Robust hedging with tradable options under price impact

Transcription

1 - Robust hedging with tradable options under price impact Arash Fahim, Florida State University joint work with Y-J Huang, DCU, Dublin March 2016, ECFM, WPI

2 practice is not robust - Pricing under a selected model P, physical measure P := sup E Q [Φ] Q M M = all martingale Q equivalent to P Super-hedging price: D := inf{a : a + ( S) T Φ P as} D = P

3 Volatility Surface: TSLA Feb 24, Implied volatiltiy Mo τ 15 Mo 1 Mo few days K ATM=$175

4 Volatility smile: TSLA Feb 24, σ imp K K τ Implied volatility at Feb 24, 2016 for Tesla stock NASDAQ:TSLA Left τ = 4 Months, middle τ = 2 weeks, and right K = $185 02

5 - Durpire 96, Gyöngy 86, Krylov 85 Suppose all vanilla call prices are given by C(K, T ) There is at most one Markov model which matches with the given call prices In this case, the volatility is given by σ 2 (t, s) = 2 C T + rkc K K 2 C KK (T,K)=(t,s) Issues and features C KK can be close to 0 Volatility surface is not sufficiently smooth Fitting a volatility surface can be done in many ways leading to different prices Markov (local vol) and hidden Markov (stochastic vol) models can be computationally feasible choices which can fit to the same volatility surface

6 Two approaches to robust - Robust (1) uncertainty Class of eligible models P: a collection of physical probability on some Polish space Ω (2) -independent Class of all martingale M consistent with price quotes on the canonical space Ω = R T + Asset price is the canonical process S t (S 1,, S T ) = S t

7 Semi-static Super-hedging - Semi-static Super-hedging Class of eligible models P Asset price S = (S 0, S 1,, S T ) Bond B t = 1 for all t For i I (finite set), ψ i is a derivatives Ψ a,η, := ( S) }{{ T +a + η } i ψ i Φ(S) P qs i I Dynamic }{{} Static The cost of super-hedging portfolio is p(a, η, ) := a + i I η ip i (η i ) price of ψ i : p i (η i ) = { a i η i > 0 b i η i < 0 [b i, a i ] is the bid-ask spread for ψ i

8 Arbitrage - Arbitrage p(a, η, ) = 0 and Ψ a,η, 0 P-qs, but P(Ψ a,η, > 0) > 0 for some P P M := {Q : Q << P for some P P, S is a Q martingale and b i E Q [ψ i ] a i } Fundamental theorem (Bouchard-Nutz 13) (Bayraktar-Zhou 14) NA P P, Q M st P << Q (or P and M have the same polar sets)

9 Duality - Super-hedging (Bouchard-Nutz 13) (Bayraktar-Zhou 14) If Φ is upper semi-analytic and NA holds: D(Φ) := inf{p(a, η, ) : Ψ a,η, Φ P qs} = sup E Q [Φ] Q M

10 Semi-static Super-hedging - Ω = R T + is the canonical space I can be infinite Semi-static Super-hedging Ψ a,u, := ( S) }{{ T +a + } Dynamic i J I is finite } {{ } Static η i ψ i Φ(S) S Ω The cost of making the above super-hedging portfolio is p(a, η, ) := a + i J η ip i (η i ), where price of ψ i : p i (η i ) = { a i η i > 0 b i η i < 0 [b i, a i ] is the bid-ask spread for ψ i

11 -independent arbitrage - Arbitrage p(a, η, ) = 0 and Ψ a,η, > 0 for some S Ω M := {Q : S is a Q martingale and b i E Q [ψ i ] a i } Fundamental theorem (Acciaio et al 2013) NA M

12 Duality - Super-hedging (Acciaio et al 2013) If Φ is continuous with linear growth, there exists a super-linear option among ψ i s and NA holds: D(Φ) := inf{p(a, η, ) : Ψ a,u, Φ S Ω} = sup E Q [Φ] Q M

13 Liquid call options - K > 0, t = 1,, T, C(t, K) = price of call option which pays (S t K) + at time t (Bid-Ask price equal) (1) K C(t, K) is nonnegative and convex, (2) lim K 0 K C(t, K) 1, (3) lim K C(t, K) = 0 Marginals C(t, K) µ t : C(t, K) = (x K) + µ t (dx) All feasible : Π := {Q : Q t µ t, t} =

14 Semi-static Super-hedging - u = (u 1,, u T ): u t linear combination of call options maturing at t Semi-static Super-hedging Ψ a,u,η, := ( S) }{{ T +a } Dynamic + u t + η i ψ i Φ(S), t i J is finite }{{} Static p(a, u, η) := a + t ut dµ t + j η jp j (η j ) S

15 Bid-ask chart for ψ i - Volume b k r k b 3 r 3 b 2 q b 1 1 r 1 a 1 a 2 r 2 Bid-ask spread c i (η) = ηp i (η) q 2 q 3 a 3 a l q l Price b 1 a 2 Slope=0 b 2 slope=a 1 η

16 Portfolio constraint - Constraint S (i) 0 S (ii) For any, S and any adapted process h with h t [0, 1] for all t = 0,, T 1, {h t t + (1 h t ) t} T 1 t=0 S (iii) For any bdd S, Q Π, and ε > 0, there exist a closed set D ε R T + and a continuous ε S such that Q(D ε ) > 1 ε and t = ε t on D ε for t = 0,, T 1

17 Duality - Super-hedging price D(Φ) := inf{p(a, u, η) : S, Ψ a,u,η, Φ S Ω} Duality when I = and no constraint of (Beiglböck et al 2013) If Φ usc continuous with linear growth, and M = {Q Π : S is a Q-martingale} D(Φ) = P (Φ) := sup E Q [Φ] Q M

