Martingale invariance and utility maximization

Martingale invariance and utility maximization Thorsten Rheinlander Jena, June 21 Thorsten Rheinlander () Martingale invariance Jena, June 21 1 / 27

Martingale invariance property Consider two ltrations F G. F is immersed in G if every F-martingale is a G-martingale. Let S be some F-adapted locally bounded semi-martingale. Typical choice: F = F S Maximizing expected exponential utility from terminal wealth: sup E exp T ϑ t ds t ϑ2θ Is it possible for an agent to achieve higher expected utility by using G-predictable strategies? What rôle does the immersion property play here? Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 2 / 27

Duality approach to utility maximization Under mild assumptions, the following key duality holds (Delbaen et al (22)): T sup E exp ϑ t ds t = exp inf H (Q, P) ϑ2θ Q 2Me Here Θ denotes a set of admissible strategies, M e the set of all equivalent martingale measures for S, and H (Q, P) is the relative entropy. The link between the two problems is provided by the minimal entropy martingale measure Q E and its density representation dq E T dp = exp c + η t ds t. Here c = H Q E, P, and η is the optimal strategy for the primal problem. Note that in general η depends on the time horizon T. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 3 / 27

Proposition: Let S be a locally bounded F-adapted process such that the entropy measure Q E (G) for S exists (and then Q E (F) exists as well). If F is immersed in G, then E (F) = E (G). Here E (H) is the density process, E t (H) = E h dq E dp H t i, relative to ltration H. Proof: A martingale measure Q for S wrt. G induces a martingale measure bq wrt. F via d bq dp = E dq dp F T. By the conditional Jensen s inequality, we have H Q, b P H (Q, P). Applying this to Q E (G) we get that Q \ E (G) has nite relative entropy, hence there is some martingale measure wrt. F which has nite relative entropy. Therefore Q E (F) exists uniquely. As E (F) S is a local (P, F)-martingale, it is by immersion also a local (P, G)-martingale. Hence E (F) is the density process of a martingale measure wrt. G, with terminal value dq E (F) /dp. As H Q E (F), P H \QE (G), P H Q E (G), P, the result follows by uniqueness of Q E (G ). Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 4 / 27

Remarks. 1) In particular, if E (G) is not F-adapted, then F cannot be immersed into G. This can be useful in case it is complicated to determine the structure of all F-martingales. 2) It is in general not true that Q E (F) = Q E (G) implies E (F) = E (G), as will be seen later on. In particular, the density of Q E (G) can be F T -measurable, whereas the density process E (G) is not F-adapted. 3) Summing up, if F is immersed in G, then no additional expected utility can be gained by employing G-predictable strategies. However, if the immersion property fails, one can in general not conclude that this leads to a utility gain. 4) To apply these results, the main task is to calculate E (G), the density process of the entropy measure wrt. G. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 5 / 27

Optimal martingale measure equation We work with a ltration G such that all G-martingales are continuous. We know that the density of the minimal entropy measure Q E can necessarily be written as dq E dp T = exp c + η t ds t for some constant c and some G-predictable process η. By the structure condition, there exists a local martingale M and a G-predictable process λ such that S = M + λ d [M], K T = T λ 2 t d [M] t <. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 6 / 27

Every martingale measure Q can be written as dq dp = E λ dm L T 1 = exp λ t dm t 2 K 1 T L T 2 [L] T = exp λ t ds t + 1 2 K 1 T L T 2 [L] T, where L is some local (P, G)-martingale strongly orthogonal to M. As we look for a candidate measure with representation as above, we would like to nd such an L as well as a constant c and a G-predictable process ψ such that we can decompose 1 T 2 K T = c + ψ t ds t + L T + 1 2 [L] T. We are looking for a solution (c, ψ, L) to this BSDE, and set η = ψ λ. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 7 / 27

We take as candidate dq T dp = exp c + η t ds t = E λ dm L Veri cation: we have to show that the stochastic exponential is a true martingale, that H Q, P has nite relative entropy, and that T η 2 t d [S] t 2 L exp(p). Here L exp (P) is the Orlicz space generated by the Young function exp (). T. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 8 / 27

