Abstract: Under the efficient market hypothesis, overwriting calls or purchasing insurance should not improve risk-adjusted portfolio returns. A proper analysis should show that if options are traded at a fair cost, the risk-reward characteristics of an option position would fall on the efficient market line. In this paper we show that, due to several limitations of mean-variance analysis, this is not the case in practice. We quantify and identify the nature of the resulting biases for performance evaluation, and explain why alternative measures such as semi variance do not help in avoiding such biases. Edhec is one of the top five BUSINESS SCHOOLs in France owing to the high quality of its academic staff (90 permanent lecturers from France and abroad) and its privileged relationship with professionals that the school has been developing since its establishment in 1906. Edhec Business School has decided to draw on its extensive knowledge of the professional environment and has therefore concentrated its research on themes that satisfy the needs of professionals. Edhec implements an active Research Policy in the field of finance. Its "Risk and Asset Management Research Centre" carries out numerous research programs in the areas of asset allocation and risk management in both the traditional and alternative investment universes. Copyright © 2003 Edhec

INTRODUCTION Derivatives and options in particular have progressively established themselves as common tools in portfolio management for asset allocation, hedging, diversification, and/or leverage. Their non-linear payoffs enable investors to create return profiles that are unachievable by simple combinations of conventional investment vehicles such as stocks and bonds. Consequently, their increased presence has been accompanied by a variety of claims regarding their ability to simultaneously enhance rewards and reduce risks. For instance, covered call writing is often presented as an efficient way to increase income of stock ownership through the received option premium and convert the prospects for uncertain future capital gains into immediate cash flows. Similarly, protective put buying allows its users to ascertain a minimum value of their portfolio, limiting downside risk while preserving the upside potential. Although desirable, this win-win suggested behavior violates the efficient market hypothesis, since risk-adjusted portfolio returns should not be improved by the systematic use of options. Aside from the specific characteristics inherent to option valuation, several authors have examined option management in a portfolio context. Unfortunately, neither the academic studies nor the nostrums for market success offered by practitioners to help investors are really conclusive. The lack of consistency among their results and their strong focus on the U.S. market does not allow for complete confidence in their conclusions. In fact, most of the option-based strategies have not yet received a careful theoretical analysis. Even worse, they have often received an analysis based on inadequate theoretical tools. Traditional portfolio performance evaluation relies on the dominating efficiency paradigm in Financial Economics, namely mean-variance analysis. It assumes a buy and hold strategy with normally distributed 2

returns over a specific time-horizon, and compares the results obtained with those of an efficient market index. These assumptions are no longer valid when options are involved in a portfolio. Hereafter, our goal is to illustrate and understand the biases of traditional performance measures when applied to portfolios containing options. As we will see, these biases are both wholly measurable and rectifiable when the option strategy is known, but cannot be solved by using alternatives performance measures such as semi-variance or downside risk. LITERATURE OVERVIEW Contrary to common belief, several studies exist on optioned portfolio performance prior to the Black-Scholes-Merton pricing formula, and even prior to the 1973 trading of listed options in Chicago. The early studies were primarily based on small samples of OTC quotes from brokers1. They generally concluded that writing options was a better strategy than buying them, and frequently observed that writing calls against a diversified portfolio would increased portfolio returns while lowering risk. Their results can be attributed to their choice of a particular time frame, to market inefficiencies and/or to the lack of an effective valuation model. After 1973, most of the studies still focused on covered call writing, but considered longer time horizons and used listed option prices as well as theoretical option premiums. Their results were mixed. On one hand, Bookbinder [1976], Kassouf [1977], Pounds [1978], Grube and Panton [1978] or Yates and Kopprasch [1980] observed that covered call writing still produced substantially larger returns than the traditional buy and hold, even though the risk level of the optioned portfolio (measured by the standard deviation of returns) was lower. On 3

