Valuation in illiquid markets

Transcription

1 Valuation in illiquid markets Ernst Eberlein Department of Mathematical Stochastics and Center for Data Analysis and Modeling (FDM) University of Freiburg Joint work with Dilip Madan, Martijn Pistorius, Wim Schoutens, and Marc Yor 8th World Congress of the Bachelier Finance Society Brussels June 2 6, 2014 c Eberlein, Uni Freiburg, 1

2 Valuation of Securities Liquid markets law of one price E Q [X] Illiquid markets ask price (upper price) bid price (lower price) c Eberlein, Uni Freiburg, 1

3 Example for a Two Price Evaluation Consider a public debt obligation Possibility of default or other causes of illiquidity: The lender (investor) has to discount the value of the loan or bond bid price The borrower (issuer of debt) does not contemplate default: For him the obligation is risk free ask price At maturity: convergence of bid and ask Consequences: Bid price will vary with the changes in the issuer s credit status Ask price will remain relatively steady Further fact: Value depends on the direction of the trade c Eberlein, Uni Freiburg, 2

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5 of Cashflows Outcome (cashflow) of a risky position: X random variable In perfectly liquid markets: pricing kernel given by a risk-neutral measure Q value of the position: E Q [X] position is acceptable if: E Q [X] 0 Real markets: Instead of a unique probability measure Q we have to consider a set of probability measures (scenarios) Q M E Q [X] 0 for all Q M or inf Q M E Q [X] 0 c Eberlein, Uni Freiburg, 4

6 Coherent Risk Measures Specification of M (test measures, generalized scenarios) Axiomatic theory of risk measures: desirable properties Monotonicity: X Y = ϱ(x) ϱ(y ) Cash invariance: ϱ(x + c) = ϱ(x) c Scale invariance: ϱ(λx) = λϱ(x), λ 0 Subadditivity: ϱ(x + Y ) ϱ(x) + ϱ(y ) Examples: Value at Risk (VaR) (not coherent) Tail-VaR (expected shortfall) Any coherent risk measure has a representation ϱ(x) = inf Q M E Q [X] c Eberlein, Uni Freiburg, 5

7 Operationalization Link between acceptability and concave distortions Assume the set M is convex and the operator ϱ(x) = inf E Q [X] = sup E Q [ X] Q M Q M is law invariant and comonotone (i.e. a spectral risk measure) concave distortion Ψ (i.e. a concave distribution function on [0, 1]) s.t. means then ϱ(x) = + + xdψ(f(x)) xdψ(f (x)) 0. The corresponding set of probability measures (the supporting set) is given by M = {Q P Ψ(P(A)) Q(A) Ψ(P(A)) (A A)} where Ψ(x) := 1 Ψ(1 x) c Eberlein, Uni Freiburg, 6

8 Distortion Ψ (x) γ = 2 γ = 10 γ = 20 γ = x c Eberlein, Uni Freiburg, 7

9 Families of Distortions (1) Consider families of distortions (Ψ γ ) γ 0 γ stress level Example: MIN VaR Ψ γ (x) = 1 (1 x) 1+γ (0 x 1, γ 0) Statistical interpretation: Let γ be an integer, then ϱ γ(x) = E(Y ) where Y law = min{x 1,..., X γ+1 } and X 1,..., X γ+1 are independent draws of X c Eberlein, Uni Freiburg, 8

10 Further examples: MAX VaR Families of Distortions (2) Ψ γ (x) = x 1 1+γ (0 x 1, γ 0) Statistical interpretation: ϱ γ(x) = E[Y ] where Y is a random variable s.t. max{y 1,..., Y γ+1 } law = X and Y 1,..., Y γ+1 are independent draws of Y. Combining MIN VaR and MAX VaR: MAX MIN VaR Ψ γ (x) = (1 (1 x) 1+γ ) 1 1+γ (0 x 1, γ 0) Interpretation: ϱ γ(x) = E[Y ] with Y s.t. max{y 1,..., Y γ+1 } law = min{x 1,..., X γ+1 } c Eberlein, Uni Freiburg, 9

