David Colwell | The University of New South Wales (original) (raw)

Papers by David Colwell

Research paper thumbnail of Non-Transferable Non-Hedgeable Executive Stock Option Pricing

Social Science Research Network, 2013

We introduce a method of valuing non-transferable non-hedgeable (NTNH) contingent claims and use ... more We introduce a method of valuing non-transferable non-hedgeable (NTNH) contingent claims and use it to price executive stock options (ESOs). We use NTNH constraints to break local co-linearity, caused by derivative assets, that renders common portfolio optimization methods ineffective. We are, thus, able to translate portfolios that include NTNH derivatives into portfolios of primary assets (only). We achieve this by replicating derivatives using primary assets, and then integrating NTNH constraints into a single rectangular constraint. Solving the constrained portfolio optimization facilitates the identification of stochastic discount factors that price any contingent claim in such portfolios. Implementing our method, we find subjective prices of NTNH European and American ESOs, both for block exercise and for a continuum of exercise rates. We identify executives' optimal exercise policies. We use these to find the objective price/cost of ESOs to firms. We then run a simulation study of price sensitivities and changes to optimal exercise policies with respect to model parameters and obtain policy implications regarding ESOs' incentivizing efficiency. We identify comparative statics results and policy implications that are sensitive to (high/low) volatility regimes. Moreover, for the first time, we demonstrate that unlike under the block exercise of European and American ESOs, with the continuous partial exercise of American ESOs, under a low volatility regime, subjective prices may be higher than objective ones.

Research paper thumbnail of Variance minimizing strategies for stochastic processes with applications to tracking stock indices

International Review of Finance, Oct 31, 2019

Research paper thumbnail of Real Options Valuation of Australian Gold Mines and Mining Companies

Social Science Research Network, 2002

ABSTRACT Conventional discounted cash flow valuation techniques are inappropriate for mining comp... more ABSTRACT Conventional discounted cash flow valuation techniques are inappropriate for mining companies because operational flexibilities are deemed an essential component of mine values. In this article we review the real options literature on the valuation of mines and their embedded option to close the mine. We use a model based on Brennan and Schwartz (1985) to empirically value Australian gold mines and mining companies. One difficulty with doing empirical research in this area is in obtaining relevant and complete data, given the nature of real assets and the fact that investments are typically private in nature. The mine data for this study is supplied by Brook Hunt mining and metal industry consultants, UK and covers the period from 1992 to 1995. The primary advantage of this data set is its consistency across different mines that cannot be matched by data sets derived from annual reports data. We find that the real options model is a useful tool for the description and valuation of operational flexibilities. However, the values of the embedded options are very sensitive to estimation errors in the input parameters of the model. While average and median closure option values are economically significant, the option values vary over a large range.

Research paper thumbnail of Martingale representation, hedging policies and contingent claims

Research paper thumbnail of Executive Stock Options Pricing with Free Wealth Weights and Continuous Partial Exercise: An Analytic Constrained Portfolio Optimization/Stochastic Discount Factor Approach

Social Science Research Network, 2011

ABSTRACT As traditional contingent claims valuation methods do not apply to non-transferable and ... more ABSTRACT As traditional contingent claims valuation methods do not apply to non-transferable and non-hedgeable contingent claims, a recent proliferation of such claims creates the need for the development of new valuation methods. Furthermore, the essential role of executive stock options (henceforth ESOs) in the current economic system makes their valuation a necessity for optimally allocating resources and incentivizing executives. In this paper, we offer a novel method of valuating non-transferable, non-hedgeable (henceforth NTNH) contingent claims and then implement this method in pricing ESOs. We find that the NTNH constraints will break the local co-linearity caused by including contingent claims in solving the portfolio optimization problem. Thus, we translate the optimization problems for portfolios that include contingent claims into optimization problems on portfolios of primary assets only by replicating the contingent claims using primary assets. We integrate the NTNH constraints into one single rectangular constraint, under which by solving the portfolio optimization problem we derive a stochastic discount factor. We then use this stochastic discount factor to price the NTNH contingent claims and implement the method in pricing ESOs. We investigate both block exercise and continuous partial exercise, and derive the first order conditions with respect to optimal exercise rates for continuous partial exercise case.

