option II 2. Multiply the future values of the option by their probabilities and discount at the risk-free rate – we determine the expected future value of the option 0.75 30 (1 0.75) 0 22.50 – we determine the present value of the option 22.50 20.45 1.1 E.28 The Risk-Neutral Valuation Method General formula: – the risk-neutral probability is the probability that makes the future expected return of an asset equal to the risk-free rate (1 r f ) d u (1 )d 1 r f ud where u= 1 + return of the underlying in the up state d= 1 + return of the underlying in the down state E.29 Multiperiod Binomial Trees • In real life it is more likely to have more than two prices for the security over a long period say of 3 months, therefore, it is better to apply to model over shorter periods of time. • Then, if we extend the model for n intervals, we can then work out the possible movements of the security in the up-state and down-state as follows: / n u e AND d = 1/u where n is the number of intervals in a year, is the S.D of the annualized stock return E.30 The Black-Scholes Model • Discrete valuation models: – finite number of future outcomes for the share price – finite number of periods • Continuous-time models: – infinite number of future outcomes for the share price – infinite number of periods • A binomial process with a large number of short periods can approximate the continuous-time process. E.31 The Black-Scholes Model C0 S N(d1 ) Ee rT N(d 2 ) Where C0 = the value of a European option at time t = 0 r = the risk-free interest rate. σ2 N(d) = Probability that a ln( S / E ) (r )T standardized, normally d1 2 T distributed, random variable will be less than or equal to d 2 d1 T d. The Black-Scholes Model allows us to value options in the real world just as we have done in the 2-state world. E.32 The Black-Scholes Model Find the value of a six-month call option on the Microsoft with an exercise price of $150 The current value of a share of Microsoft is $160 The interest rate available in the U.S. is r = 5%. The option maturity is 6 months (half of a year). The volatility of the underlying asset is 30% per annum. Before we start, note that the intrinsic value of the option is $10—our answer must be at least that amount. E.33 The Black-Scholes Model First calculate d1 and d2 ln( S / E ) (r .5σ 2 )T d1 T ln( 160 / 150) (.05 .5(0.30) 2 ).5 d1 0.5282 0.30 .5 Then, d 2 d1 T 0.52815 0.30 .5 0.31602 E.34 The Black-Scholes Model C0 S N(d1 ) Ee rT N(d 2 ) d1 0.5282 N(d1) = N(0.52815) = 0.7013 d 2 0.31602 N(d2) = N(0.31602) = 0.62401 C0 $160 0.7013 150e .05.5 0.62401 C0 $20.92 E.35 Stocks and Bonds as Options • Levered Equity is a Call Option. – The underlying asset comprise the assets of the firm. – The strike price is the payoff of the bond. • If at the maturity of their debt, the assets of the firm are greater in value than the debt, the shareholders have an in-the-money call, they will pay the bondholders and “call in” the assets of the firm. • If at the maturity of the debt the shareholders have an out-of-the-money call, they will not pay the bondholders (i.e. the shareholders will declare bankruptcy) and let the call expire. E.37 Stocks and Bonds as Options • Levered Equity is a Put Option. – The underlying asset comprise the assets of the firm. – The strike price is the payoff of the bond. • If at the maturity of their debt, the assets of the firm are less in value than the debt, shareholders have an in-the-money put. • They will put the firm to the bondholders. • If at the maturity of the debt the shareholders have an out-of-the-money put, they will not exercise the option (i.e. NOT declare bankruptcy) and let the put expire. E.38

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