11 Families of Distortions (3) Distortion used: MIN MAX VaR ) Ψ γ (x) = 1 (1 x 1+γ 1 1+γ (0 x 1, γ 0) ϱ γ(x) = E[Y ] with Y s.t. Y law = min{z 1,..., Z γ+1 }, max{z 1,..., Z γ+1 } law = X c Eberlein, Uni Freiburg, 10

12 Families of Distortions (4) Ψ γ (x) γ = 0.50 γ = 0.75 γ = 1.0 γ = x c Eberlein, Uni Freiburg, 11

13 Marking Assets and Liabilities Assets: Cash flow to be received X 0 Largest value b(x) s.t. X b(x) is acceptable b(x) = inf Q M E Q [X] Bid price or Lower price Liabilities: Cash flow to be paid out X 0 Smallest value a(x) s.t. a(x) X is acceptable a(x) = sup E Q [X] Q M Ask price or Upper price c Eberlein, Uni Freiburg, 12

14 Bid Price of a cash flow X: Ask Price of a cash flow X: Explicit Pricing b(x) = a(x) = Examples: Calls and Puts bc(k, t) = ac(k, t) = bp(k, t) = ap(k, t) = of X b(x) xdψ(f X (x)) of a(x) X K K K 0 K 0 xdψ(1 F X ( x)) ( 1 Ψ(FSt (x)) ) dx Ψ(1 F St (x))dx ( 1 Ψ(1 FSt (x)) ) dx Ψ(F St (x))dx c Eberlein, Uni Freiburg, 13

15 representation Eberlein, Glau (2013) (to appear in Applied Mathematical Finance) (L t) t 0 time-inhomogenous Lévy process (PIIAC) ( t E[exp(iξL t)] = exp 0 ) θ s(iξ)ds θ s(iξ) = ib sξ 1 2 σsξ2 + (e iξy 1 iξh(y))f s(dy) R usual integrability assumptions G t infinitesimal generator A t = G t pseudo differential operator (PDO) with symbol A t c Eberlein, Uni Freiburg, 14

16 A t(ξ) = θ t( iξ) (ξ R) (Au = F 1 (AF(u)) (u C o )) Pricing of derivatives with payoff function g (e.g. call option g(x) = (S 0 e x K ) + ) tu + A T t u = 0 (r = 0) u(0) = g Stochastic representation of the solution: u(t t, x) = E[g(L T L t + x)] c Eberlein, Uni Freiburg, 15

17 Solution of the PIDE tu + A T t u = 0 u(0) = g Desirable properties in the case of models for finance: unbounded domains (domain is the range of the process L) initial condition should cover a large range of options (literature: polynomial boundedness, Lipschitz-continuity,... ) variational solution (numerical calculation) dampening of the payoff g L 2 η := {u L 1 loc x u(x)e ηx L 2 } < u, v > L 2 η := u(x)v(x)e 2ηx dx Fourier transform with weights F η(ϕ) := e η F(ϕe η ) (ϕ L 2 η resp. ϕ S η) R Initial condition: g L 2 η c Eberlein, Uni Freiburg, 16

18 Explicit solution of the PIDE Transformation into weighted Fourier transforms Equivalent to ODE Solution or F η( tu) + F η(a T t u) = 0 F η ( L 2 η lim t 0 u(t) ) = F η(g) tf η(u(t))(ξ) + A T t (ξ iη)f η(u(t))(ξ) = 0 (a.e. ξ R) F η(u)(t = 0) = F η(g) ( T ) F η(u(t))(ξ) = F η(g)(ξ) exp A s(ξ iη)ds T t (weak solution) u(t, x) = e ηx 2π Stochastic representation R e iξ(x+iη) F η(g)e T T t A s(ξ iη)ds dξ u(t t, x) = E[g(L T L t + x)] c Eberlein, Uni Freiburg, 17

19 Theory Underlying uncertainty given by a pure jump Lévy process (X t) 0 t T Specified by: drift term α, Lévy measure k(y)dy (y 0) Example: Variance gamma Note that k(y) = C y (exp( G y )1 {y<0} + exp( M y )1 {y>0} ) R y 2 k(y)dy < Infinitesimal generator L of the process Lu(x) = α u ( x (x) + u(x + y) u(x) u ) R x (x)y k(y)dy c Eberlein, Uni Freiburg, 18

