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Steven E. Shreve Orion Hoch University Professor of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213-3890 Wean Hall 6216 (412) 268-8484 Fax: (412) 268-6380 E-mail: shreve at the address andrew.cmu.edu. |
Don't Blame the Quants, Forbes, October 7, 2008.
Model Risk, Analytics, April 2009.
Response to Pablo Triana's article "The Flawed Math of Financial Models", published by
www.quantnet.com
Errata for 2004 printing of Volume II, September 2006
More errata for 2004 printing of Volume II, July 2007
More errata for 2004 printing of Volume II, February 2008
Errata for 2008 printing of Volume I, July 2011
Errata for 2008 printing of Volume II, July 2011
"Mimicking an Ito Process" pdf file.
Abstract:
Given a multi-dimensional Ito process whose drift and diffusion terms are adapted processes, we construct a weak solution to a stochastic differential equation that matches the distribution of the Ito process at each fixed time. Moreover, we show how to match the distributions at each fixed time of functionals of the Ito process, including the running maximum and running average of one of the components of the process. A consequence of this result is that a wide variety of exotic derivative securities have the same prices when the underlying asset price is modelled by the original Ito process or the mimicking process that solves the stochastic differential equation.
"Utility Maximization" pdf file.
Abstract:
An agent invests in two types of futures contracts, whose prices are possibly correlated arithmetic Brownian motions, and invests in a money market account with a constant interest rate. The agent pays a transaction cost for trading in futures proportional to the size of the trade. She also receives utility from consumption. The agent maximizes expected infinite-horizon discounted utility from consumption. We determine the first two terms in the asymptotic expansion of the value function in the transaction cost parameter around the known value function for the case of zero transaction cost. The method of solution when the futures are uncorrelated follows a method used previously to obtain the analogous result for one risky asset. However, when the futures are correlated, a new methodology must be developed. It is suspected in this case that the value function is not twice continuously differentiable, and this prevents application of the former methodology.
Abstract:
We construct an optimal execution strategy for the purchase of a large number of shares of a financial asset over a fixed interval of time. Purchases of the asset have a nonlinear impact on price, and this is moderated over time by resilience in the limit-order book that determines the price. The limit-order book is permitted to have arbitrary shape. The form of the optimal execution strategy is to make an initial lump purchase and then purchase continuously for some period of time during which the rate of purchase is set to match the order book resiliency. At the end of this period, another lump purchase is made, and following that there is again a period of purchasing continuously at a rate set to match the order book resiliency. At the end of this second period, there is a final lump purchase. Any of the lump purchases could be of size zero. A simple condition is provided that guarantees that the intermediate lump purchase is of size zero.
Abstract:
This paper presents a heavy-traffic analysis of the behavior of a
single-server queue under an Earliest-Deadline-First (EDF)
scheduling policy, in which customers have deadlines and are served
only until their deadlines elapse. The performance of the
system is measured by the fraction of reneged work (the residual work
lost due to elapsed deadlines), which is shown to be minimized by
the EDF policy. The evolution of the lead time distribution of
customers in queue is described by a measure-valued process.
The heavy traffic limit of this (properly scaled) process
is shown to be a deterministic function of the limit of the scaled
workload process, which, in turn, is identified to be
a doubly reflected Brownian motion. This paper complements
previous work by Doytchinov, Lehoczky and Shreve on the EDF
discipline, in which customers are served to completion
even after their deadlines elapse. The fraction of reneged
work in a heavily loaded system and the fraction of late work
in the corresponding system without reneging are compared using
explicit formulas based on the heavy traffic approximations, which
are validated by simulation results.
Abstract:
A model for optimal consumption and investment is posed whose
solution is provided by the classical Merton analysis when
there is zero transaction cost. A probabilistic argument is
developed to identify the loss in value when a proportional
transaction cost is introduced. There are two sources of this
loss. The first is a loss due to "displacement'' that arises
because one cannot maintain the optimal portfolio of the
zero-transaction-cost problem. The second loss is due to
"transaction,'' a loss in capital that occurs when one adjusts
the portfolio. The first of these increases with increasing tolerance
for departure from the optimal portfolio in the
zero-transaction-cost problem, while the second decreases with
increases in this tolerance. This paper balances the marginal
costs of these two effects. The probabilistic analysis provided
here complements earlier work on a related model that
proceeded from a viscosity solution analysis of the associated
Hamilton-Jacobi-Bellman equation.
