Optimal investment and contingent claim valuation in illiquid markets
Optimal investment and contingent claim valuation in illiquid markets Teemu Pennanen King’s College London
Ari-Pekka Perkki¨o Technische Universit¨at Berlin
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Market model Optimal investment Valuation Existence of solutions Duality
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 Market model Optimal investment Valuation Existence of solutions Duality
• 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, Optimal investment and contingent claim valuation in illiquid markets, Finance and Stochastics, to appear. • Pennanen, Perkki¨ o, Stochastic programs without duality gaps, Mathematical Programming, 2012. • Pennanen, Perkki¨ o, Convex duality in optimal investment and contingent claim valuation in illiquid markets, manuscript.
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Market model Market model Optimal investment Valuation Existence of solutions Duality
Limit order book of TDC A/S on 12 January 2005 at 13:58:19.43 in Copenhagen Stock Exchange Price 238.75 238.75 238.75 238.75 238.5 238.5 238.5 238.5 238.25 238.25 238.25 238.25 238.25 . ..
Bid Quantity 140 600 3300 2000 10000 3900 15000 1500 10000 1000 3500 10000 200 . ..
Price 239 239 239 239 239 239 239 239.25 239.25 239.25 239.5 239.5 239.5 . ..
Ask Quantity 3700 1000 5000 1000 1000 2500 6600 10000 2500 3000 600 5000 800 . ..
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Market model The corresponding marginal price curve. Negative quantity corresponds to a sale. 242 241
PRICE
Market model Optimal investment Valuation Existence of solutions Duality
240 239 238 237 236 -100000
-50000
0
50000
QUANTITY
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Market model Market model Optimal investment Valuation Existence of solutions Duality
Consider a financial market where a finite set J of assets can be traded at t = 0, . . . , T . • Let (Ω, F, (Ft )Tt=0 , P ) be a filtered probability space. • The cost (in cash) of buying a portfolio x ∈ RJ at time t in state ω will be denoted by St (x, ω). • We will assume that ◦ St (·, ω) is convex with St (0, ω) = 0, ◦ St (x, ·) is Ft -measurable. • Such a sequence (St ) will be called a convex cost process.
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Market model Market model Optimal investment Valuation Existence of solutions Duality
Example 1 (Liquid markets) If s = (st )Tt=0 is an (Ft )Tt=0 -adapted RJ -valued price process, then the functions St (x, ω) = st (ω) · x define a convex cost process. Example 2 (Jouini and Kallal, 1995) If (sat )Tt=0 and (sbt )Tt=0 are (Ft )Tt=0 -adapted with sb ≤ sa , then the functions ( sat (ω)x if x ≥ 0, St (x, ω) = sbt (ω)x if x ≤ 0 define a convex cost process.
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Market model Market model Optimal investment Valuation Existence of solutions Duality
Example 3 (C ¸ etin and Rogers, 2007) If s = (st )Tt=0 is an (Ft )Tt=0 -adapted process and ψ is a lower semicontinuous convex function on R with ψ(0) = 0, then the functions St (x, ω) = x0 + st (ω)ψ(x1 ) define a convex cost process. Example 4 (Dolinsky and Soner, 2013) If s = (st )Tt=0 is (Ft )Tt=0 -adapted and Gt (x, ·) are Ft -measurable functions such that Gt (·, ω) are finite and convex, then the functions St (x, ω) = x0 + st (ω) · x1 + Gt (x1 , ω) define a convex cost process.
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Market model Market model Optimal investment Valuation Existence of solutions Duality
• We allow for portfolio constraints requiring that the portfolio held over (t, t + 1] in state ω has to belong to a set Dt (ω) ⊆ RJ . • We assume that ◦ Dt (ω) are closed and convex with 0 ∈ Dt (ω). ◦ {ω ∈ Ω | Dt (ω) ∩ U 6= ∅} ∈ Ft for every open U ⊂ RJ .