the other hand, Merton, Scholes and Gladstein [1978, 1982] obtained opposite results, producing evidence of a more favorable situation for option buyers than for option sellers. The strong U.S. bull market of 1982 heavily penalized covered call writing with respect to an indexed strategy and reduced popularity of the strategy. Research then focused on static (protective put strategies) and dynamic portfolio insurance. Both became extremely popular until the failure of the latter in the 1987 crash. Once again, interest was reduced, particularly after the publication of new studies based on more complex tools (such as stochastic dominance) that concluded that there was no real dominant strategy; see for instance Booth, Tehranian and Trennepohl [1985], Clarke [1987], or Brooks, Levy, and Yoder [1987]. However, motivated by the strong increase of stock market volatility, interest for option strategies has recently resurged. But there is still no clear evidence regarding the long-term dominance of a systematic option-based strategy on traditional buy and hold positions. The ex-post results depend on the period, the market and the strategies considered, and with a few exceptions, are often U.S. specific; see for instance Austin [1995], Lhabitant [1998], Rendleman [1999], Green and Figlewski [1999], or Ineichen [2000]. Therefore, there is a strong need to take a proper ex-ante approach to assess the expected performance of these strategies. THEORETICAL FRAMEWORK In our model, an investor is assumed to allocate his wealth among a single stock or index (henceforth stock), a long put and a long stock (protective put), or a short call and a long stock (covered call writing)2. The returns of the underlying stock are assumed to follow a 4

lognormal distribution with a mean and variance of the annual logarithmic return µ and 2. No dividends are paid on the stock, and we assume that put and call options on the stock are traded at their fair Black-Scholes values. We denote by Pt and Ct their price at time t, by K their exercise price, by T their time to maturity, and by r the annual continuously compounded rate of interest. Traditional modern portfolio theory tells us that our investor should select the strategy with the highest Sharpe [1966] ratio in other words, the strategy that is expected to give him the highest expected return per unit of risk. We will see what happens if this path is followed. By going through simple but lengthy algebraic equations, it is possible to write the moments of the return distribution for each strategy as a closed-form expression depending only of the six parameters K, T, S0, r, µ and (see appendix). Since the Sharpe ratio considers only the first two moments of a distribution (mean return and volatility), it is also possible to write the Sharpe ratio of an optioned portfolio as a closed form expression of the above mentioned parameters. This solves our strategy selection problem. Moreover, this allows us to assess the performance of each strategy for different values of the parameters µ and -- that is, for different underlying stochastic processes (i.e. different assets), different time-to-maturity (T) and degree of moneyness (K), with the latter two being controlled by the investor. EMPIRICAL RESULTS We performed various simulations. While the results would differ depending on the assumptions, they share a common nature. As a result only the results for one set of simulation (µ=10%, r=5%, =15%, T=1 year, S0= 100) will presented from this point on. 5

Figure 1 and 2 show the expected return and volatility for the three strategies as a function of the option moneyness. As one expects, in a "normal" environment, the selection of a higher strike price should increase both expected return and volatility for covered call writing. Deeply in the money covered call writing is equivalent to holding a risk-free bond, and deeply out of the money covered call writing is similar to holding a naked stock. Conversely, in the case of protective put buying, a higher strike price should decrease both expected returns and volatility. Deeply out of the money protective put buying is similar to holding a naked stock, while deeply in the money protective put buying is equivalent to holding a risk-free bond. Let us now mix both the expected return and the risk effects into a performance measure. Figure 3 shows the Sharpe ratio for the three strategies as a function of the option moneyness. When increasing the exercise price of a protective put position, we move from a pure stock to a pure risk-free bond position, and the Sharpe ratio decreases accordingly. As a consequence, the protective put has a Sharpe ratio that is always lower than the Sharpe ratio of a long stock position. When increasing the exercise price of a covered call position, we move from a pure bond to a pure stock position, and the Sharpe ratio increases accordingly. However, for a set of exercise prices that are just out of the money, the Sharpe ratio of the covered call exceeds the Sharpe ratio of the long stock position. This phenomenon is more important for long term options than for short-term ones3. The consequences for our traditional investor are twofold. First, he will not be interested in buying puts without having specific expectations on the market moves, as the protective put position is dominated. Second, his optimal strategy will consist of selling slightly out-of-themoney covered calls, since this dominates the pure stock strategy. When the stock is actually 6

