L. Campi and M. Del Vigna Weak Insider Trading and Behavioral Finance

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1 WEAK INSIDER TRADING AND BEHAVIORAL FINANCE L. Campi 1 M. Del Vigna 2 1 Université Paris XIII 2 Department of Mathematics for Economic Decisions University of Florence 5 th Florence-Ritsumeikan Workshop University of Florence March 12-13, 2013

2 OUTLINE 1 What is Insider Trading? 2 Cumulative Prospect Theory (CPT) A. Tversky and D. Kahneman, J. Risk Uncertainty, The Weak Approach F. Baudoin, Stoch. Proc. Appl., Enlargement of Filtrations J. Jeulin, Lecture Notes in Mathematics, Further Developments

3 OUTLINE 1 What is Insider Trading? 2 Cumulative Prospect Theory (CPT) A. Tversky and D. Kahneman, J. Risk Uncertainty, The Weak Approach F. Baudoin, Stoch. Proc. Appl., Enlargement of Filtrations J. Jeulin, Lecture Notes in Mathematics, Further Developments

4 OUTLINE 1 What is Insider Trading? 2 Cumulative Prospect Theory (CPT) A. Tversky and D. Kahneman, J. Risk Uncertainty, The Weak Approach F. Baudoin, Stoch. Proc. Appl., Enlargement of Filtrations J. Jeulin, Lecture Notes in Mathematics, Further Developments

5 OUTLINE 1 What is Insider Trading? 2 Cumulative Prospect Theory (CPT) A. Tversky and D. Kahneman, J. Risk Uncertainty, The Weak Approach F. Baudoin, Stoch. Proc. Appl., Enlargement of Filtrations J. Jeulin, Lecture Notes in Mathematics, Further Developments

6 OUTLINE 1 What is Insider Trading? 2 Cumulative Prospect Theory (CPT) A. Tversky and D. Kahneman, J. Risk Uncertainty, The Weak Approach F. Baudoin, Stoch. Proc. Appl., Enlargement of Filtrations J. Jeulin, Lecture Notes in Mathematics, Further Developments

7 INSIDER TRADING IN ADVANCED COUNTRIES Insider dealing: corporate insiders buy and sell stock in their own companies. Market abuse: trading a security in breach of a fiduciary duty or trust while in possession of non-public information. It includes tipping and tuyautage". Insider dealing is LEGAL in the U.S. if only if trading activity is reported to the SEC Tipping and tuyautage by banks and financial intermediaries are LEGAL (and common) in Japan, but they are ILLEGAL in the U.S. and in the U.E. Japan announced new insider trading regulations in a year s time, enforcing criminal penalties and heavy fines

8 INSIDER TRADING IN ADVANCED COUNTRIES Insider dealing: corporate insiders buy and sell stock in their own companies. Market abuse: trading a security in breach of a fiduciary duty or trust while in possession of non-public information. It includes tipping and tuyautage". Insider dealing is LEGAL in the U.S. if only if trading activity is reported to the SEC Tipping and tuyautage by banks and financial intermediaries are LEGAL (and common) in Japan, but they are ILLEGAL in the U.S. and in the U.E. Japan announced new insider trading regulations in a year s time, enforcing criminal penalties and heavy fines

9 FAMOUS AND RECENT EPISODES In , Rajat Gupta (former Goldman Sachs) tipped Raj Rajaratnam (Galleon hedge fund) for $90m of illegal profits. On November 28, 2011, Rajaratnam was sent to jail (11 years + $10m penalty). On October 25, 2012, Gupta was sent to jail (2 years + $5m penalty) In July 2012, top executives at Nomura resigned. The staff had leaked insider information about new share issues planned by Tepco, Inpex and Mizuho, in 2010 Italian managers of MPS, Saipem and RCS are currently under investigation for insider dealing and market abuse In the U.S. the FBI is investigating on the takeover of the ketchup producer H.J. Heinz Co. by Warren Buffett holding Berkeshire Hathaway (February, 2013)

