Financial Economics & Insurance

Transcription

1 Financial Economics & Insurance Albert Cohen Actuarial Sciences Program Department of Mathematics Department of Statistics and Probability A336 Wells Hall Michigan State University East Lansing MI Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

2 Course Information Syllabus to be posted on class page in first week of classes Homework assignments will posted there as well Page can be found at Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

3 Course Information Many examples within these slides are used with kind permission of Prof. Dmitry Kramkov, Dept. of Mathematics, Carnegie Mellon University. Book for course: Marcel Finan s A Discussion of Financial Economics in Actuarial Models: A Preparation for the Actuarial Exam MFE/3F. Some proofs from there will be referenced as well. Please find these notes here Some examples here will be similar to those practice questions publicly released by the SOA. Please note the SOA owns the copyright to these questions. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

4 What are financial securities? Traded Securities - price given by market. For example: Stocks Commodities Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

5 What are financial securities? Traded Securities - price given by market. For example: Stocks Commodities Non-Traded Securities - price remains to be computed. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

6 What are financial securities? Traded Securities - price given by market. For example: Stocks Commodities Non-Traded Securities - price remains to be computed. Is this always true? Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

7 What are financial securities? Traded Securities - price given by market. For example: Stocks Commodities Non-Traded Securities - price remains to be computed. Is this always true? We will focus on pricing non-traded securities. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

8 How does one fairly price non-traded securities? Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

9 How does one fairly price non-traded securities? By eliminating all unfair prices Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

10 How does one fairly price non-traded securities? By eliminating all unfair prices Unfair prices arise from Arbitrage Strategies Start with zero capital End with non-zero wealth Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

11 How does one fairly price non-traded securities? By eliminating all unfair prices Unfair prices arise from Arbitrage Strategies Start with zero capital End with non-zero wealth We will search for arbitrage-free strategies to replicate the payoff of a non-traded security Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

12 How does one fairly price non-traded securities? By eliminating all unfair prices Unfair prices arise from Arbitrage Strategies Start with zero capital End with non-zero wealth We will search for arbitrage-free strategies to replicate the payoff of a non-traded security This replication is at the heart of the engineering of financial products Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

13 More Questions Existence - Does such a fair price always exist? If not, what is needed of our financial model to guarantee at least one arbitrage-free price? Uniqueness - are there conditions where exactly one arbitrage-free price exists? Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

14 And What About... Does the replicating strategy and price computed reflect uncertainty in the market? Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

15 And What About... Does the replicating strategy and price computed reflect uncertainty in the market? Mathematically, if P is a probabilty measure attached to a series of price movements in underlying asset, is P used in computing the price? Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

16 Notation Forward Contract: Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

17 Notation Forward Contract: A financial instrument whose initial value is zero, and whose final value is derived from another asset. Namely, the difference of the final asset price and forward price: V (0) = 0, V (T ) = S(T ) F (1) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

18 Notation Forward Contract: A financial instrument whose initial value is zero, and whose final value is derived from another asset. Namely, the difference of the final asset price and forward price: V (0) = 0, V (T ) = S(T ) F (1) Value at end of term can be negative - buyer accepts this in exchange for no premium up front Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

19 Notation Interest Rate: The rate r at which money grows. Also used to discount the value today of one unit of currency one unit of time from the present Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

20 Notation Interest Rate: The rate r at which money grows. Also used to discount the value today of one unit of currency one unit of time from the present V (0) = 1, V (1) = 1 (2) 1 + r Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

21 An Example of Replication Forward Exchange Rate: There are two currencies, foreign and domestic: Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

22 An Example of Replication Forward Exchange Rate: There are two currencies, foreign and domestic: S B A = 4 is the spot exchange rate - one unit of B is worth S B A of A today (time 0) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

23 An Example of Replication Forward Exchange Rate: There are two currencies, foreign and domestic: S B A = 4 is the spot exchange rate - one unit of B is worth S B A of A today (time 0) r A = 0.1 is the domestic borrow/lend rate Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

24 An Example of Replication Forward Exchange Rate: There are two currencies, foreign and domestic: S B A = 4 is the spot exchange rate - one unit of B is worth S B A of A today (time 0) r A = 0.1 is the domestic borrow/lend rate r B = 0.2 is the foreign borrow/lend rate Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

25 An Example of Replication Forward Exchange Rate: There are two currencies, foreign and domestic: S B A = 4 is the spot exchange rate - one unit of B is worth S B A of A today (time 0) r A = 0.1 is the domestic borrow/lend rate r B = 0.2 is the foreign borrow/lend rate Compute the forward exchange rate FA B. This is the value of one unit of B in terms of A at time 1. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

26 An Example of Replication: Solution At time 1, we deliver 1 unit of B in exchange for FA B currency A. units of domestic Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

27 An Example of Replication: Solution At time 1, we deliver 1 unit of B in exchange for FA B currency A. units of domestic This is a forward contract - we pay nothing up front to achieve this. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

28 An Example of Replication: Solution At time 1, we deliver 1 unit of B in exchange for FA B currency A. units of domestic This is a forward contract - we pay nothing up front to achieve this. Initially borrow some amount foreign currency B, in foreign market to grow to one unit of B at time 1. This is achieved by the initial SA amount B (valued in domestic currency) 1+r B Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

29 An Example of Replication: Solution At time 1, we deliver 1 unit of B in exchange for FA B currency A. units of domestic This is a forward contract - we pay nothing up front to achieve this. Initially borrow some amount foreign currency B, in foreign market to grow to one unit of B at time 1. This is achieved by the initial SA amount B (valued in domestic currency) 1+r B Invest the amount currency) F B A 1+r A in domestic market (valued in domestic Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

