The Performance of Model Based Option Trading Strategies

Transcription

1 The Performance of Model Based Option Trading Strategies Bjørn Eraker Abstract This paper proposes a way to quantify expected profits and risk to model based trading strategies. A positive portfolio weight is assigned to assets which market prices exceed the price of a theoretical asset pricing model. The position is smaller, ceteris paribus, if the theoretical asset price is sensitive to model parameters subject to estimation uncertainty. Standard mean-variance analysis is used to construct optimal model based portfolio weights. In essence, these portfolio rules allow estimation risk, as well as price risk to be approximately hedged. The strategy is applied to S&P 500 index options. The model based hedging strategy generate Sharpe ratios close to one - twice that of simply writing options. The difference is still only marginally statistically significant. PRELIMINARY AND INCOMPLETE Please do not quote without the explicit consent of the author. Duke University, Department of Economics. bjorn.eraker@duke.edu. I wish to thank Tim Bollerslev and George Tauchen, as well as seminar participants in the Duke Financial Econometrics workshop, for helpful comments. The usual disclaimer applies. 1

2 1 Introduction Asset pricing models more often than not produce theoretical prices which deviate from prices observed in the market. These deviations, commonly termed pricing errors, can be given different interpretations. Most commonly, researchers look at the size of the errors to assess the relative success of the model. This reasonable strategy implicitly assumes that an underlying true model exist which would yield zero pricing errors if the econometrician could only specify the model correctly. This view rests on a strong assumption about market efficiency: There are no behavioral biases in asset prices, no market frictions such as costly trading. Investors are assumed equally well informed, they all have access to the true model and they know the values of the parameters of the true model. If, in addition, the market is assumed to be complete, deviations between model and market prices lead to arbitrage opportunities. There is, consequently, zero room for pricing errors in no-arbitrage asset pricing models. Thus, the smaller the pricing errors, the better the model. This paper takes a different view of pricing errors. We argue that the larger the pricing error, the more successful a model might be. This is because the potential profits to be made from trading on the deviations between a model and market price will increase in the size of the errors if the market is miss-pricing the assets, and the model is not. Conversely, the profits generated by trading on an incorrect model against correct market prices, will be negative. We argue accordingly that the success of a given model can be measured by its ability to produce (out-of-sample) profitable trading strategies. This view explicitly allows for the possibility that options markets are inefficient. By contrast, standard studies of the performance of option pricing models assumes that the model is indeed correct and the options market is frictionless. This joint hypothesis allow for a simple test of asset pricing models where the winning model is the model which produces the smallest pricing errors. By contrast, this paper studies the performance of simulated trading strategies formed based upon deviations between market and model prices. For a given pricing model with a known set of parameter values, we can proceed to compare its outof-sample trading potential by studying a portfolio of long positions in assets where market prices exceed model prices, and vise versa. Alternatively, and more reasonably, investment weights should be proportional to the size of the pricing errors, giving larger positions in assets with higher expected returns. This is reasonable when the parameters which enter the theoretical pricing model are known. This, unfortunately, is rarely the case, and model prices will have to be computed from estimated parameters. Since all statistically estimated parameters are subject to estimation error, it is clear that the portfolio decisions will deteriorate as estimation error increases. There- 2

3 fore, one should decrease positions in assets which theoretical prices are sensitive to estimation errors. Taking this logic one step further, this paper proposes a scheme for maximizing the expected profits while minimizing risk inherent in model prices. The procedure relies upon a Bayesian assessment of model price risk. Since model prices are functions of model parameters and state variable upon which posterior distribution functions can be estimated, either analytically or by simulation, we map the parameters/ state variables posteriors into posterior distributions on the pricing errors. These posteriors subsequently form the basis for portfolio allocation decisions. For example, using only the first two moment of posterior pricing error distributions give raise to meanvariance type allocations. The procedure is not, however, restricted to mean-variance preferences. The empirical analysis presented in this paper follows the performance of the model based portfolio rules using data on European index options trading on the S&P 500 index. Theoretical option pricing models represent an almost idealized testing ground for the model based portfolio rules because model prices can be constructed with or without conditioning upon options data. We will construct a horse race between two models, for which parameters will be estimated from data on the underlying stock index. Thus, model prices can arbitrarily deviate from observed options prices, as no attempt is made to match options data. A widely held belief is that the options markets are largely inefficient in that the reward/risk ratio to simple strategies seem too large to be consistent with equilibrium asset pricing. One such study, Coval & Shumway (2001) suggests that the returns to simply writing both put and call options earn a return of 3% a week without being subjected to systematic market risk. The proposed study will add to the findings of Coval & Shumway in some important ways: Firstly, this study uses data from while the CS study used data. Second, a careful examination of the daily options returns reveal that: Extrapolations of daily average arithmetic returns on options to annualized returns induces a huge bias in annualized return estimates. Daily return distributions are so heavily tailed that estimates of tail decay suggests that none of the moments of the return distribution exist. Therefore, the term Expected Option Returns is a misnomer altogether when returns are measured a short frequencies. This paper finds that the annual sharpe ratio from writing straddles is about 0.55 including transactions costs. This is about half of those reported in Coval & Shumway. While the strategies are constructed differently, the difference in these numbers is primarily explained by negative average returns to options writers for 3