20 Variance gamma density y C = 5 G = 1 M = x c Eberlein, Uni Freiburg, 19

21 values of GH (-0.5,100,0,1,0.1) Levy process GH Levy process with marginal densities t c Eberlein, Uni Freiburg, 20

22 Valuation of Financial Contracts Consider a contract which pays φ(x t) at time t Denote by u(x, t) its time zero value when X 0 = x u(x, t) = E [ e rt φ(x t) X 0 = x ] risk-neutral value for constant interest rate r Assume that φ is such that applies u(x, t) is the solution of the partial integro-differential equation (PIDE) u t = L(u) ru with boundary condition u(x, 0) = φ(x) c Eberlein, Uni Freiburg, 21

23 G-Expectations Using Distortions (1) Integral part of the PIDE R (u(x + y, t) u(x, t) u x(x, t)y) y 2 k(y)dy R } y {{ 2 } =:Y x,t where g(y) = y 2 k(y) y R 2 k(y)dy g is a probability density g(y)dy Define the distribution function F Yx,t (v) = g(y)dy where A(x, t, v) = {y Y x,t v} A(x,t,v) c Eberlein, Uni Freiburg, 22

24 G-Expectations Using Distortions (2) Integral part of the PIDE is now Distorted expectation R R vdf Yx,t (v) vdψ(f Yx,t (v)) which by decomposition can be written as 0 Ψ(P g (Y x,t v))dv + Define the new (distorted) operator 0 (1 Ψ(P g (Y x,t v)))dv G QV u(x) = α u 0 x (x) Ψ(P g (Y x,t v))dv+ and solve the (distorted) PIDE u t = G QV (u) ru 0 (1 Ψ(P g (Y x,t 0)))dv c Eberlein, Uni Freiburg, 23

25 Alternative G-Expectation Approach Truncation of the Lévy measure ( u(x + y, t) u(x, t) ux(x, t)y ) k(y)dy { y ɛ} Definition of a probability density h(y) via { y ɛ} ( (u(x + y, t) u(x, t) u x(x, t)y) k(y)dy { y ɛ} }{{} =:Ỹx,t ) h(y)dy where h(y) = k(y) { y ɛ} k(y)dy 1 { y ɛ} The distorted operator is now G NL u(x) = α u 0 x (x) Ψ(P h (Ỹx,t v))dv+ h (1 Ψ(P (Ỹx,t v)))dv 0 c Eberlein, Uni Freiburg, 24

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28 The discounted variance gamma model γ p(t), γ n(t) two independent standard gamma processes Driving process X(t) = b p, c p, b n, c n > 0 t Underlying discounted price process 0 t b pe rs dγ p(c ps) b ne rs dγ n(c ns) 0 scale and shape parameters of the undiscounted gamma processes M(t) = exp(x(t) + ω(t)) where exp(ω(t)) = (E[exp(X(t))]) 1 uniformly integrable martingale with a well-defined limit M( ) = exp(x( ) + ω( )) c Eberlein, Uni Freiburg, 27

29 Valuation Consider now a claim promising for very large T, F(M(T )) F(M( )), where the payout is expressed in time zero dollars (F nice function) Value of the claim at time t martingale Observe now that X( ) = X(t) + w F (t) = E[F(M( )) F t] (d) = X(t) + e rt Y t b pe ru dγ p(c pu) t b ne ru dγ n(c nu) for an independent random variable Y X( ) w F (t) = H(X(t), e rt ) c Eberlein, Uni Freiburg, 28

30 Prices (1) Martingale condition on w F (t) (write v = e rt ) rvh v + PIDE with boundary condition (H(X + y, v) H(X, v))k(y, v)dy = 0 H(X, 0) = F(exp(X( ) + ω( ))) where ( k(y, v) = cp y exp y ) ( 1 {y>0} + cn b pv y exp y ) 1 {y<0} b nv c Eberlein, Uni Freiburg, 29