10 FAMOUS AND RECENT EPISODES In , Rajat Gupta (former Goldman Sachs) tipped Raj Rajaratnam (Galleon hedge fund) for $90m of illegal profits. On November 28, 2011, Rajaratnam was sent to jail (11 years + $10m penalty). On October 25, 2012, Gupta was sent to jail (2 years + $5m penalty) In July 2012, top executives at Nomura resigned. The staff had leaked insider information about new share issues planned by Tepco, Inpex and Mizuho, in 2010 Italian managers of MPS, Saipem and RCS are currently under investigation for insider dealing and market abuse In the U.S. the FBI is investigating on the takeover of the ketchup producer H.J. Heinz Co. by Warren Buffett holding Berkeshire Hathaway (February, 2013)

11 FAMOUS AND RECENT EPISODES In , Rajat Gupta (former Goldman Sachs) tipped Raj Rajaratnam (Galleon hedge fund) for $90m of illegal profits. On November 28, 2011, Rajaratnam was sent to jail (11 years + $10m penalty). On October 25, 2012, Gupta was sent to jail (2 years + $5m penalty) In July 2012, top executives at Nomura resigned. The staff had leaked insider information about new share issues planned by Tepco, Inpex and Mizuho, in 2010 Italian managers of MPS, Saipem and RCS are currently under investigation for insider dealing and market abuse In the U.S. the FBI is investigating on the takeover of the ketchup producer H.J. Heinz Co. by Warren Buffett holding Berkeshire Hathaway (February, 2013)

12 FAMOUS AND RECENT EPISODES In , Rajat Gupta (former Goldman Sachs) tipped Raj Rajaratnam (Galleon hedge fund) for $90m of illegal profits. On November 28, 2011, Rajaratnam was sent to jail (11 years + $10m penalty). On October 25, 2012, Gupta was sent to jail (2 years + $5m penalty) In July 2012, top executives at Nomura resigned. The staff had leaked insider information about new share issues planned by Tepco, Inpex and Mizuho, in 2010 Italian managers of MPS, Saipem and RCS are currently under investigation for insider dealing and market abuse In the U.S. the FBI is investigating on the takeover of the ketchup producer H.J. Heinz Co. by Warren Buffett holding Berkeshire Hathaway (February, 2013)

13 IN SUPPORT OF INSIDER TRADING MILTON FRIEDMAN, 2003 You want more insider trading, not less. You want to give the people most likely to have knowledge about deficiencies of the company an incentive to make the public aware of that. Investors benefit from insider trading since non-public information is immediately incorporated in the price > Efficient Market Hypothesis (Fama, J. Finance, 1991) Trading where one party is more informed than the other is legal in other markets, such as real estate or commodity markets

14 IN SUPPORT OF INSIDER TRADING MILTON FRIEDMAN, 2003 You want more insider trading, not less. You want to give the people most likely to have knowledge about deficiencies of the company an incentive to make the public aware of that. Investors benefit from insider trading since non-public information is immediately incorporated in the price > Efficient Market Hypothesis (Fama, J. Finance, 1991) Trading where one party is more informed than the other is legal in other markets, such as real estate or commodity markets

15 IN SUPPORT OF INSIDER TRADING MILTON FRIEDMAN, 2003 You want more insider trading, not less. You want to give the people most likely to have knowledge about deficiencies of the company an incentive to make the public aware of that. Investors benefit from insider trading since non-public information is immediately incorporated in the price > Efficient Market Hypothesis (Fama, J. Finance, 1991) Trading where one party is more informed than the other is legal in other markets, such as real estate or commodity markets

16 CUMULATIVE PROSPECT THEORY (A. TVERSKY AND D. KAHNEMAN, 1992) Wealth Reference Point for Gains and Losses (RP) S-shaped Value Function with Loss Aversion (u +, u ) Reversed S-shaped Probability Distortions (T +, T )

17 CUMULATIVE PROSPECT THEORY (A. TVERSKY AND D. KAHNEMAN, 1992) Wealth Reference Point for Gains and Losses (RP) S-shaped Value Function with Loss Aversion (u +, u ) Reversed S-shaped Probability Distortions (T +, T )

