555 research outputs found
Optimal Mean Reversion Trading with Transaction Costs and Stop-Loss Exit
Motivated by the industry practice of pairs trading, we study the optimal
timing strategies for trading a mean-reverting price spread. An optimal double
stopping problem is formulated to analyze the timing to start and subsequently
liquidate the position subject to transaction costs. Modeling the price spread
by an Ornstein-Uhlenbeck process, we apply a probabilistic methodology and
rigorously derive the optimal price intervals for market entry and exit. As an
extension, we incorporate a stop-loss constraint to limit the maximum loss. We
show that the entry region is characterized by a bounded price interval that
lies strictly above the stop-loss level. As for the exit timing, a higher
stop-loss level always implies a lower optimal take-profit level. Both
analytical and numerical results are provided to illustrate the dependence of
timing strategies on model parameters such as transaction cost and stop-loss
level
An Optimal Multiple Stopping Approach to Infrastructure Investment Decisions
The energy and material processing industries are traditionally characterized
by very large-scale physical capital that is custom-built with long lead times
and long lifetimes. However, recent technological advancement in low-cost
automation has made possible the parallel operation of large numbers of
small-scale and modular production units. Amenable to mass-production, these
units can be more rapidly deployed but they are also likely to have a much
quicker turnover. Such a paradigm shift motivates the analysis of the combined
effect of lead time and lifetime on infrastructure investment decisions. In
order to value the underlying real option, we introduce an optimal multiple
stopping approach that accounts for operational flexibility, delay induced by
lead time, and multiple (finite/infinite) future investment opportunities. We
provide an analytical characterization of the firm's value function and optimal
stopping rule. This leads us to develop an iterative numerical scheme, and
examine how the investment decisions depend on lead time and lifetime, as well
as other parameters. Furthermore, our model can be used to analyze the critical
investment cost that makes small-scale (short lead time, short lifetime)
alternatives competitive with traditional large-scale infrastructure.Comment: 27 pages, 7 figure
Accounting for Earnings Announcements in the Pricing of Equity Options
We study an option pricing framework that accounts for the price impact of an
earnings announcement (EA), and analyze the behavior of the implied volatility
surface prior to the event. On the announcement date, we incorporate a random
jump to the stock price to represent the shock due to earnings. We consider
different distributions of the scheduled earnings jump as well as different
underlying stock price dynamics before and after the EA date. Our main
contributions include analytical option pricing formulas when the underlying
stock price follows the Kou model along with a double-exponential or Gaussian
EA jump on the announcement date. Furthermore, we derive analytic bounds and
asymptotics for the pre-EA implied volatility under various models. The
calibration results demonstrate adequate fit of the entire implied volatility
surface prior to an announcement. We also compare the risk-neutral distribution
of the EA jump to its historical distribution. Finally, we discuss the
valuation and exercise strategy of pre-EA American options, and illustrate an
analytical approximation and numerical results.Comment: 34 Page
Optimal Derivative Liquidation Timing Under Path-Dependent Risk Penalties
This paper studies the risk-adjusted optimal timing to liquidate an option at
the prevailing market price. In addition to maximizing the expected discounted
return from option sale, we incorporate a path-dependent risk penalty based on
shortfall or quadratic variation of the option price up to the liquidation
time. We establish the conditions under which it is optimal to immediately
liquidate or hold the option position through expiration. Furthermore, we study
the variational inequality associated with the optimal stopping problem, and
prove the existence and uniqueness of a strong solution. A series of analytical
and numerical results are provided to illustrate the non-trivial optimal
liquidation strategies under geometric Brownian motion (GBM) and exponential
Ornstein-Uhlenbeck models. We examine the combined effects of price dynamics
and risk penalty on the sell and delay regions for various options. In
addition, we obtain an explicit closed-form solution for the liquidation of a
stock with quadratic penalty under the GBM model.Comment: 26 pages, 11 figure
Optimal Dynamic Basis Trading
We study the problem of dynamically trading a futures contract and its
underlying asset under a stochastic basis model. The basis evolution is modeled
by a stopped scaled Brownian bridge to account for non-convergence of the basis
at maturity. The optimal trading strategies are determined from a utility
