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


 Erik Peters
 1 years ago
 Views:
Transcription
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 FlorenceRitsumeikan Workshop University of Florence March 1213, 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 nonpublic 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 nonpublic 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 nonpublic 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 nonpublic 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 nonpublic 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) Sshaped Value Function with Loss Aversion (u +, u ) Reversed Sshaped Probability Distortions (T +, T )
17 CUMULATIVE PROSPECT THEORY (A. TVERSKY AND D. KAHNEMAN, 1992) Wealth Reference Point for Gains and Losses (RP) Sshaped Value Function with Loss Aversion (u +, u ) Reversed Sshaped Probability Distortions (T +, T )
18 CUMULATIVE PROSPECT THEORY (A. TVERSKY AND D. KAHNEMAN, 1992) Wealth Reference Point for Gains and Losses (RP) Sshaped Value Function with Loss Aversion (u +, u ) Reversed Sshaped 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 rightcontinuous filtration W is a (F, P) mdimensional Brownian motion T > 0 is a fixed constant time horizon THE MARKET 1 riskfree 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 rightcontinuous filtration W is a (F, P) mdimensional Brownian motion T > 0 is a fixed constant time horizon THE MARKET 1 riskfree 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 rightcontinuous filtration W is a (F, P) mdimensional Brownian motion T > 0 is a fixed constant time horizon THE MARKET 1 riskfree 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 Pa.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 Pa.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 Pa.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 riskfree 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 NONINFORMED 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 riskfree 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 NONINFORMED 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 riskfree 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 NONINFORMED 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 noninsider (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 noninsider (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 noninsider (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 Pa.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 nonnegative. 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 Pa.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 nonnegative. 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 Pa.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 nonnegative. 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!
arxiv:1502.06681v2 [qfin.mf] 26 Feb 2015
ARBITRAGE, HEDGING AND UTILITY MAXIMIZATION USING SEMISTATIC TRADING STRATEGIES WITH AMERICAN OPTIONS ERHAN BAYRAKTAR AND ZHOU ZHOU arxiv:1502.06681v2 [qfin.mf] 26 Feb 2015 Abstract. We consider a financial
More informationSensitivity analysis of utility based prices and risktolerance wealth processes
Sensitivity analysis of utility based prices and risktolerance wealth processes Dmitry Kramkov, Carnegie Mellon University Based on a paper with Mihai Sirbu from Columbia University Math Finance Seminar,
More informationCOMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS
COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS NICOLE BÄUERLE AND STEFANIE GRETHER Abstract. In this short note we prove a conjecture posed in Cui et al. 2012): Dynamic meanvariance problems in
More informationK 1 < K 2 = P (K 1 ) P (K 2 ) (6) This holds for both American and European Options.
Slope and Convexity Restrictions and How to implement Arbitrage Opportunities 1 These notes will show how to implement arbitrage opportunities when either the slope or the convexity restriction is violated.
More informationBERKSHIRE HATHAWAY INC.
BERKSHIRE HATHAWAY INC. To: From: Re: The Directors, Executive Officers and Key Employees of Berkshire Hathaway Inc. and the Executive Officers and Key Employees of its Subsidiaries Warren E. Buffett "Insider"
More informationPricing of an Exotic Forward Contract
Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 111 Nojihigashi, Kusatsu, Shiga 5258577, Japan Email: {akahori,
More informationIn which Financial Markets do Mutual Fund Theorems hold true?
