##### Document Text Contents

Page 1

Lecture Notes in Mathematics 1847

Editors:

J.--M. Morel, Cachan

F. Takens, Groningen

B. Teissier, Paris

Page 2

Tomasz R. Bielecki Tomas Bj¤ork

Monique Jeanblanc Marek Rutkowski

Jos·e A. Scheinkman Wei Xiong

Paris-Princeton Lectures

on Mathematical Finance

2003

Editorial Committee:

R. A. Carmona, E. C‚ inlar,

I. Ekeland, E. Jouini,

J. A. Scheinkman, N. Touzi

123

Page 128

Hedging of Defaultable Claims 121

dXt = xt dt + x̂t dWt + x̃t dMt, (138)

dΘt = Θt−

(

ϑt dt + ϑ̂t dWt + ϑ̃t dMt

)

, (139)

dΨt = ψt dt + ψ̂t dWt + ψ̃t dMt, (140)

where the drifts xt, ϑt and ψt are yet to be determined. From Itô’s formula, we obtain

(recall that ξt = γt11{τ>t})

d(Vt −Xt)2 = 2(Vt −Xt)(πtσ − x̂t) dWt − 2(Vt − Xt−)x̃t dMt

+

[

(Vt −Xt− − x̃t)2 − (Vt −Xt−)2

]

dMt

+

(

2(Vt −Xt)(πtν − xt) + (πtσ − x̂t)2

+ ξt

[

(Vt −Xt − x̃t)2 − (Vt −Xt)2

])

dt,

where we denote Vt = V vt (π). The process J(π) is a martingale if and only if its

drift term k(t, πt, xt, ϑt, ψt) = 0 for every t ∈ [0, T ].

Straightforward calculations show that

k(t, πt, ϑt, xt, ψt) = ψt + Θt

[

ϑt(Vt −Xt)2

+ 2(Vt −Xt)

[

(πtν − xt) + ϑ̂t(πtσ − x̂t) + ξtx̃t

]

+ (πtσ − x̂t)2 + ξt(ϑ̃t + 1)

[

(Vt −Xt − x̃t)2 − (Vt −Xt)2

]]

.

In the first step, for any t ∈ [0, T ] we shall find π∗t such that the minimum of

k(t, πt, xt, ϑt, ψt) is attained. Subsequently, we shall choose the auxiliary processes

x = x∗, ϑ = ϑ∗ and ψ = ψ∗ in such a way that k(t, π∗t , x

∗

t , ϑ

∗

t , ψ

∗

t ) = 0. This choice

will imply that k(t, πt, x∗t , ϑ

∗

t , ψ

∗

t ) ≥ 0 for any trading strategy π and any t ∈ [0, T ].

The strategy π∗, which minimizes k(t, πt, xt, ϑt, ψt), is the solution of the following

equation:

(V vt (π) −Xt)(ν + ϑ̂tσ) + σ(πtσ − x̂t) = 0, ∀ t ∈ [0, T ].

Hence, the strategy π∗ is implicitly given by

π∗t = σ

−1x̂t − σ−2(ν + ϑ̂tσ)(V vt (π

∗) −Xt) = At −Bt(V vt (π

∗) −Xt),

where we denote

At = σ

−1x̂t, Bt = σ

−2(ν + ϑ̂tσ).

After some computations, we see that the drift term of the process J admits the

following representation:

k(t, πt, ϑt, xt, ψt) = ψt + Θt(Vt −Xt)2(ϑt − σ2B2t )

+ 2Θt(Vt −Xt)

(

σ2AtBt − ϑ̂tx̂t − ϑ̃tx̃tξt − xt

)

+ Θtξt(ϑ̃t + 1)x̃

2

t .

Page 129

122 T.R. Bielecki, M. Jeanblanc, and M. Rutkowski

From now on, we shall assume that the auxiliary processes ϑ, x and ψ are chosen as

follows:

ϑt = ϑ

∗

t = σ

2B2t ,

xt = x

∗

t = σ

2AtBt − ϑ̂tx̂t − ϑ̃tx̃tξt,

ψt = ψ

∗

t = −Θtξt(ϑ̃t + 1)x̃

2

t .

