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Optimal Timing to Purchase Options
Abstract
We study the optimal timing of derivative purchases in incomplete markets. In
our model, an investor attempts to maximize the spread between her model price
and the offered market price through optimally timing her purchase. Both the
investor and the market value the options by risk-neutral expectations but
under different equivalent martingale measures representing different market
views. The structure of the resulting optimal stopping problem depends on the
interaction between the respective market price of risk and the option payoff.
In particular, a crucial role is played by the delayed purchase premium that is
related to the stochastic bracket between the market price and the buyer's risk
premia. Explicit characterization of the purchase timing is given for two
representative classes of Markovian models: (i) defaultable equity models with
local intensity; (ii) diffusion stochastic volatility models. Several numerical
examples are presented to illustrate the results. Our model is also applicable
to the optimal rolling of long-dated options and sequential buying and selling
of options.
... Our work is closest in spirit to [32] where the delayed purchase premium concept was used to analyze the optimal timing to purchase equity European and American options under a stochastic volatility model and a defaultable stock model. In contrast, the current paper addresses the optimal timing to liquidate various credit derivatives. ...
... Remark 6.4. In a related study, Leung and Ludkovski [32] also discuss the problem of sequential buying and selling of equity options without short sale possibility and with constant interest rate. In particular, the underlying stock admits a local default intensity modeled byλ(t, S t ), a deterministic function of time t and current stock price S t . ...
... In contrast, our current model assumes stochastic default intensityλ t =λ(t, X t ) and interest rate r(t, X t ), driven by a stochastic factor vector X. Hence, our optimal stopping value functions and buying/selling strategies depend on the stochastic factor X, rather than the stock alone as in [32]. ...
This paper studies the optimal timing to liquidate defaultable securities in a general intensity-based credit risk model under stochastic interest rate. We incorporate the potential price discrepancy between the market and investors, which is characterized by risk-neutral valuation under different default risk premia specifications. To quantify the value of optimally timing to sell, we introduce the {delayed liquidation premium} which is closely related to the stochastic bracket between the market price and a pricing kernel. We analyze the optimal liquidation policy for various credit derivatives. Our model serves as the building block for the sequential buying and selling problem. We also discuss the extensions to a jump-diffusion default intensity model as well as a defaultable equity model.
... Under the utility indifference framework, we examine the non-trivial effects of risk aversion and quantity on the investor's pricing and timing decisions. In contrast, our prior work Leung and Ludkovski (2011) investigated the purchase timing where a risk-neutral investor's pricing measure differed from the market. ...
... On the other hand, when the risk aversion or quantity to buy becomes infinitesimally small, the investor will adopt the risk-neutral expectation pricing under the minimal entropy martingale measure (MEMM) (see Fritelli (2000); Fujiwara and Miyahara (2003)). This limiting case provides a link with the risk-neutral problem in Leung and Ludkovski (2011), and also explains why an investor may disagree with the market pricing measure and investigate non-trivial purchase timing. ...
... Furthermore, the zero risk-aversion limit (2.20) can be viewed as a special case of the (riskneutral) delayed purchase premium in Section 2.3 of Leung and Ludkovski (2011) where the investor's pricing measure is taken to be the MEMM Q E . In fact, Proposition 3 provides an intuitive mechanism where the investor and market measures might differ: the market reflects a risk-neutral Q * while the investor applies utility-based framework under the physical P to end up with Q E in the small-γ or small-α limit. ...
We study the problem of optimal timing to buy/sell derivatives by a risk-averse agent in incomplete markets. Adopting the exponential utility indifference valuation, we investigate this timing flexibility and the associated delayed purchase premium. This leads to a stochastic control and optimal stopping problem that combines the observed market price dynamics and the agent's risk preferences. Our results extend recent work on indifference valuation of American options, as well as the authors' first paper (Leung and Ludkovski, SIAM J. Fin. Math., 2011). In the case of Markovian models of contracts on non-traded assets, we provide analytical characterizations and numerical studies of the optimal purchase strategies, with applications to both equity and credit derivatives.
... In a related study, Leung and Ludkovski (2011) analyzed the optimal timing to buy equity European and American options under a stochastic volatility model and a defaultable stock model through the concept of delayed purchase premium. Leung and Liu (2012) studied the optimal liquidation of defaultable securities in a general intensity-based credit risk model. ...
... This amounts to changing the ess sup to ess inf in V. In a related study, Leung and Ludkovski (2011) examined the optimal purchase problem of equity options under the investor's risk-neutral pricing measure. ...
Options are widely used as a tool for investment and risk management. In a liquid market, investors have the flexibility to trade options prior to their expiration dates. This is especially important for investors with an existing option position as they can control risk exposure through timing the option trades. For effective option based portfolio management, it is imperative for any investor to determine when to liquidate an option to the market at its trading price. Prior to expiration, the investor can always sell the option immediately, or wait for a potentially better future opportunity. This chapter studies the optimal timing to liquidate an option position to the market.