In which Financial Markets do Mutual Fund Theorems hold true? W. Schachermayer M. Sîrbu E. Taflin Abstract The Mutual Fund Theorem (MFT) is considered in a general semimartingale financial market S with
More informationSimple Arbitrage. Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila. Christian Bender. Saarland University
Simple Arbitrage Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila Saarland University December, 8, 2011 Problem Setting Financial market with two assets (for simplicity) on
More informationOptimal Investment with Derivative Securities
Noname manuscript No. (will be inserted by the editor) Optimal Investment with Derivative Securities Aytaç İlhan 1, Mattias Jonsson 2, Ronnie Sircar 3 1 Mathematical Institute, University of Oxford, Oxford,
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationOn the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information
Finance 400 A. Penati  G. Pennacchi Notes on On the Efficiency of Competitive Stock Markets Where Traders Have Diverse Information by Sanford Grossman This model shows how the heterogeneous information
More informationIntroduction to ArbitrageFree Pricing: Fundamental Theorems
Introduction to ArbitrageFree Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 810, 2015 1 / 24 Outline Financial market
More informationLifeinsurancespecific optimal investment: the impact of stochastic interest rate and shortfall constraint
Lifeinsurancespecific optimal investment: the impact of stochastic interest rate and shortfall constraint An Radon Workshop on Financial and Actuarial Mathematics for Young Researchers May 3031 2007,
More informationQUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS
QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS L. M. Dieng ( Department of Physics, CUNY/BCC, New York, New York) Abstract: In this work, we expand the idea of Samuelson[3] and Shepp[,5,6] for
More information3 Introduction to Assessing Risk
3 Introduction to Assessing Risk Important Question. How do we assess the risk an investor faces when choosing among assets? In this discussion we examine how an investor would assess the risk associated
More informationPricing catastrophe options in incomplete market
Pricing catastrophe options in incomplete market Arthur Charpentier arthur.charpentier@univrennes1.fr Actuarial and Financial Mathematics Conference Interplay between finance and insurance, February 2008
More informationLecture 11 Uncertainty
Lecture 11 Uncertainty 1. Contingent Claims and the StatePreference Model 1) Contingent Commodities and Contingent Claims Using the simple twogood model we have developed throughout this course, think
More informationAn optimal transportation problem with import/export taxes on the boundary
An optimal transportation problem with import/export taxes on the boundary Julián Toledo Workshop International sur les Mathématiques et l Environnement Essaouira, November 2012..................... Joint
More informationOPTIMAL TIMING OF THE ANNUITY PURCHASES: A
OPTIMAL TIMING OF THE ANNUITY PURCHASES: A COMBINED STOCHASTIC CONTROL AND OPTIMAL STOPPING PROBLEM Gabriele Stabile 1 1 Dipartimento di Matematica per le Dec. Econ. Finanz. e Assic., Facoltà di Economia
More informationValuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution Framework
Valuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution Framework Quan Yuan Joint work with Shuntai Hu, Baojun Bian Email: candy5191@163.com
More informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationOptimal exit time from casino gambling: Why a lucky coin and a good memory matter
Optimal exit time from casino gambling: Why a lucky coin and a good memory matter Xue Dong He Sang Hu Jan Ob lój Xun Yu Zhou First version: May 24, 2013, This version: May 13, 2015 Abstract We consider
More informationIS MORE INFORMATION BETTER? THE EFFECT OF TRADERS IRRATIONAL BEHAVIOR ON AN ARTIFICIAL STOCK MARKET
IS MORE INFORMATION BETTER? THE EFFECT OF TRADERS IRRATIONAL BEHAVIOR ON AN ARTIFICIAL STOCK MARKET Wei T. Yue Alok R. Chaturvedi Shailendra Mehta Krannert Graduate School of Management Purdue University
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More informationCHAPTER 1: INTRODUCTION, BACKGROUND, AND MOTIVATION. Over the last decades, risk analysis and corporate risk management activities have
Chapter 1 INTRODUCTION, BACKGROUND, AND MOTIVATION 1.1 INTRODUCTION Over the last decades, risk analysis and corporate risk management activities have become very important elements for both financial