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Market model Market model Optimal investment Valuation Existence of solutions Duality
• Models where Dt (ω) is independent of (t, ω) have been studied e.g. in [Cvitani´c and Karatzas, 1992] and [Jouini and Kallal, 1995]. • In [Napp, 2003], Dt (ω) = {x ∈ Rd | Mt (ω)x ∈ K}, where K ⊂ RL is a closed convex cone and Mt is an Ft -measurable matrix. • General constraints have been studied in [Evstigneev, Sch¨urger and Taksar, 2004], [Rokhlin, 2005] and [Czichowsky and Schweizer, 2012].
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Optimal investment Market model Optimal investment Valuation Existence of solutions Duality
Let c ∈ M := {(ct )Tt=0 | ct ∈ L0 (Ω, Ft , P )} and consider the problem T X Vt (St (∆xt ) + ct ) over x ∈ ND minimize t=0
• ND = {(xt )Tt=0 | xt ∈ L0 (Ω, Ft , P ; RJ ), xt ∈ Dt , xT = 0}, • Vt : L0 → R are convex, nondecreasing and Vt (0) = 0. Example 5 If Vt = δL0− for t < T , the problem can be written minimize subject to
VT (ST (∆xT ) + cT ) over x ∈ ND St (∆xt ) + ct ≤ 0, t = 0, . . . , T − 1.
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Optimal investment Market model Optimal investment Valuation Existence of solutions Duality
Example 6 (Markets with a numeraire) When ˜ t (ω), St (x, ω) = x0 + S˜t (˜ x, ω) and Dt (ω) = R × D the problem can be written as minimize
VT
T X
S˜t (∆˜ xt ) +
T X t=0
t=0
ct
!
over
x ∈ ND .
When S˜t (˜ x, ω) = s˜t (ω) · x˜, T X t=0
S˜t (∆˜ xt ) =
T X t=0
s˜t · ∆˜ xt = −
T −1 X
x˜t · ∆˜ st+1 .
t=0
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Optimal investment Market model Optimal investment Valuation Existence of solutions Duality
We denote the optimal value function by T X Vt (St (∆xt ) + ct ). ϕ(c) = inf x∈ND
t=0
Note that ϕ(c) = inf V(c − d), d∈C
where V(c) :=
PT
t=0
Vt (ct ) and
C := {c ∈ M | ∃x ∈ ND : St (∆xt ) + ct ≤ 0 ∀t} is the set of claims that can be superhedged without cost.
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Optimal investment Market model Optimal investment Valuation Existence of solutions Duality
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 ∞ ).
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Optimal investment Market model Optimal investment Valuation Existence of solutions Duality
Example 8 (The classical model) Consider the classical perfectly liquid market model where C = {c ∈ M | ∃x ∈ N :
T X t=0
ct ≤
T −1 X
xt · ∆st+1 }
t=0
and C ∞ = C. We have c ∈ C ∞ ∩ (−C ∞ ) if there is an x ∈ N such that T X t=0
ct =
T −1 X
xt · ∆st+1 .
t=0
The converse holds under the no-arbitrage condition.
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Valuation of contingent claims Market model Optimal investment Valuation Existence of solutions Duality
• 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.
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Reservation value Market model Optimal investment Valuation Existence of solutions Duality
• We define the reservation value for a liability c ∈ M by π 0 (c) = inf{α ∈ R | ϕ(c − αp0 ) ≤ 0} where p0 = (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 = δM− , we have ϕ = δC and 0 (c) := inf{α ∈ R | c − αp0 ∈ C}. π 0 (c) = πsup 0 0 (c) = −πsup • Let πinf (−c).
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Reservation value Market model Optimal investment Valuation Existence of solutions Duality
Theorem 9 The reservation value π 0 is convex and 0 nondecreasing with respect to C ∞ . We have π 0 ≤ πsup and if π 0 (0) ≥ 0, then 0 0 (c) (c) ≤ π 0 (c) ≤ πsup πinf
with equalities throughout if c − αp0 ∈ C ∩ (−C) for α ∈ R. • π 0 is “translation invariant”: if c′ ∈ M is replicable with initial capital α: c′ − αp0 ∈ C ∞ ∩ (−C ∞ ), then π 0 (c + c′ ) = π 0 (c) + α. • In complete markets, c − αp0 ∈ C ∩ (−C) for some α ∈ R so π 0 (c) is independent of preferences and views. 18 / 35
Swap contracts Market model Optimal investment Valuation Existence of solutions Duality
• 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, cT ). • Claims and premiums live in the same space M = {(ct )Tt=0 | ct ∈ L0 (Ω, Ft , P ; R)}.
