Exact replication under portfolio constraints: a viability approach
|
|
- Alyson Gibbs
- 5 years ago
- Views:
Transcription
1 Exact replication under portfolio constraints: a viability approach CEREMADE, Université Paris-Dauphine Joint work with Jean-Francois Chassagneux & Idris Kharroubi
2 Motivation Complete market with no interest rate and one stock : ds t = σ(s t)dw t Price and Hedge of a European option with regular payoff h(s T ) : [ P t = E t [h(s T )] t = E t h (S T ) S ] T S t where S is the tangent process with dynamics d S t = σ (S t) S tdw t. Addition of no short sell regulatory constraints : need t 0 ] h is increasing = t = E t [h (S T ) S T S t 0 = If h is increasing, the super-replication price under no short sell constraints of h(s T ) is the replication price. In general, the super-replication price under no short sell constraints of h(s T ) is the replication price of ĥ(s T ) with ĥ the smallest increasing function above h. For which couple [model,constraints] is this property satisfied?
3 Motivation Complete market with no interest rate and one stock : ds t = σ(s t)dw t Price and Hedge of a European option with regular payoff h(s T ) : [ P t = E t [h(s T )] t = E t h (S T ) S ] T S t where S is the tangent process with dynamics d S t = σ (S t) S tdw t. Addition of no short sell regulatory constraints : need t 0 ] h is increasing = t = E t [h (S T ) S T S t 0 = If h is increasing, the super-replication price under no short sell constraints of h(s T ) is the replication price. In general, the super-replication price under no short sell constraints of h(s T ) is the replication price of ĥ(s T ) with ĥ the smallest increasing function above h. For which couple [model,constraints] is this property satisfied?
4 Motivation Complete market with no interest rate and one stock : ds t = σ(s t)dw t Price and Hedge of a European option with regular payoff h(s T ) : [ P t = E t [h(s T )] t = E t h (S T ) S ] T S t where S is the tangent process with dynamics d S t = σ (S t) S tdw t. Addition of no short sell regulatory constraints : need t 0 ] h is increasing = t = E t [h (S T ) S T S t 0 = If h is increasing, the super-replication price under no short sell constraints of h(s T ) is the replication price. In general, the super-replication price under no short sell constraints of h(s T ) is the replication price of ĥ(s T ) with ĥ the smallest increasing function above h. For which couple [model,constraints] is this property satisfied?
5 Motivation Complete market with no interest rate and one stock : ds t = σ(s t)dw t Price and Hedge of a European option with regular payoff h(s T ) : [ P t = E t [h(s T )] t = E t h (S T ) S ] T S t where S is the tangent process with dynamics d S t = σ (S t) S tdw t. Addition of no short sell regulatory constraints : need t 0 ] h is increasing = t = E t [h (S T ) S T S t 0 = If h is increasing, the super-replication price under no short sell constraints of h(s T ) is the replication price. In general, the super-replication price under no short sell constraints of h(s T ) is the replication price of ĥ(s T ) with ĥ the smallest increasing function above h. For which couple [model,constraints] is this property satisfied?
6 Agenda 1 Super-replication under portfolio constraints
7 Super-replication price The market model Portfolio process X t,x, s = x + S t = S 0 + s t t 0 uds u = x + σ(s u)dw u, 0 t T. s In addition to classical admissibility conditions, we impose t uσ(s u)dw u, 0 t s T. A K t := { A such that s K P a.s., t s T }, where K is a closed convex set. The super-replication price of h(s T ) at time t under K-constraints defines as { } pt K [h] := inf x R, A K t such that X t,x, T h(s T ) P a.s.
8 Condition at maturity : Facelift transform The super-replication price of h(s T ) at time t under K-constraints defines as { } pt K [h] := inf x R, A K t such that X t,x, T h(s T ) P a.s. At maturity T, we need T K. = Need to change the terminal condition. = Smallest function above h whose "derivatives" belong to K. Definition of the facelift operator : F K [h](x) := sup y R d h(x + y) δ K (y), x R d, where δ K : y sup z K y, z is the support function of K. F K [h] identifies as the smallest viscosity super-solution of { } min u h, inf δ K (ζ) ζ, xu = 0 ζ =1
9 Condition at maturity : Facelift transform The super-replication price of h(s T ) at time t under K-constraints defines as { } pt K [h] := inf x R, A K t such that X t,x, T h(s T ) P a.s. At maturity T, we need T K = Need to change the terminal condition. = Smallest function above h whose "derivatives" belong to K. Definition of the facelift operator : F K [h](x) := sup y R d h(x + y) δ K (y), x R d, where δ K : y sup z K y, z is the support function of K. F K [h] identifies as the smallest viscosity super-solution of { } min u h, inf ζ =1 δ K (ζ) ζ, xu = 0
10 Characterizations of the super-replication price Direct PDE characterization pt K [h] = v K [h](t, S t) where v K [h] is the unique viscosity solution of the PDE { } min L σ u, inf ζ =1 δ K (ζ) ζ, xu = 0 for t < T and u(t, x) = F K [h], with L the Dynkin operator of the diffusion S. Dual representation in terms of pricing measure : [ T ] v K [h](t, x) = sup E Qν t,x h(x t,x T ) δ K (ν s)ds ν s.t. δ K (ν)< t with Q ν the equivalent measure for which W t t νsds is a Brownian motion. 0 BSDE characterization : Minimal solution of the Z-constrained BSDE Y t = F K [h](s T ) T t Z sdw s + T t dl s, with Z t Kσ(S t),
11 The question of interest super-replicate h(s T ) under K-constraints We always have. super-replicate F K [h](s T ) under K-constraints super-replicate h(s T ) under K-constraints When do we have? replicate F K [h](s T ) without constraints In the Black Scholes model : True for intervals in dimension 1 [Broadie, Cvitanic, Soner] True for any convex set K and money or wealth proportion constraints For general local volatility model : [Our contribution] A necessary and sufficient condition for the previous property to hold for a large class of payoff functions h.
