The Equilibrium Volatility Surface

Transcription

1 GMO WORKING PAPER The Equilibrium Volatility Surface Neil Constable, GMO 5/2/2013 It is shown that the existence of a non-zero equity risk premium combined with the assumption that all investors are compensated at the same rate for taking the same risks is sufficient to explain the basic features of the implied volatility surface observed in the real world market for equity index options. In particular, we are able to demonstrate that the implied-realized volatility gap, skew and term structure are a direct consequence of investors demanding an equity-like rate of return on each unit of capital that is exposed to the downside of the equity market. Further, this framework predicts that systematic sellers of collateralized at-the-money put options ought to receive the equity risk premium over the long run and we are able to confirm that this has in fact been the case over the entire history of the listed index option market for the S&P 500. An implication of this last result is that the equity risk premium and what is known as the variance risk premium are not independent quantities but are related to each other via put-call parity.

2 Introduction Equity holders occupy the lowest rung in any firm s capital structure and therefore have historically demanded to be paid a return over and above the risk-free rate as compensation for the risk that they are taking. Because the idiosyncratic risk of any individual company can be diversified away, the only risk that they can reasonably expect to get compensated for is the common market risk shared by all stocks. This return is often referred to as the Equity Risk Premium (ERP). For a recent comprehensive overview of the ERP see ref [1]. For our purposes here, we will assert that the ERP exists and that it is non-zero. We intend to address the following question: if equity options represent contingent claims on the future market value of corporate earnings and investors in equities demand the ERP as a return on their capital, then what role does this required rate of return play in the options market? In the literature on options and volatility a non-zero expected return, such as the ERP, would be dismissed as an irrelevant drift term in the stochastic process that governs the returns of the underlying asset. This is because one of the major insights of the Black-Scholes-Merton (BSM) model [2] of option pricing is that the arbitrage-free price of an option is independent of the expected return of the underlying security. An option price can depend only on the risk-free rate of return and the expected future level of volatility of the underlying security. The ERP, it would seem, can be safely ignored. Because there is no way to actually know what the future volatility of the market will be, the BSM model cannot be used to actually price options. Instead, the BSM model is typically used as a standardized price quoting mechanism by finding the level of future volatility that corresponds to the actual market price of an option. The implied volatility, so defined, serves much the same role in the options markets as the price-to-earnings ratio does for cash equities. In the BSM model, implied volatility should be a constant. Once it is determined for one option, it should be the same for all other options with the same underlying asset. However, since Black Monday in 1987, implied volatility has exhibited a strong dependence on strike and time to expiration that is not consistent with the BSM framework. Most dramatically, out-of-the-money (OTM) short-dated put options are much more expensive (have higher implied volatility) than can be accounted for in traditional approaches. This is the often referred to volatility skew or the volatility smile. Likewise, options with expiration dates far in the future typically trade at more expensive prices than is consistent with BSM. This is referred to as volatility term structure. Together, skew and term structure form what is known as the implied volatility surface. An example of the S&P 500 implied volatility surface from March 27, 2013 is plotted in Figure 1. The surface is parameterized by the time to expiration of each option ranging from one month to two years, and the logarithm of moneyness, which ranges from -10% to +10%.

3 Figure 1. The S&P 500 Implied Volatility Surface on March 27, 2013 (Source: Bloomberg) Prior to Black Monday, this surface had been essentially flat, with no particular structure for OTM puts or OTM calls. In other words, the S&P 500 options market was pricing options in a manner that was at least consistent with the BSM framework. The appearance of a non-trivial volatility surface immediately following the largest one-day fall on record is a very good clue as to what is going on. The BSM framework is predicated on the concept of building a portfolio today that, when held until the expiration of the option, will reproduce the option s terminal value. If such a replicating portfolio exists, then the principle of no-arbitrage requires the option and the replicating portfolio to have the same value today. The logic behind this is flawless. The difficulty lies in exactly what one means by a replicating portfolio. In the case of simpler derivatives such as futures contracts, the replicating portfolio requires one to borrow money at the risk-free rate and purchase the underlying index. Both the