a market index, this implies that, ex-ante, some covered call positions form a dominating efficient frontier in the mean-variance space. Moreover, the same phenomenon will be observable when measuring performance with any mean-variance or CAPM based performance measure, such as the Treynor ratio or the Jensen alpha. Consequently, it becomes possible and easy to beat an efficient market without any timing ability: given some market parameters (µ, ), use the closed-form expression for the Sharpe ratio to find the optimal exercise price. Then, sell covered call options on the market with this exercise price and the longest maturity available. This strategy appears as superior in the mean-variance space. DISCUSSION AND EXTENSION Fortunately, this apparent dominance arises from the inadequacy of the Sharpe ratio. Despite its popularity, the Sharpe ratio suffers from a series of methodological problems. First, the Sharpe ratio ignores asymmetric reduction of the variance, therefore not presenting the whole picture. For instance, let us consider a protective put with an underlying asset's logreturns that are normally distributed4. The resulting distribution combines two effects. First, we truncate the underlying distribution at the point ST=K, and replace the left portion of the equation with a Dirac delta function having a weight equal to the probability that the stock price is lower or equal to the exercise price. Next, we translate the new distribution to the left, in order to incorporate the put premium payment influence. For the covered call strategy, we can apply the same methodology, except that we truncate the right portion and that we translate on the right, as the premium is received. As clearly evidence in Figure 4, the option strategies will have highly skewed distributions, with an asymmetric volatility reduction. Call writing truncates the right-hand side of a distribution and results in negative skewness 7

(undesirable), while put buying truncates the left-hand side of a distribution and results in positive skewness (desirable). Therefore it is natural that the compensation for risk reduction varies between the two strategies, as a reward for holding a portfolio with a skewed return distribution is required. As the Sharpe ratio ignores the asymmetric reduction of the variance, its application to optioned portfolios effectively overstates the performance of call writing and understates the performance of put buying. Second, the Sharpe ratio is a static tool, whereas the option strategies considered previously are dynamic in nature. It can be shown that the volatility of an optioned portfolio will change over time, even though the volatility of the underlying stock returns is constant (see Appendix 2). This is possible because of the time elapsing, and because of stock price variations, which result in a change of the option's delta as well. But the Sharpe ratio does not account for this changing risk over time. Nor do measures based on semi-variance or downside (static) risk, such as the Sortino [1991] ratio. Note that these observations are not only limited to portfolios containing options, but also to investment managers implementing dynamic trading rules (such as repurchasing securities after market declines, selling after market increases, etc.) as well as to some Hedge Funds whose returns exhibit non-linear option-like exposures to standard asset classes5. When the exact nature of the dynamic strategy is known, it is possible to compute the benchmark "Sharpe ratio", which differs from the market Sharpe ratio. However, this makes performance measurement strategy-dependant. Considering the wide variety of option strategies employed in the fund management industry, this can creates obvious challenges. 8

CONCLUSIONS Several managers have successfully implemented systematic option based strategies; one example would be the $700 million Gateway Fund (GATEX), who has sold covered calls since 1977, or the large number of portfolio insurance funds. Most institutions perceive these strategies as efficient. In particular, they view covered call writing as a means of augmenting portfolio returns and protective put buying as a solution to avoid downside risk. Our paper shows that these option strategies produce systematic biases when evaluated using traditional performance measures. For instance, when applied to the market portfolio, some option strategies are expected to beat the market from the point of view of the Sharpe ratio. In reality, financial theory suggests that these strategies will partially hedge market risk at the expense of a reduced portfolio return, the latter being the market price of risk reduction. Consequently, when options markets are efficient, the returns and risk reduction derived from the sale or purchase of efficiently priced options, when combined with the risk and returns of a diversified market portfolio used as a collateral, will be offsetting, and yield no excess riskadjusted portfolio returns. Observing positive excess performance when using options violates the efficient market hypothesis or implies that there is a reward for holding a portfolio with a skewed return distribution or a non-static risk profile. Therefore, beating the market is easy, but only when using inadequate performance measures! BIBLIOGRAPHY · Austin, M. (1995), "Index option overwriting: strategies and results", Derivatives Quarterly, Summer, pp. 77-84 9