It is rather clear that if the drift coefficients ϑ, x, ψ in (138)-(140) are chosen as

above, then the drift term in dynamics of J is always non-negative, and it is equal to

0 for the strategy π∗, where π∗t = At −Bt(V vt (π∗) −Xt).

Our next goal is to solve equations (138)-(140). Let us first consider equation (139).

Since ϑt = σ2B2t , it suffices to find the three-dimensional process (Θ, ϑ̂, ϑ̃) which

is a solution to the following BSDE:

dΘt = Θt

(

σ−2(ν + ϑ̂tσ)

2 dt + ϑ̂t dWt + ϑ̃t dMt

)

, ΘT = 1.

It is obvious that the processes ϑ̂ = 0, ϑ̃ = 0 and Θ, given as

Θt = exp(−θ2(T − t)), ∀ t ∈ [0, T ], (141)

solve this equation.

In the next step, we search for a three-dimensional process (X, x̂, x̃), which solves

equation (138) with xt = x∗t = σ

2At(ν/σ2) = θx̂t. It is clear that (X, x̂, x̃) is the

unique solution to the linear BSDE

dXt = θx̂t dt + x̂t dWt + x̃t dMt, XT = X.

The unique solution to this equation is Xt = EQ(X | Gt), where Q is the risk-neutral

probability measure, so that dQ = ηt dP, where

dηt = −θηt dWt, η0 = 1.

The components x̂ and x̃ are given by the integral representation of the G-martingale

X with respect to WQ and M . Notice also that since ϑ̂ = 0, the optimal portfolio π∗

is given by the feedback formula

π∗t = σ

−1(x̂t − θ(V vt (π∗) −Xt)).

Finally, since ϑ̃ = 0, we have ψt = −ξtx̃2tΘt. Therefore, we can solve explicitly the

BSDE (140) for the process Ψ . Indeed, we are now looking for a three-dimensional

process (Ψ, ψ̂, ψ̃), which is the unique solution of the BSDE

dΨt = −Θtξtx̃2t dt + ψ̂t dWt + ψ̃t dMt, ΨT = 0.

Noting that the process

Page 255

Heterogeneous Beliefs, Speculation and Trading 249

Jarrow, Robert (1980), Heterogeneous expectations, restrictions on short-sales, and

equilibrium asset prices, Journal of Finance 35, 1105-1113.

Kandel, Eugene and Neil Pearson (1995), Differential interpretation of public sig-

nals and trade in speculative markets, Journal of Political Economy 103, 831-

872.

Kogan, Leonid, Stephen Ross, Jiang Wang, and Mark Westerfield (2004), The price

impact and survival of irrational traders, Working paper, MIT.

Kyle, Albert (1985), Continuous auctions and insider trading, Econometrica 53,

1315-1336.

Kyle, Albert and Tao Lin (2003), Continuous trading with heterogeneous beliefs

and no noise trading, Working paper, Duke University.

Kyle, Albert and Albert Wang (1997), Speculation duopoly with agreement to dis-

agree: Can overconfidence survive the market test? Journal of Finance 52, 2073-

2090.

Lintner, John (1969), The aggregation of investor’s diverse judgements and prefer-

ences in purely competitive security markets, Journal of Financial and Quanti-

tative Analysis 4, 347-400.

Liptser, R. S. and A. N. Shiryayev (1977), Statistics of Random Processes, Spring-

Verlag, New York.

Lucas, Robert (1978), Asset prices in an exchange economy, Econometrica 46,

1429-1446.

Mehra, Rajnish and Edward Prescott (1985), The equity premium puzzle, Journal

of Monetary Economics 15, 145-161.

Mei, Jianping, Jose Scheinkman and Wei Xiong (2003), Speculative trading and

stock prices: An analysis of Chinese A-B share premia, Working paper, Prince-

ton University.

Milgrom, Paul and Nancy Stokey (1982), Information, trade and common knowl-

edge, Journal of Economic Theory 12, 112-128.

Miller, Edward (1977), Risk, uncertainty and divergence of opinion, Journal of Fi-

nance 32, 1151-1168.

Morris, Stephen (1996), Speculative investor behavior and learning, Quarterly

Journal of Economics 110, 1111-1133.