4 sustained period comprising about half of the current sample. The Sharpe ratio obtained from writing options over the 1999 to 2003 period is The model based strategies yield a sharpe ratio of about one. While this is twice that of simply selling all options, we basically fail to reject a null hypothesis that the model based strategy outperforms the simple sell all strategy. This is primarily due to the large uncertainty associated with estimated mean returns to holding options, which yields a test statistic with low power. The problem is similar to testing for a differences in the mean rate of return in stocks - exceptionally long sample periods are required reject differences in expected return estimates which are within range found empirically relevant. This remainder of the paper is organized as follows. The next section discusses the construction of derivatives portfolios, discusses models to be considered, and discuss the computation of posterior expected utilities. Section three describes the data set, section four presents empirical evidence on the performance. Section five concludes. 2 Optimal Derivatives Portfolios The problem of choosing an optimal portfolio of derivatives is admittedly no different from the problem of choosing an optimal portfolio of securities. Traditionally, portfolio theory has been made in reference to stock markets, and the portfolio decisions are assumed to be based upon forecasts of mean return and return risk of those stocks in question. It is fair to say that the traditional portfolio theory has assumed that the investor possesses rather vague information about the future prospects of individual securities (stocks). Even if forecasting power is weak, portfolio diversification will enhance the signal to noise ratio in the forecasts. Hence, diversification may make up for noisy forecasts. Noisy forecasts are equivalent to uncertainty about the current value of the individual stocks. The derivatives literature is characterized by an almost exactly opposite problem: While the prices of individual equity shares are at best very uncertain, the academic literature pertaining to the pricing of financial options on the very same equity shares, is typically very precise. If we take model assumptions as given, then derivatives price models produces an model exact price. Whenever the model price differs from that of the market price, the a literary interpretation of the traditional derivatives suggest that the difference is an arbitrage opportunity. Hence, an investor who is using a the Black & Scholes model, who has infinitely strong belief in his estimate market volatility, would be willing to bet his whole fortune on a deviation between the model price and the market price. 4

5 A pragmatic investor would off course never enter such a bet. There are at least three immediate reasons why: First, the arbitrage argument requires the investor to maintain a continuous riskless hedge portfolio. Trading costs, market closures, among other things, make this trading strategy infeasible, however. Even if the riskless hedge existed, chances are great the the investor would be making a mistake relative to 1) the model itself, and 2) the parameter (volatility) for which an estimate is required. In the following, we advocate a portfolio selection procedure that explicitly incorporates trading costs, discreteness of trading time, and parameter and model uncertainty. 2.1 Predictive Distributions and Bayesian Portfolio Choice A Bayesian decision maker is naturally concerned with the estimation inherent in model parameters. There is a large literature on Bayesian portfolio choice in financial markets. Early contributions focussed on finding analytic solutions analogous to standard analytical results in the classic Markowitz theory. In the classical approach, expected utility is computed conditional upon knowing the true parameter, so if w is the vector of portfolio weights, one seeks to maximize E [U(w) Θ] = U(w)p(R Θ)dR, where R is a return, Θ is the model parameter. The Bayesian decision maker by contrast, seeks to explicitly incorporate parameter uncertainty. Let p(θ Y ) denote a posterior distribution on the parameter Θ. Then the Bayesian seeks to maximize E [U(w) Y ] = U(w)p(R Θ)p(Θ Y )drdθ. The expectation is taken with respect to the predictive distribution, p(r Y ) = p(r Θ)p(Θ Y ). More generally, in the case of derivative securities, we will also have to integrate over latent variables (such as volatility) to compute the predictive distribution. In order to make the discussion precise, I discuss some models for options prices next. 2.2 The SVCJ model The SVSCJ model fall under the general class of affine jump diffusion models considered by Duffie, Pan, and Singleton (2000). The acronym stands for Stochastic 5