4 loan and the index holdings are then locked away in a drawer until the expiration date of the futures contract. The futures replicating portfolio is static. In the case of options, the BSM replicating portfolio is dynamic and is better characterized as a replicating trading strategy, usually referred to as delta-hedging. Strictly speaking, delta hedging according to the BSM model requires continuous trading, which is not feasible in any real world situation. The best one can do is approximate continuous trading by adopting a discrete trading strategy based on a daily, hourly, or minute-by-minute trading cycle. Even during normal periods discrete dynamic hedging is known to be theoretically suboptimal [3]. However, events like the crash in 1987 or the Flash Crash in 2010 are another matter altogether. When such a rare event does occur, option traders cannot possibly adjust their hedges quickly enough. A trader who is short an at-the-money (ATM) put option and is following a BSM-style delta hedging program will initially be short half a unit of the index. He will, however, find little comfort in this hedge when the market gaps down by 20%, or more, on a single day. A delta hedged put seller will always be long the market at precisely the wrong time, that is to say, when it matters. It is our contention that since 1987 these investors have demanded to be paid a rate of return commensurate with the risks they are taking. In other words, real world options are priced based on a risky expectation, not the risk-free rate. Because these investors are taking equity risk, this risky rate of return should be set by the ERP. The main point of this paper is to show that the non-trivial features of the implied volatility surface, such as skew and term structure, can be understood as a direct consequence of a rate of return greater than the risk-free rate being priced into option markets. We will argue that this risky expectation is set by the ERP. The results have several broad implications. First, there are probably far fewer genuinely independent risk premia than is generally believed. At the very least the ERP and what is known as the variance risk premium, or VRP, are not really independent quantities. We will present evidence suggesting that the VRP is related to the ERP and the risk-free rate of return in a straightforward manner. Second, the concept of risk, as something that you get compensated for bearing, is really about downside exposure, not volatility, beta, VaR, or any other statistical measure. Third, markets are much more consistently priced than is typically acknowledged by investors: we are able to derive our results by adopting the viewpoint that if the ERP is fair compensation for taking equity risk, then every dollar of exposure to the downside of the equity market ought to receive the same rate of compensation as any other dollar of exposure no matter how that exposure is achieved. In other words the implied volatility surface is a result of all investors demanding the same expected return for the same risk. A useful example to keep in mind is that of a seller of fully collateralized ATM put options on an index such as the S&P 500. Such an investor is exposed to all of the downside of the equity

5 market and none of the upside, beyond the premium collected for selling the option. The put seller has invested the same amount of capital (via his collateral) and shares all of the same risks as someone who is simply long the index. The owner of the index expects to receive the equity risk premium. Because the put seller will experience the same outcome as the pure equity investor in any market that is down between the time he establishes his position and the expiration date, he should presumably also expect to receive at least the ERP. We will find below that empirically put sellers have done exactly that, and no more. The outline of the paper is as follows. First, we will briefly review the BSM framework and its shortcomings in dealing with various features of the real world options markets. Next, we will show how put-call parity can be combined with the ERP and the BSM framework to derive the known structure of the volatility surface. Finally, our work suggests that a systematic seller of one-month ATM put options on an index such as the S&P 500 should, over time, receive the ERP as compensation, and we show via an empirical study that this has indeed been the case over the entire life of the listed index options market in the U.S. Implied Volatility in the Black-Scholes-Merton World The major drawback of the BSM model is that the future volatility of the index is not observable, or even knowable. Thus, there is no way to use the model to calculate the real market price of an option. As a result, the BSM model is usually used to convert real market prices for options into implied volatilities. These are nothing more than the level of expected future volatility that would be required for the BSM model to produce the observed market price of an option. If we take C M (K, T t) to be the market price of a call option on an equity index with price S(t),struck at K and expiring at time T, then we can write, C M (K, T t) = C BSM (S(t), K, σ I, r f, q, T t) Here C BSM ( ) is the BSM formula for the value of a European call option. σ I is the implied volatility required for this formula to produce the market price of the call option, r f is the riskfree rate of return, and q is the dividend yield. (The explicit formula for the BSM model is provided in the Appendix.) Empirically, it has been found that the BSM implied volatility is, on average, systematically higher than the subsequent realized volatility, σ R, of the index. It would seem as though the options market consistently over estimates the future volatility. The difference between the two is referred to as the implied-to-realized gap, or g σ I σ R. Another empirical fact is that despite volatility being treated as constant in the BSM model, implied volatility varies with both strike and time to expiration. Thus, the gap is observed to not be constant, but is a nontrivial function of the parameters that define the option in question:

