Lecture 16. Options and option pricing. Lecture 16 1 / 22

Transcription

1 Lecture 16 Options and option pricing Lecture 16 1 / 22

2 Introduction One of the most, perhaps the most, important family of derivatives are the options. Lecture 16 2 / 22

3 Introduction One of the most, perhaps the most, important family of derivatives are the options. In order to value them we need a stochastic model of the future behavior of the underlying. Lecture 16 2 / 22

4 Options (1) An option is a derivative in which there is some sort of choice for the owner of the derivative. Lecture 16 3 / 22

5 Options (1) An option is a derivative in which there is some sort of choice for the owner of the derivative. The two most comman types of options are the call option and the put option. Lecture 16 3 / 22

6 Options (1) An option is a derivative in which there is some sort of choice for the owner of the derivative. The two most comman types of options are the call option and the put option. Like futures, these type of derivatives are traded on many exchanges around the world. Lecture 16 3 / 22

7 Options (2) Definition A call option is the right, but not the obligation, of the owner of the option to buy an asset at a future time T for an amount K. Lecture 16 4 / 22

8 Options (2) Definition A call option is the right, but not the obligation, of the owner of the option to buy an asset at a future time T for an amount K. A put option is the right, but not the obligation, of the owner of the option to sell an asset at a future time T for an amount K. Lecture 16 4 / 22

9 Options (2) Definition A call option is the right, but not the obligation, of the owner of the option to buy an asset at a future time T for an amount K. A put option is the right, but not the obligation, of the owner of the option to sell an asset at a future time T for an amount K. Here K is known as the strike price and T as the exercise time or maturity time. Lecture 16 4 / 22

10 Options (3) The payoff at time T of a call option is given by max(s T K, 0). Lecture 16 5 / 22

11 Options (3) The payoff at time T of a call option is given by max(s T K, 0). The payoff at time T of a put option is given by max(k S T, 0). Lecture 16 5 / 22

12 Options (4) There are different types of options with respect to the exercise time. European options Lecture 16 6 / 22

13 Options (4) There are different types of options with respect to the exercise time. European options American options Lecture 16 6 / 22

14 Options (4) There are different types of options with respect to the exercise time. European options American options Bermudan options Lecture 16 6 / 22

15 Options (4) There are different types of options with respect to the exercise time. European options American options Bermudan options Other: Parisian, Canadian, Russian,... Lecture 16 6 / 22

16 Options (5) Call and put options are examples of simple derivatives Lecture 16 7 / 22

17 Options (5) Call and put options are examples of simple derivatives In this case the derivative s payoff X is a function F of the underlying S T Lecture 16 7 / 22

18 Options (5) Call and put options are examples of simple derivatives In this case the derivative s payoff X is a function F of the underlying S T : X = F (S T ). Lecture 16 7 / 22

19 Options (5) Call and put options are examples of simple derivatives In this case the derivative s payoff X is a function F of the underlying S T : X = F (S T ). There are also path-dependent derivatives: X = G((S t, 0 t T )). In this case the whole (or part) of path of the underlying is needed to know on order to calculate the payoff X. Lecture 16 7 / 22

20 Options (6) Example An up-and-in contract is a derivative in which the value of the underlying must reach a given level L in order to be active. Lecture 16 8 / 22

21 Options (6) Example An up-and-in contract is a derivative in which the value of the underlying must reach a given level L in order to be active. An up-and-in call option has payoff X = max(s T K, 0)1 ( S t L for some t [0, T ] ). Lecture 16 8 / 22

22 Options (6) Example An up-and-in contract is a derivative in which the value of the underlying must reach a given level L in order to be active. An up-and-in call option has payoff X = max(s T K, 0)1 ( S t L for some t [0, T ] ). There are also up-and-out, down-and-in and down-and-out contracts. Lecture 16 8 / 22

23 Option strategies (1) By combining basic options we can create new payoffs. Lecture 16 9 / 22

24 Option strategies (1) By combining basic options we can create new payoffs. This can be used in order to help an investor to get the payoff he wants. Lecture 16 9 / 22