18 CUMULATIVE PROSPECT THEORY (A. TVERSKY AND D. KAHNEMAN, 1992) Wealth Reference Point for Gains and Losses (RP) S-shaped Value Function with Loss Aversion (u +, u ) Reversed S-shaped Probability Distortions (T +, T )

19 PORTFOLIO OPTIMIZATION FOR A CPT TRADER (1) (Ω, F, P) is an atomless probability space F = {F t } t T is a completed right-continuous filtration W is a (F, P) m-dimensional Brownian motion T > 0 is a fixed constant time horizon THE MARKET 1 risk-free asset ds 0 (t) = S 0 (t)r(t)dt m risky assets ds i (t) = S i (t)[b i (t)dt + m j=1 σ ij(t)dw j (t)], i = 1,..., m in L 2 (Ω, F, P), continuous, adapted Hypotheses: ρ is the unique atomless pricing kernel (or SDF) Q is the unique EMM on F T

20 PORTFOLIO OPTIMIZATION FOR A CPT TRADER (1) (Ω, F, P) is an atomless probability space F = {F t } t T is a completed right-continuous filtration W is a (F, P) m-dimensional Brownian motion T > 0 is a fixed constant time horizon THE MARKET 1 risk-free asset ds 0 (t) = S 0 (t)r(t)dt m risky assets ds i (t) = S i (t)[b i (t)dt + m j=1 σ ij(t)dw j (t)], i = 1,..., m in L 2 (Ω, F, P), continuous, adapted Hypotheses: ρ is the unique atomless pricing kernel (or SDF) Q is the unique EMM on F T

21 PORTFOLIO OPTIMIZATION FOR A CPT TRADER (1) (Ω, F, P) is an atomless probability space F = {F t } t T is a completed right-continuous filtration W is a (F, P) m-dimensional Brownian motion T > 0 is a fixed constant time horizon THE MARKET 1 risk-free asset ds 0 (t) = S 0 (t)r(t)dt m risky assets ds i (t) = S i (t)[b i (t)dt + m j=1 σ ij(t)dw j (t)], i = 1,..., m in L 2 (Ω, F, P), continuous, adapted Hypotheses: ρ is the unique atomless pricing kernel (or SDF) Q is the unique EMM on F T

22 PORTFOLIO OPTIMIZATION FOR A CPT TRADER (2) (H. JIN AND X.Y. ZHOU, Math. Financ., 2008) THE OPTIMIZATION PROBLEM Given the initial wealth x 0 R and a RP = 0: Maximize V (X) := V + (X + ) V (X ) subject to E P [ρx] = x 0, X is F T -measurable and P-a.s. lower bounded V + (X + ) := + 0 T + (P{u + (X + ) > y}) dy V (X ) := + 0 T (P{u (X ) > y}) dy Idea: Split up Merge Positive part: concave max use convex duality Negative part: concave min find a corner point solution

23 PORTFOLIO OPTIMIZATION FOR A CPT TRADER (2) (H. JIN AND X.Y. ZHOU, Math. Financ., 2008) THE OPTIMIZATION PROBLEM Given the initial wealth x 0 R and a RP = 0: Maximize V (X) := V + (X + ) V (X ) subject to E P [ρx] = x 0, X is F T -measurable and P-a.s. lower bounded V + (X + ) := + 0 T + (P{u + (X + ) > y}) dy V (X ) := + 0 T (P{u (X ) > y}) dy Idea: Split up Merge Positive part: concave max use convex duality Negative part: concave min find a corner point solution

24 PORTFOLIO OPTIMIZATION FOR A CPT TRADER (2) (H. JIN AND X.Y. ZHOU, Math. Financ., 2008) THE OPTIMIZATION PROBLEM Given the initial wealth x 0 R and a RP = 0: Maximize V (X) := V + (X + ) V (X ) subject to E P [ρx] = x 0, X is F T -measurable and P-a.s. lower bounded V + (X + ) := + 0 T + (P{u + (X + ) > y}) dy V (X ) := + 0 T (P{u (X ) > y}) dy Idea: Split up Merge Positive part: concave max use convex duality Negative part: concave min find a corner point solution