We want to show that the following stochastic exponential is a martingale: = E λ dm L This could be done e.g. by a criterion due to Liptser & Shiryaev (1977), however, here we give another approach yielding nite relative entropy of the resulting measure as well. De ne a fundamental sequence of stopping times as T n = min ftj K t _ [L] t ng ^ T By Novikov s criterion, the Tn measures Q n. We have are then densities of probability T = lim n Tn a.s. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 9 / 27

Proposition: The following assertions are equivalent. h i (i) sup n E Qn K Tn + [L] Tn < (ii) sup n H (Q n, P) < (iii) T is the density of a prob. meas. Q, and H (Q, P) <. Proof: (i) () (ii) This follows by a calculation based on Girsanov s theorem. (ii) =) (iii) The sequence ( Tn ) is u.i. by de la Vallée-Poussins criterion. (iii) =) (ii) Follows from Barron s inequality. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 1 / 27

Stein & Stein model The price process S is modelled as strong solution to the SDE and the volatility process V is given as ds t = µv 2 t S t dt + σv t S t db t dv t = (m αv t ) dt + β dw t. Here µ, σ, m, α, β are positive non-zero constants and B, W are two correlated Brownian motions with correlation coe cient ρ 2 ( 1, +1). Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 11 / 27

Solving the optimal martingale measure equation and following the veri cation procedure yields that the entropy measure Q E for S relative to the ltration G generated by the BM (B, W ) is given by dq E dp = exp c T 1 ρρ 1 φ t + ψ t Vt 1 S 1 t ds t. Here φ, ψ are bounded deterministic functions which can be obtained as system of two di erential equations involving one Riccatti-type equation. In case the mean-reversion level m is non-zero, ψ will be non-zero. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 12 / 27

We choose F = F S, G = F B,W. In case ρ 2 ( 1, +1), G is the progressive enlargement of F S by sgn (V ) which is not F S -adapted. As consequence, if the mean-reversion level m is non-zero and ρ 2 ( 1, +1), then F is not immersed in G. An agent who observes sgn (V ) can take advantage of this additional information when maximizing her expected exponential utility from terminal wealth, provided ρ, m 6=. In the uncorrelated case ρ =, however, F is not immersed in G if m 6=, but since η is then equal to 1/S, the optimal investment strategy does not use the additional information and there is no bene t for the insider. Note that in this case we have Q E (G) = Q E (F), but E (G) 6= E (F). It seems to be di cult to calculate E wrt. F S in case m 6= because of the rather complicated structure of F S. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 13 / 27

Models with default Let (Ω, F, P) be a probability space, supporting a random variable modeling the time of default τ >. H t = I τt is the counting process associated with τ, H the ltration generated by H. Let F be a Brownian ltration, generated by some BM W. We assume that there is a deterministic intensity function µ such that t P (τ > t) = exp µ s ds. Let G = H _ F. We assume that F is immersed into G. In particular, W stays a Brownian motion in the larger ltration G. The G-martingale M associated with the one-jump process is given as M t = H t τ^t µ s ds. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 14 / 27

We model the price process es of a defaultable security under P as d es t /es t = a t dt + b t dw t + c t dm t The functions a, b, and c are deterministic and bounded on [, T ]. Finding a pricing measure by calibration is sometimes di cult in this framework. Therefore, we will choose martingale measures according to optimality criteria. Our main point is that the presence of two di erent noise terms dw and dm makes the analysis much more complicated compared to a scenario with only a Brownian driver. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 15 / 27

Our goal is now to calculate several optimal martingale measures, i.e. probability measures Q such that the price process es is a local Q-martingale. This is equivalent to the statement that the stochastic logarithm S = R d es/es is a local Q-martingale. Assuming the existence of an equivalent martingale measure and a structure condition, we can write the semi-martingale decomposition of S uniquely in the form S = N + λ d hni for a local martingale N and a predictable process λ; here hni denotes the predictable compensator of the quadratic variation process [N]. In our concrete market model, it is readily computed that a dn = b dw + c dm, λ = b 2 + µc 2. Moreover, we have a λ dn = b 2 (b dw + c dm), + µc2 λ N = λc M. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 16 / 27