25 Option strategies (1) By combining basic options we can create new payoffs. This can be used in order to help an investor to get the payoff he wants. Assume that you belive that a stock is going to move from its current price, but you do not know in which direction. Lecture 16 9 / 22

26 Option strategies (1) By combining basic options we can create new payoffs. This can be used in order to help an investor to get the payoff he wants. Assume that you belive that a stock is going to move from its current price, but you do not know in which direction. If this is the case, you can buy a strangle. Lecture 16 9 / 22

27 Option strategies (1) By combining basic options we can create new payoffs. This can be used in order to help an investor to get the payoff he wants. Assume that you belive that a stock is going to move from its current price, but you do not know in which direction. If this is the case, you can buy a strangle. A strangle is a a sum of a put option with strike price K 1 and a call option with strike price K 2 > K 1. Lecture 16 9 / 22

28 Option strategies (2) Other examples of option combinations include A straddle: this is a strangle with K 1 = K 2 Lecture / 22

29 Option strategies (2) Other examples of option combinations include A straddle: this is a strangle with K 1 = K 2 A bull spread: buy a call option with strike price K 1 and sell a call option with strike price K 2 > K 1. Lecture / 22

30 Option strategies (2) Other examples of option combinations include A straddle: this is a strangle with K 1 = K 2 A bull spread: buy a call option with strike price K 1 and sell a call option with strike price K 2 > K 1. A bear spread: sell a put option with strike price K 1 and buy a put with strike price K 2 > K 1. Lecture / 22

31 Option strategies (2) Other examples of option combinations include A straddle: this is a strangle with K 1 = K 2 A bull spread: buy a call option with strike price K 1 and sell a call option with strike price K 2 > K 1. A bear spread: sell a put option with strike price K 1 and buy a put with strike price K 2 > K 1. A butterfly: buy 1 call with strike price K a, sell 2 calls with strike price K and buy 1 call with strike price K + a. Lecture / 22

32 The put-call parity If we buy 1 call option and sell 1 put option both having the same strike price K and maturity time T, then the resulting payoff at T will be S T K. Lecture / 22

33 The put-call parity If we buy 1 call option and sell 1 put option both having the same strike price K and maturity time T, then the resulting payoff at T will be This is true since S T K. max(s T K, 0) max(k S T, 0) = S T K. Lecture / 22

34 The put-call parity If we buy 1 call option and sell 1 put option both having the same strike price K and maturity time T, then the resulting payoff at T will be This is true since S T K. max(s T K, 0) max(k S T, 0) = S T K. It follows from linear pricing that Price of call at 0 Price of put at 0 = S 0 K d(0, T ). Lecture / 22

35 The put-call parity If we buy 1 call option and sell 1 put option both having the same strike price K and maturity time T, then the resulting payoff at T will be This is true since S T K. max(s T K, 0) max(k S T, 0) = S T K. It follows from linear pricing that Price of call at 0 Price of put at 0 = S 0 K d(0, T ). This relation is called the put-call parity. Lecture / 22

36 Pricing options In order the find the price (or value) of an option we need to construct a stochastic model. Lecture / 22

37 Pricing options In order the find the price (or value) of an option we need to construct a stochastic model. Two popular models are The binomial model Lecture / 22

38 Pricing options In order the find the price (or value) of an option we need to construct a stochastic model. Two popular models are The binomial model The Black-Scholes(-Samuelson) model Lecture / 22

39 Single-period option pricing (1) This is the binomial model with only one time period. Lecture / 22

40 Single-period option pricing (1) This is the binomial model with only one time period. The setup is as follows. Two times: today (t = 0) and tomorrow (t = 1). Lecture / 22

41 Single-period option pricing (1) This is the binomial model with only one time period. The setup is as follows. Two times: today (t = 0) and tomorrow (t = 1). Two outcomes: up and down. Lecture / 22

