Financial Market Models. Lecture 1. One-period model of financial markets & hedging problems. Imperial College Business School

Transcription

1 Financial Market Models Lecture One-period model of financial markets & hedging problems

2 One-period model of financial markets a 4 2a 3 3a 3 a 3 -a 4 2

3 Aims of section Introduce one-period model with finite number of states and the basic asset-pricing terminology Formulate hedging problem and try to solve it Explain the concepts of complete and incomplete markets Explain the role of matrix inverse in hedging and pricing Introduce state prices and discuss two different methods for asset pricing: ) by replication 2) using a pricing kernel 3

4 Introductory example There are four assets available ) Uncertain stock price 3 scenarios Stock value tomorrow = 3, 2, or with probability /2, /6,/3 2) Risk-free asset with value tomorrow 3) Two derivative securities options on the stock 3.) Call option # struck at K =.5 3.2) Call option #2 struck at K = Task: sell option #2, and to reduce risk exposure construct a hedging portfolio consisting of stock and riskfree asset 4

5 Uncertainty in one-period model Two dates: today and tomorrow Value of all securities known today The tomorrow s payoffs are uncertain Organization of uncertainty: Finite number of scenarios Each scenario known in detail today Probability of each scenario known today 5

6 Model asset payoffs Model stock and derivative payoffs 6

7 Payoffs as vectors Risk free asset : aa = Stock : aa 2 = Options : aa 3 = , aa 4 = 2 0 7

8 Graphical representation of the payoffs Payoffs in state-space form 8

9 Create vectors in Matlab a = ones(3,) or a=[;;]; a2 = [3;2;]; a3 = [.5;0.5;0]; a4 = [2;;0]; 9

10 Scalar multiplication: Leverage Example: Buy two units of option # 2aa 3 = = 3 0 Operation on securities/vectors Matlab command: 2 aa 3 0

11 Operation on securities/vectors (cont.) Addition: Portfolios Example: Buy two units of option #, sell one unit of option #2 2aa 3 aa 4 = = 0 0 Matlab command : 2 aaa aaa

12 Matrix as a collection of securities/vectors It is common to work with several vectors at once and it is natural to form a matrix Vectors: aa = aa 2 = 3 2 aa 3 = aa 4 = 2 0 A matrix: Matlab command: A = [a a2 a3 a4] 2

13 Matrix operation: Transposition Sometimes we need a row vector rather than a column vector. Achieved by transposition of a column vector xx =, xx xx 2 xx = xx xx 2 xx nn xx nn Conversely, transposition of a row vector gives a column vector 3

14 Matrix notation Matrix notation: MM Rmm nn Where m is the number of rows and n is the number of columns The element in the i-th row and j-th column is denoted MM iiii The entire j-th column is denoted MM jj The entire i-th row is denoted MM ii 4

15 Matrix notation (cont.) MM Rmm nn MM = MM = MM MM 22 2 MM 2nn MM 2 MM nn MM mmm MM mmm MM mmmm MM MM 2 = MM MM 2 MM nn MM mm 5

16 Matrix operation: Transposition MM MM iiii MM MM jjjj MM R mm nn MM Rnn mm Example ) xx = 2) AA = , xx = , AA = 6

17 Matrix operation: Multiplication UU R mm kk aaaaaa VV R kk nn UUUU Rmm nn UV= Example UU UU 2 VV VV 2 VV nn = UU mm UU = 2 3 4, VV = UU VV UU VV 2 UU VV nn UU 2 VV UU 2 VV 2 UU 2 VV nn UU mm VV UU mm VV 2 UU mm VV nn UUUU =? 7

18 Properties of matrix multiplication Not commutative UUUU VVVV Associative UUUU WW = UU(VVVV) Transposition reverses order of multiplication! UUUU = VV UU 8

19 Matrix and its application to portfolios Example: Suppose we sell 2 units of option #, one unit of option #2, buy two units of stocks and borrow one unit of bond. What are the payoffs of this portfolio? Given: bbbbbbbb =, ssssssssss = 3 2, oooooooobbaa# = and option#2 = 2 0 Assuming prices of securities are SS bbbbbbbb =, SS ssssssssss = 3, SS oooossiibbnn# =, SS oooossiibbnn#2 = 2, what is the price of this portfolio? 9

20 Hedging Problem AAAA = bb AA denotes payoff matrix of basis assets xx denotes portfolio of basis assets bb denotes payoff of a focus asset Bank wants to issue a security bb To offset risk, it separately buys hedging portfolio xx The risk is reduced to AAAA bb If AAAA bb = 0, then the bank s position is perfectly hedged The bank will price the security at SS xx + overheads and risk premium. 20

21 Hedging Problem (cont.) Example: Given: bb = 2 0, aaa =, aaa = 3 2, aaa = Your client wants to buy a security with payoff bb, knowing there are assets aaa, aaa, aaaaaa aaa traded in the markets. What is the strategy that gives you a perfect hedge? And how much you would charge the client if SS aaa =.05 aaaaaass aa2 = 2?