12 The question of interest super-replicate h(s T ) under K-constraints We always have. super-replicate F K [h](s T ) under K-constraints super-replicate h(s T ) under K-constraints When do we have? replicate F K [h](s T ) without constraints In the Black Scholes model : True for intervals in dimension 1 [Broadie, Cvitanic, Soner] True for any convex set K and money or wealth proportion constraints For general local volatility models : [Our contribution] A necessary and sufficient condition for the previous property to hold for a large class of payoff functions h.
13 The question of interest super-replicate h(s T ) under K-constraints We always have. super-replicate F K [h](s T ) under K-constraints super-replicate h(s T ) under K-constraints When do we have? replicate F K [h](s T ) without constraints In the Black Scholes model : True for intervals in dimension 1 [Broadie, Cvitanic, Soner] True for any convex set K and money or wealth proportion constraints For general local volatility model : [Our contribution] A necessary and sufficient condition for the previous property to hold for a large class of payoff functions h.
14 Intervals in dimension 1 Dimension 1 stock : ds t = σ(s t)dw t with σ regular. Interval convex constraint K := [a, b]. Let h be a payoff function such that F K [h] is differentiable. Do we have p K t [h] = p t[f K [h]]? The unconstrained hedging strategy of F K [h] at time t is [ t := E t F K [h](s T ) S ] T, with d S t = σ (S t) S tdw t. S t = S interprets as a probability change and we can find a proba Q s. t. [ ] t := E Q t F K [h](s T ) K, 0 t T, since F K [h] is valued in the convex K. = Revisit and generalize this known result for the Black Scholes model.
15 Intervals in dimension 1 Dimension 1 stock : ds t = σ(s t)dw t with σ regular. Interval convex constraint K := [a, b]. Let h be a payoff function such that F K [h] is differentiable. Do we have p K t [h] = p t[f K [h]]? The unconstrained hedging strategy of F K [h] at time t is [ t := E t F K [h](s T ) S ] T, with d S t = σ (S t) S tdw t. S t = S interprets as a probability change and we can find a proba Q s. t. [ ] t := E Q t F K [h](s T ) K, 0 t T, since F K [h] is valued in the convex K. = Revisit and generalize this known result for the Black Scholes model.
16 Hypercubes for d stocks with separate dynamics Dimension d stock with separate dynamics : ds i t = σ i (S i t)dw t, 1 i d. Hypercube constraints K := Π d i=1[a i, b i ]. Let h be a payoff function such that F K [h] is differentiable. Do we have p K t [h] = p t[f K [h]]? The unconstrained hedging strategy of F K [h] at time t is [ ] i t := E t ( F K [h](s T )) i ST i with d S i St i t = σ i (St) i StdW i t. = Since F K [h] is valued in the hypercube K, K because [ ] [ ] S i a i = a i E T t i S i St i t b i E T t = b St i i, 0 t T, Does it generalize to any convex set or any model?
17 General convex set K and model dynamics σ Consider A model dynamics : Portfolio constraints : σ Lipschitz, differentiable and invertible K closed convex set with non empty interior Problem of interest : Is there a structural condition on the coupe [K, σ] under which for any payoff h in a given class, we have p K [h] = p[f K [h]]? First, simplified version : Is there a structural condition on the couple [K, σ] under which For any payoff h CK 1, we have p K [h] = p[h]? where CK 1 denotes the class of C 1 functions with derivatives valued in K. (i.e. regular and stable under F K )
18 General convex set K and model dynamics σ Consider A model dynamics : Portfolio constraints : σ Lipschitz, differentiable and invertible K closed convex set with non empty interior Problem of interest : Is there a structural condition on the coupe [K, σ] under which for any payoff h in a given class, we have p K [h] = p[f K [h]]? First, simplified version : Is there a structural condition on the couple [K, σ] under which For any payoff h CK 1, we have p K [h] = p[h]? where CK 1 denotes the class of Cb 1 functions with derivatives valued in K. (i.e. regular and stable under F K )
19 BSDE representation for the For any payoff h CK 1, the unconstrained price (p(t, S t)) 0 t T of h(s T ) is solution of the BSDE Y t = h(s T ) T t Z r dw r, 0 t T. The corresponding hedging strategy h t identifies to xp(t, S t) = Y t( X t) 1. Hence satisfies the (linear) BSDE : T h t = h(s T ) + t We know that h(s T ) K. d [ xσ j (S r )] Γ h r σ(s r )dr j=1 = End up on a viability problem : T t Γ h r σ(s r )dw r, For any h valued in K, does the solution h of the BSDE remains in K?
20 BSDE representation for the For any payoff h CK 1, the unconstrained price (p(t, S t)) 0 t T of h(s T ) is solution of the BSDE Y t = h(s T ) T t Z r dw r, 0 t T. The corresponding hedging strategy h t identifies to xp(t, S t) = Y t( X t) 1. Hence satisfies the (linear) BSDE : T h t = h(s T ) + t We know that h(s T ) K. d [ xσ j (S r )] Γ h r σ(s r )dr j=1 = End up on a viability problem : T t Γ h r σ(s r )dw r, For any h valued in K, does the solution h of the BSDE remains in K?
21 Viability property for BSDE [Buckdahn, Quincampoix, Rascanu] provide a Necessary and Sufficient condition for viability property on BSDEs (or PDEs) : For any terminal condition ξ K, the solution of the BSDE Y t = ξ + T t F (Y s, Z s)ds T t Z sdw s satisfies Y t K P a.s., for 0 t T. There exists C > 0 such that 2 y π K (y), F (y, z) xx[d 2 K (y)]z, z + Cd 2 K (y), (y, z) R d M d where π K and d K are the projection and distance operators on K. = This provides a sufficient condition for our problem. Is it necessary?
22 Viability property for BSDE [Buckdahn, Quincampoix, Rascanu] provide a Necessary and Sufficient condition for viability property on BSDEs (or PDEs) : For any terminal condition ξ K, the solution of the BSDE Y t = ξ + T t F (Y s, Z s)ds T t Z sdw s satisfies Y t K P a.s., for 0 t T. There exists C > 0 such that 2 y π K (y), F (y, z) xx[d 2 K (y)]z, z + Cd 2 K (y), (y, z) R d M d where π K and d K are the projection and distance operators on K. = This provides a sufficient condition for our problem. Is it necessary?