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Swap contracts Market model Optimal investment Valuation Existence of solutions Duality
• 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, cT ), we get a nonlinear version of the indifference price of [Hodges and Neuberger, 1989]. 20 / 35
Swap contracts Market model Optimal investment Valuation Existence of solutions Duality
Define the super- and subhedging swap rates, πsup (p; c) = inf{α | c−αp ∈ C ∞ }, πinf (p; c) = sup{α | αp−c ∈ C ∞ }. If C is a cone and p = (1, 0, . . . , 0), we recover the super- and 0 . 0 and πinf subhedging costs πsup Theorem 10 If π(¯ c, p; 0) ≥ 0, then πinf (p; c) ≤ πb (¯ c, p; c) ≤ π(¯ c, p; c) ≤ πsup (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 = p0 , then swap rates coincide with reservation values. 21 / 35
Swap contracts Market model Optimal investment Valuation Existence of solutions Duality
Example 11 (The classical model) Consider the classical perfectly liquid market model where C = {c ∈ M | ∃x ∈ N :
T X t=0
ct ≤
T −1 X
xt · ∆st+1 }
t=0
and C ∞ = C. The condition c − αp ∈ C ∞ ∩ (−C ∞ ) holds if there exist α ∈ R and x ∈ N such that T X t=0
ct = α
T X t=0
pt +
T −1 X
xt · ∆st+1 .
t=0
The converse holds under the no-arbitrage condition.
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Existence of solutions Market model Optimal investment Valuation Existence of solutions Duality
Given a market model (S, D), let St∞ (x, ω)
St (αx, ω) = sup α α>0
and
Dt∞ (ω)
=
\
αDt (ω).
α>0
If S is sublinear and D is conical, then S ∞ = S and D∞ = D PT Theorem 12 Assume that V(c) = E t=0 Vt (ct ), where Vt are bounded from below. If the cone L := {x ∈ ND∞ | St∞ (∆xt ) ≤ 0} is a linear space, then C is closed and ϕ is lower semicontinuous in L0 . The lower bound can be replaced by RAE; [Perkki¨o, 2014]. 23 / 35
Existence of solutions Market model Optimal investment Valuation Existence of solutions Duality
Example 13 In the classical perfectly liquid market model L = {x ∈ N | st · ∆xt ≤ 0, xT = 0}, so the linearity condition becomes the no-arbitrage condition and we recover the key lemma from [Schachermayer, 1992]. Example 14 When D ≡ RJ , 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 ∞ (x, ω) ∀x ∈ RJ , S˜t (x, ω)
0 for x ∈ / RJ− , then L = {0}. Example 16 In [C ¸ etin and Rogers, 2007], St (x, ω) = x0 + st (ω)ψ(x1 ) and St∞ (x, ω) = x0 + st (ω)ψ ∞ (x1 ). When inf ψ ′ = 0 and sup ψ ′ = ∞ we have ψ ∞ = δR− , so the condition in Example 15 holds. Example 17 If St (·, ω) = st (ω) · x for a componentwise strictly positive price process s and Dt∞ (ω) ⊆ RJ+ (infinite short selling is prohibited), then L = {0}. 25 / 35
Existence of solutions Market model Optimal investment Valuation Existence of solutions Duality
Proposition 18 Assume that ϕ is proper and lower semicontinuous. The conditions • ϕ∞ (p0 ) > 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 − αp0 ) ≤ 0} is attained for every c ∈ M.
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Existence of solutions Market model Optimal investment Valuation Existence of solutions Duality
Proposition 19 Assume that ϕ is proper and lower semicontinuous. Then, for every c¯ ∈ dom ϕ 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.