Research paper thumbnail of Do Long Rates Behave Like Short Rates - Some Australian Evidence

Social Science Research Network, 2007

ABSTRACT We estimate a general jump-diffusion model which allows us to nest many of the well-know... more ABSTRACT We estimate a general jump-diffusion model which allows us to nest many of the well-known single-factor interest rate models and their jump-diffusion extensions. We find significant differences between the dynamics of short and long rates. In particular, we find that the chance of a jump occurring in any given day is much higher for the short rates than for the long rates and that the size of the jumps is also larger. We find that jumps explain the vast majority of the total variance of daily interest rate changes for the short rates (as much as 95% over the whole period) which is much higher than for the long rates (where jumps accounted for 44%). We also estimate the CKLS model for our data and find that the only model not rejected is the CEV model, in stark contrast to previous findings. Also, we find that the levels effect for the diffusion volatility is very much greater for short rates than the long rates. We also find that adding jumps to a simple diffusion model gives a larger improvement than comes from going from a simple diffusion to the CKLS model.

Research paper thumbnail of A Jump Diffusion Model for Spot Electricity Prices and Market Price of Risk

Social Science Research Network, 2011

h i g h l i g h t s • We develop a mean-reverting jump-diffusion model for electricity spot price... more h i g h l i g h t s • We develop a mean-reverting jump-diffusion model for electricity spot prices. • This accounts for observed seasonality, time-dependent jump intensity and heteroskedastic disturbance. • We also derive closed-form pricing formula for electricity futures. • The market price of risk on the diffusion is found to be negative. • The market price of risk on jumps, on the other hand, is positive and seasonal.

Research paper thumbnail of A simple approach to valuing risky debt with constant elasticity of variance effects

Research paper thumbnail of Non-Transferable Non-Hedgeable Executive Stock Options Pricing

Social Science Research Network, 2014

Research paper thumbnail of A Markov Chain Modulated Short-Term Interest Rate Model: Inference on Central Bank Transparency

Research paper thumbnail of A Structural Model for Credit Risk with Markov Modulated Lévy Processes and Synchronous Jumps

Social Science Research Network, 2013

This paper presents a switching regime version of the Merton's structural model for the pricing o... more This paper presents a switching regime version of the Merton's structural model for the pricing of default risk. The default event depends on the total value of the rm's asset modeled by a Markov modulated Lévy process. The novelty of our approach is to consider that rm's asset jumps synchronously with a change in the regime. After a discussion of dynamics under the risk neutral measure, we present two models. In the rst one, the default occurs at bond maturity if the rm's value falls below a predetermined barrier. In the second version, the company can bankrupt at multiple predetermined discrete times. The use of a Markov chain to model switches in hidden external factors makes it possible to capture the eects of changes in trends and volatilities exhibited by default probabilities. Finally, with synchronous jumps, the rm's asset and state processes are no longer uncorrelated.

Research paper thumbnail of The Effect of Investor Category Trading Imbalances on Stock Returns

Social Science Research Network, 2007

Trading is the mechanism of the economist's 'invisible hand;' the means by which price discovery ... more Trading is the mechanism of the economist's 'invisible hand;' the means by which price discovery occurs. We use daily shareholdings data from the Australian equities clearinghouse to investigate the impact of the trading imbalances of investor categories on stock returns. Our evidence does not contradict the behavioural finance assumption that the trading of individual investors contributes to price discovery. Furthermore, we find that the trading of individual investors is distinct from that of the other investor categories. While the trading of all investor categories Granger-causes returns, returns Granger-cause trading only for the individual investor category. In the short term of up to one month, individual investors engage in feedback trading.

Research paper thumbnail of Hedging diffusion processes by local risk minimization with applications to index tracking

Journal of Economic Dynamics and Control, Jul 1, 2007

This paper extends the local risk-minimization criterion for hedging contingent claims, as introd... more This paper extends the local risk-minimization criterion for hedging contingent claims, as introduced in Fo¨llmer and Sondermann [Hedging of non-redundant contingent claims.

Research paper thumbnail of A multi-factor model with time-varying and seasonal risk premiums for the natural gas market

Energy Economics, Jul 1, 2015

In this paper, we develop a quantitative model of the US natural gas market that explores its mul... more In this paper, we develop a quantitative model of the US natural gas market that explores its multi-factor structure and its time-varying and seasonal risk premiums. With weekly spot and futures prices we show that three factors are preferred to describe the futures term structure, and the time-varying risk premiums are also significant. Moreover, we found that the market implies a seasonal risk premium with two peaks and troughs in one year, which is important to correctly price the futures by maturity month. Finally, we link this seasonal risk premium to the uncertainty of the US natural gas demand and find a positive relationship between them. These results reveal the complex aspect of the market, and may have useful applications for other commodity sectors.