6 Volatility with Correlated Jumps. In this model, the stock price, S t, is assumed to evolve according to the jump diffusion, ds t S t dv t = µ S (t)dt + V t dw S t + dj S t (1) = κ(θ V t )dt + σ V Vt dw V t + dj V t, (2) where jumps arrive with frequency constant frequency λ, and jump sizes distribute Z S t Z V t N(µ Q y + ρ J Z V t,σ 2 S) (3) Z V t exp(µ V ) (4) where V is the volatility process. The Brownian increments, dw S and dw V are correlated and E(dWt S dwt V ) = ρdt. The jump term, dj i t = Z i tdn i t, i = {S,V }, has a jump-size component Z t and a component given by a Poisson counting process N t. In the case of a common poison process, N V t = N S t, jump sizes Z S t and Z V t can be correlated. The specification of jump components determines the models to be considered. We discuss the two possible specifications next. Allowing for volatility jumps, this model generalizes models with state dependent price jumps considered by Bates (1996) and Pan (2001). A property of the previous models is that in high volatility regimes, the diffusive, Gaussian part of the system will tend to dominate the conditional distribution. As a result, as the spot volatility V t increases, short term option smiles will tend to flatten. The basic stochastic volatility (SV) model with a square root diffusion driving volatility, was initially proposed for option pricing purposes in Heston (1993). It obtains as a special case of the general model in this paper with jumps restricted to zero (dj S t = dj V t = 0). The volatility process, V t, captures serial correlation in volatility. The κ and θ parameters measure the speed of mean reversion and the (mean) level of volatility, respectively. σ is commonly referred to as the volatility of volatility. Higher values of σ V implies that the stock price distribution will have fatter tails. The correlation parameter, ρ, is typically found to be negative which implies that a fall in prices usually will be accompanied by an increase in volatility which is sometimes referred to as the leverage effect ( Black (1976)). A negative ρ implies that the conditional (on initial stock price, S t, and volatility, V t ) returns distribution is skewed to the left. 6

7 2.3 Computing Predictive Densities In the following, we describe a simple numerical scheme to estimate the predictive densities, or equivalently, a scheme that allows computation expectations of (utility) functions with respect to the predictive distribution. Let Θ (g),v (g) t denote posterior samples of the parameter vector and the latent volatility. It is assumed that such samples are available. A number of schemes involving Gibbs and Metropolis Hastings, possibly in conjunction, are available for this purpose. Assuming that the posterior density is formed based information, Y t as of time t, we wish to compute E [U(w) Y t ]. The sample mean Ê [U(w) Y t] = 1 G g U(w)g, where U(w) g denote the utility given the gth posterior draw Θ (g),v (g) t, is a simulation consistent estimator under weak regularity. 2.4 Filtering A final computational, statistical problem remains to be considered. As time progresses and market conditions change, we will typically like to re-balance the portfolio. Naturally, for any time t we can form a new posterior, and the procedure outlined above can be repeated. The computation of posterior densities by simulation is typically very time consuming and it can be useful to consider keeping the posterior for the parameter fixed, while updating the predictive density for the latent state variable, V t. This calls for solving a filtering problem, which is tantamount to designing an MCMC sampler that draws from p(y t V t, Θ)p(V t V t 1, Θ). where y t denote the t realization of derivatives data. Notice that for parameter estimation, we draw from p(y t V t, Θ)p(V t V t 1, Θ)p(V t+1 V t, Θ) Hence: We can solve the filtering problem by just removing one term from the expression used for the posterior simulation. Sampling from this density is a straightforward operation 1 and can be carried out using standard sampling schemes such as Metropolis Hastings. 1 The sampler outlined here solves essentially the same filtering problem as the algorithm known as the particle filter. Particle filtering nevertheless suffer from the unfortunate property that they will eventually reject all but one of the sampled values Θ (g) as t increases. The particle sampler, therefore, does not converge. 7

8 2.5 Optimal portfolios with Fixed Planning Horizon Having outlined how to deal with the posterior averaging, we now describe how to make portfolio decisions in the presence of trading costs. We are only considering trading costs in the form of bid/ask spreads. Other types of costs, such as brokers fees, typically only apply at the retail level. Investment firms with a reasonably large holdings face negligible additional costs. This paper is concerned with myopic investment decisions of an investor who has a fixed planning horizon. This is a simplification of the dynamic programming problem which leads to choices over portfolios and expected holding times. Let n long t and n short t denote vectors of portfolio holdings with the i th element represents the number of shares held of security i. The value of a portfolio is reasonably measured as the value obtained if positions were unloaded at prevailing market prices through market orders. So portfolio value is measure at long/bid and short/ask W t = n long t St bid + n short t St ask + K t (5) where K t is the cash holding. We can write the change in portfolio value, V t+1 = V t+1 V t, as W t+1 = n long t St+1 bid + n short t St+1 ask + K t r (6) where r represents a constant per period rate of interest. Define changes in portfolio holdings (i.e., trade) to be n t, and decompose it into shares bought n + t = n t I( n t > 0) and shares sold n t = n t I( n t < 0). The product of the n t with the indicator I() is element by element. The dynamics of the relevant state variables are W t+1 = n long t+1 St+1 bid + n short t+1 St+1 ask + K t+1, (7) K t+1 = (K t K)(1 + r), (8) n t+1 = n t + n t, (9) K = n + t S ask t Trading decisions are found by maximizing the expected utility + n t S bid t. (10) n t = argmaxe t [U(W t+τ )] s.t. n t S where the expectation is taken with respect to the predictive density for S t+τ, and S is a set of tradable securities. For example, in the empirical study below we allow trading only in securities for which there are valid bids and asks. Conditioning upon the possibility of missing values in this way is necessary because options contracts have a tendency to stop trading as the underlying price changes in such a way that 8