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Duality Market model Optimal investment Valuation Existence of solutions Duality
• Let Mp = {c ∈ M | ct ∈ Lp (Ω, Ft , P ; R)}. • The bilinear form T X c t yt hc, yi := E t=0
puts M1 and M∞ in separating duality. • The conjugate of a function f on M1 is defined by f ∗ (y) = sup {hc, yi − f (c)}. c∈M1
• If f is proper, convex and lower semicontinuous, then f (y) = sup {hc, yi − f ∗ (y)}. y∈M∞
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Duality Market model Optimal investment Valuation Existence of solutions Duality
We assume from now on that T X Vt (ct ) V(c) = E t=0
for convex random functions Vt : R × Ω → R with Vt (0) = 0. Theorem 20 If St (x, ·) ∈ L1 for all x ∈ RJ , then ϕ∗ (y) = V ∗ (y) + σC (y) PT ∗ where V (y) = E t=0 Vt∗ (yt ) and σC (y) = supc∈C hc, yi. Moreover, T X [(yt St )∗ (vt ) + σDt (E[∆vt+1 |Ft ])] σC (y) = inf 1 E v∈N
t=0
where the infimum is attained for all y ∈ M∞ .
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Duality Market model Optimal investment Valuation Existence of solutions Duality
Example 21 If St (ω, x) = st (ω) · x and Dt (ω) is a cone, C ∗ = {y ∈ M∞ | E[∆(yt+1 st+1 ) |Ft ] ∈ Dt∗ }. Example 22 If St (ω, x) = sup{s · x | s ∈ [sbt (ω), sat (ω)]} and Dt (ω) = RJ , then C ∗ = {y ∈ M∞ | ys is a martingale for some s ∈ [sb , sa ]}. Example 23 In the classical model, C ∗ consists of positive multiples of martingale densities.
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Duality Market model Optimal investment Valuation Existence of solutions Duality
Theorem 24 Assume the linearity condition, the Inada condition Vt∞ = δR− and that p0 ∈ / C ∞ and inf ϕ < 0. Then π 0 (c) = sup {hc, yi − σC (y) − σB (y) | y0 = 1} , y∈M∞
where B = {c ∈ M1 | V(c) ≤ 0}. In particular, when C is conical and V is positively homogeneous, π 0 (c) = sup {hc, yi | y ∈ C ∗ ∩ B ∗ , y0 = 1} . y∈M∞
• Extends good deal bounds to sequences of cash-flows.
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Duality Market model Optimal investment Valuation Existence of solutions Duality
Theorem 25 Assume the linearity condition, the Inada condition and that p ∈ / C ∞ and inf ϕ < ϕ(¯ c). Then π(¯ c, p; c) = sup hc, yi − σC (y) − σB(¯c) (y) hp, yi = 1 , y∈M∞
where B(¯ c) = {c ∈ M1 | V(¯ c + c) ≤ ϕ(¯ c)}. In particular, if C is conical, ∗ π(¯ c, p; c) = sup hc, yi − σB(¯c) (y) u ∈ C , hp, yi = 1 . y∈M∞
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Duality Market model Optimal investment Valuation Existence of solutions Duality
Example 26 In the classical model, with p = (1, 0, . . . , 0) and Vt = δR− for t < T , we get π(¯ c, p; c) = sup hc, yi − σA(¯c) (y) hp, yi = 1 y∈M∞ ) ( T X dQ Q = sup E (¯ ct + ct ) − σB(¯c) Et dP Q∈Q t=0 ( T ) X ∗ dQ Q (¯ ct + ct ) − α VT ( /α) − ϕ(¯ = sup sup E c) dP Q∈Q α>0 t=0 where Q is the set of absolutely continuous martingale measures; see [Biagini, Frittelli, Grasselli, 2011] for a continuous-time version. 33 / 35
Duality Market model Optimal investment Valuation Existence of solutions Duality
Theorem 27 (FTAP) Assume that S ∞ is finite-valued and that D ≡ RJ . 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, st ∈ ri dom St∗ and ys is a martingale. • In the classical linear market model, ri dom St∗ = {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∗ ⊆ ri dom St∗ .
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Summary Market model Optimal investment Valuation Existence of solutions Duality
• 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.
