COMP331/557. Chapter 6: Optimisation in Finance: Cash-Flow. (Cornuejols & Tütüncü, Chapter 3)

Transcription

1 COMP331/557 Chapter 6: Optimisation in Finance: Cash-Flow (Cornuejols & Tütüncü, Chapter 3) 159

2 Cash-Flow Management Problem A company has the following net cash flow requirements (in 1000 s of ): Month Jan Feb Mar Apr May Jun Net cash flow E.g.: In January we have to pay 150k and in March we get 200k. Initially we have no cash but the following possibilities to borrow/invest money: i ii iii a line of credit of up to 100k at an interest rate of 1% per month; in any one of the first three months, it can issue 90-day commercial paper bearing a total interest of 2% for the three-month period; excess funds can be invested at an interest rate of 0.3% per month. Task: We want to maximise the companies wealth in June, while fulfilling all payments. 160

3 Cash-Flow Management Problem Modelling as LP Decision Variables v.. wealth in June x i.. amount drawn from credit line in month i y i.. amount of commercial paper issued in month i z i.. excess funds in month i LP formulation: max v s.t. x 1 + y 1 z 1 = 150 x 2 + y x z 1 z 2 = 100 x 3 + y x z 2 z 3 = 200 x y x z 3 z 4 = 200 x y x z 4 z 5 = y x z 5 v = 300 x i 100 i x i, y i, z i 0 i 161

4 Cash-Flow Management Problem Modelling as LP cashflow.lp Maximize wealth: v Subject To Jan: x1 + y1 - z1 = 150 Feb: x2 + y x z1 - z2 = 100 Mar: x3 + y x z2 - z3 = -200 Apr: x y x z3 - z4 = 200 May: x y x z4 - z5 = -50 Jun: y x z5 - v = -300 Bounds 0 <= x1 <= <= x2 <= <= x3 <= <= x4 <= <= x5 <= 100 -Inf <= v <= Inf End 162

5 Cash-Flow Management Problem Modelling as LP Gurobi Output Solved in 5 iterations and 0.00 seconds Optimal objective e+01 v x1 0.0 y z1 0.0 x2 0.0 y z2 0.0 x3 0.0 y z x4 0.0 z4 0.0 x z5 0.0 Obj: Optimal Investment Strategy: Jan: Issue commercial paper for 150k. Feb: Issue commercial paper for 100k. Mar: Issue paper for 152k and invest 352k. Apr: Take excess to pay outgoing cashflow. May: Take a credit of 52k Jun: wealth 92k 163

6 The Fundamental Theorem of Asset Pricing (Cornuejols & Tütüncü, Chapter 4) 164

7 Derivative Securities Derivative Securities also known as contingent claims price depend on the value of another underlying security e.g. European call options European call option Gives the holder the right to buy (or sell, put option ) prescribed underlying security at expiration date (also known as maturity date) for prescribed amount (called strike price). 165

8 Example Payof 10 0 Payoff from option struck at Share of XYZ stock currently priced at $40. A month from today, we expect Figure 4.1 the Piecewise share linear price payoffto function either for adouble call optionor halve with equal probability. Consider a European call option on XYZ with strike price $50, which will expire in a month. Assume that interest rates for cash (borrowing or lending) are zero. What would be a fair price for the option? 4.1 The fundamental theorem of asset pricing European call option payoff function Share price of XYZ Figure 4.1 Piecewise linear payoff function for a call option r the share price of XYZ stock, which is currently valued at $40. A day, we expect the share price of XYZ to either double or halve, with lities. We also consider a European call option on XYZ with a strike hich will expire a month from today. The payoff function for the call Share price of XYZ We consider the share price of XYZ stock, which is currently valued at $40. A month from today, we expect the share price of XYZ to either double or halve, with equal probabilities. We also consider a European call option on XYZ with a strike price of $50 which will expire a month from today. The payoff function for the call is shown in Figure 4.1. We assume that interest rates for cash borrowing or lending are zero and that any amount of XYZ shares can be bought or sold with no commission: 80 = S 1 (u) S 0 = $40 20 = S 1 (d) (80 50) + = 30 and C 0 =? (20 50) + = 0 In Section we obtained a fair price of $10 for the option using a replication strategy identical and the no-arbitrage future payoff principle. Two same portfolios valueoftoday securities that have identical future payoffs under all possible realizations of the random states must have the same value today. In the example, the first portfolio is the option while the second one is the portfolio of a half share of XYZ and $10 in cash. Since we know the current value of the second portfolio, we can deduce the fair price of the option. To formalize this approach, we first give a definition of arbitrage: Definition 4.1 An arbitrage is a trading strategy that: has a positive initial cash flow and has no risk of a loss later (type A), or 166

9 Arbitrage Definition Arbitrage is a trading strategy that: i has positive initial cash flow and no risk of loss later (type A) ii requires no initial cash input, has no risk of loss and has positive probability of making profit in the future (type B) In the example: any price other than $10 for the option would lead to a type A arbitrage. Notes: Prices adjust quickly so that arbitrage opportunities cannot persist in the marked. Pricing arguments usually assume no arbitrage. 167

10 Replication (in slightly more general setting) Consider portfolio of shares of underlying security and B cash. Two possible outcomes: Up-state: S0 u + B R, where R = 1 + (risk-less interest rate) Down-state: S0 d + B R For what values of and B does the portfolio have the same payoffs C u 1 and C d 1 as the call option? S 0 u + B R = C u 1 S 0 d + B R = C d 1 Solving for and B gives = C u 1 C d 1 S 0 (u d) and B = uc d 1 dc u 1 R(u d) 168

