and contingent claim valuation in illiquid markets Teemu Pennanen King s College London Ari-Pekka Perkkiö Technische Universität Berlin 1 / 35
In most models of mathematical finance, there is at least one perfectly liquid asset that can be bought and sold in unlimited amounts at a fixed unit price. transaction costs (if any) are proportional to traded quantities. In practice, however, much of trading consists of exchanging sequences of cash-flows (coupon-paying bonds, dividends, swaps,...) unit prices depend nonlinearly on traded amounts. We use elementary convex analysis to extend certain fundamental theorems on optimal investment and contingent claim valuation to illiquid markets and general swap contracts. 2 / 35
Illiquidity Hodges, Neuberger, Optimal replication of contingent claims under transaction costs, Rev. Fut. Markets, 1989. Dalang, Morton, Willinger, Equivalent martingale measures and no-arbitrage in stochastic securities market models, Stoch. and Stoch. Rep., 1990. Artzner, Delbaen and Koch-Medona, Risk measures and efficient use of capital, Astin Bulletin, 2009. Hilli, Koivu, Pennanen, Cash-flow based valuation of pension liabilities, European Actuarial Journal, 2011. Pennanen, Superhedging in illiquid markets, Math. Finance, 2011 Pennanen, and contingent claim valuation in illiquid markets, Finance and Stochastics, to appear. Pennanen, Perkkiö, Stochastic programs without duality gaps, Mathematical Programming, 2012. Pennanen, Perkkiö, Convex duality in optimal investment and contingent claim valuation in illiquid markets, manuscript. 3 / 35
Limit order book of TDC A/S on 12 January 2005 at 13:58:19.43 in Copenhagen Stock Exchange Bid Ask Price Quantity Price Quantity 238.75 140 239 3700 238.75 600 239 1000 238.75 3300 239 5000 238.75 2000 239 1000 238.5 10000 239 1000 238.5 3900 239 2500 238.5 15000 239 6600 238.5 1500 239.25 10000 238.25 10000 239.25 2500 238.25 1000 239.25 3000 238.25 3500 239.5 600 238.25 10000 239.5 5000 238.25 200 239.5 800.... 4 / 35
The corresponding marginal price curve. Negative quantity corresponds to a sale. 242 241 PRICE 240 239 238 237 236-100000 -50000 0 50000 QUANTITY 5 / 35
Consider a financial market where a finite set J of assets can be traded at t = 0,...,T. Let (Ω,F,(F t ) T,P) be a filtered probability space. The cost (in cash) of buying a portfolio x R J at time t in state ω will be denoted by S t (x,ω). We will assume that S t (,ω) is convex with S t (0,ω) = 0, S t (x, ) is F t -measurable. Such a sequence (S t ) will be called a convex cost process. 6 / 35
Example 1 (Liquid markets) If s = (s t ) T is an (F t ) T -adapted R J -valued price process, then the functions S t (x,ω) = s t (ω) x define a convex cost process. Example 2 (Jouini and Kallal, 1995) If (s a t) T and (s b t) T are (F t ) T -adapted with s b s a, then the functions { s a S t (x,ω) = t(ω)x if x 0, s b t(ω)x if x 0 define a convex cost process. 7 / 35
Example 3 (Çetin and Rogers, 2007) If s = (s t ) T is an (F t ) T -adapted process and ψ is a lower semicontinuous convex function on R with ψ(0) = 0, then the functions S t (x,ω) = x 0 +s t (ω)ψ(x 1 ) define a convex cost process. Example 4 (Dolinsky and Soner, 2013) If s = (s t ) T is (F t ) T -adapted and G t (x, ) are F t -measurable functions such that G t (,ω) are finite and convex, then the functions S t (x,ω) = x 0 +s t (ω) x 1 +G t (x 1,ω) define a convex cost process. 8 / 35
