Lecture 18. More on option pricing. Lecture 18 1 / 21

Transcription

1 Lecture 18 More on option pricing Lecture 18 1 / 21

2 Introduction In this lecture we will see more applications of option pricing theory. Lecture 18 2 / 21

3 Greeks (1) The price f of a derivative depends on several parameters. Lecture 18 3 / 21

4 Greeks (1) The price f of a derivative depends on several parameters. The sensitivity of the price with respect to parameter is called a greek. Lecture 18 3 / 21

5 Greeks (1) The price f of a derivative depends on several parameters. The sensitivity of the price with respect to parameter is called a greek. Some of the most important greeks are = f S Lecture 18 3 / 21

6 Greeks (1) The price f of a derivative depends on several parameters. The sensitivity of the price with respect to parameter is called a greek. Some of the most important greeks are = f S Γ = 2 f S 2 Lecture 18 3 / 21

7 Greeks (1) The price f of a derivative depends on several parameters. The sensitivity of the price with respect to parameter is called a greek. Some of the most important greeks are = f S Γ = 2 f S 2 Θ = f t Lecture 18 3 / 21

8 Greeks (1) The price f of a derivative depends on several parameters. The sensitivity of the price with respect to parameter is called a greek. Some of the most important greeks are = f S Γ = 2 f S 2 Θ = f t V = f σ ( Vega ) Lecture 18 3 / 21

9 Greeks (2) The greeks are used in risk management. Lecture 18 4 / 21

10 Greeks (2) The greeks are used in risk management. Note that = f S, which means that x t = in the hedging portfolio of the derivative. Lecture 18 4 / 21

11 Greeks (2) The greeks are used in risk management. Note that = f S, which means that x t = in the hedging portfolio of the derivative. In the Black-Scholes model we have call = Φ(d 1 ), where again d 1 = ln(s/k) + (r f + σ 2 /2)(T t) σ. T t Lecture 18 4 / 21

12 Greeks (3) What is the Delta of a put? Lecture 18 5 / 21

13 Greeks (3) What is the Delta of a put? Use the put-call parity: Price of put + S = Price of call + Ke r f (T t). Lecture 18 5 / 21

14 Greeks (3) What is the Delta of a put? Use the put-call parity: Price of put + S = Price of call + Ke r f (T t). Taking / S on the LHS and the RHS yields put + 1 = call Lecture 18 5 / 21

15 Greeks (3) What is the Delta of a put? Use the put-call parity: Price of put + S = Price of call + Ke r f (T t). Taking / S on the LHS and the RHS yields put + 1 = call put = call 1. Lecture 18 5 / 21

16 Greeks (3) What is the Delta of a put? Use the put-call parity: Price of put + S = Price of call + Ke r f (T t). Taking / S on the LHS and the RHS yields put + 1 = call put = call 1. In the Black-Scholes model we get put = N(d 1 ) 1 Lecture 18 5 / 21

17 Greeks (3) What is the Delta of a put? Use the put-call parity: Price of put + S = Price of call + Ke r f (T t). Taking / S on the LHS and the RHS yields put + 1 = call put = call 1. In the Black-Scholes model we get put = N(d 1 ) 1 = N( d 1 ). Lecture 18 5 / 21

18 Perpetual American options (1) When the derivative is of American type, then we are allowed to choose the time at which we want to get the derivatives payoff. Lecture 18 6 / 21

19 Perpetual American options (1) When the derivative is of American type, then we are allowed to choose the time at which we want to get the derivatives payoff. In general it is a very hard problem to determine the price and the optimal time at which an American option should be exercised. Lecture 18 6 / 21

20 Perpetual American options (1) When the derivative is of American type, then we are allowed to choose the time at which we want to get the derivatives payoff. In general it is a very hard problem to determine the price and the optimal time at which an American option should be exercised. There is however one case when the problem is simplified, and that is when the option is of perpetual type. Lecture 18 6 / 21

21 Perpetual American options (1) When the derivative is of American type, then we are allowed to choose the time at which we want to get the derivatives payoff. In general it is a very hard problem to determine the price and the optimal time at which an American option should be exercised. There is however one case when the problem is simplified, and that is when the option is of perpetual type. A perpetual derivative has T =. Lecture 18 6 / 21

22 Perpetual American options (2) Let us consider the Black-Scholes model. Lecture 18 7 / 21

23 Perpetual American options (2) Let us consider the Black-Scholes model. Recall Black-Scholes equation: f t + r f S f S σ2 S 2 2 f S 2 r f f = 0. Lecture 18 7 / 21