23 Revisiting the condition of [BQR] for "regular" convex set K There exists C > 0 such that 2 y π K (y), F (y, z) xx[dk 2 (y)]z, z + CdK 2 (y), (y, z) R d M d Restriction to Polyhedral convex Denoting by n the unit normal vector to K, there exists C > 0 s.t. 2 y π K (y), F (y, z) 1 2 n(y)z, n(y)z + Cd K 2 (y), y / Int(K), z M d Restriction to Polyhedral convex There exists C > 0 s.t. y / Int(K), z M d satisfying n(y) z = 0, y π K (y), F (y, z) CdK 2 (y) Restriction to Polyhedral convex n(y), F (y, z) 0, (y, z) K M d s.t. n(y) z = 0
24 Revisiting the condition of [BQR] for "regular" convex set K There exists C > 0 such that 2 y π K (y), F (y, z) xx[dk 2 (y)]z, z + CdK 2 (y), (y, z) R d M d Restriction to Polyhedral convex Denoting by n the unit normal vector to K, there exists C > 0 s.t. 2 y π K (y), F (y, z) 1 2 n(y)z, n(y)z + Cd K 2 (y), y / Int(K), z M d Restriction to Polyhedral convex There exists C > 0 s.t. y / Int(K), z M d satisfying n(y) z = 0, y π K (y), F (y, z) CdK 2 (y) Restriction to Polyhedral convex n(y), F (y, z) 0, (y, z) K M d s.t. n(y) z = 0
25 Revisiting the condition of [BQR] for "regular" convex set K There exists C > 0 such that 2 y π K (y), F (y, z) xx[dk 2 (y)]z, z + CdK 2 (y), (y, z) R d M d Restriction to Polyhedral convex Denoting by n the unit normal vector to K, there exists C > 0 s.t. 2 y π K (y), F (y, z) 1 2 n(y)z, n(y)z + Cd K 2 (y), y / Int(K), z M d Restriction to Polyhedral convex There exists C > 0 s.t. y / Int(K), z M d satisfying n(y) z = 0, y π K (y), F (y, z) CdK 2 (y) Restriction to Polyhedral convex n(y), F (y, z) 0, (y, z) K M d s.t. n(y) z = 0
26 Revisiting the condition of [BQR] for "regular" convex set K There exists C > 0 such that 2 y π K (y), F (y, z) xx[dk 2 (y)]z, z + CdK 2 (y), (y, z) R d M d Restriction to Polyhedral convex Denoting by n the unit normal vector to K, there exists C > 0 s.t. 2 y π K (y), F (y, z) 1 2 n(y)z, n(y)z + Cd K 2 (y), y / Int(K), z M d Restriction to Polyhedral convex There exists C > 0 s.t. y / Int(K), z M d satisfying n(y) z = 0, y π K (y), F (y, z) CdK 2 (y) Restriction to Polyhedral convex n(y), F (y, z) 0, (y, z) K M d s.t. n(y) z = 0
27 Adapting the condition to our framework 2 n(y), F (y, z) 0, (y, z) K M d s.t. n(y) z = 0 rewrites d 2 n(y), [ xσ j (x)] γσ(x) = 0, (y, γ) K M d s.t. n(y) γ = 0 j=1 Condition too strong in our context. But γ is symmetric and we shall work under the condition : d n(y), [ xσ j (x)] γσ(x) = 0, (x, y, γ) R d K S d s.t. n(y) γ = 0 j=1 Technical point : What about points with multiple normal vectors? = Need to restrict to border points K with unique normal vector
28 Adapting the condition to our framework 2 n(y), F (y, z) 0, (y, z) K M d s.t. n(y) z = 0 rewrites d 2 n(y), [ xσ j (x)] γσ(x) = 0, (y, γ) K M d s.t. n(y) γ = 0 j=1 Condition too strong in our context. But γ is symmetric and we shall work under the condition : d n(y), [ xσ j (x)] γσ(x) = 0, (x, y, γ) R d K S d s.t. n(y) γ = 0 j=1 Technical point : What about points with multiple normal vectors? = Need to restrict to border points K with unique normal vector
29 Adapting the condition to our framework 2 n(y), F (y, z) 0, (y, z) K M d s.t. n(y) z = 0 rewrites d 2 n(y), [ xσ j (x)] γσ(x) = 0, (y, γ) K M d s.t. n(y) γ = 0 j=1 Condition too strong in our context. But γ is symmetric and we shall work under the condition : d n(y), [ xσ j (x)] γσ(x) = 0, (x, y, γ) R d K S d s.t. n(y) γ = 0 j=1 Technical point : What about points with multiple normal vectors? = Need to restrict to border points K with unique normal vector.
30 The main result For a closed convex set K s.t. Int K and an elliptic volatility σ, we have : For any payoff h CK 1, the hedging strategy of h(s t) belongs to K, i.e. p K (h) = p(h) d n(y), [ xσ j (x)] γσ(x) = 0, (x, y, γ) R d K S d s.t. n(y) γ = 0 j=1 This provides a structural condition on the couple [K,σ] under which portfolio restrictions have no effect on payoff functions whose derivatives satisfy the constraint.
31 Sketch of proof Half-space decomposition of K K = y K H y with H y half-space containing K and tangent to K at y Due to the linearity of the driver, we observe K is viable any half-space H y is viable = need to verify that each half-space H y with normal vector n(y) is viable iff d n(y), [ xσ j (x)] γσ(x) = 0, (x, γ) R d S d s.t. n(y) γ = 0 j=1 Focus on the dynamics of n(y), t For solution of the BSDE with T H y,ito s formula gives T d n(y), t 0 + n(y), [ xσ j (X r )] Γ r σ(x r ) dr T t j=1 t n(y), Γ r σ(x r )dw r Probability change = the condition is sufficient Terminal condition T = γ(x T x) = the condition is necessary
32 The constrained super replication problem under constraints What happens if the payoff needs to be facelifted? For any payoff h H, the hedging strategy of F K [h](s t) belongs to K, i.e. p K (h) = p(f K [h]) d n(y), [ xσ j (x)] γσ(x) = 0, (x, y, γ) R d K S d s.t. n(y) γ = 0 j=1 where H it the class of lower semi continuous, bounded from below payoffs s.t. E F K [h](s t,x T ) 2 <, (t, x) [0, T ] R d. When K is bounded, we can restrict to lower semi continuous functions.