... In practice, we numerically solve the PDE for H (N−1) (t, y; b) and use (12) to derive the indifference price p (N−1) (t, y; b). Then, we optimize over b (as in (7)) to obtain the employee's optimal static hedges b * N−1 at time t N−1 , along with the corresponding V (N−1) * (t, x, y), H (N−1) * (t, y) and p (N−1) * (t, y), which will appear in the terminal conditions for V (N−2) (t, x, y; b). ...
... The rolling decisions will lead to an optimal multiple stopping problem. This is also related to the optimal timing to purchase options [11,12] ...
This paper presents a new methodology for hedging long-term financial derivatives written on an illiquid asset. The proposed hedging strategy combines dynamic trading of a correlated liquid asset (e.g. the market index) and static positions in market-traded options such as European puts and calls. Moreover, since most market-traded options are relatively short-term, it is necessary to conduct the static hedge sequentially over time till the long-term derivative expires. This sequential static-dynamic hedging strategy leads to the study of a stochastic control problem and the associated Hamilton-Jacobi-Bellman PDEs and variational inequalities. A series of transformations allow us to simplify the problem and compute the optimal hedging strategy.
... From the perspective of an investor with no position, she is interested in determining the best time to enter the market. We study the optimal timing premium (see Leung and Ludkovski (2011), Leung and Liu (2012), Leung and Ludkovski (2012)), which plays a vital role in the optimal strategies. This premium expresses the benefit of waiting to enter as compared to initialize the position immediately. ...
This paper studies the optimal VIX futures trading problems under a regime-switching model. We consider the VIX as mean reversion dynamics with dependence on the regime that switches among a finite number of states. For the trading strategies, we analyze the timings and sequences of the investor's market participation, which leads to several corresponding coupled system of variational inequalities. The numerical approach is developed to solve these optimal double stopping problems by using projected-successive-over-relaxation (PSOR) method with Crank-Nicolson scheme. We illustrate the optimal boundaries via numerical examples of two-state Markov chain model. In particular, we examine the impacts of transaction costs and regime-switching timings on the VIX futures trading strategies.
... derivatives trading (Leung and Ludkovski (2011)), among other applications. ...
Trailing stop is a popular stop-loss trading strategy by which the investor will sell the asset once its price experiences a pre-specified percentage drawdown. In this paper, we study the problem of timing buy and then sell an asset subject to a trailing stop. Under a general linear diffusion framework, we study an optimal double stopping problem with a random path-dependent maturity. Specifically, we first derive the optimal liquidation strategy prior to a given trailing stop, and prove the optimality of using a sell limit order in conjunction with the trailing stop. Our analytic results for the liquidation problem is then used to solve for the optimal strategy to acquire the asset and simultaneously initiate the trailing stop. The method of solution also lends itself to an efficient numerical method for computing the the optimal acquisition and liquidation regions. For illustration, we implement an example and conduct a sensitivity analysis under the exponential Ornstein-Uhlenbeck model.
... For our study, the early liquidation premium turns out to be amenable to analysis and give intuitive interpretations. The related concepts of early/delayed exercise/purchase premium have been analyzed in pricing American options (see [16]) and derivatives trading ( [17]), among other applications. ...
Trailing stop is a popular stop-loss trading strategy by which the investor will sell the asset once its price experiences a pre-specified percentage drawdown. In this paper, we study the problem of timing to buy and then sell an asset subject to a trailing stop. Under a general linear diffusion framework, we study an optimal double stopping problem with a random path-dependent maturity. Specifically, we first analytically solve the optimal liquidation problem with a trailing stop, and in turn derive the optimal timing to buy the asset. Our method of solution reduces the problem of determining the optimal trading regions to solving the associated differential equations. For illustration, we implement an example and conduct a sensitivity analysis under the exponential Ornstein–Uhlenbeck model.
... A wide array of financial applications can be formulated as optimal multiple stopping problems. These include energy delivery contracts such as swing options (Carmona & Dayanik 2008, Carmona & Touzi 2008, Zeghal & Mnif 2006, derivatives liquidation (Henderson & Hobson 2011, Leung & Ludkovski 2011, 2012, real option analysis (Chiara et al. 2007, Dahlgren & Leung 2015, Dixit & Pindyck 1994, McDonald & Siegel 1985, as well as employee stock options (Grasselli & Henderson 2009, Leung & Sircar 2009a, 2009b potentially with additional reload and shout options (Dai & Kwok 2008). In many of these applications, consecutive stopping times are separated by a constant or random period. ...
... Ref. [8] analyzes the double stopping times for a risk process from the insurance company's perspective. The problem of timing to buy/sell derivatives has also been studied in Ref. [12] (European and American options) and Ref. [11] (credit derivatives). Ref. [13] studies an optimal starting-stopping problem for general Markov processes and provide the mathematical characterization of the value functions. ...
This paper studies the timing of trades under mean-reverting price dynamics subject to fixed transaction costs. We solve an optimal double stopping problem to determine the optimal times to enter and subsequently exit the market, when prices are driven by an exponential Ornstein-Uhlenbeck process. In addition, we analyze a related optimal switching problem that involves an infinite sequence of trades, and identify the conditions under which the double stopping and switching problems admit the same optimal entry and/or exit timing strategies. Among our results, we find that the investor generally enters when the price is low, but may find it optimal to wait if the current price is sufficiently close to zero. In other words, the continuation (waiting) region for entry is disconnected. Numerical results are provided to illustrate the dependence of timing strategies on model parameters and transaction costs.