Odean, Terrance (1998), Volume, volatility, price, and profit when all traders are

above average, Journal of Finance 53, 1887-1934.

Ofek, Eli and Matthew Richardson (2003), Dotcom mania: The rise and fall of

internet stock prices, Journal of Finance 58, 1113-1137.

Panageas, Stavros (2003), Speculation, overpricing and investment – theory and

empirical evidence, Job market paper, MIT.

Revuz, Daniel and Marc Yor (1999), Continuous Martingales and Brownian Mo-

tion, Springer, New York.

Rogers, L. C. G. and David Williams (1987), Diffusions, Markov Processes, and

Martingales, Volume 2: Ito Calculus, John Wiley & Sons, New York.

Sandroni, Alvaro (2000), Do markets favor agents able to make accurate predic-

tions? Econometrica 68, 1303-1341.

Page 256

250 J. Scheinkman and W. Xiong

Santos, Manuel, and Michael Woodford (1997), Rational asset pricing bubbles,

Econometrica 65, 19-57.

Scheinkman, Jose and Thaleia Zariphopoulou (2001), Optimal environmental man-

agement in the presence of irreversibilities, Journal of Economic Theory 96,

180-207.

Scheinkman, Jose and Wei Xiong (2003), Overconfidence and Speculative Bubbles,

Journal of Political Economy 111, 1183-1219.

Tirole, Jean (1982), On the possibility of speculation under rational expecations,

Econometrica 50, 1163-1181.

Varian, Hal (1989), Differences of opinion in financial markets, in Courtenay Stone

(ed.) Financial Risk: Theory, Evidence and Implications, Kluwer Academic Pub-

lishers, Boston.

Wang, Jiang (1993), A model of intertemporal asset prices under a asymmetric

information, Review of Economic Studies 60, 249-282.

Williams, Joseph (1977), Capital asset prices with heterogeneous beliefs, Journal

of Financial Economics 5, 219-239.

Zapatero, Fernando (1998), Effects of financial innovations on market volatility

when beliefs are heterogeneous, Journal of Economic Dynamics and Control

22, 597-626.

Lecture Notes in Mathematics 1847

Editors:

J.--M. Morel, Cachan

F. Takens, Groningen

B. Teissier, Paris

Page 2

Tomasz R. Bielecki Tomas Bj¤ork

Monique Jeanblanc Marek Rutkowski

Jos·e A. Scheinkman Wei Xiong

Paris-Princeton Lectures

on Mathematical Finance

2003

Editorial Committee:

R. A. Carmona, E. C‚ inlar,

I. Ekeland, E. Jouini,

J. A. Scheinkman, N. Touzi

123

Page 128

Hedging of Defaultable Claims 121

dXt = xt dt + x̂t dWt + x̃t dMt, (138)

dΘt = Θt−

(

ϑt dt + ϑ̂t dWt + ϑ̃t dMt

)

, (139)

dΨt = ψt dt + ψ̂t dWt + ψ̃t dMt, (140)

where the drifts xt, ϑt and ψt are yet to be determined. From Itô’s formula, we obtain

(recall that ξt = γt11{τ>t})

d(Vt −Xt)2 = 2(Vt −Xt)(πtσ − x̂t) dWt − 2(Vt − Xt−)x̃t dMt

+

[

(Vt −Xt− − x̃t)2 − (Vt −Xt−)2

]

dMt

+

(

2(Vt −Xt)(πtν − xt) + (πtσ − x̂t)2

+ ξt

[

(Vt −Xt − x̃t)2 − (Vt −Xt)2

])

dt,

where we denote Vt = V vt (π). The process J(π) is a martingale if and only if its

drift term k(t, πt, xt, ϑt, ψt) = 0 for every t ∈ [0, T ].

Straightforward calculations show that

k(t, πt, ϑt, xt, ψt) = ψt + Θt

[

ϑt(Vt −Xt)2

+ 2(Vt −Xt)

[

(πtν − xt) + ϑ̂t(πtσ − x̂t) + ξtx̃t

]

+ (πtσ − x̂t)2 + ξt(ϑ̃t + 1)

[

(Vt −Xt − x̃t)2 − (Vt −Xt)2

]]

.