9 the contract become far in, or far out of the money. This is an overriding concern. There are other practical limitations, which also require special considerations. One prominent problem is finding a unique solution to the portfolio problem. This may be difficult if the covariance matrix of option payoffs is singular, or near singular. For example, theoretical prices of puts and calls with the same maturity and strike are perfectly collinear, giving a singularity. Two ways of dealing with this are, 1) each option is assumed measured with idiosyncratic noise, and 2) puts and calls are only traded only as singletons or in pairs, as in straddles. In particular, the empirical study is conducted using out-of-money puts and calls, as well as at-the-money straddles. Here we impose the straddle restriction if the quasi t ratio t = τ/v 0 (X/S 1) falls within a specified range R. Here the factor τ/v 0 is the τ period variance assuming that the volatility remains at its initial value, V 0, and X/S 1 is the moneyness. This study assumes that options must be bought and sold at market posted asks and bids respectively. As the bid-ask spreads are rather large, which gives downward biases estimates of sharpe ratios for investors who can transact inside the bid/ask spread. This is true of, for example, market makers. To see how decisions differ for an investor who can transact at the mid-point of the bid/ask, we simplify the system above assuming S bid t Returns are given as, = S ask t = S t, which gives W t+1 = (n t + n t )S t+1 + (K t n t S t )(1 + r). (11) W t+1 W t = (n t + n t )S t+1 + (K t n t S t )(1 + r) n t S t + K t. (12) In this case, an investor with mean variance preferences U(R) = E(R) 1 2 γvar(r) over returns, R, will optimally revise his portfolio according to the rule, n t = 1 γ Cov(S t+1) 1 (E t S t+1 S t (1 + r)) n t. The assumption of zero transactions costs is not justified in the empirical study considered here, as the transactions costs in the form of bid/ask spreads are significant Econometrics of Sharpe Ratios The empirical study below is exclusively focusing on Sharpe portfolio measures. Consistent with the Baysian view of data analysis, one might suggest constructing posterior distributions on the Sharpe ratios. This is not possible. The out-of-sample performance study is done while explicitly averaging over estimated parameters and latent variables. Indeed, the Sharpe ratios are formed by constructing returns data 9

10 from the predictive density p(r t R [0:t 1] ) which does not depend upon the parameters. Since it is impossible to construct posterior distributions on these performance measures, we appeal to standard sampling theory and construct frequentist standard errors on the Sharpe ratios 2. 3 Data This paper uses data from optionmetrics, made available through the Wharton Research Data Services, The data set consist of closing bids and asking prices for S&P 500 cash options collected daily between January 4, 1996 and February 28, There are a total of 167,551 put options and 117,303 call options for which there are simultaneously quoted bids and asks. In the analysis which follows, only options with both bids and asks posted are used. Table 1 presents some key numbers. The mean put price is 31.2 with an intrinsic value of 12.7, giving an average premium of The mean bid-ask spread is 1.32, some 7% of the average premium. For the calls, the bid-ask spread is about 5.6% of the premium. If we assume that the fair value of the option is the mid-point of the bid and the ask, a trader who holds a position until expiration would have to have to expect in excess of 3.5% return on a put just to break even. A trader who plans to sell before expiration would have to have an expected return exceeding the 7% spread to break even. These are obviously large numbers. They indicate that any investor who is issuing market orders are at a large disadvantage relative those taking the opposite sides of these bets. The Sharpe ratios computed below, therefore, should be interpreted as 2 This point was also noted by Johannes, Polson, and Stroud (2002) in the context of dynamic trading strategies for stocks. Table 1: Data Summary Puts Calls Mid bid-ask price Intrinsic value Moneyness (X/S) Maturity bid-ask spread