33 The Necessary and sufficient condition d n(y), [ xσ j (x)] γσ(x) = 0, (x, y, γ) R d K S d s.t. n(y) γ = 0 j=1 For a fixed y, let introduce (n(y), n 2(y),..., n d (y)) an orthonormal basis of R d. The family (e kl ) 2 k l d of n(n 1)/2 elements given by e kl = n l (y) n k (y) + n k (y) n l (y), 2 k l d. is an orthonormal basis of { γ S d, s.t. n(y) γ = 0 }. The Necessary and Sufficient condition rewrites d n(y), x n k (y), σ.j (x) n l (y), σ.j (x) j=1 = 0, y K, 2 k, l d.
34 No short Sell on Asset 1 In dimension 2 No short sell on Asset 1 : n(y) = (1, 0), hence n = (0, 1) and the condition rewrites 1 [ σ σ 22 2] = 0 The quadratic variation of asset 2 does not depend on asset 1. In dimension d No short sell on Asset 1 : n(y) = (1, 0,..., 0), hence n j = (1 {i=j} ) i and the condition rewrites 1 [σ l1 σ k σ ld σ kd] = 0, 2 l k d. The quadratic covariation between other assets does not depend on asset 1.
35 Asset 1 non tradable In dimension 2 Asset 1 not tradable : n(y) = (1, 0), hence n = (0, 1) and the condition rewrites 1 [ σ σ 22 2] = 0 The quadratic variation of asset 2 does not depend on asset 1. Same conditions as for the no short sell case since only the border of the convex set K matters.
36 Bound on the number of allowed positions Bound of the form C. The convex set is a losange and we have two type of normal vectors. First n(y) = (1, 1) so that n(y) = ( 1, 1) and the condition rewrites [ 1 σ 11 σ σ 12 σ 22 2] [ + 2 σ 11 σ σ 12 σ 22 2] = 0 Second n(y) = ( 1, 1) so that n(y) = (1, 1) and the condition rewrites 1 [ σ 11 + σ σ 12 + σ 22 2] 2 [ σ 11 + σ σ 12 + σ 22 2] = 0 Conditions on quadratic variations in normal directions
37 Other applications in dimension 2 Which convex sets work for the Black Scholes model? Only the hypercube ones. Which model dynamics works for any convex set? For assets with separate dynamics, the condition is equivalent to 1σ 11 = 2σ 21 and 1σ 12 = 2σ 22. Hence, the only possible models are of the form dst 1 = σ 11 (St 1 )dbt 1 + σ 12 (St 1 )dbt 2, dst 2 = [σ 11 (St 2 ) + λ 1]dBt 1 + [σ 12 (St 2 ) + λ 2]dBt 2,
38 Conclusion Necessary and sufficient condition ensuring that in order to super-replicate under constraints, the facelifting procedure of the payoff is sufficient. We can adapt the form of the model to anticipated portfolio constraints. US options. Portfolio constraints in terms of money amount or wealth proportion? How can we compute numerically the solution whenever the condition is not satisfied?
The Self-financing Condition: Remembering the Limit Order Book
The Self-financing Condition: Remembering the Limit Order Book R. Carmona, K. Webster Bendheim Center for Finance ORFE, Princeton University November 6, 2013 Structural relationships? From LOB Models to
More informationForward Dynamic Utility
Forward Dynamic Utility El Karoui Nicole & M RAD Mohamed UnivParis VI / École Polytechnique,CMAP elkaroui@cmapx.polytechnique.fr with the financial support of the "Fondation du Risque" and the Fédération
More informationA Robust Option Pricing Problem
IMA 2003 Workshop, March 12-19, 2003 A Robust Option Pricing Problem Laurent El Ghaoui Department of EECS, UC Berkeley 3 Robust optimization standard form: min x sup u U f 0 (x, u) : u U, f i (x, u) 0,
More informationSample Path Large Deviations and Optimal Importance Sampling for Stochastic Volatility Models
Sample Path Large Deviations and Optimal Importance Sampling for Stochastic Volatility Models Scott Robertson Carnegie Mellon University scottrob@andrew.cmu.edu http://www.math.cmu.edu/users/scottrob June
More informationBasic Concepts and Examples in Finance
Basic Concepts and Examples in Finance Bernardo D Auria email: bernardo.dauria@uc3m.es web: www.est.uc3m.es/bdauria July 5, 2017 ICMAT / UC3M The Financial Market The Financial Market We assume there are
More informationAn overview of some financial models using BSDE with enlarged filtrations
An overview of some financial models using BSDE with enlarged filtrations Anne EYRAUD-LOISEL Workshop : Enlargement of Filtrations and Applications to Finance and Insurance May 31st - June 4th, 2010, Jena
More informationRMSC 4005 Stochastic Calculus for Finance and Risk. 1 Exercises. (c) Let X = {X n } n=0 be a {F n }-supermartingale. Show that.
1. EXERCISES RMSC 45 Stochastic Calculus for Finance and Risk Exercises 1 Exercises 1. (a) Let X = {X n } n= be a {F n }-martingale. Show that E(X n ) = E(X ) n N (b) Let X = {X n } n= be a {F n }-submartingale.