maximization problem under hyperbolic absolute risk aversion (HARA) risk
preferences. By analyzing the associated Hamilton-Jacobi-Bellman equation, we
derive the exact conditions under which the equation admits a solution and
solve the utility maximization explicitly. A series of numerical examples are
provided to illustrate the optimal strategies and examine the effects of model
parameters.Comment: 27 pages, 10 figure
Optimal Static Quadratic Hedging
We propose a flexible framework for hedging a contingent claim by holding
static positions in vanilla European calls, puts, bonds, and forwards. A
model-free expression is derived for the optimal static hedging strategy that
minimizes the expected squared hedging error subject to a cost constraint. The
optimal hedge involves computing a number of expectations that reflect the
dependence among the contingent claim and the hedging assets. We provide a
general method for approximating these expectations analytically in a general
Markov diffusion market. To illustrate the versatility of our approach, we
present several numerical examples, including hedging path-dependent options
and options written on a correlated asset.Comment: 33 pages, 4 figure
Pricing Derivatives with Counterparty Risk and Collateralization: A Fixed Point Approach
This paper studies a valuation framework for financial contracts subject to
reference and counterparty default risks with collateralization requirement. We
propose a fixed point approach to analyze the mark-to-market contract value
with counterparty risk provision, and show that it is a unique bounded and
continuous fixed point via contraction mapping. This leads us to develop an
accurate iterative numerical scheme for valuation. Specifically, we solve a
sequence of linear inhomogeneous PDEs, whose solutions converge to the fixed
point price function. We apply our methodology to compute the bid and ask
prices for both defaultable equity and fixed-income derivatives, and illustrate
the non-trivial effects of counterparty risk, collateralization ratio and
liquidation convention on the bid-ask spreads
The Golden Target: Analyzing the Tracking Performance of Leveraged Gold ETFs
This paper studies the empirical tracking performance of leveraged ETFs on
gold, and their price relationships with gold spot and futures. For tracking
the gold spot, we find that our optimized portfolios with short-term gold
futures are highly effective in replicating prices. The market-traded gold ETF
(GLD) also exhibits a similar tracking performance. However, we show that
leveraged gold ETFs tend to underperform their corresponding leveraged
benchmark. Moreover, the underperformance worsens over a longer holding period.
In contrast, we illustrate that a dynamic portfolio of gold futures tracks
significantly better than various static portfolios. The dynamic portfolio also
consistently outperforms the respective market-traded LETFs for different
leverage ratios over multiple years
ESO Valuation with Job Termination Risk and Jumps in Stock Price
Employee stock options (ESOs) are American-style call options that can be
terminated early due to employment shock. This paper studies an ESO valuation
framework that accounts for job termination risk and jumps in the company stock
price. Under general L\'evy stock price dynamics, we show that a higher job
termination risk induces the ESO holder to voluntarily accelerate exercise,
which in turn reduces the cost to the company. The holder's optimal exercise
boundary and ESO cost are determined by solving an inhomogeneous partial
integro-differential variational inequality (PIDVI). We apply Fourier transform
to simplify the variational inequality and develop accurate numerical methods.
Furthermore, when the stock price follows a geometric Brownian motion, we
provide closed-form formulas for both the vested and unvested perpetual ESOs.
Our model is also applied to evaluate the probabilities of understating ESO
expenses and contract termination.Comment: 28 pages, 7 figure
Dynamic Index Tracking and Risk Exposure Control Using Derivatives
We develop a methodology for index tracking and risk exposure control using
financial derivatives. Under a continuous-time diffusion framework for price
evolution, we present a pathwise approach to construct dynamic portfolios of
derivatives in order to gain exposure to an index and/or market factors that
may be not directly tradable. Among our results, we establish a general
tracking condition that relates the portfolio drift to the desired exposure
coefficients under any given model. We also derive a slippage process that
reveals how the portfolio return deviates from the targeted return. In our
multi-factor setting, the portfolio's realized slippage depends not only on the
realized variance of the index, but also the realized covariance among the
index and factors. We implement our trading strategies under a number of
models, and compare the tracking strategies and performances when using
different derivatives, such as futures and options
- …