11 Replication (in slightly more general setting) Since portfolio is worth S 0 + B today, this should also be the price for the derivative security, i.e: no arbitrage d < R < u Definition: Risk-neutral probabilities C 0 = C 1 u C 1 d + uc 1 d dc 1 u u d R(u d) = 1 ( R d R u d C 1 u + u R ) u d C 1 d p u = R d u d and p d = u R u d Remark: If price C 0 then there is arbitrage opportunity. 169

12 Generalisation of binomial (2-stage) setting Let ω 1,..., ω m be a finite set of possible states. For securities S i, i = 0,... n: S i 0.. current price (at time 0) S i 1 (ω j ).. future price (at time 1) if in state ω j S 0.. risk-less security, i.e. S0 0 = 1 and S 1 0(ω j) = R 1 for all j Risk-neutral probabilities A risk-neutral probability measure (RNPM) on the set Ω = {ω 1,..., ω m } is a vector of positive numbers p 1,..., p m such that m j=1 p j = 1 and for every security S i, i = 0,... n, S0 i = 1 m p j S i R 1(ω j ). j=1 170

13 First Fundamental Theorem of Asset Pricing Theorem 6.1 (First Fundamental Theorem of Asset Pricing). A risk-neutral probability measure exists if and only if there is no arbitrage. Proof is a simple exercise in LP duality. We will make use of the following result of Goldman and Tucker on the existence of strictly complementary optimal solutions: Theorem 6.2 (Goldman-Tucker Theorem). Consider the following primal-dual pair: min c T x s.t. Ax b max b T p s.t. A T p = c p 0 If both, the primal and the dual LP, admit a feasible solution, then there are primal and dual optimal solutions x, p such that p + (Ax b) >

14 Arbitrage Detection Using Linear Programming Scenario: Portfolio (x 1,... x n ) of European call options S 1,..., S n of same underlying security S. Payoff of portfolio: Ψ x (S 1 ) := n i=1 Ψ 70 i(s 1 )x i, where 60 Ψi (S 1 ) = (S 1 K i ) + 50, and 40 Ki is strike price of call option S i 30. cost of forming portfolio at time 0 is: 20 n i=1 S 0 i x 10 i. 0 Determine arbitrage possibility Negative cost of portfolio with non-negative payoff (type A). Zero cost and strictly positive payoff (type B). Payoff from option struck at The fundamental theorem of asset pricing European call option payoff function Share price of XYZ Figure 4.1 Piecewise linear payoff function for a call option We consider the share price of XYZ stock, which is currently valued month from today, we expect the share price of XYZ to either double or ha equal probabilities. We also consider a European call option on XYZ wit price of $50 which will expire a month from today. The payoff function fo is shown in Figure 4.1. We assume that interest rates for cash borrowing or lending are zero and amount of XYZ shares can be bought or sold with no commission: 80 = S1(u) S0 = $40 20 = S1(d) (80 50) + = and C0 =? (20 50) + = In Section we obtained a fair price of $10 for the option using tion strategy and the no-arbitrage principle. Two portfolios of securities identical future payoffs under all possible realizations of the random sta have the same value today. In the example, the first portfolio is the opti the second one is the portfolio of a half share of XYZ and $10 in cash. know the current value of the second portfolio, we can deduce the fair pr option. To formalize this approach, we first give a definition of arbitrage: Definition 4.1 An arbitrage is a trading strategy that: has a positive initial cash flow and has no risk of a loss later (type A), or requires no initial cash input, has no risk of a loss, and has a positive pro making profits in the future (type B). 172

15 Detecting arbitrage Checking for non-negative payoff: each Ψ i is piecewise-linear in S 1 with single breakpoint K i thus Ψ x is piecewise-linear in S 1 with breakpoints K 1,..., K n (assume K 1... K n ) Ψ x is non-negative in [0, ), if and only if Ψ x is Formally: non-negative at 0, non-negative at all breakpoints, and right-derivative after last breakpoint Kn is non-negative. Ψ x (0) 0 Ψ x (K j ) 0, Ψ x (K n + 1) Ψ x (K n ) 0. j, and 173

16 Detecting arbitrage Linear Program min s.t. n S0x i i (6.1) i=1 n Ψ i (0)x i 0 i=1 n Ψ i (K j )x i 0, j = 1,..., n, i=1 n (Ψ i (K n + 1) Ψ i (K n )) x i 0. i=1 174

17 Detecting arbitrage Theorem 6.3. There is no type-a arbitrage if and only if the optimal objective value of (6.1) is zero. Theorem 6.4. Suppose that there is no type-a arbitrage. Then, there is no type-b arbitrage if and only if the dual of (6.1) has a strictly positive feasible solution. 175

18 European call options - Constraint matrix Ψ i (K j ) = (K j K i ) + Constraint matrix A of (6.1) has the form A = K 2 K K 3 K 1 K 3 K K n K 1 K n K 2 K n K Theorem 6.5. Let K 1 < K 2 <... < K n denote the strike prices of European call options written on the same underlying security with the same maturity. There are no arbitrage opportunities if and only if the prices S i 0 satisfy:.. i S0 i > 0 for i = 1,..., n. ii S0 i > S 0 i+1 for i = 1,..., n 1. iii C(K i ) := S0 i defined on {K 1,..., K n } is a strictly convex function. 176

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