Research paper thumbnail of Real Options Valuation of Australian Gold Mines and Mining Companies

The Journal of Alternative Investments, Jun 30, 2003

ABSTRACT Conventional discounted cash flow valuation techniques are inappropriate for mining comp... more ABSTRACT Conventional discounted cash flow valuation techniques are inappropriate for mining companies because operational flexibilities are deemed an essential component of mine values. In this article we review the real options literature on the valuation of mines and ...

Research paper thumbnail of Discontinuous Asset Prices and Non-Attainable Contingent Claims

Mathematical Finance, Jul 1, 1993

The price of a risky asset 6 is described by a Markov diffusion with jumps. In general there may ... more The price of a risky asset 6 is described by a Markov diffusion with jumps. In general there may be many equivalent martingale measures. Contingent claims which depend on the price of [ at some time T may not be attainable, and the market may not be complete. However, ...

Research paper thumbnail of Information, Insider Trading, Executive Reload Stock Options, Incentives, and Regulation

Social Science Research Network, 2019

We introduce a theoretical model of executives with insider information (insider-executives) gran... more We introduce a theoretical model of executives with insider information (insider-executives) granted incentivizing executive stock options (ESO). We show that while insider-executives optimize their wealth, using their insider information nullies ESO incentives, misaligning their and shareholders' interests. We oer realigning methods: granting insider-executives reload stock options (RSO) and imposing blackout trading periods (blackouts). Eective blackouts keep insider-executives incentivized without being overly restrictive, i.e., without reducing their welfare below that of outsiders. We introduce RSO pricing for insider-executives and oer policy implications: reestablishing the currently out-of-favor RSO, and allowing rms, not regulators, to set blackout periods on securities they issue.

Research paper thumbnail of A Class of Stochastic Volatility HJM Interest Rate Models

Social Science Research Network, 2004

ABSTRACT This paper considers a class of stochastic volatility HJM term structure models with exp... more ABSTRACT This paper considers a class of stochastic volatility HJM term structure models with explicit finite dimensional realisations. The resulting bond market is arbitrage free but incomplete resulting in a non-unique martingale measure. Nevertheless, the market price of risk is partially determined by the forward rate drift and volatility. Numerical simulation for bond and bond option prices are included to illustrate the effect of stochastic volatility on these prices.

Research paper thumbnail of Hedging Diffusion Processes by Local Risk-Minimisation with Applications to Index Tracking

RePEc: Research Papers in Economics, Feb 1, 2004

The solution to the problem of hedging contingent claims by local risk-minimisation has been cons... more The solution to the problem of hedging contingent claims by local risk-minimisation has been considered in detail in Follmer and Sondermann (1986), Follmer and Schweizer (1991) and Schweizer (1991). However, given a stochastic process Xt and tau1 <> tau2, the strategy that is locally risk-minimising for Xtau1 is in general not locally risk-minimising for Xtau2. In the case of diffusion

Research paper thumbnail of Non-transferable non-hedgeable executive stock option pricing

Journal of Economic Dynamics and Control, Apr 1, 2015

We introduce a method of valuing non-transferable non-hedgeable (NTNH) contingent claims and use ... more We introduce a method of valuing non-transferable non-hedgeable (NTNH) contingent claims and use it to price executive stock options (ESOs). We use NTNH constraints to break local co-linearity, caused by derivative assets, that renders common portfolio optimization methods ineffective. We are, thus, able to translate portfolios that include NTNH derivatives into portfolios of primary assets (only). We achieve this by replicating derivatives using primary assets, and then integrating NTNH constraints into a single rectangular constraint. Solving the constrained portfolio optimization facilitates the identification of stochastic discount factors that price any contingent claim in such portfolios. Implementing our method, we find subjective prices of NTNH European and American ESOs, both for block exercise and for a continuum of exercise rates. We identify executives' optimal exercise policies. We use these to find the objective price/cost of ESOs to firms. We then run a simulation study of price sensitivities and changes to optimal exercise policies with respect to model parameters and obtain policy implications regarding ESOs' incentivizing efficiency. We identify comparative statics results and policy implications that are sensitive to (high/low) volatility regimes. Moreover, for the first time, we demonstrate that unlike under the block exercise of European and American ESOs, with the continuous partial exercise of American ESOs, under a low volatility regime, subjective prices may be higher than objective ones.