... For our study, the early liquidation premium turns out to be amenable to analysis and give intuitive interpretations. The related concepts of early/delayed exercise/purchase premium have been analyzed in pricing American options (see Carr et al. (1992)) and derivatives trading (Leung and Ludkovski (2011)), among other applications. ...
Trailing stop is a popular stop-loss trading strategy by which the investor will sell the asset once its price experiences a pre-specified percentage drawdown. In this paper, we study the problem of timing buy and then sell an asset subject to a trailing stop. Under a general linear diffusion framework, we study an optimal double stopping problem with a random path-dependent maturity. Specifically, we first derive the optimal liquidation strategy prior to a given trailing stop, and prove the optimality of using a sell limit order in conjunction with the trailing stop. Our analytic results for the liquidation problem is then used to solve for the optimal strategy to acquire the asset and simultaneously initiate the trailing stop. The method of solution also lends itself to an efficient numerical method for computing the the optimal acquisition and liquidation regions. For illustration, we implement an example and conduct a sensitivity analysis under the exponential Ornstein-Uhlenbeck model.
... Optimal stopping problems have different interesting features and applications. For instance, when one is considering the exercise of American options ( [14]) or choosing the sale time of an asset ( [12,9,17]). In case of real options, this is also one of the key questions, concerning the optimal investment decision (which may be exit the market, investment in other technology or any other decision). ...
In this paper we propose a formula to derive the value of a firm which is currently producing a certain
product and faces the option to exit the market, whose demand follows a geometric Brownian motion.
The problem of optimal exiting is an optimal stopping problem that can be solved using the
dynamic programming principle. This is a free-boundary problem. We propose an approximation for
the original model and, using the Implicit Function Theorem, we obtain the solution of the original
problem.
Finally we show, analytically, that the exit threshold is decreasing with the volatility as well as
the drift of the geometric Brownian motion.
... Since different LETFs share similar sources of randomness, one major concern is the price consistency of LETF options across different leverage ratios (see Ahn et al (2012) and Leung and Sircar (2012)). From the investor's perspective, it is also important to consider the optimal timing to buy or sell options on ETFs (see, for example, Leung and Ludkovski (2011)). Moreover, models that capture the connection between ETFs and the broader financial market would be very useful. ...
This paper provides a quantitative risk analysis of leveraged exchange-traded funds (LETFs) with a focus on the impact of leverage and investment horizon. From the empirical returns of several major LETFs based on the S&P 500 index, theperformanceofLETFsgenerallydeclinesastheinvestmenthorizonincreases, compared with the unleveraged ETF on the same index. The value erosion is more severe for highly leveraged ETFs. To better understand the risk impact of leverage, we introduce the admissible leverage ratio induced by a risk measure, forexample,value-at-risk(VaR)andconditionalVaR.Thisideacanhelpinvestors excludeLETFsthataredeemedtoorisky.Moreover,wealsodiscusstheconceptof admissible risk horizon so that the investor can control risk exposure by selecting an appropriate holding period. In addition, we also compute the intrahorizon risk, which leads us to evaluate a stop-loss/take-profit strategy for LETFs. Lastly, we investigate the impact of volatility exposure on the return of different LETF portfolios.
... From the perspective of an investor with no position, she is interested in determining the best time to enter the market. We study the optimal timing premium (see Leung and Ludkovski (2011), Leung and Liu (2012), Leung and Ludkovski (2012)), which plays a vital role in the optimal strategies. This premium expresses the benefit of waiting to enter as compared to initialize the position immediately. ...
This paper studies the optimal VIX futures trading problems under a regime-switching model. We consider the VIX as mean reversion dynamics with dependence on the regime that switches among a finite number of states. For the trading strategies, we analyze the timings and sequences of the investor's market participation, which leads to several corresponding coupled system of variational inequalities. The numerical approach is developed to solve these optimal double stopping problems by using projected-successive-over-relaxation (PSOR) method with Crank-Nicolson scheme. We illustrate the optimal boundaries via numerical examples of two-state Markov chain model. In particular, we examine the impacts of transaction costs and regime-switching timings on the VIX futures trading strategies.
... Karpowicz and Szajowski (2007) analyze the double stopping times for a risk process from the insurance company's perspective. The problem of timing to buy/sell derivatives has also been studied in Leung and Ludkovski (2011) (European and American options) and Leung and Liu (2012) (credit derivatives). Menaldi et al. (1996) study an optimal starting-stopping problem for general Markov processes, and provide the mathematical characterization of the value functions. ...
This paper studies the timing of trades under mean-reverting price dynamics
subject to fixed transaction costs. We solve an optimal double stopping problem
to determine the optimal times to enter and subsequently exit the market, when
prices are driven by an exponential Ornstein-Uhlenbeck process. In addition, we
analyze a related optimal switching problem that involves an infinite sequence
of trades, and identify the conditions under which the double stopping and
switching problems admit the same optimal entry and/or exit timing strategies.