11 Table 2: Descriptive Statistics for Option Returns Puts Calls mean std min max skewness kurtosis lower bounds on attainable Sharpe ratios for investors who can transact inside the bid/ ask bounds, either because they issue their own limit orders, or because they are market makers. 4 Empirical Results We start the empirical exploration by re-examining evidence of consistent high returns to options writers. Table 2 reports descriptive statistics for the options data. The data on display are changes in the midpoint of the bid and the ask of the data. Price changes are measured both in percent of the previous days price (i.e., a return). The data are recorded whenever there is both a bid and an ask posted. Multi-day returns are recorded whenever consecutive bid or ask quotes are missing. As evidenced in the table, put options changed by an average of percent per period (typically one day). The corresponding number for calls is percent. It is hard to interpret these numbers because they do not in anyway extrapolate into long term performance of option writers/ buyers. For example, the standard practice of approximating the annual return by multiplying by 252, would seem to indicate that both a call buyer and a put writer would earn a staggering 200 percent annually. This is a standard way of interpolating from daily into multi-day returns. For example, Coval and Shumway report that An at-the-money call option tends to earn an average return between 1.85 percent and 2.00 percent per week, or between 96 and 110 percent on a annualized basis. Hence, the annualized average returns reported here are on level with Coval and Shumway s estimates. They, in fact, also report daily returns which are close to those of table 2. 11

12 15,000 15,000 10,000 10,000 5,000 5, Put returns Call returns Log tails of put returns Log tails of call returns 3 Figure 1: Density histograms and estimated log - tails for daily options returns. 12

13 4.1 The problem with arithmetic returns Unfortunately, the 200 percent annual returns are not real. The problem is the sum of two one day returns do not equal the two day return. For example, a 40% increase followed by a 40% decrease in an assets value gives an average return of 0, but leaves the investor with a 16% loss. Notice that a 40% daily percentage change in the value of a call is exactly what an investor should expect, as the daily standard deviation is If a an assets realizes consecutive returns of 1% and -1%, the actual two period return is about percent, and thus close to the the arithmetic average of zero. What these numbers illustrate is that the usual practice of multiplying a daily return by the number of trading days (252) to construct an annualized return, leads to an extremely upwardly biased annualized estimates for assets with high volatility. While this is true for regular stocks with high volatilities, the situation is extreme for stock options for which the daily volatility is in excess of 20 times the underlying index volatility, as well as extremely heavy tailed. Thus we conclude, the mean daily returns in table 2 do not necessarily predict high annualized returns to options traders. This does not mean that writing options is not profitable, it only means that the daily returns are poor measures of long term performance. 4.2 Do daily expected options return exist? Figure 4 reveals another problem with return distributions - an extreme right tail. For example, of the 124 thousand observations, there are 478 put returns larger than 159% - a five standard deviation event. The maximum put and call returns are 866% and 1,595% respectively. These are 27 and 40 standard deviation events. With tails this heavy, it is an open question whether any of the moments of the return distributions actually exist. To shed some light on this, assume that the tail of the return density follow a power law; p(r c ) ax b. If b is less than -2, the expectation does not exist since x ax 2 dx diverges. Figure 4 show the log-tails of the estimated frequency histograms. Linear regression estimates of b are (0.095) and (0.1) which produces t statistics equal to 6.2 and 2.12 for the null hypothesis that the b s are greater than -2. Thus we can reject the null that moments of the daily return distributions exist. 4.3 Performance of option writers Coval and Shumway also report monthly returns to a trading strategy consisting selling straddles at market prevailing bids. To offset the effects of occasional large drops, they augment the simple sell strategy with an offsetting position in an outof-the-money put, bought at market offered prices. The strategies yield a monthly 13

14 Table 3: Performance of Options Writers The table reports Sharpe ratios for portfolios of short options positions. Trading is done through market orders, buying at the asking prices and selling at bids. Bootstrapped standard errors in parenthesis. Puts Calls All sharpe st.err. (0.057) (0.058) (0.056) sharpe st.err. (0.121) (0.123) (0.122) sharpe st.err. (0.100) (0.091) (0.092) 14