In the first step, for any t ∈ [0, T ] we shall find π∗t such that the minimum of

k(t, πt, xt, ϑt, ψt) is attained. Subsequently, we shall choose the auxiliary processes

x = x∗, ϑ = ϑ∗ and ψ = ψ∗ in such a way that k(t, π∗t , x

∗

t , ϑ

∗

t , ψ

∗

t ) = 0. This choice

will imply that k(t, πt, x∗t , ϑ

∗

t , ψ

∗

t ) ≥ 0 for any trading strategy π and any t ∈ [0, T ].

The strategy π∗, which minimizes k(t, πt, xt, ϑt, ψt), is the solution of the following

equation:

(V vt (π) −Xt)(ν + ϑ̂tσ) + σ(πtσ − x̂t) = 0, ∀ t ∈ [0, T ].

Hence, the strategy π∗ is implicitly given by

π∗t = σ

−1x̂t − σ−2(ν + ϑ̂tσ)(V vt (π

∗) −Xt) = At −Bt(V vt (π

∗) −Xt),

where we denote

At = σ

−1x̂t, Bt = σ

−2(ν + ϑ̂tσ).

After some computations, we see that the drift term of the process J admits the

following representation:

k(t, πt, ϑt, xt, ψt) = ψt + Θt(Vt −Xt)2(ϑt − σ2B2t )

+ 2Θt(Vt −Xt)

(

σ2AtBt − ϑ̂tx̂t − ϑ̃tx̃tξt − xt

)

+ Θtξt(ϑ̃t + 1)x̃

2

t .

Page 129

122 T.R. Bielecki, M. Jeanblanc, and M. Rutkowski

From now on, we shall assume that the auxiliary processes ϑ, x and ψ are chosen as

follows:

ϑt = ϑ

∗

t = σ

2B2t ,

xt = x

∗

t = σ

2AtBt − ϑ̂tx̂t − ϑ̃tx̃tξt,

ψt = ψ

∗

t = −Θtξt(ϑ̃t + 1)x̃

2

t .

It is rather clear that if the drift coefficients ϑ, x, ψ in (138)-(140) are chosen as

above, then the drift term in dynamics of J is always non-negative, and it is equal to

0 for the strategy π∗, where π∗t = At −Bt(V vt (π∗) −Xt).

Our next goal is to solve equations (138)-(140). Let us first consider equation (139).

Since ϑt = σ2B2t , it suffices to find the three-dimensional process (Θ, ϑ̂, ϑ̃) which

is a solution to the following BSDE:

dΘt = Θt

(

σ−2(ν + ϑ̂tσ)

2 dt + ϑ̂t dWt + ϑ̃t dMt

)

, ΘT = 1.

It is obvious that the processes ϑ̂ = 0, ϑ̃ = 0 and Θ, given as

Θt = exp(−θ2(T − t)), ∀ t ∈ [0, T ], (141)

solve this equation.

In the next step, we search for a three-dimensional process (X, x̂, x̃), which solves

equation (138) with xt = x∗t = σ

2At(ν/σ2) = θx̂t. It is clear that (X, x̂, x̃) is the

unique solution to the linear BSDE

dXt = θx̂t dt + x̂t dWt + x̃t dMt, XT = X.

The unique solution to this equation is Xt = EQ(X | Gt), where Q is the risk-neutral

probability measure, so that dQ = ηt dP, where

dηt = −θηt dWt, η0 = 1.

The components x̂ and x̃ are given by the integral representation of the G-martingale

X with respect to WQ and M . Notice also that since ϑ̂ = 0, the optimal portfolio π∗

is given by the feedback formula

π∗t = σ

−1(x̂t − θ(V vt (π∗) −Xt)).

Finally, since ϑ̃ = 0, we have ψt = −ξtx̃2tΘt. Therefore, we can solve explicitly the

BSDE (140) for the process Ψ . Indeed, we are now looking for a three-dimensional

process (Ψ, ψ̂, ψ̃), which is the unique solution of the BSDE

dΨt = −Θtξtx̃2t dt + ψ̂t dWt + ψ̃t dMt, ΨT = 0.

