Discussion of “Optimal Exercise of Executive Stock Options and Implications for Firm Cost”*

By Jennifer Carpenter, Richard Stanton

and

Nancy Wallace

* Thanks to Andrei Malenko and Paul Pfleiderer for helpful discussions on this discussion.

Taking the Rational Theory a Bit Further

In the model, executives choose an optimal exercise strategy, where they can dynamically trade a stock (subject to a short-sale constraint), a market index, and a riskless asset.

While executives are typically restricted from shorting their own stock (or from similar strategies involving derivatives on their own stock), they are able to trade other stocks.

What if they can also trade other stocks? (i.e., hire a professional portfolio manager to do this)

The paper shows that as the correlation with the available risky portfolio increases towards 1, the exercise strategy approaches that of a tradable option and the value of the option converges to the tradable option value.

How close can we come?

Taking the Rational Theory a Bit Further

I tried this with a sample of 10 S&P 500 technology companies (with market values under $10 billion) and volatilities near 0.50.

I searched for the portfolio among 65 S&P 500 technology stocks that maximized correlation.

Taking the Rational Theory a Bit Further

Company Name Volatility Correlation with S&P500

Correlation with stock ptf

Altera Corp. 0.4924 0.6187 0.8585

Citrix Systems 0.4776 0.5070 0.6294

Jabil Cirquit 0.4596 0.6020 0.7536

LSI Corp. 0.5485 0.5654 0.7435

Micron Technology 0.4906 0.5414 0.7435

National Semiconductor

0.4912 0.5991 0.8611

Novellus Systems 0.4717 0.6304 0.8865

QLogic Corp 0.4869 0.5619 0.7771

Teradyne 0.5402 0.6027 0.8356

Xilinx 0.4771 0.6365 0.8718

Average 0.4936 0.5865 0.7961

Taking the Rational Theory a Bit Further

On average, while the correlation with the S&P500 is around 58%, the correlation with the stock portfolio is around 80%. On several, it is almost 90%.

Thus, using table 5a, the executive option would be worth about 75% of its fully tradable value if the executive could only trade the S&P500, but would be worth about 95% of its fully tradable value if the executive could trade other stocks.

This is probably an understatement, since I only considered a universe of 65 stocks.

This is very similar to the intuition of Merton’s 1998 AER article that speaks about dynamic hedging of options on illiquid assets through finding the traded asset portfolio that minimizes tracking error.

Implications ofDynamic Hedging Argument

1. This would imply that the problem is most severe for stocks with the greatest idiosyncratic risk.

2. While we typically think about the incentives of managers to control volatility, we may also think about managers attempting to lower idiosyncratic risk.

3. Given that in actuality we know that managers exercise early, they are probably not doing this!

The Odd Case of

the Split Continuation Region The authors provide an example where for low stock prices,

the option is exercised at an upper trigger (the typical result), but for high stock prices, the option is exercised at a lower trigger.

Here is the underlying utility function:

Numerically, it seems that for c=0, the exercise region has a single boundary. But for c=0.0001, it has two boundaries.

What is the intuition? (I’m not sure, but here is a guess…)

cWA

WWU

A

1)(

1

The Odd Case of

the Split Continuation Region While for c=0, relative risk aversion is constant, for c>0,

relative risk aversion is decreasing in wealth.

.)('

,)(

2

12

cW

cWAWR

cW

AWWR

A

A

A

A

The Odd Case of

the Split Continuation Region When the stock price is high, the option is a significant fraction

of total wealth. Thus, if the stock price falls by a sufficient amount, the agent’s risk aversion increases enough so as to justify exercising the risky option and moving into a diversified portfolio.

When the stock price is low, the option is a small fraction of overall wealth, and thus the decreasing risk aversion effect is overwhelmed by the usual benefits of exercise at an upper trigger.

Notice the impact of even a small value of c. When the stock price is high and total wealth is 3, then R’(W) = -4.13. When the stock price is low and total wealth is 1, then R’(W) = -0.01.

Consideration of Sequential Exercise

What would happen if the executive was not constrained to exercise all options simultaneously?

The exercise region could be determined recursively. The first option (or block) would be exercised at an optimal

trigger, as determined in this paper. Conditional on this strategy, the second option (or block) would

be exercised later, since the executive now is more diversified than before, and can better tolerate option risk.

This recursion would continue until the final option (or block). Presumably, the value of the overall option portfolio would be

closer to the tradable option value than is the case in the simultaneous exercise version.