30 An Example of Replication: Solution Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

31 An Example of Replication: Solution This results in the initial value V (0) = Since the initial value is 0, this means F B A = S B A F B A 1 + r A S B A 1 + r B (3) 1 + r A = (4) 1 + r B Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

32 Outline 1 Continuous Model-Probability Expected Values Application of Option Greeks 2 Continuous Model-Ito Calculus Brownian Motion BSM Examples More Exotic Options Bond Pricing and Stochastic Calculus Volatility and Simulation 3 PDE Methods Barrier Options Asian Options Lookback Options Heat Equation Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

33 Black Scholes Pricing using Underlying Asset In the next section, we will derive the following solutions to the Black-Scholes PDE: V C (S, t) = e r(t t) Ẽ [(S T K) + S t = S] = Se δ(t t) N(d 1 ) Ke r(t t) N(d 2 ) V P (S, t) = e r(t t) Ẽ [(K S T ) + S t = S] = Ke r(t t) N( d 2 ) Se δ(t t) N( d 1 ) ( ) ln S K + (r δ σ2 )(T t) d 1 = σ T t d 2 = d 1 σ T t N(x) = 1 x e z2 2 dz. 2π Notice that V C (S, t) V P (S, t) = Se δ(t t) Ke r(t t). Question: What underlying model of stock evolution leads to this value? How can we support such a probability measure? Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163 (5)

34 Normal Random Variables We say that X is a Normal Random Variable with parameters µ, σ 2 if f X (x) = 1 2πσ e (x µ)2 2σ 2 for < x < (6) Furthermore, we say that Z is a Standard Normal Random Variable if it is Normal with parameters µ = 0, σ = 1. Consider z = x µ σ 1 f X (x)dx = e (x µ)2 2σ 2 dx 2πσ = = 1 2π e (z)2 2 dz f Z (z)dz (7) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

35 Normal Random Variables We user the notation X N(µ, σ 2 ) and Z N(0, 1). By our transformation above, it can be seen that if Z N(0, 1), then X = µ + σ Z N(µ, σ 2 ). We can see this via Φ(z) := F Z (z) = P[Z z] = P[ X µ x µ σ σ ] = P[X x] =: F X (x) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

36 Normal Random Variables We user the notation X N(µ, σ 2 ) and Z N(0, 1). By our transformation above, it can be seen that if Z N(0, 1), then X = µ + σ Z N(µ, σ 2 ). We can see this via Φ(z) := F Z (z) = P[Z z] = P[ X µ x µ σ σ ] = P[X x] =: F X (x) ( ) 1 x µ σ f Z = d σ dx F Z (z) = d dx F X (x) = f X (x) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

37 Normal Random Variables We user the notation X N(µ, σ 2 ) and Z N(0, 1). By our transformation above, it can be seen that if Z N(0, 1), then X = µ + σ Z N(µ, σ 2 ). We can see this via Φ(z) := F Z (z) = P[Z z] = P[ X µ x µ σ σ ] = P[X x] =: F X (x) ( ) 1 x µ σ f Z = d σ dx F Z (z) (8) = d dx F X (x) = f X (x) f Z (z) = 1 e (z)2 2 2π Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

38 Convolutions and Sums of Independent Random Variables f X +Y (a) = d da = = ( ) F X (a y)f Y (y)dy d da F X (a y)f Y (y)dy by cty of f X (Leibniz) f X (a y)f Y (y)dy (9) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

39 Sums of Normal Random Variables For a sequence {c i } n i=1 of real numbers, we have for a sequence of correlated normal random variables {X i } n i=1, where the X i N(µ i σ i ) with covariance ρ ij that ( n n n n c i X i N µ i, c i c j ρ ij σ i σ j ). (10) i=1 i=1 i=1 j=1 Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

40 Lognormal Random Variables We say that Y LN(µ, σ) is Lognormal if ln(y ) N(µ, σ 2 ). As sums of normal random variables remain normal, products of lognormal random variables remain lognormal. Recall that the moment-generating function of X N(µ, σ 2 ) µ + σn(0, 1) is M X (t) = E[e tx ] = e µt+ 1 2 σ2 t 2 (11) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

41 Lognormal Random Variables We say that Y LN(µ, σ) is Lognormal if ln(y ) N(µ, σ 2 ). As sums of normal random variables remain normal, products of lognormal random variables remain lognormal. Recall that the moment-generating function of X N(µ, σ 2 ) µ + σn(0, 1) is If Y = e µ+σz, then, it can be seen that M X (t) = E[e tx ] = e µt+ 1 2 σ2 t 2 (11) E[Y n ] = E[e nx ] = e µn+ 1 2 σ2 n 2 (12) and f Y (y) = ( 1 σ 2πy exp ) (ln(y) µ)2 2σ 2 (13) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

42 Stock Evolution and Lognormal Random Variables One application of lognormal distributions is their use in modeling the evolution of asset prices S. If we assume a physical measure P with α the expected return on the stock under the physical measure, then ln ( St S 0 ) ( = N (α δ 1 ) 2 σ2 )t, σ 2 t S t = S 0 e (α δ 1 2 σ2 )t+σ tz (14) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

43 Stock Evolution and Lognormal Random Variables We can use the previous facts to show E[S t ] = S 0 e (α δ)t ( ln S 0 K P[S t > K] = N + (α δ ) 0.5σ2 )t σ. t (15) Note that under the risk-neutral measure P, we exchange α with r, the risk-free rate: Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

44 Stock Evolution and Lognormal Random Variables We can use the previous facts to show E[S t ] = S 0 e (α δ)t ( ln S 0 K P[S t > K] = N + (α δ ) 0.5σ2 )t σ. t (15) Note that under the risk-neutral measure P, we exchange α with r, the risk-free rate: Ẽ[S t ] = S 0 e (r δ)t P[S t > K] = N(d 2 ). (16) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