6 g = g(k, T t) = σ I (K, T t) σ R This function of strike and time to expiration is known as the implied volatility surface. For each option with a strike K and time to expiration T-t, there is a unique value of implied volatility for which the BSM model yields the observed market price. Referring to our example of a seller of a fully collateralized ATM put option, the question about implied volatility should not be what value of implied volatility allows one to recover the market price of an option from the BSM framework? Rather, the question should be what level of implied volatility, and hence market price for the option, is required so that in expectation the put seller will get the ERP? As we will show, if participants in the options market expect to get paid an equity-like rate of return on the portion of their capital that is exposed to downside equity market risk, then they will price options in such a way as to generate a non-trivial implied volatility surface that has both skew and term structure reminiscent of the real world. The Equity Risk Premium and the Volatility Surface In order to get started, we recall a very basic fact about (European-style) options. At all times market prices for options must be consistent with put-call parity 1 : S(t) = C M (K, T t) P M (K, T t) + Ke r f (T t) Here, S(t) is the price of an equity index at time t and T is assumed to be some later time corresponding to the expiration date of the options. Put-call parity says that it is completely equivalent to own either a portfolio made up of one unit of the index itself, or to invest in a portfolio that is long one call option struck at K, short one put option struck at K, and long $K of a risk-free asset. The realized returns to both portfolios must be identical between the time of purchase and the expiration of the options. This basic relationship will drive all of our analysis below. The statement that there is a non-zero ERP is equivalent to saying that, in equilibrium, there is a non-zero expected excess return for the equity index. The expected price of the index at time T is therefore, <S(T) >= S(t)e μ(t t) Here μ = ERP + r f is the instantaneous total rate of return of the index expressed in annualized units and angled brackets are being used to represent the expectation value of a 1 For simplicity we assume here that the index does not pay a dividend.

7 quantity at a fixed time. Throughout this paper all quantities will be assumed to be annualized unless otherwise stated. The expected profit, ρ, coming from owning the index is therefore, < ρ(t) > < S(T) > S(t) = S(t)(e μt 1) Using put-call parity, an alternate expression for the expected profit can be written formally as, < ρ(t) > = < C(0, K) > C M (T t, K) + P M (T t, K) < P(0, K) > +K 1 e r f (T t) In this equation the expectation value, < C(0, K) >, represents the expected value, at expiration, of a call option on the index. The expectation value here is calculated under the assumption that at time T the price of the index is log-normally distributed with a mean value of (μ σ R 2 )(T t) and standard deviation σ 2 R T t. For our purposes σ R is assumed to be some estimate of the future realized volatility of the index such as the trailing realized volatility. Note however that μ and σ R could in principle be any forecasted return and level of future volatility. As shown in the appendix, the expected profit from buying (selling) a call (put) option on an index that does not pay a dividend can be written entirely in terms of BSM-like formulae as follows [4,5], < ρ C (K, T t, g) > < C(0, K) > C M (K, T t) = C BSM S(t)e μ(t t), K, σ R, 0,0, T t C BSM (S(t), K, σ R + g, r f, 0, T t) < ρ P (K, T t, g) > < P(0, K) > +P M (K, T t) = P BSM S(t)e μ(t t), K, σ R, 0,0, T t + P BSM (S(t), K, σ R + g, r f, 0, T t) Here, we have written the implied volatility as the expected realized volatility plus the expected implied-to-realized gap. Note that the gap, g = g(k, T t), is implicitly a function of both strike and time to expiry. The expected implied-realized gap is also often referred to as the variance risk premium or VRP. We will have more to say about the ultimate source of this risk premium in the following sections. We can now explore the implications of our claim that the expected profit on any investment ought to be proportional to the amount of downside risk being taken. For strikes that match or exceed the current price of the index, buying call options has no downside equity market risk and so should generate no return. However, selling a fully collateralized put struck at or above the current index price is fully exposed to the entire downside of the market and should therefore earn an equity rate of return. Meanwhile, buying call options with strikes below the spot price of the index exposes the investor to the downside between the current price and the