24 Perpetual American options (2) Let us consider the Black-Scholes model. Recall Black-Scholes equation: f t + r f S f S σ2 S 2 2 f S 2 r f f = 0. When T = in the Black-Scholes mode, the term f / t disappears from the equation due to time-homogeneity Lecture 18 7 / 21

25 Perpetual American options (2) Let us consider the Black-Scholes model. Recall Black-Scholes equation: f t + r f S f S σ2 S 2 2 f S 2 r f f = 0. When T = in the Black-Scholes mode, the term f / t disappears from the equation due to time-homogeneity, and we get an ODE instead of a PDE: Lecture 18 7 / 21

26 Perpetual American options (2) Let us consider the Black-Scholes model. Recall Black-Scholes equation: f t + r f S f S σ2 S 2 2 f S 2 r f f = 0. When T = in the Black-Scholes mode, the term f / t disappears from the equation due to time-homogeneity, and we get an ODE instead of a PDE: r f Sf σ2 S 2 f r f f = 0. Lecture 18 7 / 21

27 Perpetual American options (2) Let us consider the Black-Scholes model. Recall Black-Scholes equation: f t + r f S f S σ2 S 2 2 f S 2 r f f = 0. When T = in the Black-Scholes mode, the term f / t disappears from the equation due to time-homogeneity, and we get an ODE instead of a PDE: r f Sf σ2 S 2 f r f f = 0. To solve this equation, we look for solutions on the form with a R. f (S) = S a Lecture 18 7 / 21

28 Perpetual American options (3) Inserting this into the ODE yields r f SaS a σ2 S 2 a(a 1)S a 2 r f S a = 0 Lecture 18 8 / 21

29 Perpetual American options (3) Inserting this into the ODE yields r f SaS a σ2 S 2 a(a 1)S a 2 r f S a = 0 [ S a r f a + 1 ] 2 σ2 a(a 1) r f = 0. Lecture 18 8 / 21

30 Perpetual American options (3) Inserting this into the ODE yields r f SaS a σ2 S 2 a(a 1)S a 2 r f S a = 0 [ S a r f a + 1 ] 2 σ2 a(a 1) r f = 0. ) ( a 2 2rf + σ 2 1 a 2r f σ 2 = 0. Lecture 18 8 / 21

31 Perpetual American options (3) Inserting this into the ODE yields r f SaS a σ2 S 2 a(a 1)S a 2 r f S a = 0 [ S a r f a + 1 ] 2 σ2 a(a 1) r f = 0. ) ( a 2 2rf + σ 2 1 The solution to this equation is a 2r f σ 2 = 0. a 1 = 1 and a 2 = 2r f σ 2 Lecture 18 8 / 21

32 Perpetual American options (4) Hence, the solution to the time-homogenuous Black-Scholes equation is given by f (S) = A 1 S + A 2 S 2r f /σ 2. for A 1, A 2 R. Lecture 18 9 / 21

33 Perpetual American options (4) Hence, the solution to the time-homogenuous Black-Scholes equation is given by f (S) = A 1 S + A 2 S 2r f /σ 2. for A 1, A 2 R. How do we determine the constants A 1 and A 2? Lecture 18 9 / 21

34 Perpetual American options (4) Hence, the solution to the time-homogenuous Black-Scholes equation is given by f (S) = A 1 S + A 2 S 2r f /σ 2. for A 1, A 2 R. How do we determine the constants A 1 and A 2? They will depend on the derivative s payoff. Lecture 18 9 / 21

35 The perpetual American put option (1) Let us consider a put option with strike price K: F (S(t)) = max(k S(t), 0). Lecture / 21

36 The perpetual American put option (1) Let us consider a put option with strike price K: F (S(t)) = max(k S(t), 0). Note that there is no given exercise time we have to choose that time as well. Lecture / 21

37 The perpetual American put option (1) Let us consider a put option with strike price K: F (S(t)) = max(k S(t), 0). Note that there is no given exercise time we have to choose that time as well. Now consider the general solution f (S) = A 1 S + A 2 S 2r f /σ 2. Lecture / 21

38 The perpetual American put option (1) Let us consider a put option with strike price K: F (S(t)) = max(k S(t), 0). Note that there is no given exercise time we have to choose that time as well. Now consider the general solution f (S) = A 1 S + A 2 S 2r f /σ 2. Since lim F (S) = 0 S Lecture / 21

39 The perpetual American put option (1) Let us consider a put option with strike price K: F (S(t)) = max(k S(t), 0). Note that there is no given exercise time we have to choose that time as well. Now consider the general solution f (S) = A 1 S + A 2 S 2r f /σ 2. Since we must have lim F (S) = 0, S lim f (S) = 0 S Lecture / 21