More informationLecture 7: Computation of Greeks
Lecture 7: Computation of Greeks Ahmed Kebaier kebaier@math.univ-paris13.fr HEC, Paris Outline 1 The log-likelihood approach Motivation The pathwise method requires some restrictive regularity assumptions
More informationOptimal trading strategies under arbitrage
Optimal trading strategies under arbitrage Johannes Ruf Columbia University, Department of Statistics The Third Western Conference in Mathematical Finance November 14, 2009 How should an investor trade
More informationRobust Portfolio Choice and Indifference Valuation
and Indifference Valuation Mitja Stadje Dep. of Econometrics & Operations Research Tilburg University joint work with Roger Laeven July, 2012 http://alexandria.tue.nl/repository/books/733411.pdf Setting
More informationChapter 15: Jump Processes and Incomplete Markets. 1 Jumps as One Explanation of Incomplete Markets
Chapter 5: Jump Processes and Incomplete Markets Jumps as One Explanation of Incomplete Markets It is easy to argue that Brownian motion paths cannot model actual stock price movements properly in reality,
More informationModel-independent bounds for Asian options
Model-independent bounds for Asian options A dynamic programming approach Alexander M. G. Cox 1 Sigrid Källblad 2 1 University of Bath 2 CMAP, École Polytechnique University of Michigan, 2nd December,
More informationModel-independent bounds for Asian options
Model-independent bounds for Asian options A dynamic programming approach Alexander M. G. Cox 1 Sigrid Källblad 2 1 University of Bath 2 CMAP, École Polytechnique 7th General AMaMeF and Swissquote Conference
More informationExam Quantitative Finance (35V5A1)
Exam Quantitative Finance (35V5A1) Part I: Discrete-time finance Exercise 1 (20 points) a. Provide the definition of the pricing kernel k q. Relate this pricing kernel to the set of discount factors D
More informationEconomathematics. Problem Sheet 1. Zbigniew Palmowski. Ws 2 dw s = 1 t
Economathematics Problem Sheet 1 Zbigniew Palmowski 1. Calculate Ee X where X is a gaussian random variable with mean µ and volatility σ >.. Verify that where W is a Wiener process. Ws dw s = 1 3 W t 3
More informationAnumericalalgorithm for general HJB equations : a jump-constrained BSDE approach
Anumericalalgorithm for general HJB equations : a jump-constrained BSDE approach Nicolas Langrené Univ. Paris Diderot - Sorbonne Paris Cité, LPMA, FiME Joint work with Idris Kharroubi (Paris Dauphine),
More informationABOUT THE PRICING EQUATION IN FINANCE
ABOUT THE PRICING EQUATION IN FINANCE Stéphane CRÉPEY University of Evry, France stephane.crepey@univ-evry.fr AMAMEF at Vienna University of Technology 17 22 September 2007 1 We derive the pricing equation
More informationFunctional vs Banach space stochastic calculus & strong-viscosity solutions to semilinear parabolic path-dependent PDEs.
Functional vs Banach space stochastic calculus & strong-viscosity solutions to semilinear parabolic path-dependent PDEs Andrea Cosso LPMA, Université Paris Diderot joint work with Francesco Russo ENSTA,
More informationHedging under Arbitrage
Hedging under Arbitrage Johannes Ruf Columbia University, Department of Statistics Modeling and Managing Financial Risks January 12, 2011 Motivation Given: a frictionless market of stocks with continuous
More informationRohini Kumar. Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque)
Small time asymptotics for fast mean-reverting stochastic volatility models Statistics and Applied Probability, UCSB (Joint work with J. Feng and J.-P. Fouque) March 11, 2011 Frontier Probability Days,
More informationWeak Reflection Principle and Static Hedging of Barrier Options
Weak Reflection Principle and Static Hedging of Barrier Options Sergey Nadtochiy Department of Mathematics University of Michigan Apr 2013 Fields Quantitative Finance Seminar Fields Institute, Toronto
More information25857 Interest Rate Modelling
25857 UTS Business School University of Technology Sydney Chapter 20. Change of Numeraire May 15, 2014 1/36 Chapter 20. Change of Numeraire 1 The Radon-Nikodym Derivative 2 Option Pricing under Stochastic
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Simulating Stochastic Differential Equations Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More informationMartingale Approach to Pricing and Hedging
Introduction and echniques Lecture 9 in Financial Mathematics UiO-SK451 Autumn 15 eacher:s. Ortiz-Latorre Martingale Approach to Pricing and Hedging 1 Risk Neutral Pricing Assume that we are in the basic
More informationHedging under arbitrage
Hedging under arbitrage Johannes Ruf Columbia University, Department of Statistics AnStAp10 August 12, 2010 Motivation Usually, there are several trading strategies at one s disposal to obtain a given
More informationArbitrage Bounds for Volatility Derivatives as Free Boundary Problem. Bruno Dupire Bloomberg L.P. NY
Arbitrage Bounds for Volatility Derivatives as Free Boundary Problem Bruno Dupire Bloomberg L.P. NY bdupire@bloomberg.net PDE and Mathematical Finance, KTH, Stockholm August 16, 25 Variance Swaps Vanilla
More informationPAPER 211 ADVANCED FINANCIAL MODELS
MATHEMATICAL TRIPOS Part III Friday, 27 May, 2016 1:30 pm to 4:30 pm PAPER 211 ADVANCED FINANCIAL MODELS Attempt no more than FOUR questions. There are SIX questions in total. The questions carry equal
More informationValuation of derivative assets Lecture 6
Valuation of derivative assets Lecture 6 Magnus Wiktorsson September 14, 2017 Magnus Wiktorsson L6 September 14, 2017 1 / 13 Feynman-Kac representation This is the link between a class of Partial Differential
More informationM5MF6. Advanced Methods in Derivatives Pricing
Course: Setter: M5MF6 Dr Antoine Jacquier MSc EXAMINATIONS IN MATHEMATICS AND FINANCE DEPARTMENT OF MATHEMATICS April 2016 M5MF6 Advanced Methods in Derivatives Pricing Setter s signature...........................................