31 Rewrite the PIDE where rvh v = + Prices (2) (H(X + y, v) H(X, v)) + y 2 k(y, v)dy df QV (y) y 2 1 F QV (a) = + y 2 k(y, v)dy a Bid price is the solution of the distorted PIDE y 2 k(y, v)dy rvh v = + (H(X + y, v) H(X, v)) + y 2 k(y, v)dy dψ γ (F QV (y)) y 2 Ask price: Negative of the bid price of the negative cash flow c Eberlein, Uni Freiburg, 30

32 Implementation Details Risk neutral parameters from S & P 500 r = b p = c p = Actually solved for a PIDE in M(t): b n = c n = ( ( G(M(v), v) = M(v) exp ω( ) ω ln v )) φ Y ( iv) r c Eberlein, Uni Freiburg, 31

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35 Two Price Valuation of (1) Cumulated loss process L(t): e.g. compound Poisson (arrival rate λ) Loss sizes are iid γ-distributed (scale and shape parameters ζ and κ) Consider the value process in time zero dollars V (t) = E t [ Let X(t) be the discounted losses to date Rewrite 0 X(t) = 0 t e rs dl(s) = X(t) + e rt 0 ] e rs dl(s) e rs dl(s) where Y is an independent copy of 0 e rs dl(s) t e r(s t) dl(s) (d) = X(t) + e rt Y V (t) = H(X(t), e rt ) c Eberlein, Uni Freiburg, 34

36 Two Price Valuation of (2) Applying Itô s formula and using the martingale condition (where we replace t by v = e rt ) rvh v = 0 (H(x + w, v) H(X, v))k(w, v)dw where k(w, v) is related to the Lévy system for X(t) k(w, v) = λ ( ζ ) κw ( κ 1 exp ζ ) Γ(κ) v v w Risk neutral price is the solution of this PIDE Bid price is the solution of the distorted PIDE How to distort a measure integral? c Eberlein, Uni Freiburg, 35

37 Measure Distortions (1) Consider a possibly infinite measure µ with tails of finite measure and Rewrite this as m = 0 m = + µ(v(y) x)dx + v(y)µ(dy) < Distorted measure integrals 0 m = Γ +(µ(v(y) x))dx µ(v(y) > x)dx Γ (µ(v(y) > x))dx for functions Γ +, Γ : R + R + Γ ±(0) = 0, monotone increasing, resp. concave and convex, bounded below and above by the identity function c Eberlein, Uni Freiburg, 36

38 Measure Distortions (2) Natural Candidates: Γ +(x) = x + α(1 e cx ) 1 1+γ + Γ (x) = x β c (1 e c(1+γ )x ) (Γ + derived from maxvar, Γ from minvar) c Eberlein, Uni Freiburg, 37

39 Measure Distortions y Γ+ Γ x c Eberlein, Uni Freiburg, 38

40 Bid Price for the Discounted Cumulated Loss Process Distorted measure integral for positive variables m = Rewrite this as (integration by parts) m = 0 0 Γ (µ(χ > x))dx xdγ (µ(χ > x)) Now choose χ(y) = H(X + y, v) H(X, v), µ(dy) = k(y, v)dy Bid price is the solution of rvh v = 0 xdγ (µ(χ > x)) For the ask price one has to consider the integral rvh v = 0 xdγ +(µ(χ > x)) c Eberlein, Uni Freiburg, 39

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42 Eberlein, E., Gehrig, T., Madan, D.: Pricing to acceptability: With applications to valuing one s own credit risk. The Journal of Risk 15(1) (2012) Eberlein, E., Madan, D.: Unbounded liabilities, capital reserve requirements and the taxpayer put option. Quantitative Finance 12 (2012) Eberlein, E., Madan, D., Pistorius, M., Schoutens, W., Yor, M.: Two price economies in continuous time. Annals of Finance 10 (2014) Eberlein, E., Madan, D., Pistorius, M., Yor, M.: Bid and ask prices as non-linear continuous time G-expectations based on distortions. Mathematics and Financial Economics (2014) (to appear) c Eberlein, Uni Freiburg, 41