Among our results, we find that the investor generally enters when the price is
low, but may find it optimal to wait if the current price is sufficiently close
to zero. In other words, the continuation (waiting) region for entry is
disconnected. Numerical results are provided to illustrate the dependence of
timing strategies on model parameters and transaction costs.
... A wide array of financial applications can be formulated as optimal multiple stopping problems. These include energy delivery contracts such as swing options [12,13,51], derivatives liquidation [25,36,37], real option analysis [15,17,19,44], as well as employee stock options [24,38,39] potentially with additional reload and shout options [18]. In many of these applications, consecutive stopping times are separated by a constant or random period. ...
We study an optimal multiple stopping problem driven by a spectrally negative Levy process. The stopping times are separated by constant refraction times, and the discount rate can be positive or negative. The computation involves a distribution of the Levy process at a constant horizon and hence the solutions in general cannot be attained analytically. Motivated by the maturity randomization (Canadization) technique by Carr (1998), we approximate the refraction times by i.i.d. Erlang random variables. In addition, fitting random jumps to phase-type distributions, our method involves repeated integrations with respect to the resolvent measure written in terms of the scale function of the underlying Levy process. We derive a recursive algorithm to compute the value function in closed form, and sequentially determine the optimal exercise thresholds. A series of numerical examples are provided to compare our analytic formula to results from Monte Carlo simulation.
... Our path-dependent risk penalization model can also be viewed as an alternative way to incorporate the investor's risk sensitivity in option liquidation/exercise timing problems, as compared to the utility maximization/indifference pricing approach (Henderson and Hobson, 2011;Leung and Ludkovski, 2012;. On the other hand, Leung and Ludkovski (2011) investigate the optimal timing to buy equity European and American options without risk penalty under incomplete markets, where the investor is assumed to select risk-neutral pricing measure different from the market's. Leung and Liu (2013) also discuss the timing to sell an option under the GBM model without any risk penalty, which is a special example of our model. ...
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.
... Since different LETFs share similar sources of randomness, one major concern is the price consistency of LETF options across different leverage ratios (see Ahn et al (2012) and Leung and Sircar (2012)). From the investor's perspective, it is also important to consider the optimal timing to buy or sell options on ETFs (see, for example, Leung and Ludkovski (2011)). Moreover, models that capture the connection between ETFs and the broader financial market would be very useful. ...
This paper provides a quantitative risk analysis of leveraged ETFs (LETFs) with a focus on the impact of leverage and investment horizon. From the empirical returns of several major LETFs based on the S&P 500 index, the performance of LETFs generally declines as investment horizon increases, compared to the reference index. The value erosion is more severe for highly leveraged ETFs. To better understand the risk impact of leverage, we introduce the admissible leverage ratio induced by a risk measure, for example, value-at-risk (VaR) and conditional VaR. This idea can help investors exclude LETFs that are deemed too risky. Moreover, we also discuss the concept of admissible risk horizon so that the investor can control risk exposure by selecting an appropriate holding period. In addition, we also compute the intra-horizon risk, which leads us to evaluate a stop-loss/take-profit strategy for LETFs. Lastly, we investigate the impact of volatility exposure on the return of different LETF portfolios.
... Ekström et al. [13] investigated the optimal liquidation of a call spread when the investor's belief on the volatility differs from the implied volatility. Our work is closest in spirit to [24] where the delayed purchase premium concept was used to analyze the optimal timing to buy European and American options under price discrepancy. On the other hand, the problem of optimal liquidation involving price impacts has been studied in [1,26,27], among others. ...
This paper studies the optimal timing to liquidate credit derivatives in a
general intensity-based credit risk model under stochastic interest rate. We
incorporate the potential price discrepancy between the market and investors,
which is characterized by risk-neutral valuation under different default risk
premia specifications. We quantify the value of optimally timing to sell
through the concept of delayed liquidation premium, and analyze the associated
probabilistic representation and variational inequality. We illustrate the
optimal liquidation policy for both single-named and multi-named credit
derivatives. Our model is extended to study the sequential buying and selling
problem with and without short-sale constraint.
This paper studies an optimal trading problem that incorporates the trader's market view on the terminal asset price distribution and uninformative noise embedded in the asset price dynamics. We model the underlying asset price evolution by an exponential randomized Brownian bridge (rBb) and consider various prior distributions for the random endpoint. We solve for the optimal strategies to sell a stock, call, or put, and analyze the associated delayed liquidation premia. We solve for the optimal trading strategies numerically and compare them across different prior beliefs. Among our results, we find that disconnected continuation/exercise regions arise when the trader prescribe a two-point discrete distribution and double exponential distribution.
Optimal Design of Combined Contingent Claims: Theory and Applications.