Minimal martingale measure The minimal martingale measure (see Schweizer (1995)), commonly referred to as bp, is characterized by the property that P-martingales strongly orthogonal to N under P remain martingales under bp. Its density is given by the Doléans-Dade stochastic exponential T 1 T E λ dn = exp λ t b t dw t λ 2 t bt 2 dt 2 T <tt (1 λ t c t M t ). As there is one single jump of M with jump size one, the density of the minimal martingale measure gets negative with non-zero probability in case a jump occurs at τ T and λ τ c τ > 1. Therefore, bp is in general a signed measure, and we conclude that the minimal martingale measure is not a good choice in our situation. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 17 / 27

Linear Esscher measure For every strategy ϑ such that R ϑ ds is exponentially special, we de ne the modi ed Laplace cumulant process K S (ϑ) as exponential compensator of R ϑ ds. In particular, ϑ = exp ϑ ds K S (ϑ) is a local martingale with ϑ = 1. In case ϑ is a martingale on [, T ], it is the density process of a probability measure P ϑ. If we can nd ϑ such that S is a local P ϑ -martingale, P ϑ is called the linear Esscher measure. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 18 / 27

By Kallsen & Shiryaev (21), we have to solve the equation DK X (ϑ) =. This translates into the equation µ a = b 2 ϑ+cµe ϑc By the boundedness of the co-e cients, a bounded solution always exists and gives a martingale measure by the above exponential tilting due to a veri cation result by Protter & Shimbo (28). The linear Esscher measure is stable wrt. stopping, and coincides with the minimal Hellinger measure as studied by Choulli & Stricker (25). Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 19 / 27

Minimal entropy martingale measure Entropy measure Q E : Its density can always be written in the form dq E T dp = exp c + φ t ds t for some predictable process φ such that R φ ds is a Q E -martingale. In general, a martingale measure Q for S has a density of the form dq dp = E λ dn + L, where L, L =, is a local martingale strongly orthogonal to N. According to martingale representation we can write L as dl = b L dw + c L dm for some predictable processes b L, c L. The orthogonality relation hn, Li = yields bb L + cc L µ 2 =. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 2 / 27 T

Equating the two di erent representations, we obtain the optimal martingale measure equation 1 c = φa T 2 λ2 b 2 1 + µλc 2 b2 L µc L φµc dt + ( φb λb + b L ) dw T + log (1 (λ τ c τ c L (τ) φ τ c τ )) I τt. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 21 / 27

Assuming for now formally that there exists a smooth function u u(t, h) (where t 2 [, T ] and h 2 f, 1g) such that we write log (1 (λ τ c τ c L (τ) φ τ c τ )) = u (t, H t ) u (t, H t ), u (T, h) =, h 2 f, 1g, log (1 (λ τ c τ c L (τ) φ τ c τ )) = u (τ, 1) u (τ, ) = fu(t, 1) u (τ, 1) + u (τ, ) u (, )g = u (, ) T t u (t, H t) dt u (, ). Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 22 / 27

We work with the hypothesis that the integrals over the dt-terms as well as the dw -terms vanish at maturity T. This yields the two equations φa + 1 2 λ2 b 2 µλc + 1 2 b2 L + µc L + φµc + u t =, φb + λb b L =. Inserting the orthogonality relation, we solve for φ as φ = λ + cc Lµ 2 b 2. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 23 / 27

Moreover, we get, with the notation t u := u (t, H t ) u (t, H t ), that Introducing c L = exp ( u + φc) + λc 1 = exp u λc + c2 µ 2 g (t, u t ) := φa + 1 2 λ2 b 2 b 2 c L + λc 1. µλc + 1 2 b2 L + µc L + φµc, t we derive the system of two ordinary di erential equations, u t + g (t, u t) =, u (T, ) = u(t, 1) =. Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 24 / 27

The density process E associated with Q E de ned by t E = dqe dp Gt = E λ u b u bu L dw u + λ u cu S + c L u dm u is the density process of the minimal entropy martingale measure. Veri cation proceeds along the lines of Rheinländer & Steiger (26). t Thorsten Rheinlander (London School of Economics) Martingale invariance Jena, June 21 25 / 27

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