Research paper thumbnail of Non-Transferable Non-Hedgeable Executive Stock Option Pricing

Social Science Research Network, 2013

We introduce a method of valuing non-transferable non-hedgeable (NTNH) contingent claims and use ... more We introduce a method of valuing non-transferable non-hedgeable (NTNH) contingent claims and use it to price executive stock options (ESOs). We use NTNH constraints to break local co-linearity, caused by derivative assets, that renders common portfolio optimization methods ineffective. We are, thus, able to translate portfolios that include NTNH derivatives into portfolios of primary assets (only). We achieve this by replicating derivatives using primary assets, and then integrating NTNH constraints into a single rectangular constraint. Solving the constrained portfolio optimization facilitates the identification of stochastic discount factors that price any contingent claim in such portfolios. Implementing our method, we find subjective prices of NTNH European and American ESOs, both for block exercise and for a continuum of exercise rates. We identify executives' optimal exercise policies. We use these to find the objective price/cost of ESOs to firms. We then run a simulation study of price sensitivities and changes to optimal exercise policies with respect to model parameters and obtain policy implications regarding ESOs' incentivizing efficiency. We identify comparative statics results and policy implications that are sensitive to (high/low) volatility regimes. Moreover, for the first time, we demonstrate that unlike under the block exercise of European and American ESOs, with the continuous partial exercise of American ESOs, under a low volatility regime, subjective prices may be higher than objective ones.

Research paper thumbnail of Variance minimizing strategies for stochastic processes with applications to tracking stock indices

International Review of Finance, Oct 31, 2019

Research paper thumbnail of Real Options Valuation of Australian Gold Mines and Mining Companies

Social Science Research Network, 2002

ABSTRACT Conventional discounted cash flow valuation techniques are inappropriate for mining comp... more ABSTRACT Conventional discounted cash flow valuation techniques are inappropriate for mining companies because operational flexibilities are deemed an essential component of mine values. In this article we review the real options literature on the valuation of mines and their embedded option to close the mine. We use a model based on Brennan and Schwartz (1985) to empirically value Australian gold mines and mining companies. One difficulty with doing empirical research in this area is in obtaining relevant and complete data, given the nature of real assets and the fact that investments are typically private in nature. The mine data for this study is supplied by Brook Hunt mining and metal industry consultants, UK and covers the period from 1992 to 1995. The primary advantage of this data set is its consistency across different mines that cannot be matched by data sets derived from annual reports data. We find that the real options model is a useful tool for the description and valuation of operational flexibilities. However, the values of the embedded options are very sensitive to estimation errors in the input parameters of the model. While average and median closure option values are economically significant, the option values vary over a large range.

Research paper thumbnail of Martingale representation, hedging policies and contingent claims

Research paper thumbnail of Executive Stock Options Pricing with Free Wealth Weights and Continuous Partial Exercise: An Analytic Constrained Portfolio Optimization/Stochastic Discount Factor Approach

Social Science Research Network, 2011

ABSTRACT As traditional contingent claims valuation methods do not apply to non-transferable and ... more ABSTRACT As traditional contingent claims valuation methods do not apply to non-transferable and non-hedgeable contingent claims, a recent proliferation of such claims creates the need for the development of new valuation methods. Furthermore, the essential role of executive stock options (henceforth ESOs) in the current economic system makes their valuation a necessity for optimally allocating resources and incentivizing executives. In this paper, we offer a novel method of valuating non-transferable, non-hedgeable (henceforth NTNH) contingent claims and then implement this method in pricing ESOs. We find that the NTNH constraints will break the local co-linearity caused by including contingent claims in solving the portfolio optimization problem. Thus, we translate the optimization problems for portfolios that include contingent claims into optimization problems on portfolios of primary assets only by replicating the contingent claims using primary assets. We integrate the NTNH constraints into one single rectangular constraint, under which by solving the portfolio optimization problem we derive a stochastic discount factor. We then use this stochastic discount factor to price the NTNH contingent claims and implement the method in pricing ESOs. We investigate both block exercise and continuous partial exercise, and derive the first order conditions with respect to optimal exercise rates for continuous partial exercise case.