45 Stock Evolution and Lognormal Random Variables The next challenge is to construct a process S t that possesses the above properties as well as continuity of paths. Click here for a neat article relating actuarial reserving to option pricing! Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

46 Stock Evolution and Lognormal Random Variables Risk managers are also interested in Conditional Tail Expectations (CTE s) of random variables: ] E [X 1 {X >k} CTE X (k) := E[X X > k] =. (17) P[X > k] Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

47 Stock Evolution and Lognormal Random Variables In our case, E[S t S t > K] = E [ S 0 e (α δ 1 2 σ2 )t+σ tz 1 { S 0 e (α δ 1 2 σ2 )t+σ } tz >K ] = S 0 e (α δ)t N [ P N S 0 e (α δ 1 2 σ2 )t+σ tz > K ) ( ln S 0 K +(α δ+0.5σ 2 )t σ t ( ln S 0 K +(α δ 0.5σ 2 )t σ t ) ] Ẽ[S t S t > K] = S 0 e (r δ)t N(d 1) N(d 2 ) (18) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

48 Stock Evolution and Lognormal Random Variables In fact, we can use this CTE framework to solve for the European Call option price in the Black-Scholes framework, where P 0 [A] = P[A S 0 = S] and [ V C (S, 0) := e rt Ẽ ] (S T K) + S 0 = S = e rt Ẽ 0 [S T K S T > K = e rt Ẽ 0 [S T S T > K ] P 0 [S T > K] ] P 0 [S T > K] Ke rt P 0 [S T > K] = e rt Se (r δ)t N(d 1) N(d 2 ) N(d 2) Ke rt N(d 2 ) = Se δt N(d 1 ) Ke rt N(d 2 ). (19) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

49 Black Scholes Pricing using Prepaid Forwards In order to apply the previous formulae to a myriad of underlying assets, we rewrite in terms of prepaid forwards: ( ) ln Se δ(t t) + 1 Ke d 1 (S, t) = r(t t) 2 σ2 (T t) σ T t ( F S ) ln t,t + 1 Ft,T K 2 σ2 (T t) (20) = σ T t d 2 = d 1 σ T t. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

50 Black Scholes Analysis: Option Greeks For any option price V (S, t), define its various sensitivities as follows: = V S Γ = S = 2 V S 2 ν = V σ Θ = V t ρ = V r Ψ = V δ. These are known accordingly as the Option Greeks. (21) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

51 Black Scholes Analysis: Option Greeks Straightforward partial differentiation leads to C = e δ(t t) N(d 1 ) P = e δ(t t) N( d 1 ) Γ C = Γ P = e δ(t t) N (d 1 ) σs T t ν C = ν P = Se δ(t t) T tn (d 1 ) (22) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

52 Black Scholes Analysis: Option Greeks as well as.. ρ C = (T t)ke r(t t) N(d 2 ) ρ P = (T t)ke r(t t) N( d 2 ) Ψ C = (T t)se δ(t t) N(d 1 ) Ψ P = (T t)se δ(t t) N( d 1 ). (23) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

53 Black Scholes Analysis: Option Greeks as well as.. ρ C = (T t)ke r(t t) N(d 2 ) ρ P = (T t)ke r(t t) N( d 2 ) Ψ C = (T t)se δ(t t) N(d 1 ) Ψ P = (T t)se δ(t t) N( d 1 ). What do the signs of the Greeks tell us? (23) HW: Compute Θ for puts and calls. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

54 Portfolio Sensitivity Analysis For a portfolio of M options, each with weighting λ i and M i=1 λ i = 1: Greek(Portfolio) = M λ i Greek(i th asset). (24) i=1 Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

55 Portfolio Sensitivity Analysis For a portfolio of M options, each with weighting λ i and M i=1 λ i = 1: Greek(Portfolio) = M λ i Greek(i th asset). (24) i=1 If P(i) is the price of a portfolio of income streams: P(i) = n k=1 P k(i). D = (1 + i) P (i) n k=1 = (1 + i) P k (i) P(i) P(i) n P k = (1 + i) (i) n P(i) = (1 + i) = n D k q k k=1 k=1 k=1 P k (i) P(i) P k (i) P k (i) (25) and so the portfolio duration is the weighted average of the individual durations. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

56 Portfolio Sensitivity Analysis For a portfolio of M options, each with weighting λ i and M i=1 λ i = 1: Greek(Portfolio) = M λ i Greek(i th asset). (24) i=1 If P(i) is the price of a portfolio of income streams: P(i) = n k=1 P k(i). D = (1 + i) P (i) n k=1 = (1 + i) P k (i) P(i) P(i) n P k = (1 + i) (i) n P(i) = (1 + i) = n D k q k k=1 k=1 k=1 P k (i) P(i) P k (i) P k (i) (25) and so the portfolio duration is the weighted average of the individual durations. What if the interest rate i is a random variable itself? Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

57 Option Elasticity Define Consequently, Ω(S, t) := lim ɛ 0 = V (S+ɛ,t) V (s,t) V (S,t) S+ɛ S S S V (S, t) lim ɛ 0 = S V (S, t). V (S + ɛ, t) V (s, t) S + ɛ S (26) Ω C (S, t) = Ω P (S, t) = C S V C (S, t) = P S V P (S, t) 0. Se δ(t t) Se δ(t t) Ke r(t t) N(d 2 ) 1 (27) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

58 Option Elasticity Theorem The volatility of an option is the option elasticity times the volatility of the stock: σ option = σ stock Ω. (28) The proof comes from Finan: Consider the strategy of hedging a portfolio of shorting an option and purchasing = V S shares. The initial and final values of this portfolio are Initally: V (S(t), t) (S(t), t) S(t) Finally: V (S(T ), T ) (S(t), t) S(T ) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