· Black, F. and M. Scholes (1973), "The pricing of options and corporate liabilities", Journal of Political Economy 81, May-June, pp. 637-654 · Boness, J.A. (1964), "Some evidence on the profitability of trading in put and call options", in P.H. Cootner (ed.): The random character of stock market prices, MIT Press, Cambridge, MAss., pp. 475-496 · Bookbinder, A.I.A. (1976): "Security options strategy", Programmed Press, New-York · Bookstaber, R.M. and R.G. Clarke (1983): "An algorithm to calculate the return distribution of portfolios with option positions", Management Science 29, pp. 419-429 · Bookstaber, R.M. and R.G. Clarke (1984): "Option portfolio strategies: measurement and evaluation", Journal of Business 57, pp. 469-492 · Bookstaber, R.M. and R.G. Clarke (1985): "Problems in evaluating the performance of portfolios with options", Financial Analysts Journal, January/February, pp. 48-62 · Booth, J.R., A. Tehranian and G.L. Trennepohl (1985): "Efficiency analysis and option portfolio selection", Journal of Financial and quantitative analysis 4, pp. 435-450 · Brooks, R., H. Levy and J. Yoder (1987): "Using stochastic dominance to evaluate the performance of portfolios with options", Financial Analysts Journal, March/April, pp. 7982 · Clarke, R.G. (1987): "Stochastic dominance properties of option strategies", Advances in Futures and Options Research, vol. 2, JAI Press, pp. 1-18 · Ferguson, R. (1987): " A comparison of the mean-variance and long-term return characteristics of three investment strategies ", Financial Analysts Journal, July/August, pp. 55-66 · Green T.C. and S. Figlewski (1999), "Market risk and model risk for a financial institution writing options", Journal of Finance, August 10

· Grube, R.C. and D.B. Panton (1978): "How well do filter-rule strategies work for options?", Journal of Portfolio Management, Winter, pp. 52-57 · Ineichen A. M. (2000), "Systematic stock option writing does not work", Derivatives Quarterly, Spring, pp. 50-59 · Jensen, M.C. (1968): "The performance of mutual funds in the period 1945-1964", Journal of Finance 23, pp. 389-416 · Kassouf, S.T. (1977): "Option pricing: theory and practice", The Institute for quantitative research in Finance, Columbia University, Spring seminar, Palm Springs, California · Kruizenga, R.J. (1964): "Profit returns from purchasing puts and calls", in P.H. Cootner (ed.): The random character of stock market prices, MIT Press, Cambridge, Mass., pp. 392-411 · Lhabitant, F.S. (1997): "On the abuse of expected utility approximations for portfolio selection and portfolio performance", Working Paper, 14th International Conference of the French Finance Association, June · Lhabitant, F.S. (1998): "Enhancing portfolio performance using options strategies: why beating the market is easy", in "European Research Symposium Proceedings", Chicago Board of Trade, pp. 149-214 · Malkiel, B.G. and R.E. Quandt (1969): "Strategies and rational decisions in the securities options market", The MIT Press, Cambridge, Massachusetts · Malkiel, B.G. (1972): "Trading in options: what are the best strategies?", Commercial and Financial Chronicle, December, pp. 1-12 · Merton, R.C. (1973): "Theory of rational option pricing", Bell Journal of Economics and Management Science 4, pp. 141-183 · Merton, R.C., M. Scholes and M.L. Gladstein (1978): "The returns and risks of alternative call option portfolio investment strategies", Journal of Business 51, pp. 183-242 11