We allow for portfolio constraints requiring that the portfolio held over (t,t+1] in state ω has to belong to a set D t (ω) R J. We assume that D t (ω) are closed and convex with 0 D t (ω). {ω Ω D t (ω) U } F t for every open U R J. 9 / 35
Models where D t (ω) is independent of (t,ω) have been studied e.g. in [Cvitanić and Karatzas, 1992] and [Jouini and Kallal, 1995]. In [Napp, 2003], D t (ω) = {x R d M t (ω)x K}, where K R L is a closed convex cone and M t is an F t -measurable matrix. General constraints have been studied in [Evstigneev, Schürger and Taksar, 2004], [Rokhlin, 2005] and [Czichowsky and Schweizer, 2012]. 10 / 35
Let c M := {(c t ) T c t L 0 (Ω,F t,p)} and consider the problem T minimize V t (S t ( x t )+c t ) over x N D N D = {(x t ) T x t L 0 (Ω,F t,p;r J ), x t D t, x T = 0}, V t : L 0 R are convex, nondecreasing and V t (0) = 0. Example 5 If V t = δ L 0 for t < T, the problem can be written minimize V T (S T ( x T )+c T ) over x N D subject to S t ( x t )+c t 0, t = 0,...,T 1. 11 / 35
Example 6 (Markets with a numeraire) When S t (x,ω) = x 0 + S t ( x,ω) and D t (ω) = R D t (ω), the problem can be written as ( T minimize V T S t ( x t )+ T c t ) over x N D. When S t ( x,ω) = s t (ω) x, T S t ( x t ) = T s t x t = T 1 x t s t+1. 12 / 35
We denote the optimal value function by T ϕ(c) = inf V t (S t ( x t )+c t ). x N D Note that ϕ(c) = inf d C V(c d), where V(c) := T V t(c t ) and C := {c M x N D : S t ( x t )+c t 0 t} is the set of claims that can be superhedged without cost. 13 / 35
The recession cone C = {c M c+αc C c C, α > 0} of C consists of claims that can be superhedged without cost in unlimited amounts. If C is a cone, then C = C. Lemma 7 The function ϕ : M R is convex and ϕ( c+c) ϕ( c) c M, c C. In particular, ϕ is constant on the linear space C ( C ). 14 / 35
Example 8 (The classical model) Consider the classical perfectly liquid market model where C = {c M x N : T c t T 1 x t s t+1 } and C = C. We have c C ( C ) if there is an x N such that T c t = T 1 x t s t+1. The converse holds under the no-arbitrage condition. 15 / 35
of contingent claims In incomplete markets, the hedging argument for valuation of contingent claims has two natural generalizations: reservation value: How much capital do we need to cover our liabilities at an acceptable level of risk? indifference price: What is the least price we can sell a financial product for without increasing our risk? The former is important in accounting, financial reporting and supervision (and in the Black Scholes Merton model). The latter is more relevant in trading. In complete markets, reservation values and indifference prices coincide. 16 / 35
Reservation value We define the reservation value for a liability c M by π 0 (c) = inf{α R ϕ(c αp 0 ) 0} where p 0 = (1,0,...,0). π 0 can be interpreted much like a risk measure in [Artzner, Delbaen, Eber and Heath, 1999]. However, we have not assumed the existence of a cash-account so π 0 is defined on sequences of cash-flows. If V = δ M, we have ϕ = δ C and π 0 (c) = π 0 sup(c) := inf{α R c αp 0 C}. Let π 0 inf (c) = π0 sup( c). 17 / 35
Reservation value Theorem 9 The reservation value π 0 is convex and nondecreasing with respect to C. We have π 0 π 0 sup and if π 0 (0) 0, then π 0 inf(c) π 0 (c) π 0 sup(c) with equalities throughout if c αp 0 C ( C) for α R. π 0 is translation invariant : if c M is replicable with initial capital α: c αp 0 C ( C ), then π 0 (c+c ) = π 0 (c)+α. In complete markets, c αp 0 C ( C) for some α R so π 0 (c) is independent of preferences and views. 18 / 35
