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OPTIONS - EXAMPLES

OPTIONS - EXAMPLES. SOME STRATEGIES. COVERED STRATEGIES : Take a position in the option and the underlying stock. SPREAD STRATEGIES : Take a position in 2 or more options of the same type (A spread ). COMBINATION STRATEGIES : Take a position in a mixture of calls and puts.

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OPTIONS - EXAMPLES

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  1. OPTIONS - EXAMPLES MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  2. SOME STRATEGIES • COVERED STRATEGIES: Take a position in the option and the underlying stock. • SPREAD STRATEGIES: Take a position in 2 or more options of the same type (A spread). • COMBINATION STRATEGIES: Take a position in a mixture of calls and puts. MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  3. TYPES OF STRATEGIES • STANDARD SYMBOLS: • C = current call price, P = current put price • S0 = current stock price, ST = stock price at time T • T = time to maturity • X = exercise price (or K in some books) • P = profit from strategy • STAKES: • NC = number of calls • NP = number of puts • NS = number of shares of stock MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  4. Types of Strategies • These symbols imply the following: • NC or NP or NS > 0 implies buying (going long) • NC or NP or NS < 0 implies selling (going short) • Recall the PROFIT EQUATIONS • Profit equation for calls held to expiration • P = NC[Max(0,ST - X) – Cexp(rT)] • For buyer of one call (NC = 1) this implies • P = Max(0,ST - X) - Cexp(rT) • For seller of one call (NC = -1) this implies • P = -Max(0,ST - X) + Cexp(rT) MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  5. Types of Strategies The Profit Equations (continued) • Profitequation for puts held to expiration • P= NP[Max(0,X - ST) - Pexp(rT)] • For buyer of one put (NP = 1) this implies P= Max(0,X - ST) - Pexp(rT) • For seller of one put (NP = -1) this implies P= -Max(0,X - ST) + Pexp(rT) MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  6. Types of Strategies • The Profit Equations (continued) • Profit equation for stock • P = NS[ST - S0] • For buyer of one share (NS = 1) this implies P= ST - S0 • For short seller of one share (NS = -1) this implies P= -ST + S0 MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  7. Positions in an Option & the Underlying Profit Profit K ST ST K (a) (b) Profit Profit K K ST ST (c) (d) MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  8. Bull Spread Using Calls Bull Spread Using Calls: Buying a call option on a stock with a particular strike price and selling a call option on the same stock with a higher strike price. Payoff from a Bull Spread: MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  9. Bull Spread Using Calls Ex: An investor buys $3 a call with a strike price of $30 and sells for $1 a call with a strike price of $35. Payoff from a Bull Spread: MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  10. Bull Spread Using Calls Profit ST K1 • K2 MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  11. Profit K1 K2 ST Bull Spread Using Puts MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  12. Profit K1 K2 ST Bear Spread Using Puts-buying one put with a strike price of K2 and selling one put with a strike price of K1 MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  13. Bear Spread Using Calls Bear Spread: Buying a call option on a stock with a particular strike price and selling a call option on the same stock with a lower strike price. MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  14. Bear Spread Using Calls Example: An investor buys a call for $1 with a strike price of $35 and sells for $3 a call with a strike price of $30. MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  15. Bear Spread Using Calls Profit K1 K2 ST MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  16. Box Spread • A combination of a bull call spread and a bear put spread • If all options are European a box spread is worth the present value of the difference between the strike prices • If they are American this is not necessarily so. MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  17. Butterfly Spread Using Calls • Butterfly Spread: buying a call option with a relative low strike price, K1,, buying a call option with a relative high strike price. K3, and selling two call options with a strike price halfway in between, K2. MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  18. Butterfly Spread Using Calls • Example: Call option prices on a $61 stock are: $10 for a $55 strike, $7 for a $60 strike, and $5 for a $65 strike. The investor could create a butterfly spread by buying one call with $55 strike price, buying a call with a $65 strike price, and selling two calls with a $60 strike price. MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  19. Butterfly Spread Using Calls Profit K1 K2 K3 ST MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  20. Butterfly Spread Using Puts Profit K1 K2 K3 ST MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  21. Calendar Spread Using Calls Profit ST K MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  22. Calendar Spread Using Puts Profit ST K MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  23. A Straddle Combination Straddle: Buying a call and a put with the same strike price and expiration Date. MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  24. A Straddle Combination Example: An investor buying a call and a put with a strike price of $70 and an expiration date in 3 months. Suppose the call costs $4 and the put $3. MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  25. A Straddle Combination Profit K ST MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  26. Strip & StrapStrip: combining one long call with two long putsStrap: combining two long calls with one long put Profit Profit K ST K ST Strip Strap MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  27. A Strangle Combinationbuying one call with a strike price of K2 and buying one put with a strike price of K1 Profit K1 K2 ST MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  28. BINOMIAL MODELS - EXAMPLES • A stock price is currently $20 • In three months it will be either $22 or $18 Stock Price = $22 Stock price = $20 Stock Price = $18 MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  29. A Call Option A 3-month call option on the stock has a strike price of 21. Stock Price = $22 Option Price = $1 Stock price = $20 Option Price=? Stock Price = $18 Option Price = $0 MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  30. 22D – 1 18D Setting Up a Riskless Portfolio • Consider the Portfolio: long D shares short 1 call option • Portfolio is riskless when 22D – 1 = 18D or D = 0.25 MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  31. Valuing the Portfolio(Risk-Free Rate is 12%) • The riskless portfolio is: long 0.25 shares short 1 call option • The value of the portfolio in 3 months is 22 ´ 0.25 – 1 = 4.50 • The value of the portfolio today is 4.5e – 0.12´0.25 = 4.3670 MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  32. Valuing the Option • The portfolio that is long 0.25 shares short 1 option is worth 4.367 • The value of the shares is 5.000 (= 0.25 ´ 20 ) • The value of the option is therefore 0.633 (= 5.000 – 4.367 ) MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  33. Generalization A derivative lasts for time T and is dependent on a stock S(1+a)=Su ƒu S ƒ S(1-a)=Sd ƒd MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  34. Generalization(continued) • Consider the portfolio that is long D shares and short 1 derivative • The portfolio is riskless when SuD– ƒu= SdD – ƒd or SuD – ƒu SdD – ƒd MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  35. Generalization(continued) • Value of the portfolio at time Tis SuD – ƒu • Value of the portfolio today is (SuD – ƒu )e–rT • Another expression for the portfolio value today is SD – f • Hence ƒ = SD – (SuD – ƒu)e–rT MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  36. Generalization(continued) • Substituting for D we obtain ƒ = [ pƒu + (1 – p )ƒd ]e–rT where MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  37. Risk-Neutral Valuation ƒ = [ p ƒu + (1 – p )ƒd ]e-rT The variables p and (1– p ) can be interpreted as the risk-neutral probabilities of up and down movements The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate Su ƒu S ƒ Sd ƒd p (1– p ) MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  38. Original Example Revisited Su = 22 ƒu = 1 • Since p is a risk-neutral probability 20e0.12 ´0.25 = 22p + 18(1 – p ); p = 0.6523 p S ƒ Sd = 18 ƒd = 0 (1– p ) MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  39. Su = 22 ƒu = 1 0.6523 S ƒ Sd = 18 ƒd = 0 0.3477 Valuing the Option The value of the option is e–0.12´0.25 [0.6523´1 + 0.3477´0] = 0.633 MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  40. Estimating p One way of matching the volatility is to set where s is the volatility andDt is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  41. 24.2 22 19.8 20 18 16.2 A Two-Step Example • Each time step is 3 months • K=21, r=12% MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  42. Valuing a Call Option 24.2 3.2 D • Value at node B = e–0.12´0.25(0.6523´3.2 + 0.3477´0) = 2.0257 • Value at node A = e–0.12´0.25(0.6523´2.0257 + 0.3477´0) = 1.2823 22 B 19.8 0.0 20 1.2823 2.0257 A E 18 C 0.0 16.2 0.0 F MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  43. 72 0 D 60 B 48 4 50 4.1923 1.4147 A E 40 C 9.4636 32 20 F A Put Option Example; K=52 K = 52, Dt = 1yr r = 5% MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  44. Behaviorof Stock Prices MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  45. Categorization of Stochastic Processes • Discrete time; discrete variable • Discrete time; continuous variable • Continuous time; discrete variable • Continuous time; continuous variable MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  46. Modeling Stock Prices • We can use any of the four types of stochastic processes to model stock prices • The continuous time, continuous variable process proves to be the most useful for the purposes of valuing derivative securities MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  47. Markov Processes • In a Markov process future movements in a variable depend only on where we are, not the history of how we got where we are • We will assume that stock prices follow Markov processes MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  48. Weak-Form Market Efficiency • The assertion is that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices. In other words technical analysis does not work. • A Markov process for stock prices is clearly consistent with weak-form market efficiency MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  49. Example of a Discrete Time Continuous Variable Model • A stock price is currently at $40 • At the end of 1 year it is considered that it will have a probability distribution of(40,10) where f(m,s) is a normal distributionwith mean m and standard deviation s. MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

  50. Questions • What is the probability distribution of the stock price at the end of 2 years? • ½ years? • ¼ years? • Dt years? Taking limits we have defined a continuous variable, continuous time process MILJAN KNEŽEVIĆ, BANJA LUKA - AUGUST 2010

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