Smooth pasting as rate of return equalization

doi:10.1016/j.econlet.2005.05.028

Economics Letters

Volume 89, Issue 2, November 2005, Pages 200-206

https://doi.org/10.1016/j.econlet.2005.05.028 Get rights and content

Abstract

We further elucidate the smooth pasting condition behind optimal early exercise of options. It is easy to show that smooth pasting implies rate of return equalization between the option and the levered position that results from exercise. This yields new economic insights into the optimal early exercise condition that the option holder faces.

Introduction

The smooth pasting (or high-contact) condition associated with option and real options decisions has generated considerable interest because of the optimality of early exercise. It is well known that smooth pasting is a first-order condition for optimum; proposed by Samuelson (1965), proven by Merton (1973), discussed by Dumas (1991) and several others. Brekke and Øksendal (1991) also show that the condition is sufficient under weak constraints. Nonetheless, smooth pasting remains somewhat mysterious to both economists and practitioners and it is apparently not very useful for many except theorists. The popular real options introduction by Dixit and Pindyck (1994) saves the discussion of smooth pasting for a quite technical appendix, and no simple rules of thumb seem to exist for practitioners.

Dixit et al. (1999) bridged some of the gap between theory and practice using an analogy between optimal exercise of investment options of the McDonald and Siegel (1986) type and application of standard market power models. Optimal investment can be characterized by an elasticity-based premium, analogous to the markup price chosen by a profit-maximizing monopolist.

We provide another, more intuitive and natural, explanation of the phenomenon; that of rate of return equalization between the option and its levered payoff. This allows a larger audience to appreciate and implement smooth pasting techniques in a wider variety of situations. We also relate results to the elasticity-based rules introduced by Dixit et al. (1999) and Sødal (1998). The results are illustrated here using geometric Brownian motion but are also valid for other diffusions.

Section snippets

Rates of return

Geometric Brownian diffusions can be written in the Risk Neutral Q or Real World P; having drift r − δ or μ − δ, respectively (r, δ, μ, σ represent the continuous risk free, dividend, project return and volatility rates) $\frac{d S}{S} = (r - δ) d t + σ d W^{Q}$ $\frac{d S}{S} = (μ - δ) d t + σ d W^{P} .$

Local changes dC in the call price C (puts can also be analyzed) are given by the Ito expansion, furthermore no arbitrage requires that Risk Neutral expectations E^Q[dC] of these changes must be risk free (or the hedged position yields the risk

Smooth pasting

The rate of return can be investigated at the point of optimal early call exercise S¯ (where S¯ > X, the exercise price). The two conditions necessary for this are value matching (payoff compensates for option termination) and smooth pasting (slope equality between option and payoff functions) $C (\bar{S}) = \bar{S} - X$ ${\frac{\partial C}{\partial S} |}_{S = \bar{S}} = 1 .$

Thus at the critical exercise boundary S = S¯ the call return r_C is $r_{C} (\bar{S}) = r + \frac{\bar{S}}{C (\bar{S})} (μ - r) = \frac{μ \bar{S} - r X}{\bar{S} - X} = r_{PO}$ which is also r_PO, the return rate of the levered payoff S¯ − X (as a fraction

Relationship to other approaches

The result that the return on the option equates the return on the net payoff is closely related to other findings on smooth pasting. Dixit et al. (1999) argues that the optimal exercise of a perpetual call option consists of maximizing the expected net present value $C = \max_{\bar{S}} D (S, \bar{S}) (\bar{S} - X)$ $D (S, \bar{S}) = E^{P} [e^{- ρ T}] = E^{Q} [e^{- r T}]$ where T is the (random) first-hitting time from the current value of the project, S, up to the value S¯ at which the option is exercised. The objective is to maximize the expected,

Dividend yields and an analytic approximation

The dividend yield δ is important since when zero, American calls become zero dividend Black Scholes calls; early exercise, smooth pasting and rate of return equalization are all ruled out (r is key for the put). Without an opportunity cost of waiting, early exercise never occurs. Thus it is important to understand the yield, δ in determining option returns, since without it equalization is impossible.

The fractional amounts of stock and borrowing required in the replicating portfolio are often

Intuition and implications

The results above have economic implications and intuition, particularly for real option situations where it is difficult to evaluate the option value function explicitly. Optimal early exercise of real options is driven by two conditions, no loss (gain) of value on exercise and rate of return equalization.

This provides a second, more intuitive, condition to managers other than smooth pasting, which may be difficult to evaluate for some pricing problems. Do managers think that the rate of return

References (12)

B. Dumas
Super contact and related optimality conditions
Journal of Economic Dynamics and Control
(1991)
G. Barone-Adesi et al.
Efficient analytic approximation of American option values
Journal of Finance
(1987)
F. Black et al.
The pricing of options and corporate liabilities
Journal of Political Economy
(1973)
K.A. Brekke et al.
The high contact principle as a sufficiency condition for optimal stopping
A. Dixit et al.
Investment Under Uncertainty
(1994)
A. Dixit et al.
A markup interpretation of optimal investment rules
Economic Journal
(1999)

There are more references available in the full text version of this article.

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It is also worth pointing out that according to the results of Theorem 3.1 the value function satisfies the smooth pasting condition in both cases. As was established in Shackleton and Sødal (2005), this guarantees the equalization of the rate of return of the firm both prior and after the irreversible decision has been exercised (see also Dias and Shackleton, 2011). It is worth emphasizing that the conclusions of Theorem 4.1 are important from the point of view of other related stochastic control problems as well.
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Smooth pasting as rate of return equalization

Abstract

Introduction

Section snippets

Rates of return

Smooth pasting

Relationship to other approaches

Dividend yields and an analytic approximation

Intuition and implications

Journal of Economic Dynamics and Control

Efficient analytic approximation of American option values

Journal of Finance

The pricing of options and corporate liabilities

Journal of Political Economy

The high contact principle as a sufficiency condition for optimal stopping

Investment Under Uncertainty

A markup interpretation of optimal investment rules

Economic Journal