22 Hedging Problem (cont.) Solution: 22

23 Hedging Complications If inverse of AA exists: xx = AA bb is a perfect hedge and the solution is also unique Now consider xx = No solution, Inverse does not exist 2 xx = 0 Solution exists, inverse does not Consider

24 Linear independence and redundant assets Some complications in the hedging problem are caused by redundant assets An asset is redundant if it can be replicated by other assets Securities AA,, AA nn are linearly independent if none of them is a portfolio payoff of the remaining n- securities 24

25 Dimension of marketed subspace Marketed subspace = all portfolio payoffs Ax generated by all possible basis assets x Mathematically SSSSSSSS(AA,, AA nn ) Each linearly independent asset adds a new dimension to the marketed subspace Maximum number of lin. ind. assets = dimension of marketed subspace Dimensionality Theorem: The lin. ind. assets can be chosen in many ways, but their number is always the same = dimension 25

26 Hedging in a complete market without redundant basis assets Complete market # of linearly independent basis assets = # of states Theorem: Suppose we have m states and a complete market with m basis assets. Then the payoff matrix is invertible and the hedging portfolio for any focus asset b is given by xx = AA bb Suppose prices of basis assets are stored in vector S. Under frictionless trading the only possible price of the focus asset equals SS xx = SS AA bb 26

27 Find dimension of marketed subspace Concepts Suppose AA,, AA kk are linearly independent. Then either AA kk+ is redundant or AA,, AA kk+ are linearly independent With m states, there cannot be more than m linearly independent assets. How we do it Sort securities into two baskets Linearly independent Redundant 27

28 Example Which of the following assets are linearly independent and redundant assets? Is this market complete? AA =, AA 2 = 3 2, AA 3 = , AA 4 =

29 Dimension and matrix rank Maximum number of linearly independent columns in a matrix A is called rank, r(a) If A is a payoff matrix of basis assets then r(a) = dimension of marketed subspace Facts about rank rr AA AA = rr AA rr AAAA min rr AA, rr BB rr AA = rr(aa ) If AA is mm nn then rr AA min mm, nn When rr AA = min mm, nn, we say A has full rank Square matrices with full rank are invertible (non-singular matrices) 29

30 Identity matrix and Arrow- Debreu securities Arrow-Debreu (elementary) security for state j, denoted ee jj, has payoff in state j and payoff zero in all other states With m scenarios stacking all elementary securities into a matrix gives an m x m identity matrix 30

31 Matrix Inverse For every square matrix A with full rank there is a matrix BB such that AAAA = AAAA = II Matrix BB is unique, it is called the inverse to matrix AA and it is commonly denoted by AA Facts: If CC, DD are invertible then (CCCC) = DD CC (AA ) = AA 3

32 Inverse matrix and hedging portfolios Interpretation of inverse matrix Let split AAAA = II by columns: This looks like AAAA = ee jj Therefore jj ttt column of AA gives replicating portfolio to Arrow-Debreu security ee jj Example: if AA = AA = replicated by portfolio and knowing that, show that Arrow-Debreu security ee jj can be 32

33 State prices State price j (ψψ jj ) is the price of an Arrow-Debreu security ee jj Vector ψψ is called the state price vector Assuming complete market and no redundant assets State price vector ψψ = SS AA, where SS is the vector of prices of the basis assets and A is the matrix of the payoffs of the basis assets Proof : 33

34 Pricing formulae Suppose A is invertible (square, full rank) Complete market, no redundant basis assets Perfect hedge AAAA = bb Two ways to find the price of focus asset b. By replication: Focus asset price = SS xx 2. Using state price (pricing kernel): Focus asset price = ψψ bb Proof of the second formula: 34

35 Example What is the price of bb = prices S= By replication: 2 0 given basis assets AA = with By state price: 35

36 Hedging in an incomplete market a 4 2a 3 3a 3 a 3 -a 4 36

37 Aims of this section Explain how to compute replicating portfolios in an incomplete market 37

38 Hedging formula for incomplete markets, no redundant basis assets r(a) = n < m, fewer assets than states Ax = b () multiply by A * from the left A * Ax = A * b (2) A * A is square with full rank, it has an inverse x = (A * A) - A * b (3) x solves (2) but does it solve ()? Hedging error = Ax-b = A(A * A) - A * b-b If hedging error = 0 then solution exists and is given by (3), otherwise there is no solution 38

39 Example: Incomplete markets without redundant basis assets Is there a perfect hedge of a focus asset b considering that we can trade portfolio A? 39

40 Hedging problem with redundant basis assets Redundant basis assets do not affect existence of a solution, they merely add free parameters to the solution A = A A 2, A 2 = A C, r(a)=r(a ), x = x x 2 If A is square, market is complete x = (A ) - b Cx 2 If A is not square, market is not complete x = (A * A ) - A * b Cx 2 Hedging error = A (A * A ) - A * b b x 2 represents the free parameters in the solution Example 2.3 pp

41 The Least-Squares Hedge In practice most markets are incomplete rr AA = nn < mm, fewer assets than number of states This means we cannot always find a perfect hedge (replication) for a focus asset bb mathematically, AAAA = bb cannot always be solved. Instead, we would like to find the best approximate hedge according to some criterion 4

42 The Least-Squares Hedge Replication error : εε = AAAA bb Criterion: Minimize the Sum of Squared Replication Errors SSSSSSSS = εε 2 + εε εε mm 2 Optimal hedge = least-squares hedging portfolio xx = (AA AA) AA bb 42

43 Least Squares Hedge Example Focus asset bb = 2 3 Basis assets: AA = 0 aaaaaa AA 2 = 0 0 What is the least-squares hedge of the focus asset? Solution: 43

44 Least squares hedge - Geometry Optimal criteria Minimize εε εε (square length of vector εε) Point of minimal distance of AAAA from bb: - εε must be orthogonal to all vectors in marketed subspace A - Mathematically AA εε = 0 - AA bb AAAA = 0 xx = (AA AA) AA bb 44

45 Minimizing the Expected Squared Replication Error (ESRE) Not all scenarios are equally likely: we need criterion putting more weight on likely scenarios: EESSSSSS = pp εε 2 + pp 2 εε pp mm εε mm 2 The optimal hedge is : xx = (AA AA ) AA bb where AA ii pp ii AA ii and bb ii pp ii bb ii 45

46 Working out the minimal ESRE hedge Proof Transform ESRE into SSRE 46

47 Example 47