More informationThe Discrete Binomial Model for Option Pricing
The Discrete Binomial Model for Option Pricing Rebecca Stockbridge Program in Applied Mathematics University of Arizona May 4, 2008 Abstract This paper introduces the notion of option pricing in the context
More informationMoreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing
More informationFund Manager s Portfolio Choice
Fund Manager s Portfolio Choice Zhiqing Zhang Advised by: Gu Wang September 5, 2014 Abstract Fund manager is allowed to invest the fund s assets and his personal wealth in two separate risky assets, modeled
More informationThe BlackScholes pricing formulas
The BlackScholes pricing formulas Moty Katzman September 19, 2014 The BlackScholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock
More informationLECTURE 15: AMERICAN OPTIONS
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These
More informationSome Research Problems in Uncertainty Theory
Journal of Uncertain Systems Vol.3, No.1, pp.310, 2009 Online at: www.jus.org.uk Some Research Problems in Uncertainty Theory aoding Liu Uncertainty Theory Laboratory, Department of Mathematical Sciences
More informationThe Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees
The Effect of Management Discretion on Hedging and Fair Valuation of Participating Policies with Maturity Guarantees Torsten Kleinow HeriotWatt University, Edinburgh (joint work with Mark Willder) Marketconsistent
More informationMartingale Pricing Applied to Options, Forwards and Futures
IEOR E4706: Financial Engineering: DiscreteTime Asset Pricing Fall 2005 c 2005 by Martin Haugh Martingale Pricing Applied to Options, Forwards and Futures We now apply martingale pricing theory to the
More informationOptimization under uncertainty: modeling and solution methods
Optimization under uncertainty: modeling and solution methods Paolo Brandimarte Dipartimento di Scienze Matematiche Politecnico di Torino email: paolo.brandimarte@polito.it URL: http://staff.polito.it/paolo.brandimarte
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationWhat Is a Good Risk Measure: Bridging the Gaps between Robustness, Subadditivity, Prospect Theory, and Insurance Risk Measures
What Is a Good Risk Measure: Bridging the Gaps between Robustness, Subadditivity, Prospect Theory, and Insurance Risk Measures C.C. Heyde 1 S.G. Kou 2 X.H. Peng 2 1 Department of Statistics Columbia University
More informationOptimisation Problems in NonLife Insurance
Frankfurt, 6. Juli 2007 1 The de Finetti Problem The Optimal Strategy De Finetti s Example 2 Minimal Ruin Probabilities The HamiltonJacobiBellman Equation Two Examples 3 Optimal Dividends Dividends in
More informationSTOCK LOANS. XUN YU ZHOU Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong 1.
Mathematical Finance, Vol. 17, No. 2 April 2007), 307 317 STOCK LOANS JIANMING XIA Center for Financial Engineering and Risk Management, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
More informationThe Meaning of Market Efficiency
The Meaning of Market Efficiency Robert Jarrow Martin Larsson February 23, 211 Abstract Fama (197) defined an efficient market as one in which prices always fully reflect available information. This paper
More informationLECTURE 9: A MODEL FOR FOREIGN EXCHANGE
LECTURE 9: A MODEL FOR FOREIGN EXCHANGE 1. Foreign Exchange Contracts There was a time, not so long ago, when a U. S. dollar would buy you precisely.4 British pounds sterling 1, and a British pound sterling
More informationSukanto Bhattacharya School of Information Technology Bond University, Australia. Abstract
Computational Exploration of Investor Utilities Underlying a Portfolio Insurance Strategy Dr. M. Khoshnevisan School of Accounting & Finance Griffith University, Australia & Dr. Florentin Smarandache University
More informationECON 422A: FINANCE AND INVESTMENTS
ECON 422A: FINANCE AND INVESTMENTS LECTURE 10: EXPECTATIONS AND THE INFORMATIONAL CONTENT OF SECURITY PRICES MuJeung Yang Winter 2016 c 2016 MuJeung Yang OUTLINE FOR TODAY I) Informational Efficiency:
More informationMertonBlackScholes model for option pricing. Peter Denteneer. 22 oktober 2009
MertonBlackScholes model for option pricing Instituut{Lorentz voor Theoretische Natuurkunde, LION, Universiteit Leiden 22 oktober 2009 With inspiration from: J. Tinbergen, T.C. Koopmans, E. Majorana,
More informationIMPLEMENTING ARROWDEBREU EQUILIBRIA BY TRADING INFINITELYLIVED SECURITIES