Swap contracts In a swap contract, an agent receives a sequence p M of premiums and delivers a sequence c M of claims. Examples: Swaps with a fixed leg : p = (1,...,1), random c. In credit derivatives (CDS, CDO,...) and other insurance contracts, both p and c are random. Traditionally in mathematical finance, p = (1,0,...,0) and c = (0,...,0,c T ). Claims and premiums live in the same space M = {(c t ) T c t L 0 (Ω,F t,p;r)}. 19 / 35
Swap contracts If we already have liabilities c M, then π( c,p;c) := inf{α R ϕ( c+c αp) ϕ( c)} gives the least swap rate that would allow us to enter a swap contract without worsening our financial position. Similarly, π b ( c,p;c) := sup{α R ϕ( c c+αp) ϕ( c)} = π( c,p; c) gives the greatest swap rate we would need on the opposite side of the trade. When p = (1,0,...,0) and c = (0,...,0,c T ), we get a nonlinear version of the indifference price of [Hodges and Neuberger, 1989]. 20 / 35
Swap contracts Define the super- and subhedging swap rates, π sup (p;c) = inf{α c αp C }, π inf (p;c) = sup{α αp c C }. If C is a cone and p = (1,0,...,0), we recover the super- and subhedging costs π 0 sup and π 0 inf. Theorem 10 If π( c,p;0) 0, then π inf (p;c) π b ( c,p;c) π( c,p;c) π sup (p;c) with equalities if c αp C ( C ) for some α R. Agents with identical views, preferences and financial position have no reason to trade with each other. Prices are independent of such subjective factors when c αp C ( C ) for some α R. If in addition, p = p 0, then swap rates coincide with reservation values. 21 / 35
Swap contracts Example 11 (The classical model) Consider the classical perfectly liquid market model where C = {c M x N : T c t T 1 x t s t+1 } and C = C. The condition c αp C ( C ) holds if there exist α R and x N such that T c t = α T p t + T 1 x t s t+1. The converse holds under the no-arbitrage condition. 22 / 35
Given a market model (S,D), let S t (x,ω) = sup α>0 S t (αx,ω) α and D t (ω) = α>0αd t (ω). If S is sublinear and D is conical, then S = S and D = D Theorem 12 Assume that V(c) = E T V t(c t ), where V t are bounded from below. If the cone L := {x N D S t ( x t ) 0} is a linear space, then C is closed and ϕ is lower semicontinuous in L 0. The lower bound can be replaced by RAE; [Perkkiö, 2014]. 23 / 35
Example 13 In the classical perfectly liquid market model L = {x N s t x t 0, x T = 0}, so the linearity condition becomes the no-arbitrage condition and we recover the key lemma from [Schachermayer, 1992]. Example 14 When D R J, the linearity condition becomes the robust no-arbitrage condition: there exists a positively homogeneous arbitrage-free cost process S with S t (x,ω) S t (x,ω) x R J, S t (x,ω) < S t (x,ω) x / lins t (,ω); see [Schachermayer, 2004]. 24 / 35
The linearity condition can hold even under arbitrage. Example 15 If S t (x,ω) > 0 for x / R J, then L = {0}. Example 16 In [Çetin and Rogers, 2007], S t (x,ω) = x 0 +s t (ω)ψ(x 1 ) and S t (x,ω) = x 0 +s t (ω)ψ (x 1 ). When infψ = 0 and supψ = we have ψ = δ R, so the condition in Example 15 holds. Example 17 If S t (,ω) = s t (ω) x for a componentwise strictly positive price process s and D t (ω) R J + (infinite short selling is prohibited), then L = {0}. 25 / 35
Proposition 18 Assume that ϕ is proper and lower semicontinuous. The conditions ϕ (p 0 ) > 0, π 0 (0) >, π 0 (c) > for all c M, are equivalent and imply that π 0 is proper and lower semicontinuous on M and that the infimum π 0 (c) = inf{α ϕ(c αp 0 ) 0} is attained for every c M. 26 / 35