In “Combined Custom Hedging: Optimal Design, Noninsurable Exposure, and Operational Risk Management”, Paolo Guiotto and Andrea Roncoroni develop a normative framework for the optimal design, value assessment, and operations management integration of financial derivatives. Most business and operating revenues entail a mix of financially insurable and noninsurable risk. A risk-averse firm may face them by positioning in a pair of financial derivatives with optimal bespoke payoff functions; one claim is written on the insurable term, and the other claim is written on any observable index exhibiting correlation to the noninsurable term. On a theoretical ground, the authors 1) state the problem in a general setup and prove existence and uniqueness of the optimal pair of combined claims, 2) show that the optimal payoff functions satisfy a Fredholm integral equation, and 3) assess the incremental benefit the firm obtains by switching from the optimal single-claim custom hedge to the optimal combined custom hedge they propose. On an experimental ground, they show that 1) the optimal combined custom hedge would be empirically relevant for a highly risk-averse firm facing a market shock shown during the first period of the COVID-19 pandemic in 2020, 2) integration with the optimal procurement in a generalized newsvendor model leads to a significant improvement in both risk and return, and: 3) this gain can be traded off for a substantial enhancement in operational flexibility.
This paper studies the optimal VIX futures trading problems under a regime-switching model. We consider the VIX as mean reversion dynamics with dependence on the regime that switches among a finite number of states. For the trading strategies, we analyze the timings and sequences of the investor’s market participation, which leads to several corresponding coupled system of variational inequalities. The numerical approach is developed to solve these optimal double stopping problems by using projected-successive-over-relaxation (PSOR) method with Crank–Nicolson scheme. We illustrate the optimal boundaries via numerical examples of two-state Markov chain model. In particular, we examine the impacts of transaction costs and regime-switching timings on the VIX futures trading strategies.
This paper studies an optimal trading problem that incorporates the trader’s market view on the terminal asset price distribution and uninformative noise embedded in the asset price dynamics. We model the underlying asset price evolution by an exponential randomized Brownian bridge (rBb) and consider various prior distributions for the random endpoint. We solve for the optimal strategies to sell a stock, call, or put, and analyze the associated delayed liquidation premia. We solve for the optimal trading strategies numerically and compare them across different prior beliefs. Among our results, we find that disconnected continuation/exercise regions arise when the trader prescribe a two-point discrete distribution and double exponential distribution.
This paper studies an optimal trading problem that incorporates the trader's market view on the terminal asset price distribution and uninformative noise embedded in the asset price dynamics. We model the underlying asset price evolution by an exponential randomized Brownian bridge (rBb) and consider various prior distributions for the random endpoint. We solve for the optimal strategies to sell a stock, call, or put, and analyze the associated delayed liquidation premia. We solve for the optimal trading strategies numerically and compare them across different prior beliefs. Among our results, we find that disconnected continuation/exercise regions arise when the trader prescribe a two-point discrete distribution and double exponential distribution.
We study the perpetual American put option in a general jump-to-default model, deriving an explicit expression for the price of the option. We find that in some cases the optimal stopping boundary vanishes and thus it is not optimal to exercise the option before default occurs. Precise conditions for when this situation arises are given. Furthermore we present a necessary and sufficient condition for convexity of the option price, and also show that a nonincreasing intensity is sufficient, but not necessary, to have convexity. From this we also get conditions for when option prices are monotone in the model parameters.
Data from an emerging market were used to determine when to sell or hold a stock for a single model of a stock whose price is assumed to be a geometric Brownian motion in which the jump Markov process changes back and forth between positive and negative values.
In an incomplete market framework, contingent claims are of particular interest since they improve the market efficiency. This paper addresses the problem of market completeness when trading in contingent claims is allowed. We extend recent results by Bajeux and Rochet (1996) in a stochastic volatility model to the case where the asset price and its volatility variations are correlated. We also relate the ability of a given contingent claim to complete the market to the convexity of its price function in the current asset price. This allows us to state our results for general contingent claims by examining the convexity of their “admissible arbitrage prices.”
This paper considers the American put option valuation in a jump-diffusion model and relates this optimal-stopping problem
to a parabolic integro-differential free-boundary problem, with special attention to the behavior of the optimal-stopping
boundary. We study the regularity of the American option value and obtain in particular a decomposition of the American put
option price as the sum of its counterpart European price and the early exercise premium. Compared with the Black-Scholes
(BS) [5] model, this premium has an additional term due to the presence of jumps. We prove the continuity of the free boundary
and also give one estimate near maturity, generalizing a recent result of Barleset al. [3] for the BS model. Finally, we study the effect of the market price of jump risk and the intensity of jumps on the American
put option price and its critical stock price.
This study develops a real options approach for analyzing the optimal risk adoption policy in an environment where the adoption means a switch from one stochastic flow representation into another. We establish that increased volatility does not necessarily decelerate investment, as predicted by the standard literature on real options, once the underlying volatility of the state variable is made endogenous. We prove that for a decision maker with a convex (concave) objective function, increased post-adoption volatility increases (decreases) the expected cumulative present value of the post-adoption profit flow, which consequently decreases (increases) the option value of waiting and, therefore, accelerates (decelerates) current investment.