More informationUtility indifference valuation for non-smooth payoffs on a market with some non tradable assets
Utility indifference valuation for non-smooth payoffs on a market with some non tradable assets - Joint work with G. Benedetti (Paris-Dauphine, CREST) - Luciano Campi Université Paris 13, FiME and CREST
More informationPAPER 27 STOCHASTIC CALCULUS AND APPLICATIONS
MATHEMATICAL TRIPOS Part III Thursday, 5 June, 214 1:3 pm to 4:3 pm PAPER 27 STOCHASTIC CALCULUS AND APPLICATIONS Attempt no more than FOUR questions. There are SIX questions in total. The questions carry
More informationViability, Arbitrage and Preferences
Viability, Arbitrage and Preferences H. Mete Soner ETH Zürich and Swiss Finance Institute Joint with Matteo Burzoni, ETH Zürich Frank Riedel, University of Bielefeld Thera Stochastics in Honor of Ioannis
More informationAMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationOn Using Shadow Prices in Portfolio optimization with Transaction Costs
On Using Shadow Prices in Portfolio optimization with Transaction Costs Johannes Muhle-Karbe Universität Wien Joint work with Jan Kallsen Universidad de Murcia 12.03.2010 Outline The Merton problem The
More informationEquity correlations implied by index options: estimation and model uncertainty analysis
1/18 : estimation and model analysis, EDHEC Business School (joint work with Rama COT) Modeling and managing financial risks Paris, 10 13 January 2011 2/18 Outline 1 2 of multi-asset models Solution to
More informationSPDE and portfolio choice (joint work with M. Musiela) Princeton University. Thaleia Zariphopoulou The University of Texas at Austin
SPDE and portfolio choice (joint work with M. Musiela) Princeton University November 2007 Thaleia Zariphopoulou The University of Texas at Austin 1 Performance measurement of investment strategies 2 Market
More informationContinuous-time Stochastic Control and Optimization with Financial Applications
Huyen Pham Continuous-time Stochastic Control and Optimization with Financial Applications 4y Springer Some elements of stochastic analysis 1 1.1 Stochastic processes 1 1.1.1 Filtration and processes 1
More informationCHAPTER 12. Hedging. hedging strategy = replicating strategy. Question : How to find a hedging strategy? In other words, for an attainable contingent
CHAPTER 12 Hedging hedging dddddddddddddd ddd hedging strategy = replicating strategy hedgingdd) ddd Question : How to find a hedging strategy? In other words, for an attainable contingent claim, find
More informationRobust Trading of Implied Skew
Robust Trading of Implied Skew Sergey Nadtochiy and Jan Obłój Current version: Nov 16, 2016 Abstract In this paper, we present a method for constructing a (static) portfolio of co-maturing European options
More informationOptimal robust bounds for variance options and asymptotically extreme models
Optimal robust bounds for variance options and asymptotically extreme models Alexander Cox 1 Jiajie Wang 2 1 University of Bath 2 Università di Roma La Sapienza Advances in Financial Mathematics, 9th January,
More informationLecture IV Portfolio management: Efficient portfolios. Introduction to Finance Mathematics Fall Financial mathematics
Lecture IV Portfolio management: Efficient portfolios. Introduction to Finance Mathematics Fall 2014 Reduce the risk, one asset Let us warm up by doing an exercise. We consider an investment with σ 1 =
More informationGreek parameters of nonlinear Black-Scholes equation
International Journal of Mathematics and Soft Computing Vol.5, No.2 (2015), 69-74. ISSN Print : 2249-3328 ISSN Online: 2319-5215 Greek parameters of nonlinear Black-Scholes equation Purity J. Kiptum 1,
More informationGeneralized Affine Transform Formulae and Exact Simulation of the WMSV Model
On of Affine Processes on S + d Generalized Affine and Exact Simulation of the WMSV Model Department of Mathematical Science, KAIST, Republic of Korea 2012 SIAM Financial Math and Engineering joint work
More informationNon-semimartingales in finance
Non-semimartingales in finance Pricing and Hedging Options with Quadratic Variation Tommi Sottinen University of Vaasa 1st Northern Triangular Seminar 9-11 March 2009, Helsinki University of Technology
More informationRisk Measures and Optimal Risk Transfers
Risk Measures and Optimal Risk Transfers Université de Lyon 1, ISFA April 23 2014 Tlemcen - CIMPA Research School Motivations Study of optimal risk transfer structures, Natural question in Reinsurance.
More informationDoubly reflected BSDEs with jumps and generalized Dynkin games
Doubly reflected BSDEs with jumps and generalized Dynkin games Roxana DUMITRESCU (University Paris Dauphine, Crest and INRIA) Joint works with M.C. Quenez (Univ. Paris Diderot) and Agnès Sulem (INRIA Paris-Rocquecourt)
More informationMartingale Transport, Skorokhod Embedding and Peacocks
Martingale Transport, Skorokhod Embedding and CEREMADE, Université Paris Dauphine Collaboration with Pierre Henry-Labordère, Nizar Touzi 08 July, 2014 Second young researchers meeting on BSDEs, Numerics
More informationLecture 3: Review of mathematical finance and derivative pricing models
Lecture 3: Review of mathematical finance and derivative pricing models Xiaoguang Wang STAT 598W January 21th, 2014 (STAT 598W) Lecture 3 1 / 51 Outline 1 Some model independent definitions and principals
More informationThe Black-Scholes Equation using Heat Equation
The Black-Scholes Equation using Heat Equation Peter Cassar May 0, 05 Assumptions of the Black-Scholes Model We have a risk free asset given by the price process, dbt = rbt The asset price follows a geometric
More informationLECTURE 4: BID AND ASK HEDGING
LECTURE 4: BID AND ASK HEDGING 1. Introduction One of the consequences of incompleteness is that the price of derivatives is no longer unique. Various strategies for dealing with this exist, but a useful
More informationMATH3075/3975 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS
MATH307/37 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS School of Mathematics and Statistics Semester, 04 Tutorial problems should be used to test your mathematical skills and understanding of the lecture material.