40 The perpetual American put option (1) Let us consider a put option with strike price K: F (S(t)) = max(k S(t), 0). Note that there is no given exercise time we have to choose that time as well. Now consider the general solution f (S) = A 1 S + A 2 S 2r f /σ 2. Since we must have lim F (S) = 0, S lim f (S) = 0 A 1 = 0. S Lecture / 21

41 The perpetual American put option (2) We want to exercise the put option at the first time t such that max(k S(t), 0) }{{} f (t, S(t)) }{{}. value of exercising at t value of not exercising at t Lecture / 21

42 The perpetual American put option (2) We want to exercise the put option at the first time t such that max(k S(t), 0) }{{} f (t, S(t)) }{{}. value of exercising at t value of not exercising at t But it is never optimal to exercise the option when max(k S(t), 0) = 0. Lecture / 21

43 The perpetual American put option (3) Let S c solve the equation Here c stands for critical K S c = f (S c ) = A 2 S 2r f /σ 2 c. Lecture / 21

44 The perpetual American put option (3) Let S c solve the equation K S c = f (S c ) = A 2 S 2r f /σ 2 c. Here c stands for critical and S c is the critical level at which we choose to use the option when the stock price equals (or is less than) S c. Lecture / 21

45 The perpetual American put option (3) Let S c solve the equation K S c = f (S c ) = A 2 S 2r f /σ 2 c. Here c stands for critical and S c is the critical level at which we choose to use the option when the stock price equals (or is less than) S c. Note that we need to find S c ; this is a part of the solution. Lecture / 21

46 The perpetual American put option (3) Let S c solve the equation K S c = f (S c ) = A 2 S 2r f /σ 2 c. Here c stands for critical and S c is the critical level at which we choose to use the option when the stock price equals (or is less than) S c. Note that we need to find S c ; this is a part of the solution. One can show that we also have smooth fit. This means that the value function f (S) meets the payoff function max(k S, 0) tangentially. Lecture / 21

47 The perpetual American put option (4) In formulas the smooth fit condition means that f (S c ) = d (K S) ds S=Sc Lecture / 21

48 The perpetual American put option (4) In formulas the smooth fit condition means that f (S c ) = d (K S) ds S=Sc 2r f σ 2 A 2S 2r f /σ2 1 c = 1. Lecture / 21

49 The perpetual American put option (4) In formulas the smooth fit condition means that f (S c ) = d (K S) ds S=Sc 2r f σ 2 A 2S 2r f /σ2 1 c = 1. We now have a system of equations with two equations and two unknowns Lecture / 21

50 The perpetual American put option (4) In formulas the smooth fit condition means that f (S c ) = d (K S) ds S=Sc 2r f σ 2 A 2S 2r f /σ2 1 c = 1. We now have a system of equations with two equations and two unknowns: K S c = A 2 S 2r f /σ 2 c 1 = 2r f σ 2 A 2S 2r f /σ2 1 c Lecture / 21

51 The perpetual American put option (5) It is not hard to solve the system of equations, and doing this we get ( ) S 2rf /σ 2 f (S) = (K S c ), S c where S c = K 2r f 2r f + σ 2. Lecture / 21

52 The perpetual American call option Why didn t we study the perpetual American call option? Lecture / 21

53 The perpetual American call option Why didn t we study the perpetual American call option? The reason is that one can show that it is never optimal to exercise this option. Lecture / 21

54 The perpetual American call option Why didn t we study the perpetual American call option? The reason is that one can show that it is never optimal to exercise this option. In fact, using the general solution f (S) = A 1 S + A 2 S 2r f /σ 2 we must have A 2 = 0 in this case otherwise f (0) = ±. Lecture / 21

55 The perpetual American call option Why didn t we study the perpetual American call option? The reason is that one can show that it is never optimal to exercise this option. In fact, using the general solution f (S) = A 1 S + A 2 S 2r f /σ 2 we must have A 2 = 0 in this case otherwise f (0) = ±. The smooth fit condition will then imply f (S) = S and S c =. Lecture / 21

56 Risk-neutral valuation One can show that the principle of risk-neutral valuation (or risk-neutral pricing) holds generally. Lecture / 21

57 Risk-neutral valuation One can show that the principle of risk-neutral valuation (or risk-neutral pricing) holds generally. For a simple European derivative this means that the price is given by f (t, S(t)) = e r f (T t) Ê[F (S(T )) S t ]. Lecture / 21

58 Risk-neutral valuation One can show that the principle of risk-neutral valuation (or risk-neutral pricing) holds generally. For a simple European derivative this means that the price is given by f (t, S(t)) = e r f (T t) Ê[F (S(T )) S t ]. When calculating expected values Ê( ), we use the dynamics ds(t) = r f S(t)dt + σs(t)dẑ(t). Lecture / 21