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Ari-Pekka Perkki¨o Technische Universit¨at Berlin
1 / 35
Market model Optimal investment Valuation Existence of solutions Duality
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 Market model Optimal investment Valuation Existence of solutions Duality
• 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, Optimal investment and contingent claim valuation in illiquid markets, Finance and Stochastics, to appear. • Pennanen, Perkki¨ o, Stochastic programs without duality gaps, Mathematical Programming, 2012. • Pennanen, Perkki¨ o, Convex duality in optimal investment and contingent claim valuation in illiquid markets, manuscript.
3 / 35
Market model Market model Optimal investment Valuation Existence of solutions Duality
Limit order book of TDC A/S on 12 January 2005 at 13:58:19.43 in Copenhagen Stock Exchange Price 238.75 238.75 238.75 238.75 238.5 238.5 238.5 238.5 238.25 238.25 238.25 238.25 238.25 . ..
Bid Quantity 140 600 3300 2000 10000 3900 15000 1500 10000 1000 3500 10000 200 . ..
Price 239 239 239 239 239 239 239 239.25 239.25 239.25 239.5 239.5 239.5 . ..
Ask Quantity 3700 1000 5000 1000 1000 2500 6600 10000 2500 3000 600 5000 800 . ..
4 / 35
Market model The corresponding marginal price curve. Negative quantity corresponds to a sale. 242 241
PRICE
Market model Optimal investment Valuation Existence of solutions Duality
240 239 238 237 236 -100000
-50000
0
50000
QUANTITY
5 / 35
Market model Market model Optimal investment Valuation Existence of solutions Duality
Consider a financial market where a finite set J of assets can be traded at t = 0, . . . , T . • Let (Ω, F, (Ft )Tt=0 , P ) be a filtered probability space. • The cost (in cash) of buying a portfolio x ∈ RJ at time t in state ω will be denoted by St (x, ω). • We will assume that ◦ St (·, ω) is convex with St (0, ω) = 0, ◦ St (x, ·) is Ft -measurable. • Such a sequence (St ) will be called a convex cost process.
6 / 35
Market model Market model Optimal investment Valuation Existence of solutions Duality
Example 1 (Liquid markets) If s = (st )Tt=0 is an (Ft )Tt=0 -adapted RJ -valued price process, then the functions St (x, ω) = st (ω) · x define a convex cost process. Example 2 (Jouini and Kallal, 1995) If (sat )Tt=0 and (sbt )Tt=0 are (Ft )Tt=0 -adapted with sb ≤ sa , then the functions ( sat (ω)x if x ≥ 0, St (x, ω) = sbt (ω)x if x ≤ 0 define a convex cost process.
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Market model Market model Optimal investment Valuation Existence of solutions Duality
Example 3 (C ¸ etin and Rogers, 2007) If s = (st )Tt=0 is an (Ft )Tt=0 -adapted process and ψ is a lower semicontinuous convex function on R with ψ(0) = 0, then the functions St (x, ω) = x0 + st (ω)ψ(x1 ) define a convex cost process. Example 4 (Dolinsky and Soner, 2013) If s = (st )Tt=0 is (Ft )Tt=0 -adapted and Gt (x, ·) are Ft -measurable functions such that Gt (·, ω) are finite and convex, then the functions St (x, ω) = x0 + st (ω) · x1 + Gt (x1 , ω) define a convex cost process.
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Market model Market model Optimal investment Valuation Existence of solutions Duality
• We allow for portfolio constraints requiring that the portfolio held over (t, t + 1] in state ω has to belong to a set Dt (ω) ⊆ RJ . • We assume that ◦ Dt (ω) are closed and convex with 0 ∈ Dt (ω). ◦ {ω ∈ Ω | Dt (ω) ∩ U 6= ∅} ∈ Ft for every open U ⊂ RJ .