15 return of about 3-4%, with a Sharpe ratio of about 0.3. The corresponding annualized numbers are 12-16%, and 1. These are on the order of twice the historically attainable Sharpe ratio from being long the underlying S&P 500 index. Table 3 present evidence of the annual Sharpe ratio earned by writing options in over the 1996 to 2003 period. The Sharpe ratio earned over the whole sample is This compares to about 0.2 for the S&P 500 over the same period. Thus, we conclude that writing options, although not as attractive as reported for the 1988 to 1996 period, still beats buying the S&P 500. The table also reveals that the reduction in the attainable Sharpe ratio from writing options over the whole sample, is likely due to relatively cheap options markets over the 1996 to 1999 period. The subsequent period again yield a Sharpe measure exceeding one. Figure 4.3 depicts the performance of options writers through four key time series. The bottom plot shows the portfolio value of an options writer. As can be seen, the option writer makes a positive profit on average, however, there is a long period of dismal returns characterizing the 96 to 99 period. The Sharpe ratio earned by writing options over this period is -0.5, as shown in table 3. The losses can perhaps naively be characterized as losses on call options which ended in-the-money during this period with a substantial increase in the value of the S&P 500. Another indicator of option valuation levels is given by the Black & Scholes implied volatilities. The second plot in figure 4.3 shows implied volatility of short dated options as well as spot volatility obtained by the filtering algorithm described above. The third plot shows the difference between these two. Casual observation from these graphs reveal a few interesting patterns: First, implied volatility is consistently higher than spot volatility. Second, during periods in which the difference between the volatilities (henceforth volatility spread ) increases, option writers tend to loose money. For example, over the period from Jan 1996 to about Sept 1998 (first vertical line in the graph), options writers are loosing money. Over the same period the spread is increasing from negative to about These numbers are annualized, and correspond to a difference of opinion of about 1.5% daily standard deviation. Put in perspective, this roughly means that the option market seems to believe that the daily standard deviation is about 2.5%, while filtered spot volatility is close to the historical average for the S&P 500 of about 1% daily. Obviously, these deviations are large and may suggest that the market at the time was overvaluing the options. The following period from Fall of 1998 to Spring 2001 proved very profitable for option writers, as evidenced by the bottom plot in figure 4.3. If we accept that the volatility spread is a rough measure of option overpricing, the size of the spread at the beginning of the period, as well as the fact that the spread shrunk from 0.25 to about 0.1, are two possible explanations for the large returns over this period (the Sharpe ratio is about 1.5). 15

16 Index level Filtered Volatility and Implied BS volatility x Difference in BS and filtered spot volatility Portfolio Value from option writing Figure 2: Time series of index levels, returns, volatility, and profits from writing options. 16

17 The vertical lines in the plot mark periods in which the option portfolio is loosing relatively much over a short period. During most of these periods, it is possible to make a visual link between the reduction in option portfolio values, and changes in either the volatility spread, or the stock price. For example, during the spring and fall of 2001, both the S&P index and the volatility spread dropped (increased) sharply, leading to sharp declines in the portfolio of short options. 4.4 Model Estimation Estimation of the SVSCJ model and the SV model has been discussed in length in Eraker, Johannes, and Polson (2003), and in Eraker (2004). Table 4 reproduces results from Eraker (2004) on summary statistics of MCMC samples from a posterior density formed based on daily returns data from 1970 to The results in table have previously been reported in Eraker (2004). They were obtained prior to the current study, and results which follows are therefore true out-of-sample results. The results in table 4 have been thoroughly discussed elsewhere, and readers are referred to the above mentioned papers for an interpretation of the estimates. 4.5 Performance of Hedging Strategies Consistent with the exposition section 2.5, we construct optimized portfolios and study the subsequent out-of-sample performance of the strategy. The portfolios are chosen under a few additional constraints, which were as follows: First, as mentioned above, there has to be both bids and asks posted. Second, the minimum maturity considered here is 12 trading days. Third, we only trade in at-the-money straddles, out-of-money puts, and out-of-money calls. This allows for three types of bets on the probability distribution of returns. OTM puts allow bets on the left tail, OTM calls allow bets on the right tail, while ATM straddles basically allow for volatility bets. This prior restriction is meant to reduce the dimensionality of the optimization problem, while presumably still spanning the space of possible bets. A second advantage is that we never consider offsetting positions in puts and calls with same strike. If we did, the covariance matrix of one-step ahead option prices would be singular, as put-call parity gives perfect negative correlation between puts and calls. Table 5 gives the results of the out-of-sample trading simulation with the full parameter and price/ volatility hedge, as well as only price/ volatility hedging. In constructing the latter, we keep the parameter vector Θ fixed at the posterior mean, rather than averaging. The price and volatility is simulated as with the full predictive density. The difference between the price/volatility hedge, and the parameter/price/volatility hedge serve to give the marginal improvement in the hedging 17

18 Table 4: Parameter Estimates The table reports posterior means and standard deviations (in parenthesis), and 99% credibility intervals (in square brackets) for parameters in the jump diffusion models based on returns data only. Parameter estimates were obtained using the estimation procedure in Eraker, Johannes, and Polson (2003) using 5307 time-series observations of the S&P 500 index from Jan 1970 to Dec SV SVYJ SVCJ a (0.011) (0.011) (0.011) [0.000, 0.051] [0.000, 0.050] [0.004, 0.056] θ (0.098) (0.122) (0.078) [0.692, 1.163] [0.590, 1.221] [0.397, 0.750] κ (0.005) (0.006) (0.003) [0.008, 0.030] [0.004, 0.022] [0.009, 0.023] ρ (0.056) (0.065) (0.073) [-0.500, ] [-0.601, ] [-0.616, ] σ V (0.011) (0.011) (0.012) [0.082, 0.137] [0.061, 0.104] [0.030, 0.078] µ y (2.486) (2.523) [ , 1.281] [ , 2.436] ρ J (1.459) [-3.580, 3.833] σ y (1.697) (1.272) [3.714, ] [2.880, 9.295] µ V (0.381) [0.681, 2.523] λ (0.001) (0.001) [0.001, 0.006] [0.001, 0.007] 18