· Merton, R.C., M. Scholes and M.L. Gladstein (1982): "The returns and risks of alternative put option portfolio investment strategies", Journal of Business 55, pp. 1-55 · Mitchell, M., and T. Pulvino (2000), "Characteristics of Risk in Risk Arbitrage," Working Paper, Harvard Business School and Kellogg Graduate School of Management. · Pounds, H.M. (1978): "Covered call option writing: strategies and results", Journal of Portfolio Management, Winter, pp. 31-42 · Rendleman R.J. Jr. (1999), "Option investing from a risk return perspective", The Journal of Portfolio Management, May · Rosett, R.N. (1967): " Estimating the utility of wealth from call options data ", in D. Hester and J. Tobin (eds.), "Risk aversion and portfolio choice ", Wiley, New-York, pp. 154-169 · Sortino, F. and R. van der Meer (1991), "Downside risk", Journal of Portfolio Management, Summer, pp. 27-31 · Sharpe, W.F. (1966): "Mutual fund performance", Journal of Business 39, pp. 119-138 · Treynor, J.L. (1965): "How to rate management of investment funds", Harvard Business Review 43, pp. 63-75 · Yates, J.W. and R.W. Kopprasch (1980): "Writing covered call options: profits and risks", Journal of Portfolio Management, Fall, pp. 74-79 12

APPENDIX 1 The methodology for computing the moments of the return distribution for an optioned portfolio is detailed in Lhabitant (1998). In particular, for the moments of order one (mean return) and two (variance), we have:

For the long spot (S) returns: E(R S ) = eµT - 1 2 (R S ) = e2µT (e2T - 1)

For the protective put (PP) returns:

E(R PP ) =

KL(0) + U (1) S0 + P0

-1

2 (R PP )

=

K 2L(0)

+

U (2) (S0

- (KL(0) + P0 )2

+

U (1) ) 2

For the covered call (CC) returns:

E(R CC )

=

KU (0) + L(1) S0 - C0

-1

2 (R CC )

=

K 2U(0)

+

L(2) (S0

- (KU (0) - C0)2

+

L (1) ) 2

13

The functions U(n) and L(n) are called nth order upper and lower partial moments of the stock price and are equal to

L (n)

=

S0n

e

n

(µ+(n

-1)0.52

)T

N

ln

K S0

- µ +

(2n T

- 1)2 2

T

and

U(n)

ln S0

=

S0n

e

n (µ+(n-1)0.52

)T

N

K

+

µ

+

(2n T

- 1)2 2

T

where N() denotes the cumulative normal function.

APPENDIX 2 To obtain the instantaneous volatility of the (continuously compounded) returns of an optionbased portfolio, one needs to apply Ito's lemma to the value of the portfolio. For a protective put position, for instance, we obtain that its conditional volatility is equal to: (1 + P ) St St + Pt where P is the delta of the put option. Even when is constant, this expression will change over time, particularly when the option is at the money.

14

E(R)

Long stock

0.1 Covered call 0.09

0.08

0.07

0.06

Protective put

K 60 80 100 120 140 160 180 200

Figure 1: Expected returns for the three strategies

Volatility

Long stock

0.025 0.02

Covered call

0.015

0.01

0.005

Protective put

K 60 80 100 120 140 160 180 200

Figure 2: Expected volatility for the strategies

15

Sharpe ratio Long stock

0.3

0.25

0.2

0.15

0.1 Covered call 0.05

Protective put

K 80 100 120 140 160 180 200

Figure 3: SHARPE ratios for the strategies

16

Probability

protective put

long stock

Probability long stock

Return (%) covered call

Return (%) Figure 4: Log-return distribution of a protective put and a covered call Notes [1] See for instance Kruizenga [1964], Boness [1964], Rosett [1967], Malkiel and Quandt [1969] or Malkiel [1972]. [2] Other option strategies could easily be added here. However, these three strategies are the most popular among institutional investors. [3] The remark is still valid if the underlying asset log-returns are not normally distributed. 17

[4] Note that considering skewness in the performance measure does not solve anything. For instance, Lhabitant [1997] provides examples in which rational risk-averse investors select a portfolio with the lowest mean, highest volatility, and lowest skewness. Considering kurtosis will again only move us one step forward. All higher order moments should be considered. [5] See for instance Michell and Pulvino [2000], who evidence that that "risk arbitrage" strategy payoffs are similar to those obtained from writing an uncovered put option on the market. 18

FS Lhabitant

Human capital investment and poverty reduction over generations: A case from the rural Philippines, 1979-2003, 47 pages, 0.52 Mb

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