42 Single-period option pricing (1) This is the binomial model with only one time period. The setup is as follows. Two times: today (t = 0) and tomorrow (t = 1). Two outcomes: up and down. The probability of up and down is p and 1 p respectively. Lecture / 22

43 Single-period option pricing (1) This is the binomial model with only one time period. The setup is as follows. Two times: today (t = 0) and tomorrow (t = 1). Two outcomes: up and down. The probability of up and down is p and 1 p respectively. Two assets: one riskless and one risky. Lecture / 22

44 Single-period option pricing (2) The riskless asset has price 1 today and price 1 + r f tomorrow. Lecture / 22

45 Single-period option pricing (2) The riskless asset has price 1 today and price 1 + r f tomorrow r f 1 + r f t = 0 t = 1 Lecture / 22

46 Single-period option pricing (3) The risky asset has price S today and either us or ds tomorrow. Lecture / 22

47 Single-period option pricing (3) The risky asset has price S today and either us or ds tomorrow. S us ds t = 0 t = 1 Lecture / 22

48 Single-period option pricing (3) The risky asset has price S today and either us or ds tomorrow. S us ds t = 0 t = 1 We assume that d < u. Lecture / 22

49 Single-period option pricing (4) In order to rule out arbitrage opportunities we must have d < 1 + r f < u Lecture / 22

50 Single-period option pricing (4) In order to rule out arbitrage opportunities we must have d < 1 + r f < u Note the strict inequalities here. Lecture / 22

51 Single-period option pricing (4) In order to rule out arbitrage opportunities we must have d < 1 + r f < u Note the strict inequalities here. The reason for this is that if 1 + r f u, then we can create an arbitrage opprortunity by selling a stock and putting the money in the bank. Lecture / 22

52 Single-period option pricing (4) In order to rule out arbitrage opportunities we must have d < 1 + r f < u Note the strict inequalities here. The reason for this is that if 1 + r f u, then we can create an arbitrage opprortunity by selling a stock and putting the money in the bank. If 1 + r f d, then borrowing money in the bank and buying stocks for it will create an arbitrage opprtunity Lecture / 22

53 Single-period option pricing (5) Now let us try to find the price today of a derivative that has payoff C u in the up state and C d in the down state. Lecture / 22

54 Single-period option pricing (5) Now let us try to find the price today of a derivative that has payoff C u in the up state and C d in the down state. To solve this problem we create a replicating portfolio. Lecture / 22

55 Single-period option pricing (5) Now let us try to find the price today of a derivative that has payoff C u in the up state and C d in the down state. To solve this problem we create a replicating portfolio. Assume that we can construct a portfolio of x number of stocks and y units of money in the bank. Lecture / 22

56 Single-period option pricing (5) Now let us try to find the price today of a derivative that has payoff C u in the up state and C d in the down state. To solve this problem we create a replicating portfolio. Assume that we can construct a portfolio of x number of stocks and y units of money in the bank. This portfolio will have the following payoff: In the up state: x us + y (1 + r f ) In the down state: x ds + y (1 + r f ). Lecture / 22

57 Single-period option pricing (6) To find the replicating portfolio, we solve the following linear system of equations: { xus + y(1 + rf ) = C u The solution is given by xds + y(1 + r f ) = C d. { x = C u C d y = (u d)s uc d dc u (1+r f )(u d) Lecture / 22

58 Single-period option pricing (6) To find the replicating portfolio, we solve the following linear system of equations: { xus + y(1 + rf ) = C u The solution is given by xds + y(1 + r f ) = C d. { x = C u C d y = (u d)s uc d dc u (1+r f )(u d) Hence, the value, or price, of the derivative with payoff (C u, C d ) is given by V = x S + y Lecture / 22

59 Single-period option pricing (6) To find the replicating portfolio, we solve the following linear system of equations: { xus + y(1 + rf ) = C u The solution is given by xds + y(1 + r f ) = C d. { x = C u C d y = (u d)s uc d dc u (1+r f )(u d) Hence, the value, or price, of the derivative with payoff (C u, C d ) is given by V = x S + y = C u C d u d + uc d dc u (1 + r f )(u d) Lecture / 22