More informationAnalytical formulas for local volatility model with stochastic. Mohammed Miri
Analytical formulas for local volatility model with stochastic rates Mohammed Miri Joint work with Eric Benhamou (Pricing Partners) and Emmanuel Gobet (Ecole Polytechnique Modeling and Managing Financial
More informationHow to hedge Asian options in fractional Black-Scholes model
How to hedge Asian options in fractional Black-Scholes model Heikki ikanmäki Jena, March 29, 211 Fractional Lévy processes 1/36 Outline of the talk 1. Introduction 2. Main results 3. Methodology 4. Conclusions
More informationPricing in markets modeled by general processes with independent increments
Pricing in markets modeled by general processes with independent increments Tom Hurd Financial Mathematics at McMaster www.phimac.org Thanks to Tahir Choulli and Shui Feng Financial Mathematics Seminar
More informationPolynomial processes in stochastic portofolio theory
Polynomial processes in stochastic portofolio theory Christa Cuchiero University of Vienna 9 th Bachelier World Congress July 15, 2016 Christa Cuchiero (University of Vienna) Polynomial processes in SPT
More informationMarkets with convex transaction costs
1 Markets with convex transaction costs Irina Penner Humboldt University of Berlin Email: penner@math.hu-berlin.de Joint work with Teemu Pennanen Helsinki University of Technology Special Semester on Stochastics
More information13.3 A Stochastic Production Planning Model
13.3. A Stochastic Production Planning Model 347 From (13.9), we can formally write (dx t ) = f (dt) + G (dz t ) + fgdz t dt, (13.3) dx t dt = f(dt) + Gdz t dt. (13.33) The exact meaning of these expressions
More informationPricing early exercise contracts in incomplete markets
Pricing early exercise contracts in incomplete markets A. Oberman and T. Zariphopoulou The University of Texas at Austin May 2003, typographical corrections November 7, 2003 Abstract We present a utility-based
More informationRobust Portfolio Decisions for Financial Institutions
Robust Portfolio Decisions for Financial Institutions Ioannis Baltas 1,3, Athanasios N. Yannacopoulos 2,3 & Anastasios Xepapadeas 4 1 Department of Financial and Management Engineering University of the
More informationHedging with Life and General Insurance Products
Hedging with Life and General Insurance Products June 2016 2 Hedging with Life and General Insurance Products Jungmin Choi Department of Mathematics East Carolina University Abstract In this study, a hybrid
More informationOption pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard
Option pricing in the stochastic volatility model of Barndorff-Nielsen and Shephard Indifference pricing and the minimal entropy martingale measure Fred Espen Benth Centre of Mathematics for Applications
More informationIndifference fee rate 1
Indifference fee rate 1 for variable annuities Ricardo ROMO ROMERO Etienne CHEVALIER and Thomas LIM Université d Évry Val d Essonne, Laboratoire de Mathématiques et Modélisation d Evry Second Young researchers
More informationSTOCHASTIC INTEGRALS
Stat 391/FinMath 346 Lecture 8 STOCHASTIC INTEGRALS X t = CONTINUOUS PROCESS θ t = PORTFOLIO: #X t HELD AT t { St : STOCK PRICE M t : MG W t : BROWNIAN MOTION DISCRETE TIME: = t < t 1
More informationA new approach for scenario generation in risk management
A new approach for scenario generation in risk management Josef Teichmann TU Wien Vienna, March 2009 Scenario generators Scenarios of risk factors are needed for the daily risk analysis (1D and 10D ahead)
More informationHedging Credit Derivatives in Intensity Based Models
Hedging Credit Derivatives in Intensity Based Models PETER CARR Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU Stanford
More informationAmerican Foreign Exchange Options and some Continuity Estimates of the Optimal Exercise Boundary with respect to Volatility
American Foreign Exchange Options and some Continuity Estimates of the Optimal Exercise Boundary with respect to Volatility Nasir Rehman Allam Iqbal Open University Islamabad, Pakistan. Outline Mathematical
More informationMARTINGALES AND LOCAL MARTINGALES
MARINGALES AND LOCAL MARINGALES If S t is a (discounted) securtity, the discounted P/L V t = need not be a martingale. t θ u ds u Can V t be a valid P/L? When? Winter 25 1 Per A. Mykland ARBIRAGE WIH SOCHASIC
More informationOptimal Asset Allocation with Stochastic Interest Rates in Regime-switching Models
Optimal Asset Allocation with Stochastic Interest Rates in Regime-switching Models Ruihua Liu Department of Mathematics University of Dayton, Ohio Joint Work With Cheng Ye and Dan Ren To appear in International
More informationDrawdowns, Drawups, their joint distributions, detection and financial risk management
Drawdowns, Drawups, their joint distributions, detection and financial risk management June 2, 2010 The cases a = b The cases a > b The cases a < b Insuring against drawing down before drawing up Robust
More informationVolatility Smiles and Yield Frowns
Volatility Smiles and Yield Frowns Peter Carr NYU IFS, Chengdu, China, July 30, 2018 Peter Carr (NYU) Volatility Smiles and Yield Frowns 7/30/2018 1 / 35 Interest Rates and Volatility Practitioners and
More informationBROWNIAN MOTION Antonella Basso, Martina Nardon
BROWNIAN MOTION Antonella Basso, Martina Nardon basso@unive.it, mnardon@unive.it Department of Applied Mathematics University Ca Foscari Venice Brownian motion p. 1 Brownian motion Brownian motion plays
More informationFinance II. May 27, F (t, x)+αx f t x σ2 x 2 2 F F (T,x) = ln(x).