Research paper thumbnail of Do Long Rates Behave Like Short Rates - Some Australian Evidence

Social Science Research Network, 2007

ABSTRACT We estimate a general jump-diffusion model which allows us to nest many of the well-know... more ABSTRACT We estimate a general jump-diffusion model which allows us to nest many of the well-known single-factor interest rate models and their jump-diffusion extensions. We find significant differences between the dynamics of short and long rates. In particular, we find that the chance of a jump occurring in any given day is much higher for the short rates than for the long rates and that the size of the jumps is also larger. We find that jumps explain the vast majority of the total variance of daily interest rate changes for the short rates (as much as 95% over the whole period) which is much higher than for the long rates (where jumps accounted for 44%). We also estimate the CKLS model for our data and find that the only model not rejected is the CEV model, in stark contrast to previous findings. Also, we find that the levels effect for the diffusion volatility is very much greater for short rates than the long rates. We also find that adding jumps to a simple diffusion model gives a larger improvement than comes from going from a simple diffusion to the CKLS model.

Research paper thumbnail of A Jump Diffusion Model for Spot Electricity Prices and Market Price of Risk

Social Science Research Network, 2011

h i g h l i g h t s • We develop a mean-reverting jump-diffusion model for electricity spot price... more h i g h l i g h t s • We develop a mean-reverting jump-diffusion model for electricity spot prices. • This accounts for observed seasonality, time-dependent jump intensity and heteroskedastic disturbance. • We also derive closed-form pricing formula for electricity futures. • The market price of risk on the diffusion is found to be negative. • The market price of risk on jumps, on the other hand, is positive and seasonal.

Research paper thumbnail of A simple approach to valuing risky debt with constant elasticity of variance effects

Research paper thumbnail of Non-Transferable Non-Hedgeable Executive Stock Options Pricing

Social Science Research Network, 2014

Research paper thumbnail of A Markov Chain Modulated Short-Term Interest Rate Model: Inference on Central Bank Transparency

Research paper thumbnail of A Structural Model for Credit Risk with Markov Modulated Lévy Processes and Synchronous Jumps

Social Science Research Network, 2013

This paper presents a switching regime version of the Merton's structural model for the pricing o... more This paper presents a switching regime version of the Merton's structural model for the pricing of default risk. The default event depends on the total value of the rm's asset modeled by a Markov modulated Lévy process. The novelty of our approach is to consider that rm's asset jumps synchronously with a change in the regime. After a discussion of dynamics under the risk neutral measure, we present two models. In the rst one, the default occurs at bond maturity if the rm's value falls below a predetermined barrier. In the second version, the company can bankrupt at multiple predetermined discrete times. The use of a Markov chain to model switches in hidden external factors makes it possible to capture the eects of changes in trends and volatilities exhibited by default probabilities. Finally, with synchronous jumps, the rm's asset and state processes are no longer uncorrelated.

Research paper thumbnail of The Effect of Investor Category Trading Imbalances on Stock Returns

Social Science Research Network, 2007

Trading is the mechanism of the economist's 'invisible hand;' the means by which price discovery ... more Trading is the mechanism of the economist's 'invisible hand;' the means by which price discovery occurs. We use daily shareholdings data from the Australian equities clearinghouse to investigate the impact of the trading imbalances of investor categories on stock returns. Our evidence does not contradict the behavioural finance assumption that the trading of individual investors contributes to price discovery. Furthermore, we find that the trading of individual investors is distinct from that of the other investor categories. While the trading of all investor categories Granger-causes returns, returns Granger-cause trading only for the individual investor category. In the short term of up to one month, individual investors engage in feedback trading.

Research paper thumbnail of Hedging diffusion processes by local risk minimization with applications to index tracking

Journal of Economic Dynamics and Control, Jul 1, 2007

This paper extends the local risk-minimization criterion for hedging contingent claims, as introd... more This paper extends the local risk-minimization criterion for hedging contingent claims, as introduced in Fo¨llmer and Sondermann [Hedging of non-redundant contingent claims.

Research paper thumbnail of A multi-factor model with time-varying and seasonal risk premiums for the natural gas market

Energy Economics, Jul 1, 2015

In this paper, we develop a quantitative model of the US natural gas market that explores its mul... more In this paper, we develop a quantitative model of the US natural gas market that explores its multi-factor structure and its time-varying and seasonal risk premiums. With weekly spot and futures prices we show that three factors are preferred to describe the futures term structure, and the time-varying risk premiums are also significant. Moreover, we found that the market implies a seasonal risk premium with two peaks and troughs in one year, which is important to correctly price the futures by maturity month. Finally, we link this seasonal risk premium to the uncertainty of the US natural gas demand and find a positive relationship between them. These results reveal the complex aspect of the market, and may have useful applications for other commodity sectors.