9 / 35
Market model Market model Optimal investment Valuation Existence of solutions Duality
• Models where Dt (ω) is independent of (t, ω) have been studied e.g. in [Cvitani´c and Karatzas, 1992] and [Jouini and Kallal, 1995]. • In [Napp, 2003], Dt (ω) = {x ∈ Rd | Mt (ω)x ∈ K}, where K ⊂ RL is a closed convex cone and Mt is an Ft -measurable matrix. • General constraints have been studied in [Evstigneev, Sch¨urger and Taksar, 2004], [Rokhlin, 2005] and [Czichowsky and Schweizer, 2012].
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Optimal investment Market model Optimal investment Valuation Existence of solutions Duality
Let c ∈ M := {(ct )Tt=0 | ct ∈ L0 (Ω, Ft , P )} and consider the problem T X Vt (St (∆xt ) + ct ) over x ∈ ND minimize t=0
• ND = {(xt )Tt=0 | xt ∈ L0 (Ω, Ft , P ; RJ ), xt ∈ Dt , xT = 0}, • Vt : L0 → R are convex, nondecreasing and Vt (0) = 0. Example 5 If Vt = δL0− for t < T , the problem can be written minimize subject to
VT (ST (∆xT ) + cT ) over x ∈ ND St (∆xt ) + ct ≤ 0, t = 0, . . . , T − 1.
11 / 35
Optimal investment Market model Optimal investment Valuation Existence of solutions Duality
Example 6 (Markets with a numeraire) When ˜ t (ω), St (x, ω) = x0 + S˜t (˜ x, ω) and Dt (ω) = R × D the problem can be written as minimize
VT
T X
S˜t (∆˜ xt ) +
T X t=0
t=0
ct
!
over
x ∈ ND .
When S˜t (˜ x, ω) = s˜t (ω) · x˜, T X t=0
S˜t (∆˜ xt ) =
T X t=0
s˜t · ∆˜ xt = −
T −1 X
x˜t · ∆˜ st+1 .
t=0
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Optimal investment Market model Optimal investment Valuation Existence of solutions Duality
We denote the optimal value function by T X Vt (St (∆xt ) + ct ). ϕ(c) = inf x∈ND
t=0
Note that ϕ(c) = inf V(c − d), d∈C
where V(c) :=
PT
t=0
Vt (ct ) and
C := {c ∈ M | ∃x ∈ ND : St (∆xt ) + ct ≤ 0 ∀t} is the set of claims that can be superhedged without cost.
13 / 35
Optimal investment Market model Optimal investment Valuation Existence of solutions Duality
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 ∞ ).
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Optimal investment Market model Optimal investment Valuation Existence of solutions Duality
Example 8 (The classical model) Consider the classical perfectly liquid market model where C = {c ∈ M | ∃x ∈ N :
T X t=0
ct ≤
T −1 X
xt · ∆st+1 }
t=0
and C ∞ = C. We have c ∈ C ∞ ∩ (−C ∞ ) if there is an x ∈ N such that T X t=0
ct =
T −1 X
xt · ∆st+1 .
t=0
The converse holds under the no-arbitrage condition.
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Valuation of contingent claims Market model Optimal investment Valuation Existence of solutions Duality
• 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.
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Reservation value Market model Optimal investment Valuation Existence of solutions Duality
• We define the reservation value for a liability c ∈ M by π 0 (c) = inf{α ∈ R | ϕ(c − αp0 ) ≤ 0} where p0 = (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 = δM− , we have ϕ = δC and 0 (c) := inf{α ∈ R | c − αp0 ∈ C}. π 0 (c) = πsup 0 0 (c) = −πsup • Let πinf (−c).
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Reservation value Market model Optimal investment Valuation Existence of solutions Duality
Theorem 9 The reservation value π 0 is convex and 0 nondecreasing with respect to C ∞ . We have π 0 ≤ πsup and if π 0 (0) ≥ 0, then 0 0 (c) (c) ≤ π 0 (c) ≤ πsup πinf
with equalities throughout if c − αp0 ∈ C ∩ (−C) for α ∈ R. • π 0 is “translation invariant”: if c′ ∈ M is replicable with initial capital α: c′ − αp0 ∈ C ∞ ∩ (−C ∞ ), then π 0 (c + c′ ) = π 0 (c) + α. • In complete markets, c − αp0 ∈ C ∩ (−C) for some α ∈ R so π 0 (c) is independent of preferences and views. 18 / 35
Swap contracts Market model Optimal investment Valuation Existence of solutions Duality
• 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, cT ). • Claims and premiums live in the same space M = {(ct )Tt=0 | ct ∈ L0 (Ω, Ft , P ; R)}.