25 THE SOLUTION OF THE PROBLEM THEOREM (H. JIN AND X.Y. ZHOU, 2008) If u ( ) is strictly concave at 0, then the optimal solution is ( ) X = (u +) 1 λ ρ T + (F(ρ)) I ρ c x + x 0 E P [ρi ρ>c ] I ρ>c Gain(ω) I ρ c Loss I ρ>c The realization of a gain or a loss only depends on the terminal state of the pricing kernel ρ Since x + x 0, the trader possibly takes a leverage to finance a shortfall x + x 0 The payoff is a combination of two binary options and resembles a lottery ticket

26 THE SOLUTION OF THE PROBLEM THEOREM (H. JIN AND X.Y. ZHOU, 2008) If u ( ) is strictly concave at 0, then the optimal solution is ( ) X = (u +) 1 λ ρ T + (F(ρ)) I ρ c x + x 0 E P [ρi ρ>c ] I ρ>c Gain(ω) I ρ c Loss I ρ>c The realization of a gain or a loss only depends on the terminal state of the pricing kernel ρ Since x + x 0, the trader possibly takes a leverage to finance a shortfall x + x 0 The payoff is a combination of two binary options and resembles a lottery ticket

27 THE SOLUTION OF THE PROBLEM THEOREM (H. JIN AND X.Y. ZHOU, 2008) If u ( ) is strictly concave at 0, then the optimal solution is ( ) X = (u +) 1 λ ρ T + (F(ρ)) I ρ c x + x 0 E P [ρi ρ>c ] I ρ>c Gain(ω) I ρ c Loss I ρ>c The realization of a gain or a loss only depends on the terminal state of the pricing kernel ρ Since x + x 0, the trader possibly takes a leverage to finance a shortfall x + x 0 The payoff is a combination of two binary options and resembles a lottery ticket

28 WEAK INSIDER TRADING (F. BAUDOIN, 2002) (Ω, F, Q, F) is an atomless probability space 1 risk-free asset with price S 0 1 m risky assets whose prices S(t) = (S 1 (t),..., S m (t)) L 2 (Ω, F, Q, F)-martingales, continuous, adapted Extra information on the r.v. Y (related to the stock prices) INFORMATION OF THE NON-INFORMED TRADER Q the unique EMM Q Y the law of Y under Q INFORMATION OF THE INSIDER Q and Q Y ν the law of Y under P (the unknown historical probability)

29 WEAK INSIDER TRADING (F. BAUDOIN, 2002) (Ω, F, Q, F) is an atomless probability space 1 risk-free asset with price S 0 1 m risky assets whose prices S(t) = (S 1 (t),..., S m (t)) L 2 (Ω, F, Q, F)-martingales, continuous, adapted Extra information on the r.v. Y (related to the stock prices) INFORMATION OF THE NON-INFORMED TRADER Q the unique EMM Q Y the law of Y under Q INFORMATION OF THE INSIDER Q and Q Y ν the law of Y under P (the unknown historical probability)

30 WEAK INSIDER TRADING (F. BAUDOIN, 2002) (Ω, F, Q, F) is an atomless probability space 1 risk-free asset with price S 0 1 m risky assets whose prices S(t) = (S 1 (t),..., S m (t)) L 2 (Ω, F, Q, F)-martingales, continuous, adapted Extra information on the r.v. Y (related to the stock prices) INFORMATION OF THE NON-INFORMED TRADER Q the unique EMM Q Y the law of Y under Q INFORMATION OF THE INSIDER Q and Q Y ν the law of Y under P (the unknown historical probability)

31 THE MINIMAL PROBABILITY Hypothesis: ν Q Y with density ξ := dν dq Y DEFINITION (F. BAUDOIN, 2002) The probability measure Q ν defined on (Ω, F T ) by Q ν (A) := Q(A Y = y)ν( dy), A F T R is called the minimal probability associated with the weak information (Y, ν). Why minimal? E := {µ Q on Ω s.t. the law of Y under µ is ν}