59 Interest rate derivatives (1) In general, money in the bank grows according to db(t) = r(t)b(t)dt with B(0) = 1. Lecture / 21

60 Interest rate derivatives (1) In general, money in the bank grows according to We can solve this equation: db(t) = r(t)b(t)dt with B(0) = 1. t B(t) = e 0 r(s)ds. Lecture / 21

61 Interest rate derivatives (1) In general, money in the bank grows according to We can solve this equation: db(t) = r(t)b(t)dt with B(0) = 1. t B(t) = e 0 r(s)ds. So far we have assumed that r(t) = r f, i.e. a constant, but we can as well assume that r(t) is a stochastic process. Lecture / 21

62 Interest rate derivatives (1) In general, money in the bank grows according to We can solve this equation: db(t) = r(t)b(t)dt with B(0) = 1. t B(t) = e 0 r(s)ds. So far we have assumed that r(t) = r f, i.e. a constant, but we can as well assume that r(t) is a stochastic process. As when valuing options on stocks we can use either use a discrete time model or a continuous time model. Lecture / 21

63 Interest rate derivatives (2) When using discrete time models, it is usual to use trinomial trees Lecture / 21

64 Interest rate derivatives (2) When using discrete time models, it is usual to use trinomial trees: u r t r t r t d r t t t + 1 Lecture / 21

65 Interest rate derivatives (3) In continuous time models, we use Ito diffusions to model the interest rate r(t). Lecture / 21

66 Interest rate derivatives (3) In continuous time models, we use Ito diffusions to model the interest rate r(t). One can show that the value at time t [0, T ] of a zero-coupon bond with maturity time T is given by P(t, T ) = (e Ê ) T t r(s)ds r t. Lecture / 21

67 Interest rate derivatives (3) In continuous time models, we use Ito diffusions to model the interest rate r(t). One can show that the value at time t [0, T ] of a zero-coupon bond with maturity time T is given by P(t, T ) = (e Ê ) T t r(s)ds r t. Note that we use risk-neutral probabilities in this formula. Lecture / 21

68 Interest rate derivatives (3) In continuous time models, we use Ito diffusions to model the interest rate r(t). One can show that the value at time t [0, T ] of a zero-coupon bond with maturity time T is given by P(t, T ) = (e Ê ) T t r(s)ds r t. Note that we use risk-neutral probabilities in this formula. Let us now lok at some examples of models. Lecture / 21

69 Interest rate derivatives (4) The Vasicek model: dr(t) = a(b r(t))dt + σdẑ(t) Lecture / 21

70 Interest rate derivatives (4) The Vasicek model: dr(t) = a(b r(t))dt + σdẑ(t) The Cox-Ingersoll-Ross (CIR) model: dr(t) = a(b r(t))dt + σ r(t)dẑ(t). Lecture / 21

71 Interest rate derivatives (4) The Vasicek model: dr(t) = a(b r(t))dt + σdẑ(t) The Cox-Ingersoll-Ross (CIR) model: dr(t) = a(b r(t))dt + σ r(t)dẑ(t). The Ho-Lee model: dr(t) = θ(t)dt + σdẑ(t) Lecture / 21

72 Interest rate derivatives (4) The Vasicek model: dr(t) = a(b r(t))dt + σdẑ(t) The Cox-Ingersoll-Ross (CIR) model: dr(t) = a(b r(t))dt + σ r(t)dẑ(t). The Ho-Lee model: dr(t) = θ(t)dt + σdẑ(t) The Hull-White model: d(r) = [θ(t) ar(t)]dt + σdẑ(t). Lecture / 21

73 Interest rate derivatives (4) The Vasicek model: dr(t) = a(b r(t))dt + σdẑ(t) The Cox-Ingersoll-Ross (CIR) model: dr(t) = a(b r(t))dt + σ r(t)dẑ(t). The Ho-Lee model: dr(t) = θ(t)dt + σdẑ(t) The Hull-White model: d(r) = [θ(t) ar(t)]dt + σdẑ(t). Lecture / 21

74 Interest rate derivatives (5) There is also an equation similar to the Black-Scholes equation in the interest rate derivatives case. Lecture / 21

75 Interest rate derivatives (5) There is also an equation similar to the Black-Scholes equation in the interest rate derivatives case. If dr(t) = µ(r(t), t)dt + σ(r(t), t)dẑ(t) Lecture / 21

76 Interest rate derivatives (5) There is also an equation similar to the Black-Scholes equation in the interest rate derivatives case. If dr(t) = µ(r(t), t)dt + σ(r(t), t)dẑ(t) then the price f (r(t), t) of a derivative satifies the PDE f t + µ(r, t) f r σ2 (r, t) 2 f rf = 0. r 2 Lecture / 21