Research paper thumbnail of Real Options Valuation of Australian Gold Mines and Mining Companies

The Journal of Alternative Investments, Jun 30, 2003

ABSTRACT Conventional discounted cash flow valuation techniques are inappropriate for mining comp... more ABSTRACT Conventional discounted cash flow valuation techniques are inappropriate for mining companies because operational flexibilities are deemed an essential component of mine values. In this article we review the real options literature on the valuation of mines and ...

Research paper thumbnail of Discontinuous Asset Prices and Non-Attainable Contingent Claims

Mathematical Finance, Jul 1, 1993

The price of a risky asset 6 is described by a Markov diffusion with jumps. In general there may ... more The price of a risky asset 6 is described by a Markov diffusion with jumps. In general there may be many equivalent martingale measures. Contingent claims which depend on the price of [ at some time T may not be attainable, and the market may not be complete. However, ...

Research paper thumbnail of Information, Insider Trading, Executive Reload Stock Options, Incentives, and Regulation

Social Science Research Network, 2019

We introduce a theoretical model of executives with insider information (insider-executives) gran... more We introduce a theoretical model of executives with insider information (insider-executives) granted incentivizing executive stock options (ESO). We show that while insider-executives optimize their wealth, using their insider information nullies ESO incentives, misaligning their and shareholders' interests. We oer realigning methods: granting insider-executives reload stock options (RSO) and imposing blackout trading periods (blackouts). Eective blackouts keep insider-executives incentivized without being overly restrictive, i.e., without reducing their welfare below that of outsiders. We introduce RSO pricing for insider-executives and oer policy implications: reestablishing the currently out-of-favor RSO, and allowing rms, not regulators, to set blackout periods on securities they issue.

Research paper thumbnail of A Class of Stochastic Volatility HJM Interest Rate Models

Social Science Research Network, 2004

ABSTRACT This paper considers a class of stochastic volatility HJM term structure models with exp... more ABSTRACT This paper considers a class of stochastic volatility HJM term structure models with explicit finite dimensional realisations. The resulting bond market is arbitrage free but incomplete resulting in a non-unique martingale measure. Nevertheless, the market price of risk is partially determined by the forward rate drift and volatility. Numerical simulation for bond and bond option prices are included to illustrate the effect of stochastic volatility on these prices.

Research paper thumbnail of Hedging Diffusion Processes by Local Risk-Minimisation with Applications to Index Tracking

RePEc: Research Papers in Economics, Feb 1, 2004

The solution to the problem of hedging contingent claims by local risk-minimisation has been cons... more The solution to the problem of hedging contingent claims by local risk-minimisation has been considered in detail in Follmer and Sondermann (1986), Follmer and Schweizer (1991) and Schweizer (1991). However, given a stochastic process Xt and tau1 <> tau2, the strategy that is locally risk-minimising for Xtau1 is in general not locally risk-minimising for Xtau2. In the case of diffusion

Research paper thumbnail of Non-transferable non-hedgeable executive stock option pricing

Journal of Economic Dynamics and Control, Apr 1, 2015

We introduce a method of valuing non-transferable non-hedgeable (NTNH) contingent claims and use ... more We introduce a method of valuing non-transferable non-hedgeable (NTNH) contingent claims and use it to price executive stock options (ESOs). We use NTNH constraints to break local co-linearity, caused by derivative assets, that renders common portfolio optimization methods ineffective. We are, thus, able to translate portfolios that include NTNH derivatives into portfolios of primary assets (only). We achieve this by replicating derivatives using primary assets, and then integrating NTNH constraints into a single rectangular constraint. Solving the constrained portfolio optimization facilitates the identification of stochastic discount factors that price any contingent claim in such portfolios. Implementing our method, we find subjective prices of NTNH European and American ESOs, both for block exercise and for a continuum of exercise rates. We identify executives' optimal exercise policies. We use these to find the objective price/cost of ESOs to firms. We then run a simulation study of price sensitivities and changes to optimal exercise policies with respect to model parameters and obtain policy implications regarding ESOs' incentivizing efficiency. We identify comparative statics results and policy implications that are sensitive to (high/low) volatility regimes. Moreover, for the first time, we demonstrate that unlike under the block exercise of European and American ESOs, with the continuous partial exercise of American ESOs, under a low volatility regime, subjective prices may be higher than objective ones.