60 Single-period option pricing (7) Note that There exists a unique value of every derivative Lecture / 22

61 Single-period option pricing (7) Note that There exists a unique value of every derivative The probabilities of up and down moves does not enter the valuation formula. Lecture / 22

62 Single-period option pricing (7) Note that There exists a unique value of every derivative The probabilities of up and down moves does not enter the valuation formula. Let q = 1 + r f d. u d It follows from the condition d < 1 + r f < u that q (0, 1). Lecture / 22

63 Single-period option pricing (8) Using this q we can write V = C u C d u d + uc d dc u (1 + r f )(u d) Lecture / 22

64 Single-period option pricing (8) Using this q we can write V = C u C d u d + uc d dc u (1 + r f )(u d) 1 = (1 + r f )C u (1 + r f )C d + uc d dc u 1 + r f u d Lecture / 22

65 Single-period option pricing (8) Using this q we can write V = C u C d u d + uc d dc u (1 + r f )(u d) 1 = (1 + r f )C u (1 + r f )C d + uc d dc u 1 + r f u d [ rf d = C u + u (1 + r ] f ) C d 1 + r f u d u d Lecture / 22

66 Single-period option pricing (8) Using this q we can write V = C u C d u d + uc d dc u (1 + r f )(u d) 1 = (1 + r f )C u (1 + r f )C d + uc d dc u 1 + r f u d [ rf d = C u + u (1 + r ] f ) C d 1 + r f u d u d 1 = (qc u + (1 q)c d ) 1 + r f Lecture / 22

67 Single-period option pricing (8) Using this q we can write V = C u C d u d + uc d dc u (1 + r f )(u d) 1 = (1 + r f )C u (1 + r f )C d + uc d dc u 1 + r f u d [ rf d = C u + u (1 + r ] f ) C d 1 + r f u d u d 1 = (qc u + (1 q)c d ) 1 + r f 1 = Ê(C). 1 + r f Lecture / 22

68 Single-period option pricing (8) Using this q we can write V = C u C d u d + uc d dc u (1 + r f )(u d) 1 = (1 + r f )C u (1 + r f )C d + uc d dc u 1 + r f u d [ rf d = C u + u (1 + r ] f ) C d 1 + r f u d u d 1 = (qc u + (1 q)c d ) 1 + r f 1 = Ê(C). 1 + r f Again, Ê means expected value with respect to the risk-neutral (q, 1 q)-probabilities. Lecture / 22

69 Multiperiod models (1) Now add more time steps, and assume the standard lattice (=recombining tree) model with independent ups and downs. Lecture / 22

70 Multiperiod models (1) Now add more time steps, and assume the standard lattice (=recombining tree) model with independent ups and downs. In this case we can think of the model as a lot af one-period models glued together. Lecture / 22

71 Multiperiod models (2) Using risk-neutral probabilities the random variable S T can be written S T = S 0 u X d T X Lecture / 22

72 Multiperiod models (2) Using risk-neutral probabilities the random variable S T can be written S T = S 0 u X d T X, where X Bin(T, q). Lecture / 22

73 Multiperiod models (2) Using risk-neutral probabilities the random variable S T can be written where S T = S 0 u X d T X, X Bin(T, q). Hence, the value at time 0 of a European derivative with payoff function F and exercise time T is V = 1 (1 + r f ) T E Q [F (S T )] Lecture / 22

74 Multiperiod models (2) Using risk-neutral probabilities the random variable S T can be written where S T = S 0 u X d T X, X Bin(T, q). Hence, the value at time 0 of a European derivative with payoff function F and exercise time T is V = = 1 (1 + r f ) T E Q [F (S T )] 1 (1 + r f ) T T F k=0 ( S 0 u k d T k) ( ) T q k (1 q) T k. k Lecture / 22