Proposition 19 Assume that ϕ is proper and lower semicontinuous. Then, for every c domϕ and p M, the conditions ϕ (p) > 0, π( c,p;0) >, π( c,p;c) > for all c M, are equivalent and imply that π( c,p; ) is proper and lower semicontinuous on M and that the infimum π( c,p;c) = inf{α ϕ( c+c αp) ϕ( c)} is attained for every c M. 27 / 35
Let M p = {c M c t L p (Ω,F t,p;r)}. The bilinear form T c,y := E c t y t puts M 1 and M in separating duality. The conjugate of a function f on M 1 is defined by f (y) = sup c M 1 { c,y f(c)}. If f is proper, convex and lower semicontinuous, then f(y) = sup y M { c,y f (y)}. 28 / 35
We assume from now on that T V(c) = E V t (c t ) for convex random functions V t : R Ω R with V t (0) = 0. Theorem 20 If S t (x, ) L 1 for all x R J, then ϕ (y) = V (y)+σ C (y) where V (y) = E T V t (y t ) and σ C (y) = sup c C c,y. Moreover, T σ C (y) = inf v N 1E [(y t S t ) (v t )+σ Dt (E[ v t+1 F t ])] where the infimum is attained for all y M. 29 / 35
Example 21 If S t (ω,x) = s t (ω) x and D t (ω) is a cone, C = {y M E[ (y t+1 s t+1 ) F t ] D t}. Example 22 If S t (ω,x) = sup{s x s [s b t(ω),s a t(ω)]} and D t (ω) = R J, then C = {y M ys is a martingale for some s [s b,s a ]}. Example 23 In the classical model, C consists of positive multiples of martingale densities. 30 / 35
Theorem 24 Assume the linearity condition, the Inada condition V t = δ R and that p 0 / C and infϕ < 0. Then π 0 (c) = sup y M { c,y σ C (y) σ B (y) y 0 = 1}, where B = {c M 1 V(c) 0}. In particular, when C is conical and V is positively homogeneous, π 0 (c) = sup y M { c,y y C B, y 0 = 1}. Extends good deal bounds to sequences of cash-flows. 31 / 35
Theorem 25 Assume the linearity condition, the Inada condition and that p / C and infϕ < ϕ( c). Then π( c,p;c) = sup y M { c,y σc (y) σ B( c) (y) p,y = 1 }, where B( c) = {c M 1 V( c+c) ϕ( c)}. In particular, if C is conical, π( c,p;c) = sup y M { c,y σb( c) (y) u C, p,y = 1 }. 32 / 35
Example 26 In the classical model, with p = (1,0,...,0) and V t = δ R for t < T, we get { π( c,p;c) = sup c,y σa( c) (y) p,y = 1 } y M { ) } = sup Q Q T ( E Q dq ( c t +c t ) σ B( c) E t dp T = sup sup Q Q α>0e Q{ ( c t +c t ) α [ V T( dq dp /α) ϕ( c) ] } where Q is the set of absolutely continuous martingale measures; see [Biagini, Frittelli, Grasselli, 2011] for a continuous-time version. 33 / 35
Theorem 27 (FTAP) Assume that S is finite-valued and that D R J. Then the following are equivalent 1. S satisfies the robust no-arbitrage condition. 2. There is a strictly consistent price system: adapted processes y and s such that y > 0, s t ridoms t and ys is a martingale. In the classical linear market model, ridoms t = {1, s t } so we recover the Dalang Morton Willinger theorem. The robust no-arbitrage condition means that there exists a sublinear arbitrage-free cost process S with dom S t ridoms t. 34 / 35
Summary Financial contracts often involve sequences of cash-flows. Reservation values and indifference swap rates/prices can (and should) be derived from hedging arguments. In practice (incomplete markets), valuations are subjective: they depend on views, risk preferences, trading expertise and the current financial position of an agent. Much of classical asset pricing theory can be extended to convex models of illiquid markets. The mathematics and computational techniques for hedging and pricing in illiquid markets combine techniques from stochastics and convex analysis. 35 / 35