19 Table 5: Model Based Investment and Hedging Performance The table reports realized Sharpe ratios for portfolios of options formed conditional upon either the stochastic volatility model (SV), or the Stochastic Volatility with Correlated Jumps (SVCJ) model. Portfolios maximize expected quadratic (meanvariance) utility. Sharpe ratios are compared using fixed parameters and initial volatility, versus hedging of the estimation uncertainty of both. Trading is done through market orders, buying at the asking prices and selling at bids. Bootstrapped standard errors in parenthesis. Panel A: Sharpe Ratios SV SVCJ Price and Volatility Hedging sharpe st. err. (0.056) (0.056) Price, volatility, and Parameter Hedging sharpe st. err. (0.057) (0.057) Panel B: t-statistics Parameter Hedging Model Comparison (SVCJ vs SV) Price/vol Hedge Full Hedge

20 Table 6: Significance of Model Based Strategies The table reports t statistics for tests of differences in Sharpe ratios generated by model based trading strategies vs. simply selling all options. SV SVCJ Price and Volatility Hedging Price, volatility, and Parameter Hedging performance due to parameter hedging. For the price/volatility hedge, we also keep the spot volatility fixed at its posterior mean. There is considerable variation in the filtered spot volatility, and short dated options are particularly sensitive to the spot volatility estimate. There are several noteworthy points to be made studying the numbers in table 5. First, the Sharpe ratios attained using only price and volatility hedging are and 1.07 for the SV and SVCJ models, respectively. This difference produces a t-statistic of 1.92 in favor of the SVCJ model. Thus, we conclude that if parameters are kept constant, the SVCJ model is preferable. The performance of the full hedging strategy involving the both price and volatility as well as the estimation uncertainty in the parameter and volatility are 0.94 and 0.96 for the respective models. In either case, we fail to reject the null hypothesis that hedging the estimation uncertainty produces significant improvements. Indeed, for the SVCJ model, the Sharpe ratio generated by the full hedging strategy is lower that when only the state variables are hedged. The difference could well be due to random fluctuations in the data as the test of a significant difference between the two fails to produce evidence against the null hypothesis. In a final consideration of the performance of the model based hedging strategies, we compare the numbers of table 5 to the basic sell all strategy in table 3. The results of these pairwise comparisons are presented in table 6. The t ratios presented 20

21 in this table test the null hypothesis that Sharpe ratios of the model based trading strategies are no larger than from simply writing options unconditionally. As can be seen from the table, the SVCJ model with price and volatility hedging produces evidence against the null hypothesis which is marginally significant at the 5% level using a one sided test (t = vs ). The results, however, in no way produce particularly strong evidence against the null hypothesis. The lack of power is to be expected given the relatively small sample size. 4.6 Can volatility and jump risk premia explain high returns? In this section we investigate whether appropriate adjustment for risk related to stochastic volatility and jumps can explain the large returns to options writers. In a classic definition of factor risk, a risk factor is said to be priced if an assets return s sensitivity to the risk factor (factor loading) determine the assets expected rate of return. A necessary condition for a factor to be priced, therefore, is that the asset return is correlated with the risk factor. In our case, we are first and foremost relevant to see if the stochastic volatility factor is priced. To investigate this in a rather reduced form fashion, we regress Wt/W i t, the return on trading strategy i, on change in spot volatility V t, and stock return, S/S. If none of the regression coefficients appear positive, we can conclude that price and volatility risk do not explain the large returns Market and Volatility Risk Evidence presenting the sensitivity of trading returns to S& P 500 returns and volatility changes is presented in table 7. These regressions should be interpreted in a Bayesian way, and the estimates represent posterior means of linear regression coefficients, averaged against the filtering distribution of spot volatility. As can be seen from the table, there is basically no relationship between the portfolio returns, and the candidate risk factors. Thus, we conclude, the large returns on these trading strategies are not due to exposure to systematic risk factors such as market risk or volatility risk Jump Risk We finally consider jump risks as the source of large options premias. There is really no way in which we can effectively argue that there does not exist a jump fear 21