59 Option Elasticity Proof. If this portfolio is self-financing and arbitrage-free requirement, then e r(t t)( ) V (S(t), t) (S(t), t) S(t) = V (S(T ), T ) (S(t), t) S(T ). It follows that for κ := e r(t t), (29) V (S(T ), T ) V (S(t), t) = κ 1 + V (S(t), t) [ ] V (S(T ), T ) V (S(t), t) Var V (S(t), t) σ option = σ stock Ω. [ S(T ) S(t) S(t) = Ω 2 Var [ + 1 κ S(T ) S(t) S(t) ] ] Ω (30) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

60 Option Elasticity If γ is the expected rate of return on an option with value V, α is the expected rate of return on the underlying stock, and r is of course the risk free rate, then the following equation holds: ( ) γ V (S, t) = α (S, t) S + r V (S, t) (S, t) S. (31) In terms of elasticity Ω, this reduces to Risk Premium(Option) := γ r = (α r)ω. (32) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

61 Option Elasticity If γ is the expected rate of return on an option with value V, α is the expected rate of return on the underlying stock, and r is of course the risk free rate, then the following equation holds: ( ) γ V (S, t) = α (S, t) S + r V (S, t) (S, t) S. (31) In terms of elasticity Ω, this reduces to Risk Premium(Option) := γ r = (α r)ω. (32) Furthermore, we have the Sharpe Ratio for an asset as the ratio of risk premium to volatility: Sharpe(Stock) = (α r) σ = (α r)ω σω = Sharpe(Call). (33) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

62 Option Elasticity If γ is the expected rate of return on an option with value V, α is the expected rate of return on the underlying stock, and r is of course the risk free rate, then the following equation holds: ( ) γ V (S, t) = α (S, t) S + r V (S, t) (S, t) S. (31) In terms of elasticity Ω, this reduces to Risk Premium(Option) := γ r = (α r)ω. (32) Furthermore, we have the Sharpe Ratio for an asset as the ratio of risk premium to volatility: (α r) (α r)ω Sharpe(Stock) = = = Sharpe(Call). (33) σ σω HW Sharpe Ratio for a put? How about elasticity for a portfolio of options? Now read about Calendar Spreads, Implied Volatility, and Perpetual American Options. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

63 Example: Hedging Under a standard framework, assume you write a 4 yr European Call option a non-dividend paying stock with the following: S 0 = 10 = K σ = 0.2 r = Calculate the initial number of shares of the stock for your hedging program. (34) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

64 Example: Hedging Recall It follows that C = e δ(t t) N(d 1 ) ( ) ln S K + (r δ σ2 )(T t) d 1 = σ T t d 2 = d 1 σ T t. (35) ( ) C = N 0.4 = (36) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

65 Example: Risk Analysis Assume that an option is written on an asset S with the following information: The expected rate of return on the underlying asset is The expected rate of return on a riskless asset is The volatility on the underlying asset is ( ) V (S, t) = e 0.05(10 t) S 2 e S Compute Ω(S, t) and the Sharpe Ratio for this option. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

66 Example: Risk Analysis By definition, Ω(S, t) = S V (S, t) = S V (S,t) S V (S, t) = S d ds (S 2 e S ) (S 2 e S = S (2SeS + S 2 e S ) ) S 2 e S = 2 + S. (37) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

67 Example: Risk Analysis By definition, Ω(S, t) = S V (S, t) = S V (S,t) S V (S, t) = S d ds (S 2 e S ) (S 2 e S = S (2SeS + S 2 e S ) ) S 2 e S = 2 + S. Furthermore, since Ω = 2 + S 2, we have (37) Sharpe = = (38) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

68 Example: Black Scholes Pricing Consider a portfolio of options on a non-dividend paying stock S that consists of a put and a call, both with strike K = 5 = S 0. What is the Γ for this option as well as the option value at time 0 if the time to expiration is T = 4, r = 0.02, σ = 0.2. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

69 Example: Black Scholes Pricing Consider a portfolio of options on a non-dividend paying stock S that consists of a put and a call, both with strike K = 5 = S 0. What is the Γ for this option as well as the option value at time 0 if the time to expiration is T = 4, r = 0.02, σ = 0.2. In this case, V = V C + V P Γ = 2 S 2 ( V C + V P) = 2Γ C. (39) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

70 Example: Black Scholes Pricing Consequently, d 1 = 0.4 and d 2 = = 0, and so V (5, 0) = V C (5, 0) + V P (5, 0) ( ) = 5 N(d 1 ) + e 4r N( d 2 ) e 4r N(d 2 ) N( d 1 ) ( ) = 5 N(0.4) + e 4r N(0) e 4r N(0) N( 0.4) = Γ(5, 0) = 2N (0.4) = N (0.4) = 1 2π e 0.5 (0.4)2 = (40) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

71 Homework From Finan: Problems 27.1, 27.2, 27.4, 27.8 Problems 28.2, 28.3, 28.4, 28.6, 28.7, 28.12, Problems 29.1, 29.2, 29.3, 29.4 Problems 30.3, 30.4, 30.6, Problems 31.2, 31.3, 31.4, 31.5, 31.6, 31.7 Problems Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

72 Market Making On a periodic basis, a Market Maker, services the option buyer by rebalancing the portfolio designed to replicate the payoff written into the option contract. Define V i = Option Value i periods from inception i = Delta required i periods from inception P i = i S i V i (41) Rebalancing at time i requires an extra ( i+1 i ) shares. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

73 Market Making On a periodic basis, a Market Maker, services the option buyer by rebalancing the portfolio designed to replicate the payoff written into the option contract. Define V i = Option Value i periods from inception i = Delta required i periods from inception P i = i S i V i = Cost of Strategy Rebalancing at time i requires an extra ( i+1 i ) shares. (42) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