Noting that the process

Page 255

Heterogeneous Beliefs, Speculation and Trading 249

Jarrow, Robert (1980), Heterogeneous expectations, restrictions on short-sales, and

equilibrium asset prices, Journal of Finance 35, 1105-1113.

Kandel, Eugene and Neil Pearson (1995), Differential interpretation of public sig-

nals and trade in speculative markets, Journal of Political Economy 103, 831-

872.

Kogan, Leonid, Stephen Ross, Jiang Wang, and Mark Westerfield (2004), The price

impact and survival of irrational traders, Working paper, MIT.

Kyle, Albert (1985), Continuous auctions and insider trading, Econometrica 53,

1315-1336.

Kyle, Albert and Tao Lin (2003), Continuous trading with heterogeneous beliefs

and no noise trading, Working paper, Duke University.

Kyle, Albert and Albert Wang (1997), Speculation duopoly with agreement to dis-

agree: Can overconfidence survive the market test? Journal of Finance 52, 2073-

2090.

Lintner, John (1969), The aggregation of investor’s diverse judgements and prefer-

ences in purely competitive security markets, Journal of Financial and Quanti-

tative Analysis 4, 347-400.

Liptser, R. S. and A. N. Shiryayev (1977), Statistics of Random Processes, Spring-

Verlag, New York.

Lucas, Robert (1978), Asset prices in an exchange economy, Econometrica 46,

1429-1446.

Mehra, Rajnish and Edward Prescott (1985), The equity premium puzzle, Journal

of Monetary Economics 15, 145-161.

Mei, Jianping, Jose Scheinkman and Wei Xiong (2003), Speculative trading and

stock prices: An analysis of Chinese A-B share premia, Working paper, Prince-

ton University.

Milgrom, Paul and Nancy Stokey (1982), Information, trade and common knowl-

edge, Journal of Economic Theory 12, 112-128.

Miller, Edward (1977), Risk, uncertainty and divergence of opinion, Journal of Fi-

nance 32, 1151-1168.

Morris, Stephen (1996), Speculative investor behavior and learning, Quarterly

Journal of Economics 110, 1111-1133.

Odean, Terrance (1998), Volume, volatility, price, and profit when all traders are

above average, Journal of Finance 53, 1887-1934.

Ofek, Eli and Matthew Richardson (2003), Dotcom mania: The rise and fall of

internet stock prices, Journal of Finance 58, 1113-1137.

Panageas, Stavros (2003), Speculation, overpricing and investment – theory and

empirical evidence, Job market paper, MIT.

Revuz, Daniel and Marc Yor (1999), Continuous Martingales and Brownian Mo-

tion, Springer, New York.

Rogers, L. C. G. and David Williams (1987), Diffusions, Markov Processes, and

Martingales, Volume 2: Ito Calculus, John Wiley & Sons, New York.

Sandroni, Alvaro (2000), Do markets favor agents able to make accurate predic-

tions? Econometrica 68, 1303-1341.

Page 256

250 J. Scheinkman and W. Xiong

Santos, Manuel, and Michael Woodford (1997), Rational asset pricing bubbles,

Econometrica 65, 19-57.

Scheinkman, Jose and Thaleia Zariphopoulou (2001), Optimal environmental man-

agement in the presence of irreversibilities, Journal of Economic Theory 96,

180-207.

Scheinkman, Jose and Wei Xiong (2003), Overconfidence and Speculative Bubbles,

Journal of Political Economy 111, 1183-1219.

Tirole, Jean (1982), On the possibility of speculation under rational expecations,

Econometrica 50, 1163-1181.

Varian, Hal (1989), Differences of opinion in financial markets, in Courtenay Stone

(ed.) Financial Risk: Theory, Evidence and Implications, Kluwer Academic Pub-

lishers, Boston.

Wang, Jiang (1993), A model of intertemporal asset prices under a asymmetric

information, Review of Economic Studies 60, 249-282.

Williams, Joseph (1977), Capital asset prices with heterogeneous beliefs, Journal

of Financial Economics 5, 219-239.

Zapatero, Fernando (1998), Effects of financial innovations on market volatility

when beliefs are heterogeneous, Journal of Economic Dynamics and Control

22, 597-626.