18 Monge-Kantorovich theory of optimal - Optimal duality If Φ usc continuous with linear growth { inf u t dµ t : } u t Φ S Ω t t = sup E Q [Φ] Q Π

19 Constraint and illiquid option - Penalty terms {A Q t } t ( upper variation process for constraint S) : A Q t+1 AQ t = ess sup Q t E Q [S t+1 S t F t ] S Then, for any S, ( S) t A Q t and E Q [A Q T ] = sup S E Q [( S) T ] is a Q-supermartingale Illiquid options with price impact E Q I := sup sup η j (E Q [ψ j ] p j (η j )) J I(finite) η j J

20 Duality result - Duality (F-Huang 2014) If Φ usc continuous with linear growth, D S = sup E Q [Φ A Q T ] EQ I Q Π =: P S

21 No arbitrage - -independent arbitrage p(a, u, η) = 0 and Ψ a,u,η, > 0 S Ω P I,S := {Q Π {( S) t } is a Q-supermartingale S and b i E Q [ψ i ] a i } Fundamental theorem (F-Huang 14) NA P I,S Remark M I,S P I,S Π Not necessarily M I,S = P I,S

22 M P Π - Remark For all S (1) if Q Q I,S, {( S) t A Q t } t is a local Q-supermartingale (2) if Q P I,S, {( S) t } t is a local Q-supermartingale (3) If S, locality is removed (4) If 1, 1 S I,S, then P I,S = M I,S (5) If S I,S contains all bounded strategies, M I,S = P I,S

23 No-unbounded profit condition - Unbounded profit For all z R +, there exists a S, u, and η with p(a, u, η) = z and Ψ a,u,η, > 0 x Ω Q := {Q Π : E Q [A Q T ] <, EQ I < }, P Q Fundamental theorem (F-Huang 14) No unbounded profit Q I,S Remark: Imagine P I,S = (There is an arbitrage), but Q I,S Then, D = P > and arbitrage is limited by a certain amount of profit

24 - Short-selling constraint Relative-drawdown constraint P = Q t is bounded by two functions of S t St No constraint Q = Π M = P = Q Non-tradable asset 0, P = Q = Π

25 - Gamma constraint S Γ := {{ t } T 1 t=0 : t t 1 Γ, t = 0,, T 1}, 1 0 (or any other constant) Constraint S with bounded strategies (i) 0 S (ii) (Boundedness) S, is bounded (iii) (Continuous approximation) S can be approximated by continuous strategies in S in the sense described before

26 Superhedging duality for bounded - Penalty term C Q = sup S E Q [( S) T ] Q S := {Q Π C Q < } C Q and Q S are analogous to EQ [A Q T ] and Q S for convex constraint Duality (F-Huang 14) Φ(x) usc with linear growth D S (Φ) = P S(Φ) := sup E Q [Φ] C Q Q Q

27 No arbitrage - Fundamental theorem (F-Huang 14) NA P := {Q Q C Q = 0} = Example: Gamma constraint M = P

28 - Definition A mapping ρ : X R is called a convex risk measure if the following conditions are satisfied for all Φ, Φ X : Monotonicity: If Φ Φ, then ρ(φ) ρ(φ ) Translation Invariance: If m R, then ρ(φ + m) = ρ(φ) m Convexity: If 0 λ 1, then ρ(λφ + (1 λ)φ ) λρ(φ) + (1 λ)ρ(φ ) Superhedging as a risk measure ρ S (Φ) := D S ( Φ) X := {Φ ρ S (Φ) < }

29 - Theorem (F-Huang 14) Suppose Q S ( ) ρ S (Φ) = sup Q Π E Q [ Φ] α (Q), where the penalty function α is given by { α E Q [A Q T (Q) := ] + EQ I if Q Q I,S,, otherwise Moreover, for any α : Π R { } such that (29) holds (with α replaced by α), we have α (Q) α(q) for all Q Π

30 Bibliography - Early developments: (1) Hobson, David The Skorokhod embedding problem and model-independent bounds for option prices Paris-Princeton Lectures on Mathematical Finance 2010 Springer Berlin Heidelberg, Discrete-time: (2) Dolinsky, Yan, and H Mete Soner Robust hedging with proportional transaction costs Finance and Stochastics 182 (2014): (3) Acciaio, et al A MODELFREE VERSION OF THE FUNDAMENTAL THEOREM OF ASSET PRICING AND THE SUPERREPLICATION THEOREM Mathematical Finance (2013) (4) Beiglbck, Mathias, Pierre Henry-Labordre, and Friedrich Penkner -independent bounds for option pricesa mass transport approach Finance and Stochastics 173 (2013): Continuous-time: (5) Dolinsky, Yan, and H Mete Soner Martingale optimal transport and robust hedging in continuous time Probability Theory and Related Fields (2014): (6) Hou, Zhaoxu, and Jan Obloj On robust hedging duality in continuous time arxiv preprint arxiv: (2015) (7) Dolinsky, Yan, and H Mete Soner Convex duality with transaction costs arxiv preprint arxiv: (2015) APA uncertainty: (8) Arbitrage and Duality in Non-dominated Discrete-Time s Annals of Applied Probability, Vol 25, No 2, pp , 2015 (9) Bayraktar, Erhan, and Zhou Zhou On arbitrage and duality under model uncertainty and portfolio Available at SSRN (2014) (10) Bayraktar, Erhan, and Zhou Zhou Arbitrage, hedging and utility maximization using semi-static trading strategies with American options Available at SSRN (2015)

31 - Finance theory consists of a set of concepts that help you to organize your thinking about how to allocate resources over time and a set of quantitative models to help you evaluate alternatives, make decisions, and implement them Finance, Z Bodie and R Merton, Prentice Hall 2000 Thank you for you attention