74 Market Making Define Then S i = S i+1 S i P i = P i+1 P i (43) i = i+1 i P i = Net Cash Flow = i S i V i rp i ) (44) = i S i V i r ( i S i V i Under what conditions is the Net Flow = 0? Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

75 Market Making For a continuous rate r, we can see that if := V S, P t = t S t V t, dv := V (S t + ds t, t + dt) V (S t, t) Θdt + ds t Γ (ds t) 2 dp t = t ds t dv t rp t dt t ds t (Θdt + ds t + 12 ) Γ (ds t) 2 r ( t S t V t ) dt ) (Θdt + r( S t V (S t, t))dt + 12 Γ [ds t] 2. (45) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

76 Market Making If dt is small, but not infinitessimally small, then on a periodic basis given the evolution of S t, the periodic jump in value from S t S t + ds t may be known exactly and correspond to a non-zero jump in Market Maker profit dp t. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

77 Market Making If dt is small, but not infinitessimally small, then on a periodic basis given the evolution of S t, the periodic jump in value from S t S t + ds t may be known exactly and correspond to a non-zero jump in Market Maker profit dp t. If ds t ds t = σ 2 S 2 t dt, then if we sample continuously and enforce a zero net-flow, we retain the BSM PDE for all relevant (S, t): V ( t + r S V ) S V σ2 S 2 2 V S 2 = 0 V (S, T ) = G(S) for final time payoff G(S). (46) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

78 Note: Delta-Gamma Neutrality vs Bond Immunization In an actuarial analysis of cashflow, a company may wish to immunize its portfolio. This refers to the relationship between a non-zero value for the second derivative with respect to interest rate of the (deterministic) cashflow present value and the subsequent possibility of a negative PV. This is similar to the case of market maker with a non-zero Gamma. In the market makers cash flow, a move of ds in the stock corresponds to a move 1 2 Γ(dS)2 in the portfolio value. In order to protect against large swings in the stock causing non-linear effects in the portfolio value, the market maker may choose to offset positions in her present holdings to maintain Gamma Neutrality or she wish to maintain Delta Neutrality, although this is only a linear effect. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

79 Option Greeks and Analysis - Some Final Comments It is important to note the similarities between Market Making and Actuarial Reserving. In engineering the portfolio to replicate the payoff written into the contract, the market maker requires capital. The idea of Black Scholes Merton pricing is that the portfolio should be self-financing. One should consider how this compares with the capital required by insurers to maintain solvency as well as the possibility of obtaining reinsurance. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

80 Homework From Finan: Problems , 35.7, 35.8 Problems Problems Problems Problems Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

81 More Practice Consider an economy where : The current exchange rate is x 0 = $ yen. A four-year dollar-denominated European put option on yen with a strike price of 0.008$ sells for $. The continuously compounded risk-free interest rate on dollars is 3%. The continuously compounded risk-free interest rate on yen is 1.5%. Compute the price of a 4 year dollar-denominated European call option on yens with a strike price of 0.008$. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

82 More Practice Consider an economy where : The current exchange rate is x 0 = $ yen. A four-year dollar-denominated European put option on yen with a strike price of 0.008$ sells for $. The continuously compounded risk-free interest rate on dollars is 3%. The continuously compounded risk-free interest rate on yen is 1.5%. Compute the price of a 4 year dollar-denominated European call option on yens with a strike price of 0.008$. ANSWER: By put call parity, and the Black Scholes formula, with the asset S as the exchange rate, and the foreign risk-free rate r f = δ, V C (x 0, 0) = V P (x 0, 0) + x 0 e r f T Ke rt = e e = (47) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

83 More Practice An investor purchases a 1 year, 50 strike European Call option on a non-dividend paying stock by borrowing at the risk-free rate r. The investor paid V C (S 0, 0) = 10. Six months later, the investor finds out that the Call option has increased in value by one: V C (S 0.05, 0.5) = 11. Assuming (σ, r) = (0.2, 0.02). Should she close out her position after 6 months? Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

84 More Practice An investor purchases a 1 year, 50 strike European Call option on a non-dividend paying stock by borrowing at the risk-free rate r. The investor paid V C (S 0, 0) = 10. Six months later, the investor finds out that the Call option has increased in value by one: V C (S 0.05, 0.5) = 11. Assuming (σ, r) = (0.2, 0.02). Should she close out her position after 6 months? ANSWER: Simply put, her profit if she closes out after 6 months is So, yes, she should liquidate her position e = (48) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

85 More Practice Consider a 1 year at the money European Call option on a non-dividend paying stock. If you are told that C = 0.65, and the economy bears a 1% rate, can you estimate the volatility σ? Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

86 More Practice Consider a 1 year at the money European Call option on a non-dividend paying stock. If you are told that C = 0.65, and the economy bears a 1% rate, can you estimate the volatility σ? ANSWER: By definition, ( r + C = e δt 1 N(d 1 ) = N ( = N σ 2 σ2 2 σ2 σ ) = 0.65 ) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

87 More Practice Consider a 1 year at the money European Call option on a non-dividend paying stock. If you are told that C = 0.65, and the economy bears a 1% rate, can you estimate the volatility σ? ANSWER: By definition, σ2 σ ( r + C = e δt 1 N(d 1 ) = N ( = N σ = σ2 2 σ2 σ ) = 0.65 ) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

88 More Practice Consider a 1 year at the money European Call option on a non-dividend paying stock. If you are told that C = 0.65, and the economy bears a 1% rate, can you estimate the volatility σ? ANSWER: By definition, ( r + C = e δt 1 N(d 1 ) = N ( = N σ 2 σ σ2 = σ σ {0.0269, }. 2 σ2 σ ) = 0.65 ) (49) More information is needed to choose from the two roots computed above. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