The minimal entropy martingale measures (MEMMs) for geometric Lévy processes are investigated. It is shown that, under a
quite mild condition, the MEMMs can be defined and furthermore represented explicitly. Furthermore, it is shown that the MEMM
price is the limit of the utility indifference price as the risk aversion parameter tends to 0.
The study of dynamic coherent risk measures and risk adjusted values as introduced by Artzner, Delbaen, Eber, Heath and Ku, leads to a property called fork convexity, rectangularity or m-stability. We give necessary and sufficient conditions for a closed convex set of measures to be fork convex. Since the set of martingale measures for price processes is m-stable, this leads to a characterisation of closed convex sets that can be obtained as the set of risk neutral measures in an arbitrage free model of security prices. We also relate the property of m-stability with the validity of Bell-man's principle. It turns out that the stability property investigated in this paper is equivalent to properties known as time-consistency and rectangularity as used in multiprior Bayesian decision theory. Finally we characterise the set of risk neutral measures for continuous price processes.
The validity of the classic Black-Scholes option pricing formula depends on the capability of investors to follow a dynamic portfolio strategy in the stock that replicates the payoff structure to the option. The critical assumption required for such a strategy to be feasible, is that the underlying stock return dynamics can be described by a stochastic process with a continuous sample path. In this paper, an option pricing formula is derived for the more-general case when the underlying stock returns are generated by a mixture of both continuous and jump processes. The derived formula has most of the attractive features of the original Black-Scholes formula in that it does not depend on investor preferences or knowledge of the expected return on the underlying stock. Moreover, the same analysis applied to the options can be extended to the pricing of corporate liabilities.
Abstract Incomplete information is at the heart of information-based credit risk models. In this paper, we rigorously deflne incomplete information with the notion of \delayed flltrations". We characterize two distinct types of delayed information, continuous and discrete: the flrst generated by a time change of flltrations and the second by flnitely many marked point processes. This notion unifles the noisy information in Du‐e and Lando (2001) and the partial information in Collin-Dufresne et al. (2004), under which structural models are translated into reduced-form intensity-based models. We illustrate through a simple example the importance of this notion of delayed information, as well as the potential pitfall for abusing the Laplacian approximation techniques for calculating the intensity process in an information-based model. The authors thank the Associate Editor and the two anonymous referees for their careful reading, constructive
Following the economic rationale of Peskir and Samee (2008a,b), we present a new class of Asian options where the holder enjoys the early exercise feature of American options whereupon his payoff (deliverable immediately) is the ‘best prediction’ of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Inherent in this is a protection feature that is key to the British Asian option. Should the option holder believe the true drift of the stock price to be unfavorable (based upon the observed price movements), he can substitute the true drift with the contract drift and minimize his losses. The practical implications of this protection feature are most remarkable, as not only is the option holder afforded a unique protection against unfavorable stock price movements (covering the ability to sell in a liquid option market completely endogenously), but also when the stock price movements are favorable he will generally receive high returns. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterized as the unique solution to a nonlinear integral equation. Using these results we perform a financial analysis of the British Asian option that leads to the conclusions above and shows that with the contract drift properly selected the British Asian option becomes a very attractive alternative to the classic (European) Asian option.
An incomplete market driven by a pair of Wiener and Poisson processes is considered. The range of European and American claim prices is determined.
The problem of pricing contingent claims or options from the price dynamics of certain securities is well understood in the context of a complete financial market. This paper studies the same problem in an incomplete market. When the market is incomplete, prices cannot be derived from the absence of arbitrage, since it is not possible to replicate the payoff of a given contingent claim by a controlled portfolio of the basic securities. In this situation, there is a price range for the actual market price of the contingent claim. The maximum and minimum prices are studied using stochastic control methods. The main result of this work is the determination that the maximum price is the smallest price that allows the seller to hedge completely by a controlled portfolio of the basic securities. A similar result is obtained for the minimum price (which corresponds to the purchase price).
We present a new put option where the holder enjoys the early exercise feature of American options whereupon his payoff (deliverable immediately) is the ‘best prediction’ of the European payoff under the hypothesis that the true drift of the stock price equals a contract drift. Inherent in this is a protection feature which is key to the British put option. Should the option holder believe the true drift of the stock price to be unfavourable (based upon the observed price movements) he can substitute the true drift with the contract drift and minimize his losses. The practical implications of this protection feature are most remarkable as not only can the option holder exercise at or above the strike price to a substantial reimbursement of the original option price (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favourable he will generally receive higher returns at a lesser price. We derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterized as the unique solution to a nonlinear integral equation. Using these results we perform a financial analysis of the British put option that leads to the conclusions above and shows that with the contract drift properly selected the British put option becomes a very attractive alternative to the classic American put.
We study the problem of hedging early exercise (American) options with respect to exponential utility within a general incomplete market model. This leads us to construct a duality formula involving relative entropy minimization and optimal stopping. We further consider claims with multiple exercises, and static-dynamic hedges of American claims with other European and American options. The problem is important for accurate valuation of employee Stock Options (ESOs), and we demonstrate this in a standard diffusion model. We find that incorporating static hedges with market-traded options induces the holder to delay exercises, and increases the ESO cost to the firm.