IMPLEMENTING ARROWDEBREU EQUILIBRIA BY TRADING INFINITELYLIVED SECURITIES Kevin X.D. Huang and Jan Werner DECEMBER 2002 RWP 0208 Research Division Federal Reserve Bank of Kansas City Kevin X.D. Huang
More informationCS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options
CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common
More informationLecture 05: MeanVariance Analysis & Capital Asset Pricing Model (CAPM)
Lecture 05: MeanVariance Analysis & Capital Asset Pricing Model (CAPM) Prof. Markus K. Brunnermeier Slide 051 Overview Simple CAPM with quadratic utility functions (derived from stateprice beta model)
More information9 Basics of options, including trading strategies
ECG590I Asset Pricing. Lecture 9: Basics of options, including trading strategies 1 9 Basics of options, including trading strategies Option: The option of buying (call) or selling (put) an asset. European
More informationOptimal casino betting: why lucky coin and good memory are important
Optimal casino betting: why lucky coin and good memory are important Xue Dong He Sang Hu Jan Ob lój Xun Yu Zhou First version: May 24, 2013, This version: December 7, 2014 Abstract We consider the dynamic
More informationOptions, preblack Scholes
Options, preblack Scholes Modern finance seems to believe that the option pricing theory starts with the foundation articles of Black, Scholes (973) and Merton (973). This is far from being true. Numerous
More informationLecture 15. Ranking Payoff Distributions: Stochastic Dominance. FirstOrder Stochastic Dominance: higher distribution
Lecture 15 Ranking Payoff Distributions: Stochastic Dominance FirstOrder Stochastic Dominance: higher distribution Definition 6.D.1: The distribution F( ) firstorder stochastically dominates G( ) if
More informationLecture 4: Derivatives
Lecture 4: Derivatives School of Mathematics Introduction to Financial Mathematics, 2015 Lecture 4 1 Financial Derivatives 2 uropean Call and Put Options 3 Payoff Diagrams, Short Selling and Profit Derivatives
More informationOn characterization of a class of convex operators for pricing insurance risks
On characterization of a class of convex operators for pricing insurance risks Marta Cardin Dept. of Applied Mathematics University of Venice email: mcardin@unive.it Graziella Pacelli Dept. of of Social
More informationLecture 15. Sergei Fedotov. 20912  Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 6
Lecture 15 Sergei Fedotov 20912  Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 6 Lecture 15 1 BlackScholes Equation and Replicating Portfolio 2 Static
More informationAsset Pricing. Chapter IV. Measuring Risk and Risk Aversion. June 20, 2006
Chapter IV. Measuring Risk and Risk Aversion June 20, 2006 Measuring Risk Aversion Utility function Indifference Curves U(Y) tangent lines U(Y + h) U[0.5(Y + h) + 0.5(Y h)] 0.5U(Y + h) + 0.5U(Y h) U(Y
More informationPHOENIX NEW MEDIA LIMITED STATEMENT OF POLICIES GOVERNING MATERIAL, NONPUBLIC INFORMATION AND THE PREVENTION OF INSIDER TRADING
PHOENIX NEW MEDIA LIMITED STATEMENT OF POLICIES GOVERNING MATERIAL, NONPUBLIC INFORMATION AND THE PREVENTION OF INSIDER TRADING Adopted on [ ], 2011 and effective conditional and immediately upon commencement
More informationPortfolio Optimization Part 1 Unconstrained Portfolios
Portfolio Optimization Part 1 Unconstrained Portfolios John Norstad jnorstad@northwestern.edu http://www.norstad.org September 11, 2002 Updated: November 3, 2011 Abstract We recapitulate the singleperiod
More informationLecture 10: Market Efficiency
Lecture 10: Market Efficiency Prof. Markus K. Brunnermeier Overview Efficiency concepts EMH implies Martingale Property Evidence I: Return Predictability Mispricing versus Riskfactor Informational (market)
More informationChapter 1: Financial Markets and Financial Derivatives
Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange
More informationMental Accounting for Multiple Outcomes: Theoretical Results and an Empirical Study
Mental Accounting for Multiple Outcomes: Theoretical Results and an Empirical Study Martín Egozcue Department of Economics, University of Montevideo, and FCS Universidad de la Republica del Uruguay, Montevideo,
More informationIntroduction to Mathematical Finance
Introduction to Mathematical Finance Martin Baxter Barcelona 11 December 2007 1 Contents Financial markets and derivatives Basic derivative pricing and hedging Advanced derivatives 2 Banking Retail banking
More informationLecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 7: Bounds on Options Prices Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Option Price Quotes Reading the