89 Ideas to Review The definition of the Black-Scholes pricing formulae for European puts and calls. What are the Greeks? Given a specific option, could you compute the Greeks? What is the Option Elasticity? How is it useful? How about the Sharpe ratio of an option? Can you compute the Elasticity and Sharpe ration of a given option? Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

90 Ideas to Review What is Delta Hedging? Can you replicate the example on p.417? If the Delta and Gamma values of an option are known, can you calculate the change in option value given a small change in the underlying asset value? How does this correspond the Market Maker s profit? Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

91 Brownian Motion Consider a probability space (Ω, F, P) and a process (W t, F t ) that lives on it, where F t represents all the information about {W u } 0 u t. Assume that our pair satisfies, for s, t 0 and {A i } n i=1 F P[W 0 = 0] = 1 P[W t dx] = 1 2πt e x2 2t dx Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

92 Brownian Motion Consider a probability space (Ω, F, P) and a process (W t, F t ) that lives on it, where F t represents all the information about {W u } 0 u t. Assume that our pair satisfies, for s, t 0 and {A i } n i=1 F P[W 0 = 0] = 1 P[W t dx] = 1 2πt e x2 2t dx P[lim t s W t = W s ] = 1 Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

93 Brownian Motion Consider a probability space (Ω, F, P) and a process (W t, F t ) that lives on it, where F t represents all the information about {W u } 0 u t. Assume that our pair satisfies, for s, t 0 and {A i } n i=1 F P[W 0 = 0] = 1 P[W t dx] = 1 2πt e x2 2t dx P[lim t s W t = W s ] = 1 P[W t+s W s A F s ] = P[W t A] for all A F P[ n i=1 { Wti W ti 1 A i } ] = Π n i=1 P[W ti W ti 1 A i ] Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

94 Brownian Motion Consider a probability space (Ω, F, P) and a process (W t, F t ) that lives on it, where F t represents all the information about {W u } 0 u t. Assume that our pair satisfies, for s, t 0 and {A i } n i=1 F P[W 0 = 0] = 1 P[W t dx] = 1 2πt e x2 2t dx P[lim t s W t = W s ] = 1 P[W t+s W s A F s ] = P[W t A] for all A F P[ n i=1 { Wti W ti 1 A i } ] = Π n i=1 P[W ti W ti 1 A i ] What s so hard about that? Take a Z N(0, 1) and define X t = tz (50) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

95 Quadratic Variation Clearly, the last quantity, also known as independent increments, is what makes Brownian motion truly special. We can use this property to define other, related properties. The first is the notion of quadratic variation. Simply put, and so, for an i.i.d. N(0, 1) sequence {Z i } n i=1 W t+ t W t W t (51) n ( ) 2 n ( ) 2 Wtj+1 W tj Wtj+1 t j j=1 j=1 (52) n ( ) 2 tj+1 t j Z j j=1 Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

96 Quadratic Variation Assuming we have an even partition of [0, T ], with t j+1 t j = T n, then n j=1 ( Wtj+1 W tj ) 2 T n n j=1 To see why, recall that, for γ = 1 1 2t, ( ) n M χ 2 n (t) = E[e t Z 2 ] E[e t Z 2 ] = 1 2π e tx2 e x2 2 dx (Z j ) 2 = T χ2 n n T. (53) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

97 Quadratic Variation Assuming we have an even partition of [0, T ], with t j+1 t j = T n, then n j=1 ( Wtj+1 W tj ) 2 T n n j=1 To see why, recall that, for γ = 1 1 2t, ( ) n M χ 2 n (t) = E[e t Z 2 ] E[e t Z 2 ] = 1 2π (Z j ) 2 = T χ2 n n T. (53) e tx2 e x2 2 dx = 1 2π e x 2 2γ 2 dx Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

98 Quadratic Variation Assuming we have an even partition of [0, T ], with t j+1 t j = T n, then n j=1 ( Wtj+1 W tj ) 2 T n n j=1 To see why, recall that, for γ = 1 1 2t, ( ) n M χ 2 n (t) = E[e t Z 2 ] E[e t Z 2 ] = 1 2π = 1 1 2t (Z j ) 2 = T χ2 n n T. (53) e tx2 e x2 2 dx = 1 2π e x 2 2γ 2 dx Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

99 Quadratic Variation Assuming we have an even partition of [0, T ], with t j+1 t j = T n, then n j=1 ( Wtj+1 W tj ) 2 T n n j=1 To see why, recall that, for γ = 1 1 2t, ( ) n M χ 2 n (t) = E[e t Z 2 ] E[e t Z 2 ] = 1 2π = M χ 2 n n (t) = 1 1 2t (Z j ) 2 = T χ2 n n T. (53) e tx2 e x2 2 dx = 1 2π 1 (1 2 tn )n e t = E[e t 1 ]. e x 2 2γ 2 dx (54) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

100 k th order Variation, where k > 2 Assuming we have an even partition of [0, T ], with t j+1 t j = T n, then n ( ( ) k T Wtj+1 W tj n j=1 n ( ) E k Wtj+1 W tj T k 2 j=1 ) k 2 n n k 2 n (Z j ) k j=1 0 (55) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

101 Quadratic Variation We can use this to define integration against Brownian Motion. This is defined as T 0 f (W t, t)dw t = lim n n f (W jh, jh) (W jh+h W jh ) (56) In general, we can define the stochastic differential equation i=1 dx t = µ(x t, t)dt + σ(x t, t)dw t (57) and the accompanying Ito Equation for a new, stochastic calculus based on the relationship dw t dw t = dt. (58) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

102 Black-Scholes-Merton Analysis Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

103 Black-Scholes-Merton Analysis Independent of the model for evolution of underlying asset, pricing must be arbitrage-free Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