32 THE MINIMAL PROBABILITY Hypothesis: ν Q Y with density ξ := dν dq Y DEFINITION (F. BAUDOIN, 2002) The probability measure Q ν defined on (Ω, F T ) by Q ν (A) := Q(A Y = y)ν( dy), A F T R is called the minimal probability associated with the weak information (Y, ν). Why minimal? E := {µ Q on Ω s.t. the law of Y under µ is ν}

33 THE MINIMAL PROBABILITY Hypothesis: ν Q Y with density ξ := dν dq Y DEFINITION (F. BAUDOIN, 2002) The probability measure Q ν defined on (Ω, F T ) by Q ν (A) := Q(A Y = y)ν( dy), A F T R is called the minimal probability associated with the weak information (Y, ν). Why minimal? E := {µ Q on Ω s.t. the law of Y under µ is ν}

34 WHY MINIMAL? U : R + R + is a strictly concave utility function X is the terminal wealth of the insider Insider s problem: max Π admissible E??? [U(X)] THEOREM (F. BAUDOIN, 2002) inf µ E sup E µ [U(X)] = Π admissible sup E ν [U(X)] Π admissible The optimal terminal wealth is ˆX = (U ) 1 (λξ(y ) 1 ), where λ is the associated Lagrange multiplier. PROOF. Use the martingale approach + convex duality.

35 WHY MINIMAL? U : R + R + is a strictly concave utility function X is the terminal wealth of the insider Insider s problem: max Π admissible E??? [U(X)] THEOREM (F. BAUDOIN, 2002) inf µ E sup E µ [U(X)] = Π admissible sup E ν [U(X)] Π admissible The optimal terminal wealth is ˆX = (U ) 1 (λξ(y ) 1 ), where λ is the associated Lagrange multiplier. PROOF. Use the martingale approach + convex duality.

36 WHY MINIMAL? U : R + R + is a strictly concave utility function X is the terminal wealth of the insider Insider s problem: max Π admissible E??? [U(X)] THEOREM (F. BAUDOIN, 2002) inf µ E sup E µ [U(X)] = Π admissible sup E ν [U(X)] Π admissible The optimal terminal wealth is ˆX = (U ) 1 (λξ(y ) 1 ), where λ is the associated Lagrange multiplier. PROOF. Use the martingale approach + convex duality.

37 WHY MINIMAL? U : R + R + is a strictly concave utility function X is the terminal wealth of the insider Insider s problem: max Π admissible E??? [U(X)] THEOREM (F. BAUDOIN, 2002) inf µ E sup E µ [U(X)] = Π admissible sup E ν [U(X)] Π admissible The optimal terminal wealth is ˆX = (U ) 1 (λξ(y ) 1 ), where λ is the associated Lagrange multiplier. PROOF. Use the martingale approach + convex duality.

38 THE PROBLEM FOR A WEAK CPT INSIDER THE OPTIMIZATION PROBLEM Maximize V ν (X) := V ν +(X + ) V ν (X ) subject to E ν [ 1 ξ(y ) X ] = x 0 X is F T -measurable and Q ν -a.s. lower bounded V ν +(X + ) := + 0 T + (Q ν {u + (X + ) > y}) dy V ν (X ) := + 0 T (Q ν {u (X ) > y}) dy The solving technique is the same, replace ρ with 1 ξ(y ) and P with Q ν

39 THE PROBLEM FOR A WEAK CPT INSIDER THE OPTIMIZATION PROBLEM Maximize V ν (X) := V ν +(X + ) V ν (X ) subject to E ν [ 1 ξ(y ) X ] = x 0 X is F T -measurable and Q ν -a.s. lower bounded V ν +(X + ) := + 0 T + (Q ν {u + (X + ) > y}) dy V ν (X ) := + 0 T (Q ν {u (X ) > y}) dy The solving technique is the same, replace ρ with 1 ξ(y ) and P with Q ν

40 THE SOLUTION FOR A WEAK CPT INSIDER THEOREM (L. CAMPI AND M. D V, 2011) The optimal solution for a CPT insider is ( X ν = (u +) 1 λ ν ξ(y ) 1 ) T + (F ν (ξ(y ) 1 I )) ξ(y ) 1 c x + x 0 1 E ν [ξ(y ) 1 I ξ(y ) 1 c ]I ξ(y ) 1 >c COMPARISON RESULTS ˆX = (U ) 1 (λξ(y ) 1 ) vs X ν = Gain(ω) I A Loss I A C where A F T depends on the final prices and on (Y, ν) CPT insider selects a combination of two binary options A CPT insider obtains more prospect value than a CPT non-insider (L. Campi and M. D V, 2011)