22 Table 7: Trading Return Sensitivity to S&P 500 returns and Volatility The table reports posterior means and standard deviations for the regression of returns from trading strategies on S&P returns and volatility changes. Volatility factor loading incorporate first step estimation uncertainty. Const S&P return volatility changes A: Price and Volatility Hedging SV (0.0222) (1.8224) (0.1291) SVCJ (0.0077) (0.6337) (0.0449) B: Price, Volatility, and Parameter Hedging SV (0.0020) (0.1655) (0.0117) SVCJ (0.0092) (0.7575) (0.0537) 22

23 consistent with rational behavior which will explain the large premium to writing options. To make this argument precise, suppose we write down a partial equilibrium model where the pricing kernel ξ t follow the same exponential affine jump diffusion as the S t in equation (1). Let µ ξ y,λ ξ etc., denote the parameters which govern the dynamics of ξ. The dynamics of stock prices under the risk neutral measure will under these assumptions again be an affine jump diffusion within the same class. The jump frequency is λ Q = λ + λ ξ, the mean price jump is µ Q y = µ y + µ ξ y, among others. Thus, pricing kernel with the same exponential affine jump diffusion process as the stock price itself, may yield arbitrarily greater jump intensities and jump sizes under the risk neutral measure. Thus there is almost no way in which we can rule out the possibility that there exists a pricing kernel with dynamics which provide significant jump premiums. If such pricing kernel exist, then we must conclude that the premiums are consistent with rational behavior. [INCOMPLETE] 5 Conclusion This paper has studied the performance of model based trading strategies. There are two purposes to this. First, the this paper describes a normative scheme for constructing portfolios of derivative securities which maximize returns and minimize risk. Risk is given a wide interpretation, and include the standard sources of risk, including the possibility of changing prices of the underlying security, as well as changes in volatility. Schemes designed to hedge this changes in state variables such as stock price and stock price volatility, have been studied previously. In order to hedge for example volatility risk within the context of a stochastic volatility model, one need to take offsetting positions in other options. In practice, to sell an overpriced option, a minimum variance type hedging scheme calls for taking an offsetting position in another option. The profitability of the trading strategy relies on whether the hedge position can be put on without a loss of expected profit. This is so only if the hedge option is selling at fair value and there are no transactions costs. If both options are overvalued, it is no longer clear that selling the first and buying the second to hedge is profitable. Rather, we argue, the potential profit should be measured against the potential loss in a standard risk/ return tradeoff analysis. Thus, an optimal portfolio of derivative securities maximize the expected profit given as the difference between market and theoretical prices, while minimizing the risk of the position. While the risks related to state variables such as prices and volatility are well understood, this paper goes a step further and asks if it is possible to hedge estimation risks associated with parameters and latent variables, notably volatility, which enter the computation of theoretical prices. The optimal portfolio idea relayed above 23

24 extends straightforwardly to a situation in which uncertainty about parameters and latent variables can be expressed through Bayesian posterior simulations and simulation filters. In this case, expected utilities can be computed with respect to the Bayesian predictive densities which average out estimation uncertainty. A second main objective in this paper is to study the empirical performance of hedging schemes using historical options prices of S&P 500 options. Previous work has documented large premiums to writing options contracts over the 1988 to 1995 period. This suggest that options markets are not efficient, in that on average, buyers of options overpay relative to a reasonable, risk adjusted fair value. This paper presents empirical evidence which seem supportive of this view. Sharpe ratios earned by options writers are large, in particular when considering that there is little or no market risk associated with such positions. Other risk factors seem unlikely to explain the findings. It is hard to interpret the evidence presented in this paper as not being substantially at odds with risk based asset pricing theories. This warrants a final important comparison with previous evidence reported in the literature, among others by the current author, suggesting that by and large, models do a fine job of fitting option prices. In this paper, purposely, we did not try explicitly or implicitly to minimize option pricing errors. Papers such as Bates (2000), Pan (2002), and Eraker (2004) all try to match observed options prices as closely as possible. This is accomplished in two ways: First, parameters which determine the average prices (such as θ in the Heston model) of options can be increased to match the overall price level. Second, the estimated spot volatility can be increased accordingly. It is easy to justify the likelihood functions which give raise to a minimization of the pricing errors, as there simply is no way of identifying the model parameters from options data without this assumption. The assumption is effectively a joint hypothesis of 1) the model being true and pricing errors distribute according to a specified form, and 2) markets are efficient. This paper concludes that the second hypothesis may indeed be violated. The jump diffusion models in question do a fine job of capturing the dynamics of stock returns, as demonstrated in previous work by this author and in numerous other studies of similar models. Thus, assuming that the model is true, we should expect that the theoretical option prices closely mimic option market prices without any attempt to actually fit those data. This is a tougher test than a joint fitting exercise, because it is really out-of-sample. Evidence presented here suggest that the data do not pass this test, and we conclude accordingly that the level premiums observed in S&P 500 options, as well as intertemporal variations in these premiums, are hard to reconcile with rational asset pricing theory. 24

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