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Swap contracts Market model Optimal investment Valuation Existence of solutions Duality
• 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, cT ), we get a nonlinear version of the indifference price of [Hodges and Neuberger, 1989]. 20 / 35
Swap contracts Market model Optimal investment Valuation Existence of solutions Duality
Define the super- and subhedging swap rates, πsup (p; c) = inf{α | c−αp ∈ C ∞ }, πinf (p; c) = sup{α | αp−c ∈ C ∞ }. If C is a cone and p = (1, 0, . . . , 0), we recover the super- and 0 . 0 and πinf subhedging costs πsup Theorem 10 If π(¯ c, p; 0) ≥ 0, then πinf (p; c) ≤ πb (¯ c, p; c) ≤ π(¯ c, p; c) ≤ πsup (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 = p0 , then swap rates coincide with reservation values. 21 / 35
Swap contracts Market model Optimal investment Valuation Existence of solutions Duality
Example 11 (The classical model) Consider the classical perfectly liquid market model where C = {c ∈ M | ∃x ∈ N :
T X t=0
ct ≤
T −1 X
xt · ∆st+1 }
t=0
and C ∞ = C. The condition c − αp ∈ C ∞ ∩ (−C ∞ ) holds if there exist α ∈ R and x ∈ N such that T X t=0
ct = α
T X t=0
pt +
T −1 X
xt · ∆st+1 .
t=0
The converse holds under the no-arbitrage condition.
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Existence of solutions Market model Optimal investment Valuation Existence of solutions Duality
Given a market model (S, D), let St∞ (x, ω)
St (αx, ω) = sup α α>0
and
Dt∞ (ω)
=
\
αDt (ω).
α>0
If S is sublinear and D is conical, then S ∞ = S and D∞ = D PT Theorem 12 Assume that V(c) = E t=0 Vt (ct ), where Vt are bounded from below. If the cone L := {x ∈ ND∞ | St∞ (∆xt ) ≤ 0} is a linear space, then C is closed and ϕ is lower semicontinuous in L0 . The lower bound can be replaced by RAE; [Perkki¨o, 2014]. 23 / 35
Existence of solutions Market model Optimal investment Valuation Existence of solutions Duality
Example 13 In the classical perfectly liquid market model L = {x ∈ N | st · ∆xt ≤ 0, xT = 0}, so the linearity condition becomes the no-arbitrage condition and we recover the key lemma from [Schachermayer, 1992]. Example 14 When D ≡ RJ , 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 ∞ (x, ω) ∀x ∈ RJ , S˜t (x, ω)
0 for x ∈ / RJ− , then L = {0}. Example 16 In [C ¸ etin and Rogers, 2007], St (x, ω) = x0 + st (ω)ψ(x1 ) and St∞ (x, ω) = x0 + st (ω)ψ ∞ (x1 ). When inf ψ ′ = 0 and sup ψ ′ = ∞ we have ψ ∞ = δR− , so the condition in Example 15 holds. Example 17 If St (·, ω) = st (ω) · x for a componentwise strictly positive price process s and Dt∞ (ω) ⊆ RJ+ (infinite short selling is prohibited), then L = {0}. 25 / 35
Existence of solutions Market model Optimal investment Valuation Existence of solutions Duality
Proposition 18 Assume that ϕ is proper and lower semicontinuous. The conditions • ϕ∞ (p0 ) > 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 − αp0 ) ≤ 0} is attained for every c ∈ M.
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Existence of solutions Market model Optimal investment Valuation Existence of solutions Duality
Proposition 19 Assume that ϕ is proper and lower semicontinuous. Then, for every c¯ ∈ dom ϕ 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.