Finance II May 27, 25 1.-15. All notation should be clearly defined. Arguments should be complete and careful. 1. (a) Solve the boundary value problem F (t, x)+αx f t x + 1 2 σ2 x 2 2 F (t, x) x2 =, F
More informationUtility Indifference Pricing and Dynamic Programming Algorithm
Chapter 8 Utility Indifference ricing and Dynamic rogramming Algorithm In the Black-Scholes framework, we can perfectly replicate an option s payoff. However, it may not be true beyond the Black-Scholes
More informationEuropean option pricing under parameter uncertainty
European option pricing under parameter uncertainty Martin Jönsson (joint work with Samuel Cohen) University of Oxford Workshop on BSDEs, SPDEs and their Applications July 4, 2017 Introduction 2/29 Introduction
More informationThe Uncertain Volatility Model
The Uncertain Volatility Model Claude Martini, Antoine Jacquier July 14, 008 1 Black-Scholes and realised volatility What happens when a trader uses the Black-Scholes (BS in the sequel) formula to sell
More informationFinancial Derivatives Section 5
Financial Derivatives Section 5 The Black and Scholes Model Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un. of
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
More informationMean-Variance Hedging under Additional Market Information
Mean-Variance Hedging under Additional Market Information Frank hierbach Department of Statistics University of Bonn Adenauerallee 24 42 53113 Bonn, Germany email: thierbach@finasto.uni-bonn.de Abstract
More informationModern Methods of Option Pricing
Modern Methods of Option Pricing Denis Belomestny Weierstraß Institute Berlin Motzen, 14 June 2007 Denis Belomestny (WIAS) Modern Methods of Option Pricing Motzen, 14 June 2007 1 / 30 Overview 1 Introduction
More informationStochastic Dynamical Systems and SDE s. An Informal Introduction
Stochastic Dynamical Systems and SDE s An Informal Introduction Olav Kallenberg Graduate Student Seminar, April 18, 2012 1 / 33 2 / 33 Simple recursion: Deterministic system, discrete time x n+1 = f (x
More informationKØBENHAVNS UNIVERSITET (Blok 2, 2011/2012) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours
This question paper consists of 3 printed pages FinKont KØBENHAVNS UNIVERSITET (Blok 2, 211/212) Naturvidenskabelig kandidateksamen Continuous time finance (FinKont) TIME ALLOWED : 3 hours This exam paper
More informationParametric Inference and Dynamic State Recovery from Option Panels. Nicola Fusari
Parametric Inference and Dynamic State Recovery from Option Panels Nicola Fusari Joint work with Torben G. Andersen and Viktor Todorov July 2012 Motivation Under realistic assumptions derivatives are nonredundant
More informationDerivatives Pricing and Stochastic Calculus
Derivatives Pricing and Stochastic Calculus Romuald Elie LAMA, CNRS UMR 85 Université Paris-Est Marne-La-Vallée elie @ ensae.fr Idris Kharroubi CEREMADE, CNRS UMR 7534, Université Paris Dauphine kharroubi
More informationRisk minimizing strategies for tracking a stochastic target
Risk minimizing strategies for tracking a stochastic target Andrzej Palczewski Abstract We consider a stochastic control problem of beating a stochastic benchmark. The problem is considered in an incomplete
More informationArbitrage of the first kind and filtration enlargements in semimartingale financial models. Beatrice Acciaio
Arbitrage of the first kind and filtration enlargements in semimartingale financial models Beatrice Acciaio the London School of Economics and Political Science (based on a joint work with C. Fontana and
More informationLecture 8: The Black-Scholes theory
Lecture 8: The Black-Scholes theory Dr. Roman V Belavkin MSO4112 Contents 1 Geometric Brownian motion 1 2 The Black-Scholes pricing 2 3 The Black-Scholes equation 3 References 5 1 Geometric Brownian motion
More informationThe stochastic calculus
Gdansk A schedule of the lecture Stochastic differential equations Ito calculus, Ito process Ornstein - Uhlenbeck (OU) process Heston model Stopping time for OU process Stochastic differential equations
More informationOptimal Execution: II. Trade Optimal Execution
Optimal Execution: II. Trade Optimal Execution René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Purdue June 21, 212 Optimal Execution
More informationLecture 4. Finite difference and finite element methods
Finite difference and finite element methods Lecture 4 Outline Black-Scholes equation From expectation to PDE Goal: compute the value of European option with payoff g which is the conditional expectation
More informationLocal Volatility Dynamic Models
René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Columbia November 9, 27 Contents Joint work with Sergey Nadtochyi Motivation 1 Understanding
More informationShort-time-to-expiry expansion for a digital European put option under the CEV model. November 1, 2017
Short-time-to-expiry expansion for a digital European put option under the CEV model November 1, 2017 Abstract In this paper I present a short-time-to-expiry asymptotic series expansion for a digital European
More informationAdvanced topics in continuous time finance
Based on readings of Prof. Kerry E. Back on the IAS in Vienna, October 21. Advanced topics in continuous time finance Mag. Martin Vonwald (martin@voni.at) November 21 Contents 1 Introduction 4 1.1 Martingale.....................................
More informationHedging of Contingent Claims under Incomplete Information
Projektbereich B Discussion Paper No. B 166 Hedging of Contingent Claims under Incomplete Information by Hans Föllmer ) Martin Schweizer ) October 199 ) Financial support by Deutsche Forschungsgemeinschaft,
More informationMath 416/516: Stochastic Simulation
Math 416/516: Stochastic Simulation Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 13 Haijun Li Math 416/516: Stochastic Simulation Week 13 1 / 28 Outline 1 Simulation
More informationLévy models in finance
Lévy models in finance Ernesto Mordecki Universidad de la República, Montevideo, Uruguay PASI - Guanajuato - June 2010 Summary General aim: describe jummp modelling in finace through some relevant issues.
More informationLimited liability, or how to prevent slavery in contract theory
Limited liability, or how to prevent slavery in contract theory Université Paris Dauphine, France Joint work with A. Révaillac (INSA Toulouse) and S. Villeneuve (TSE) Advances in Financial Mathematics,
More informationValuation of performance-dependent options in a Black- Scholes framework
Valuation of performance-dependent options in a Black- Scholes framework Thomas Gerstner, Markus Holtz Institut für Numerische Simulation, Universität Bonn, Germany Ralf Korn Fachbereich Mathematik, TU
More informationOption Pricing with Delayed Information
Option Pricing with Delayed Information Mostafa Mousavi University of California Santa Barbara Joint work with: Tomoyuki Ichiba CFMAR 10th Anniversary Conference May 19, 2017 Mostafa Mousavi (UCSB) Option
More informationOptimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models
Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models José E. Figueroa-López 1 1 Department of Statistics Purdue University University of Missouri-Kansas City Department of Mathematics
More informationSensitivity Analysis on Long-term Cash flows
Sensitivity Analysis on Long-term Cash flows Hyungbin Park Worcester Polytechnic Institute 19 March 2016 Eastern Conference on Mathematical Finance Worcester Polytechnic Institute, Worceseter, MA 1 / 49
More information