41 THE SOLUTION FOR A WEAK CPT INSIDER THEOREM (L. CAMPI AND M. D V, 2011) The optimal solution for a CPT insider is ( X ν = (u +) 1 λ ν ξ(y ) 1 ) T + (F ν (ξ(y ) 1 I )) ξ(y ) 1 c x + x 0 1 E ν [ξ(y ) 1 I ξ(y ) 1 c ]I ξ(y ) 1 >c COMPARISON RESULTS ˆX = (U ) 1 (λξ(y ) 1 ) vs X ν = Gain(ω) I A Loss I A C where A F T depends on the final prices and on (Y, ν) CPT insider selects a combination of two binary options A CPT insider obtains more prospect value than a CPT non-insider (L. Campi and M. D V, 2011)

42 THE SOLUTION FOR A WEAK CPT INSIDER THEOREM (L. CAMPI AND M. D V, 2011) The optimal solution for a CPT insider is ( X ν = (u +) 1 λ ν ξ(y ) 1 ) T + (F ν (ξ(y ) 1 I )) ξ(y ) 1 c x + x 0 1 E ν [ξ(y ) 1 I ξ(y ) 1 c ]I ξ(y ) 1 >c COMPARISON RESULTS ˆX = (U ) 1 (λξ(y ) 1 ) vs X ν = Gain(ω) I A Loss I A C where A F T depends on the final prices and on (Y, ν) CPT insider selects a combination of two binary options A CPT insider obtains more prospect value than a CPT non-insider (L. Campi and M. D V, 2011)

43 ENLARGEMENT OF FILTRATIONS Let G be a F -measurable random variable. Define G := {G t } t [0,T ], G t = F t σ(g). Hypotheses: P[G F t ](ω) P[G ] t [0, T ], P a.s.; r(t) 0. THEOREM (J. JACOD, 1985, & J. AMENDINGER, 2000) Define Q G ρ (A) := dp, A G T, pt x = dp[g dx F T ] dp[g dx]. p A T G Q G is a Martingale Preserving Probability Measure (MPPM) Then: Stock prices are (G, Q G )-martingales; Q G is the unique (modulo G 0 ) MPPM and implies market completeness for the insider (i.e. a Martingale Representation Theorem on (Ω, F, Q G, G)).

44 ENLARGEMENT OF FILTRATIONS Let G be a F -measurable random variable. Define G := {G t } t [0,T ], G t = F t σ(g). Hypotheses: P[G F t ](ω) P[G ] t [0, T ], P a.s.; r(t) 0. THEOREM (J. JACOD, 1985, & J. AMENDINGER, 2000) Define Q G ρ (A) := dp, A G T, pt x = dp[g dx F T ] dp[g dx]. p A T G Q G is a Martingale Preserving Probability Measure (MPPM) Then: Stock prices are (G, Q G )-martingales; Q G is the unique (modulo G 0 ) MPPM and implies market completeness for the insider (i.e. a Martingale Representation Theorem on (Ω, F, Q G, G)).

45 ENLARGEMENT OF FILTRATIONS Let G be a F -measurable random variable. Define G := {G t } t [0,T ], G t = F t σ(g). Hypotheses: P[G F t ](ω) P[G ] t [0, T ], P a.s.; r(t) 0. THEOREM (J. JACOD, 1985, & J. AMENDINGER, 2000) Define Q G ρ (A) := dp, A G T, pt x = dp[g dx F T ] dp[g dx]. p A T G Q G is a Martingale Preserving Probability Measure (MPPM) Then: Stock prices are (G, Q G )-martingales; Q G is the unique (modulo G 0 ) MPPM and implies market completeness for the insider (i.e. a Martingale Representation Theorem on (Ω, F, Q G, G)).