104 Black-Scholes-Merton Analysis Independent of the model for evolution of underlying asset, pricing must be arbitrage-free Achieve this via replication of derivative up to end of term of contract Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

105 Black-Scholes-Merton Analysis Independent of the model for evolution of underlying asset, pricing must be arbitrage-free Achieve this via replication of derivative up to end of term of contract Mathematical tools we require are now more involved than linear algebra; namely Ito Calculus and Partial Differential Equations Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

106 Black-Scholes-Merton Analysis Independent of the model for evolution of underlying asset, pricing must be arbitrage-free Achieve this via replication of derivative up to end of term of contract Mathematical tools we require are now more involved than linear algebra; namely Ito Calculus and Partial Differential Equations To enable this analysis, assume the following for the model (Ω, F, P) for which W t is a standard Brownian motion. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

107 Black-Scholes-Merton Analysis Independent of the model for evolution of underlying asset, pricing must be arbitrage-free Achieve this via replication of derivative up to end of term of contract Mathematical tools we require are now more involved than linear algebra; namely Ito Calculus and Partial Differential Equations To enable this analysis, assume the following for the model (Ω, F, P) for which W t is a standard Brownian motion. ds t = S t (µdt + σdw t ), a Geometric Brownian Motion models the asset evolution Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

108 Black-Scholes-Merton Analysis Independent of the model for evolution of underlying asset, pricing must be arbitrage-free Achieve this via replication of derivative up to end of term of contract Mathematical tools we require are now more involved than linear algebra; namely Ito Calculus and Partial Differential Equations To enable this analysis, assume the following for the model (Ω, F, P) for which W t is a standard Brownian motion. ds t = S t (µdt + σdw t ), a Geometric Brownian Motion models the asset evolution For a function f (x, t) C 2,1 ( R 2 R ) and Y t f (W t, t), Ito Calculus gives us: Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

109 Black-Scholes-Merton Analysis Independent of the model for evolution of underlying asset, pricing must be arbitrage-free Achieve this via replication of derivative up to end of term of contract Mathematical tools we require are now more involved than linear algebra; namely Ito Calculus and Partial Differential Equations To enable this analysis, assume the following for the model (Ω, F, P) for which W t is a standard Brownian motion. ds t = S t (µdt + σdw t ), a Geometric Brownian Motion models the asset evolution For a function f (x, t) C 2,1 ( R 2 R ) and Y t f (W t, t), Ito Calculus gives us: dy t = ( f t (W t, t) f xx(w t, t) ) dt + f x (W t, t)dw t Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

110 Black-Scholes-Merton Analysis Independent of the model for evolution of underlying asset, pricing must be arbitrage-free Achieve this via replication of derivative up to end of term of contract Mathematical tools we require are now more involved than linear algebra; namely Ito Calculus and Partial Differential Equations To enable this analysis, assume the following for the model (Ω, F, P) for which W t is a standard Brownian motion. ds t = S t (µdt + σdw t ), a Geometric Brownian Motion models the asset evolution For a function f (x, t) C 2,1 ( R 2 R ) and Y t f (W t, t), Ito Calculus gives us: dy t = ( f t (W t, t) f xx(w t, t) ) dt + f x (W t, t)dw t Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

111 Black-Scholes-Merton Analysis In general, for Z t = g(x t, t), with dx t = µ(x t, t)dt + σ(x t, t)dw t, we have dz t = g t (X t, t)dt g xx(x t, t)dx t dx t + g x dx t ( = g t (X t, t) + µ(x t, t)g x (X t, t) + 1 ) 2 σ(x t, t) 2 g xx (X t, t) dt (59) + σ(x t, t)g x (X t, t)dw t Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

112 Black-Scholes-Merton Analysis In general, for Z t = g(x t, t), with dx t = µ(x t, t)dt + σ(x t, t)dw t, we have dz t = g t (X t, t)dt g xx(x t, t)dx t dx t + g x dx t ( = g t (X t, t) + µ(x t, t)g x (X t, t) + 1 ) 2 σ(x t, t) 2 g xx (X t, t) dt (59) + σ(x t, t)g x (X t, t)dw t Using the above assumptions, we arrive at the conclusion that we must construct a portfolio X t that matches the value V t of the derivative we wish to price at all times t T, where T is the term of the contract. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

113 Black-Scholes-Merton Analysis We match Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

114 Black-Scholes-Merton Analysis We match dx t = dv t X T = V T = G(S T ) (60) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

115 Black-Scholes-Merton Analysis We match dx t = dv t X T = V T = G(S T ) where G(S) is the payoff of the contract at time T if the value of the underlying asset S T = S. (60) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

116 Black-Scholes-Merton Analysis To achieve this, we recognize that X t = V t dx t = dv t and by the assumption V = V (S t, t), Ito provides: Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

117 Black-Scholes-Merton Analysis To achieve this, we recognize that X t = V t dx t = dv t and by the assumption V = V (S t, t), Ito provides: dx t = r (X t t S t ) dt + t ds t = r (V t t S t ) dt + t ds t ( V dv t = t (S t, t) σ2 St 2 2 ) V S 2 (S t, t) dt + V S (S t, t)ds t. (61) This is due to the fact that ds t = µs t dt + σs t dw t (ds t ) 2 = σ 2 S 2 t dt. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

118 Black Scholes PDE Matching terms implies t = V (S, t) S V t (S, t) σ2 S 2 2 V (S, t) = r S 2 V (S, T ) = G(S) ( V (S, t) S V ) (S, t) S (62) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