More information7: The CRR Market Model
Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The CoxRossRubinstein
More informationOn MarketMaking and DeltaHedging
On MarketMaking and DeltaHedging 1 Market Makers 2 MarketMaking and BondPricing On MarketMaking and DeltaHedging 1 Market Makers 2 MarketMaking and BondPricing What to market makers do? Provide
More informationAmerican Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options
American Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Early Exercise Since American style options give the holder the same rights as European style options plus
More informationFair Valuation and Hedging of Participating LifeInsurance Policies under Management Discretion
Fair Valuation and Hedging of Participating LifeInsurance Policies under Management Discretion Torsten Kleinow Department of Actuarial Mathematics and Statistics and the Maxwell Institute for Mathematical
More informationINSIDER TRADING POLICY
INSIDER TRADING POLICY In the normal course of business, officers, directors and employees of this company may come into possession of material nonpublic information about the company, its business or
More informationFactors Affecting Option Prices
Factors Affecting Option Prices 1. The current stock price S 0. 2. The option strike price K. 3. The time to expiration T. 4. The volatility of the stock price σ. 5. The riskfree interest rate r. 6. The
More informationEconomics 2020a / HBS 4010 / HKS API111 Fall 2011 Practice Problems for Lectures 1 to 11
Economics 2020a / HBS 4010 / HKS API111 Fall 2011 Practice Problems for Lectures 1 to 11 LECTURE 1: BUDGETS AND REVEALED PREFERENCE 1.1. Quantity Discounts and the Budget Constraint Suppose that a consumer
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000
More informationLecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing
Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Key concept: Ito s lemma Stock Options: A contract giving its holder the right, but not obligation, to trade shares of a common
More informationInterpreting KullbackLeibler Divergence with the NeymanPearson Lemma
Interpreting KullbackLeibler Divergence with the NeymanPearson Lemma Shinto Eguchi a, and John Copas b a Institute of Statistical Mathematics and Graduate University of Advanced Studies, Minamiazabu
More informationAmerican Option Pricing with Transaction Costs
American Option Pricing with Transaction Costs Valeri. I. Zakamouline Department of Finance and Management Science Norwegian School of Economics and Business Administration Helleveien 30, 5045 Bergen,
More informationUniversal Portfolios With and Without Transaction Costs
CS8B/Stat4B (Spring 008) Statistical Learning Theory Lecture: 7 Universal Portfolios With and Without Transaction Costs Lecturer: Peter Bartlett Scribe: Sahand Negahban, Alex Shyr Introduction We will
More informationEssays in Financial Mathematics
Essays in Financial Mathematics Essays in Financial Mathematics Kristoffer Lindensjö Dissertation for the Degree of Doctor of Philosophy, Ph.D. Stockholm School of Economics, 2013. Dissertation title:
More informationA Guide to the Insider Buying Investment Strategy
Mar03 Aug03 Jan04 Jun04 Nov04 Apr05 Sep05 Feb06 Jul06 Dec06 May07 Oct07 Mar08 Aug08 Jan09 Jun09 Nov09 Apr10 Sep10 Mar03 Jul03 Nov03 Mar04 Jul04 Nov04 Mar05 Jul05 Nov05 Mar06
More informationStock Loans in Incomplete Markets
Applied Mathematical Finance, 2013 Vol. 20, No. 2, 118 136, http://dx.doi.org/10.1080/1350486x.2012.660318 Stock Loans in Incomplete Markets MATHEUS R. GRASSELLI* & CESAR GÓMEZ** *Department of Mathematics
More informationBlackScholes Option Pricing Model
BlackScholes Option Pricing Model Nathan Coelen June 6, 22 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change,
More informationFINANCIAL ECONOMICS OPTION PRICING
OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.
More informationTHE BLACKSCHOLES MODEL AND EXTENSIONS
THE BLACSCHOLES MODEL AND EXTENSIONS EVAN TURNER Abstract. This paper will derive the BlackScholes pricing model of a European option by calculating the expected value of the option. We will assume that
More informationMoral Hazard. Itay Goldstein. Wharton School, University of Pennsylvania
Moral Hazard Itay Goldstein Wharton School, University of Pennsylvania 1 PrincipalAgent Problem Basic problem in corporate finance: separation of ownership and control: o The owners of the firm are typically
More informationWho Should Sell Stocks?