Abstract:
An explicit formula for the Skorokhod map $\Gamma_{0,a}$
on $[0,a]$ for $a>0$ is provided and related to similar
formulas in the literature. Specifically, it is shown
that on the space $D[0,\infty)$ of right-continuous
functions with left limits taking values in $\mathbb{R}$,
$$
\Gamma_{0,a}(\psi)(t) = \psi (t)
-\left[\big(\psi(0)-a\big)^+
\wedge\inf_{u\in[0,t]}\psi(u)\right]
\vee
\sup_{s \in [0,t]} \left[ (\psi(s) - a)
\wedge
\inf_{u \in [s,t]} \psi(u)\right]
$$
is the unique function taking values in $[0,a]$ that is
obtained from $\psi$ by minimal ``pushing'' at the
endpoints $0$ and $a$. An application of this result to
real-time queues with reneging is outlined.
Abstract:
The Skorokhod map is a convenient tool for constructing solutions to
stochastic differential equations with reflecting boundary conditions.
In this work, an explicit formula for the Skorokhod map $\Gamma_{0,a}$
on $[0,a]$ for any $a>0$ is derived.
Specifically, it is shown that on the
space $D[0,\infty)$ of right-continuous functions
with left limits taking values in R,
$\Gamma_{0,a} = \Lambda_a \circ \Gamma_0$, where
$\Lambda_a$ mapping $D[0,\infty)$ into itself is defined by
$$
\Lambda_a(\phi)(t)
=\phi(t)-\sup_{s\in[0,t]}[(\phi(s)-a)^+ \wedge \inf_{u\in [s,t]}\phi(u)]
$$
and $\Gamma_0$ mapping $D[0,\infty)$ into itself is the Skorokhod map
on $[0,\infty)$. In addition, properties of $\Lambda_a$ are developed
and comparison properties of $\Gamma_{0,a}$ are established.
Abstract:
A firm issues a convertible bond.
At each subsequent time,
the bondholder must decide
whether to continue to hold the bond,
thereby collecting
coupons, or to convert it to stock.
The bondholder wishes to choose a conversion
strategy to maximize the bond value. Subject to
some restrictions, the bond can be called by the
issuing firm, which presumably acts to maximize the
equity value of the firm by minimizing the bond value.
This creates a two-person game. We show that
if the coupon rate is below the interest rate
times the call price, then conversion should
precede call. On the other hand, if the dividend
rate times the call price is below the coupon rate,
call should precede conversion.
In either case, the game reduces to a problem of optimal
stopping.
"Finite Maturity Convertible Bonds" pdf file.
Abstract:
A random variable, representing the final position
of a trading strategy, is deemed acceptable
if under each of a variety of
probability measures its expectation
dominates a floor associated with the measure.
The set of random variables representing
pre-final positions from which it is possible
to trade to final acceptability is characterized.
In particular, the set of initial capitals
from which one can trade to final acceptability
is shown to be a closed half-line
{x;x\geq a}.
Methods for computing a are provided,
and the application of these ideas to
derivative security pricing is developed.
Abstract:
A firm issues a convertible bond.
At each subsequent time,
the bondholder must decide
whether to continue to hold the bond,
thereby collecting
coupons, or to convert it to stock.
The firm may at any time call the bond.
Because calls and conversions usually occur
far from maturity, we model this situation
with a perpetual convertible bond, i.e,
a convertible coupon-paying bond without maturity.
This model admits a relatively simple solution,
under which the value of the
perpetual convertible bond, as a function of
the value of the underlying
firm, is determined by a nonlinear
ordinary differential equation.
"PerpetualConvertibleBonds" pdf file.
Abstract:
This paper presents a second-order heavy traffic analysis of a single
server queue that processes customers having deadlines using the
earliest-deadline-first scheduling policy. For such systems, referred
to as {\em real-time queueing systems}, performance is measured by the
fraction of customers who meet their
deadline, rather than more traditional performance measures such as
customer delay, queue length, or server utilization. To model such
systems, one must keep track of customer lead times (the time
remaining until a customer deadline elapses) or equivalent
information. This paper reviews the earlier heavy traffic analysis of
such systems that provided approximations to the system's behavior.
The main result of this paper is the development of a second-order
analysis that gives the accuracy of the approximations and the rate of
convergence of the sequence of real-time queueing systems to
its heavy traffic limit.
"Accuracy of State Space Collapse" pdf file.