119 Black Scholes PDE Matching terms implies t = V (S, t) S V t (S, t) σ2 S 2 2 V (S, t) = r S 2 V (S, T ) = G(S) ( V (S, t) S V ) (S, t) S This is the famous B-S-M PDE formulation for European option pricing, with payoff G(S). The question now - how do we solve it?! Note: For ds t = µs t dt + σs t dw t, we have as the solution S t = S u e (µ 1 2 σ2 )(t u)+σ(w t W u) S u e (µ 1 2 σ2 )(t u)+σw t u S u e (µ 1 2 σ2 )(t u)+σ t uz. (62) (63) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

120 Our First Arbitrage Opportunity Consider two perfectly correlated assets X, Y whose evolution is modeled by Assume wlog that 1 σ 1 dx t = µ 1 X t dt + σ 1 X t dw t dy t = µ 2 Y t dt + σ 2 Y t dw t (64) σ 1 σ 2 > 1 σ 2. Design a portfolio that consists of 1 going long σ 1 X t units of X 1 short σ 2 Y t units of Y, borrowing 1 σ 1 1 σ 2 at continuous rate r. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

121 Our First Arbitrage Opportunity Borrowing the amount at time t means that the total net output at time t is nothing, but the evolution of our portfolio is 1 dx t 1 ( 1 dy t 1 ) ( µ1 r rdt = σ 1 X t σ 2 Y t σ 1 σ 2 σ 1 Unless the respective Sharpe Ratios are equivalent, ie unless µ 1 r σ 1 µ ) 2 r dt (65) σ 2 = µ 2 r σ 2 (66) one could make a deterministic profit with zero upfront capital. Some more on Statistical Arbitrage and some neat code on pairs trading Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

122 Homework From Finan: Problems 58.1, 58.2 Problems 59.1, 59.2, 59.7 Problems 60.1, 60.2, 60.4, 60.5, 60.6 Problems 61.2, 61.7 Problems 62.1, 62.2, 62.3, 62.4, 62.8, Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

123 Spring 2009 Q10 Consider two perfectly correlated, non-dividend paying assets X, Y whose evolution is modeled by dx t = 0.08X t dt + 0.2X t dw t dy t = Y t dt 0.25Y t dw t (67) An investor wishes to synthesize a risk-free asset by allocating 1000 between X and Y. How much should she initially invest in the X? Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

124 Spring 2009 Q10 If she goes long 5 X t units of X for every 4 Y t units of Y she goes long, then her portfolio has a deterministic growth rate. Symbolically, dp t = X t dx t + Y t dy t Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

125 Spring 2009 Q10 If she goes long 5 X t units of X for every 4 Y t units of Y she goes long, then her portfolio has a deterministic growth rate. Symbolically, dp t = X t dx t + Y t dy t = 5 X t (0.08X t dt + 0.2X t dw t ) + 4 Y t (0.0925Y t dt 0.25Y t dw t ) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

126 Spring 2009 Q10 If she goes long 5 X t units of X for every 4 Y t units of Y she goes long, then her portfolio has a deterministic growth rate. Symbolically, dp t = X t dx t + Y t dy t = 5 X t (0.08X t dt + 0.2X t dw t ) + 4 Y t (0.0925Y t dt 0.25Y t dw t ) (68) = 0.77dt So, for every 9 units she spends initially, she has 5 invested in X. It follows that if she spends 1000 initially, is invested in X to obtain a risk free asset. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

127 Spring 2009 Q18 Consider two perfectly correlated, non-dividend paying assets X, Y whose evolution is modeled by X t = X 0 e 0.1t+0.2Wt with a constant risk-free rate r for all t 0. Y t = Y 0 e 0.125t+0.3Wt (69) If you are constrained to a non-arbitrage market, solve for r. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

128 Spring 2009 Q18 Recall that for constant µ, σ, ds t = µs t dt + σs t dw t S t = S 0 e (µ 1 2 σ2 )t+σw t. (70) For stock X, σ 1 = 0.2 and µ 1 = = For stock Y, σ 2 = 0.3 and µ 2 = = Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

129 Spring 2009 Q18 Recall that for constant µ, σ, ds t = µs t dt + σs t dw t S t = S 0 e (µ 1 2 σ2 )t+σw t. (70) For stock X, σ 1 = 0.2 and µ 1 = = For stock Y, σ 2 = 0.3 and µ 2 = = Since the two assets are perfectly correlated, their Sharpe ratios are equal: µ 1 r = µ 2 r σ 1 σ 2 r = (71) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

130 Practice Aug 2010 Q13 Let W t be a Brownian motion and define Which of these has zero drift? X t = 2W t 2 Y t = W 2 t t Z t = t 2 W t 2 t 0 sw s ds (72) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

131 Practice Aug 2010 Q24 Consider the SDE dx t = λ (α X t ) dt + σdw t (73) where λ, α, σ > 0 and X 0 are known. Solve for X t. Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

132 Practice Aug 2010 Q24 Consider the SDE dx t = λ (α X t ) dt + σdw t (73) where λ, α, σ > 0 and X 0 are known. Solve for X t. Answer: Using integrating factor e λt, we compute ) d (e λt X t = αλe λt dt + σe λt dw t e λt X t X 0 = α(e λt 1) + σ t 0 e λs dw s t X t = X 0 e λt + α(1 e λt ) + σe λt e λs dw s. 0 (74) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163

133 Practice Aug 2010 Q24 Consider the SDE dx t = λ (α X t ) dt + σdw t (73) where λ, α, σ > 0 and X 0 are known. Solve for X t. Answer: Using integrating factor e λt, we compute ) d (e λt X t = αλe λt dt + σe λt dw t e λt X t X 0 = α(e λt 1) + σ t 0 e λs dw s t X t = X 0 e λt + α(1 e λt ) + σe λt e λs dw s. Bonus: Can we compute lim t X t in some meaningful way? 0 (74) Albert Cohen (MSU) Math 458: Financial Economics & Insurance MSU / 163