Who Should Sell Stocks? Ren Liu joint work with Paolo Guasoni and Johannes MuhleKarbe ETH Zürich ImperialETH Workshop on Mathematical Finance 2015 1 / 24 Merton s Problem (1969) Frictionless market consisting
More informationChoice Under Uncertainty
Decision Making Under Uncertainty Choice Under Uncertainty Econ 422: Investment, Capital & Finance University of ashington Summer 2006 August 15, 2006 Course Chronology: 1. Intertemporal Choice: Exchange
More information10. Proximal point method
L. Vandenberghe EE236C Spring 201314) 10. Proximal point method proximal point method augmented Lagrangian method MoreauYosida smoothing 101 Proximal point method a conceptual algorithm for minimizing
More informationThe BlackScholes Model
The BlackScholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The BlackScholes Model Options Markets 1 / 19 The BlackScholesMerton
More informationA LogRobust Optimization Approach to Portfolio Management
A LogRobust Optimization Approach to Portfolio Management Dr. Aurélie Thiele Lehigh University Joint work with Ban Kawas Research partially supported by the National Science Foundation Grant CMMI0757983
More informationDynamic Hybrid Products in Life Insurance: Assessing the Policyholders Viewpoint
Dynamic Hybrid Products in Life Insurance: Assessing the Policyholders Viewpoint Samos, May 2014 Alexander Bohnert *, Patricia Born, Nadine Gatzert * * FriedrichAlexanderUniversity of ErlangenNürnberg,
More informationBlackScholes Equation for Option Pricing
BlackScholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there
More informationLECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS
LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS Robert S. Pindyck Massachusetts Institute of Technology Cambridge, MA 02142 Robert Pindyck (MIT) LECTURES ON REAL OPTIONS PART II August, 2008 1 / 50
More informationτ θ What is the proper price at time t =0of this option?
Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min
More informationMonte Carlo Methods in Finance
Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction
More informationValuing Stock Options: The BlackScholesMerton Model. Chapter 13
Valuing Stock Options: The BlackScholesMerton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The BlackScholesMerton Random Walk Assumption
More information15.401 Finance Theory
Finance Theory MIT Sloan MBA Program Andrew W. Lo Harris & Harris Group Professor, MIT Sloan School Lectures 10 11 11: : Options Critical Concepts Motivation Payoff Diagrams Payoff Tables Option Strategies
More informationBubbles and futures contracts in markets with shortselling constraints
Bubbles and futures contracts in markets with shortselling constraints Sergio Pulido, Cornell University PhD committee: Philip Protter, Robert Jarrow 3 rd WCMF, Santa Barbara, California November 13 th,
More informationFinancial Market Efficiency and Its Implications
Financial Market Efficiency: The Efficient Market Hypothesis (EMH) Financial Market Efficiency and Its Implications Financial markets are efficient if current asset prices fully reflect all currently available
More informationAn Introduction to Exotic Options
An Introduction to Exotic Options Jeff Casey Jeff Casey is entering his final semester of undergraduate studies at Ball State University. He is majoring in Financial Mathematics and has been a math tutor
More informationRiskminimization for life insurance liabilities
Riskminimization for life insurance liabilities Francesca Biagini Mathematisches Institut Ludwig Maximilians Universität München February 24, 2014 Francesca Biagini USC 1/25 Introduction A large number
More informationDecomposition of life insurance liabilities into risk factors theory and application
Decomposition of life insurance liabilities into risk factors theory and application Katja Schilling University of Ulm March 7, 2014 Joint work with Daniel Bauer, Marcus C. Christiansen, Alexander Kling
More informationECG590I Asset Pricing. Lecture 2: Present Value 1
ECG59I Asset Pricing. Lecture 2: Present Value 1 2 Present Value If you have to decide between receiving 1$ now or 1$ one year from now, then you would rather have your money now. If you have to decide
More informationIs a Brownian motion skew?
Is a Brownian motion skew? Ernesto Mordecki Sesión en honor a Mario Wschebor Universidad de la República, Montevideo, Uruguay XI CLAPEM  November 2009  Venezuela 1 1 Joint work with Antoine Lejay and
More information