46 ENLARGEMENT OF FILTRATIONS Let G be a F -measurable random variable. Define G := {G t } t [0,T ], G t = F t σ(g). Hypotheses: P[G F t ](ω) P[G ] t [0, T ], P a.s.; r(t) 0. THEOREM (J. JACOD, 1985, & J. AMENDINGER, 2000) Define Q G ρ (A) := dp, A G T, pt x = dp[g dx F T ] dp[g dx]. p A T G Q G is a Martingale Preserving Probability Measure (MPPM) Then: Stock prices are (G, Q G )-martingales; Q G is the unique (modulo G 0 ) MPPM and implies market completeness for the insider (i.e. a Martingale Representation Theorem on (Ω, F, Q G, G)).

47 THE SOLUTION FOR AN INSIDER At t = 0, the insider knows, ω by ω, G (related to stock prices) THE OPTIMIZATION PROBLEM FOR THE INSIDER Maximize V (X) = V + (X + ) V (X ) subject to [ ] E P ρ X = x pt G 0, X is G T -measurable and P-a.s. lower bounded The solving technique is the same, just replace ρ with ρ pt G PROPOSITION (M. D V, 2011) The insider s gain is non-negative. PROOF. For the insider, the feasible set is larger.

48 THE SOLUTION FOR AN INSIDER At t = 0, the insider knows, ω by ω, G (related to stock prices) THE OPTIMIZATION PROBLEM FOR THE INSIDER Maximize V (X) = V + (X + ) V (X ) subject to [ ] E P ρ X = x pt G 0, X is G T -measurable and P-a.s. lower bounded The solving technique is the same, just replace ρ with ρ pt G PROPOSITION (M. D V, 2011) The insider s gain is non-negative. PROOF. For the insider, the feasible set is larger.

49 THE SOLUTION FOR AN INSIDER At t = 0, the insider knows, ω by ω, G (related to stock prices) THE OPTIMIZATION PROBLEM FOR THE INSIDER Maximize V (X) = V + (X + ) V (X ) subject to [ ] E P ρ X = x pt G 0, X is G T -measurable and P-a.s. lower bounded The solving technique is the same, just replace ρ with ρ pt G PROPOSITION (M. D V, 2011) The insider s gain is non-negative. PROOF. For the insider, the feasible set is larger.

50 FURTHER DEVELOPMENTS (PH.D. THESIS) A similar technique applies to Dual Theory of Choice V DT (X) = + 0 T (1 F X (x))dx (M. Yaari, Econometrica, 1987) Goal Reaching V GR (X) = P(X B) (S. Browne, Adv. Appl. Probab., 1999) Partially Informed Trader using filtering techniques (T. Björk et al., Math. Method. Oper. Res., 2010) CPT Trader under EMM

51 FURTHER DEVELOPMENTS (PH.D. THESIS) A similar technique applies to Dual Theory of Choice V DT (X) = + 0 T (1 F X (x))dx (M. Yaari, Econometrica, 1987) Goal Reaching V GR (X) = P(X B) (S. Browne, Adv. Appl. Probab., 1999) Partially Informed Trader using filtering techniques (T. Björk et al., Math. Method. Oper. Res., 2010) CPT Trader under EMM

52 FURTHER DEVELOPMENTS (PH.D. THESIS) A similar technique applies to Dual Theory of Choice V DT (X) = + 0 T (1 F X (x))dx (M. Yaari, Econometrica, 1987) Goal Reaching V GR (X) = P(X B) (S. Browne, Adv. Appl. Probab., 1999) Partially Informed Trader using filtering techniques (T. Björk et al., Math. Method. Oper. Res., 2010) CPT Trader under EMM

53 FURTHER DEVELOPMENTS (PH.D. THESIS) A similar technique applies to Dual Theory of Choice V DT (X) = + 0 T (1 F X (x))dx (M. Yaari, Econometrica, 1987) Goal Reaching V GR (X) = P(X B) (S. Browne, Adv. Appl. Probab., 1999) Partially Informed Trader using filtering techniques (T. Björk et al., Math. Method. Oper. Res., 2010) CPT Trader under EMM

54 Thank you!

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