In this paper, we try to develop a comprehensive theory of risk management for illiquid trading instruments and exotics by examining the consequences of a quasi–static hedging strategy. In contrast to a static hedging strategy, in which an initial hedge once executed is kept in place for the life of the trade, and a dynamic hedging strategy, in which hedges are frequently adjusted over the life of the trade, a quasi–static hedging strategy utilizes hedge adjustments but tries to minimize the frequency. Almost all the examples studied in the framework introduced here take this minimization to the extreme by limiting hedge adjustments to at most one during the life of a trade. We examine the application of this approach to long–dated forwards, long–dated options and exotic options such as cliquet and barriers. The model we present for barriers is a new generation of the Derman–Ergener–Kani approach which combines the flexibility of the approach with a sizable increase in model independence.
We consider the problem facing a risk averse agent who seeks to liquidate or exercise a portfolio of (infinitely divisible) perpetual American style options on a single underlying asset. The optimal liquidation strategy is of threshold form and can be characterized explicitly as the solution of a calculus of variations problem. Apart from a possible initial exercise of a tranche of options, the optimal behavior involves liquidating the portfolio in infinitesimal amounts, but at times which are singular with respect to calendar time. We consider a number of illustrative examples involving CRRA and CARA utility, stocks, and portfolios of options with different strikes, and a model where the act of exercising has an impact on the underlying asset price.
We derive alternative representations of the McKean equation for the value of the American put option. Our main result decomposes the value of an American put option into the corresponding European put price and the early exercise premium. We then represent the European put price in a new manner. This representation allows us to alternatively decompose the price of an American put option into its intrinsic value and time value, and to demonstrate the equivalence of our results to the McKean equation.
We study optimal hedging of barrier options, using a combination of a static position in vanilla options and dynamic trading of the underlying asset. The problem reduces to computing the Fenchel–Legendre transform of the utility-indifference price as a function of the number of vanilla options used to hedge. Using the well-known duality between exponential utility and relative entropy, we provide a new characterization of the indifference price in terms of the minimal entropy measure, and give conditions guaranteeing differentiability and strict convexity in the hedging quantity, and hence a unique solution to the hedging problem. We discuss computational approaches within the context of Markovian stochastic volatility models.
This book addresses problems in financial mathematics of pricing and hedging derivative securities in an environment of uncertain and changing market volatility. These problems are important to investors from large trading institutions to pension funds. It presents mathematical and statistical tools that exploit the bursty nature of market volatility. The mathematics is introduced through examples and illustrated with simulations and the modeling approach that is described is validated and tested on market data. The material is suitable for a one semester course for graduate students who have had exposure to methods of stochastic modeling and arbitrage pricing theory in finance. It is easily accessible to derivatives practitioners in the financial engineering industry.
Finance is one of the fastest growing areas in the modern banking and corporate world. This, together with the sophistication of modern financial products, provides a rapidly growing impetus for new mathematical models and modern mathematical methods; the area is an expanding source for novel and relevant ‘real-world’ mathematics. In this book the authors describe the modelling of financial derivative products from an applied mathematician’s viewpoint, from modelling through analysis to elementary computation. A unified approach to modelling derivative products as partial differential equations is presented, using numerical solutions where appropriate. Some mathematics is assumed, but clear explanations are provided for material beyond elementary calculus, probability, and algebra. Over 140 exercises are included. This volume will become the standard introduction to this exciting new field for advanced undergraduate students.
We develop a theory of optimal stopping under Knightian uncertainty. A suitable martingale theory for multiple priors is derived that extends the classical dynamic programming or Snell envelope approach to multiple priors. We relate the multiple prior theory to the classical setup via a minimax theorem. In a multiple prior version of the classical model of independent and identically distributed random variables, we discuss several examples from microeconomics, operation research, and finance. For monotone payoffs, the worst-case prior can be identified quite easily with the help of stochastic dominance arguments. For more complex payoff structures like barrier options, model ambiguity leads to stochastic changes in the worst-case beliefs. Copyright 2009 The Econometric Society.
LetM(X) be the family of all equivalent local martingale measuresQ for some locally boundedd-dimensional processX, andV be a positive process. The main result of the paper (Theorem 2.1) states that the processV is a supermartingale whateverQVtV_t
whereH is an integrand forX, andC is an adapted increasing process. We call such a representationoptional because, in contrast to the Doob-Meyer decomposition, it generally exists only with an adapted (optional) processC. We apply this decomposition to the problem of hedging European and American style contingent claims in the setting ofincomplete security markets.
We study the problem of hedging early exercise (American) options with respect to exponential utility within a general incomplete market model. This leads us to construct a duality formula involving relative entropy minimization and optimal stopping. We further consider claims with multiple exercises, and static-dynamic hedges of American claims with other European and American options. The problem is important for accurate valuation of employee stock options (ESOs), and we demonstrate this in a standard diffusion model. We find that incorporating static hedges with market-traded options induces the holder to delay exercises and increases the ESO cost to the firm.