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Duality Market model Optimal investment Valuation Existence of solutions Duality
• Let Mp = {c ∈ M | ct ∈ Lp (Ω, Ft , P ; R)}. • The bilinear form T X c t yt hc, yi := E t=0
puts M1 and M∞ in separating duality. • The conjugate of a function f on M1 is defined by f ∗ (y) = sup {hc, yi − f (c)}. c∈M1
• If f is proper, convex and lower semicontinuous, then f (y) = sup {hc, yi − f ∗ (y)}. y∈M∞
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Duality Market model Optimal investment Valuation Existence of solutions Duality
We assume from now on that T X Vt (ct ) V(c) = E t=0
for convex random functions Vt : R × Ω → R with Vt (0) = 0. Theorem 20 If St (x, ·) ∈ L1 for all x ∈ RJ , then ϕ∗ (y) = V ∗ (y) + σC (y) PT ∗ where V (y) = E t=0 Vt∗ (yt ) and σC (y) = supc∈C hc, yi. Moreover, T X [(yt St )∗ (vt ) + σDt (E[∆vt+1 |Ft ])] σC (y) = inf 1 E v∈N
t=0
where the infimum is attained for all y ∈ M∞ .
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Duality Market model Optimal investment Valuation Existence of solutions Duality
Example 21 If St (ω, x) = st (ω) · x and Dt (ω) is a cone, C ∗ = {y ∈ M∞ | E[∆(yt+1 st+1 ) |Ft ] ∈ Dt∗ }. Example 22 If St (ω, x) = sup{s · x | s ∈ [sbt (ω), sat (ω)]} and Dt (ω) = RJ , then C ∗ = {y ∈ M∞ | ys is a martingale for some s ∈ [sb , sa ]}. Example 23 In the classical model, C ∗ consists of positive multiples of martingale densities.
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Duality Market model Optimal investment Valuation Existence of solutions Duality
Theorem 24 Assume the linearity condition, the Inada condition Vt∞ = δR− and that p0 ∈ / C ∞ and inf ϕ < 0. Then π 0 (c) = sup {hc, yi − σC (y) − σB (y) | y0 = 1} , y∈M∞
where B = {c ∈ M1 | V(c) ≤ 0}. In particular, when C is conical and V is positively homogeneous, π 0 (c) = sup {hc, yi | y ∈ C ∗ ∩ B ∗ , y0 = 1} . y∈M∞
• Extends good deal bounds to sequences of cash-flows.
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Duality Market model Optimal investment Valuation Existence of solutions Duality
Theorem 25 Assume the linearity condition, the Inada condition and that p ∈ / C ∞ and inf ϕ < ϕ(¯ c). Then π(¯ c, p; c) = sup hc, yi − σC (y) − σB(¯c) (y) hp, yi = 1 , y∈M∞
where B(¯ c) = {c ∈ M1 | V(¯ c + c) ≤ ϕ(¯ c)}. In particular, if C is conical, ∗ π(¯ c, p; c) = sup hc, yi − σB(¯c) (y) u ∈ C , hp, yi = 1 . y∈M∞
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Duality Market model Optimal investment Valuation Existence of solutions Duality
Example 26 In the classical model, with p = (1, 0, . . . , 0) and Vt = δR− for t < T , we get π(¯ c, p; c) = sup hc, yi − σA(¯c) (y) hp, yi = 1 y∈M∞ ) ( T X dQ Q = sup E (¯ ct + ct ) − σB(¯c) Et dP Q∈Q t=0 ( T ) X ∗ dQ Q (¯ ct + ct ) − α VT ( /α) − ϕ(¯ = sup sup E c) dP Q∈Q α>0 t=0 where Q is the set of absolutely continuous martingale measures; see [Biagini, Frittelli, Grasselli, 2011] for a continuous-time version. 33 / 35
Duality Market model Optimal investment Valuation Existence of solutions Duality
Theorem 27 (FTAP) Assume that S ∞ is finite-valued and that D ≡ RJ . 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, st ∈ ri dom St∗ and ys is a martingale. • In the classical linear market model, ri dom St∗ = {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∗ ⊆ ri dom St∗ .
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Summary Market model Optimal investment Valuation Existence of solutions Duality
• 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.
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