Let χ be a family of stochastic processes on a given filtered probability space (Ω, "F", ("F""t")"t" is an element of "T", "P") with "T"⊆R+. Under the assumption that the set "M""e" of equivalent martingale measures for χ is not empty, we give sufficient conditions for the existence of a unique equivalent martingale measure that minimizes the relative entropy, with respect to "P", in the class of martingale measures. We then provide the characterization of the density of the minimal entropy martingale measure, which suggests the equivalence between the maximization of expected exponential utility and the minimization of the relative entropy. Copyright Blackwell Publishers, Inc. 2000.
We present a general model for default time, making precise the role of the intensity process, and showing that this process allows for a knowledge of the conditional distribution of the default only "before the default". This lack of information is crucial while working in a multi-default setting. In a single default case, the knowledge of the intensity process does not allow to compute the price of defaultable claims, except in the case where immersion property is satisfied. We propose in this paper the density approach for default time. The density process will give a full characterization of the links between the default time and the reference filtration, in particular "after the default time". We also investigate the description of martingales in the full filtration in terms of martingales in the reference filtration, and the impact of Girsanov transformation on the density and intensity processes, and also on the immersion property.
This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk. As an example, and as our main motivation, we investigate option pricing under the class of q-optimal pricing measures. Using the ordering result, we prove comparison theorems between option prices under the minimal martingale, minimal entropy and variance-optimal pricing measures. As a concrete example, we specialise to a variant of the Heston model. For this example we are able to deduce that option prices are decreasing in the parameter q.
In this paper we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for conves payoffs the option price is increasing in the the jump-risk parameter. We apply this result to deduce general inequalities comparing the prices of contingent claims under various martingale measures which have been propsed in the literature as candidate pricing measures. Our proods are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities.
We solve in closed form a parsimonious extension of the Black-Scholes-Merton model with bankruptcy where the hazard rate of bankruptcy is a negative power of the stock price. Combining a scale change and a measure change, the model dynamics is reduced to a linear stochastic differential equation whose solution is a diffusion process that plays a central role in the pricing of Asian options. The solution is in the form of a spectral expansion associated with the diffusion infinitesimal generator. The latter is closely related to the Schrödinger operator with Morse potential. Pricing formulas for both corporate bonds and stock options are obtained in closed form. Term credit spreads on corporate bonds and implied volatility skews of stock options are closely linked in this model, with parameters of the hazard rate specification controlling both the shape of the term structure of credit spreads and the slope of the implied volatility skew. Our analytical formulas are easy to implement and should prove useful to researchers and practitioners in corporate debt and equity derivatives markets.
Recent advances in the theory of credit risk allow the use of standard term structure machinery for default risk modeling and estimation. The empirical literature in this area often interprets the drift adjustments of the default intensity's diffusion state variables as the only default risk premium. We show that this interpretation implies a restriction on the form of possible default risk premia, which can be justified through exact and approximate notions of ``diversifiable default risk.'' The equivalence between the empirical and martingale default intensities that follows from diversifiable default risk greatly facilitates the pricing and management of credit risk. We emphasize that this is not an equivalence in distribution, and illustrate its importance using credit spread dynamics estimated in Duffee (1999). We also argue that the assumption of diversifiability is implicitly used in certain existing models of mortgage-backed securities.
The aim of this paper is to study the minimal entropy and variance-optimal martingale measures for stochastic volatility models. In particular, for a diffusion model where the asset price and volatility are correlated, we show that the problem of determining the q-optimal measure can be reduced to finding a solution to a representation equation. The minimal entropy measure and variance-optimal measure are seen as the special cases q = 1 and q = 2 respectively. In the case where the volatility is an autonomous diffusion we give a stochastic representation for the solution of this equation. If the correlation p between the traded asset and the autonomous volatility satisfies ρ 2 < 1/q, and if certain smoothness and boundedness conditions on the parameters are satisfied, then the q-optimal measure exists. If ρ 2 ≥ 1/q, then the q-optimal measure may cease to exist beyond a certain time horizon. As an example we calculate the q-optimal measure explicitly for the Heston model.
Risk management using quasistatic hedging Optimal risk adoption: A real options approach
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Allen, S. and O. Padovani, 2002: 277–336. Risk management using quasistatic hedging. Economic Notes, 31 (2), Alvarez, L. and R. Stenbacka, 2004: Optimal risk adoption: A real options approach. Economic Theory, 23, 123–148
Derivatives in Financial Markets with Stochastic Fritelli, M., 2000: The minimal entropy martingale measure and the valuation problem in incomplete markets The minimal entropy martingale measures for geometric L´ evy processes
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The Mathematics of Financial Derivatives Redistribution subject to SIAM license or copyright
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P. Wilmott, S. Howison, and J. Dewynne, The Mathematics of Financial Derivatives, Cambridge University Press, Cambridge, UK, 1995. Downloaded 11/21/14 to 158.42.28.33. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
The British Asian option. Sequential Anal
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Peskir, G., K. Glover, and F. Samee, 2009: The British Asian option. Sequential Anal. to appear.