8 strike. Likewise, selling put options at lower strikes has downside risk, but less than an ATM put and so should receive an equity return only on the capital required to collateralize the option. With this in mind, for K S(t) we decompose the put-call parity expected profit relationship into separate call and put pieces as: While for K < S(t) we have, < ρ C (K, T t, g) > + K S(t) 1 e r f (T t) = 0 < ρ P (K, T t, g) > +S(t) 1 e r f (T t) = S(t)(e μ(t t) 1) < ρ C (K, T t, g) >= (S(t) K)(e μ(t t) 1) < ρ P (K, T t, g) > + K 1 e r f (T t) = K e μ(t t) 1 In each regime the sum of the two equations is the expected profit from holding the index, as required by put-call parity. What we have done is break up the put-call parity relationship into independent put and call portfolios and required that the short put portfolio be fully collateralized. The claim we are making is that in each regime the put and call equations should be separately true. In other words, put-call parity should be satisfied in a very specific way. If each component is individually satisfied, then put-call parity is also satisfied provided that the solutions are consistent. What we will find is that for each strike and expiration there is a single value of implied volatility that simultaneously satisfies both the put and call equations. We will now solve for the expected implied-to-realized gap that is consistent with the equations above. In other words, we are asking, what is the required level of implied-to-realized gap in each regime such that a put seller and a call buyer have the appropriate level of expected return to compensate them for the equity risk they are taking? We will call this the equilibrium implied volatility. In order to invert the above equations one must revert to a numerical procedure, however the calculation is straightforward as it only involves Gaussian integrals. For any given level of expected future volatility we perform this calculation for a range of strikes and times to expiration and extract the required value of g, which is then added back to the expected level of volatility to generate a surface of implied volatility. As a matter of consistency we note that one obtains the same implied volatility surface from the puts and calls separately. In other words, the implied volatilities derived in this way also satisfy put-call parity. As shown in Figure 2, the equilibrium implied volatility surface has all of the generic features of the actual implied volatility surface for equities. The surface has been parameterized by time to expiration, measured in years and the logarithm of moneyness. Skew and term structure are

9 both present and are reasonably similar to that actually observed in the market. In particular, we find a steep skew for short-dated OTM puts and a mildly upward sloping skew for deep OTM calls. Figure 2. The Equilibrium Volatility Surface In the example shown, we have assumed a baseline level of expected realized market volatility of 8%, an ERP of 5%, and a risk-free rate of 0.5%. These were chosen to be roughly in line with the prevailing conditions on March 27, 2013 so as to allow for a reasonable comparison with the actual volatility surface on that date shown in Figure 1. The similarity is obvious, but there are some notable differences. Specifically, 1-month skew is steeper in the equilibrium model presented here, however it is within the upper end of the range of observed volatility skews on other dates. We should note that in the K < S(t) regime we have assumed that a fully collateralized short put portfolio should earn the full equity rate of return even though it is not exposed to the downside of the market between the current price of the index and the strike. The results here should therefore be interpreted as an upper bound on the implied volatility of OTM put options. In most circumstances the market is likely to only demand a fraction of the ERP as

10 compensation for exposure to a limited amount of the downside of the return distribution. This can be accommodated in the framework we have set out here by generalizing the decomposition of put-call parity above for K < S(t) as follows: < ρ C (K, T t, g) >= S(t) e μ(t t) 1 K e η(t t) 1 < ρ P (K, T t, g) > + K 1 e r f (T t) = K e η(t t) 1 Here, η = η(k, T t) is a rate of return that is dependent on time horizon and strike in such a way that η(k < S, T t) μ and η(k S, T t) = μ. This function represents the rate of return that fully collateralized OTM put sellers would be entitled to due to the fact that they are only exposed to large downside moves in the equity market, i.e., gap risk. There is no obvious choice for what functional form η should take, however it can always be inferred from the actual market prices of OTM put options at any given point in time. What should be clear is that any non-trivial form of η corresponds to a market that prices options as risky assets and as a result will generate skew and term structure. Further, choosing η = 0 leads to a volatility surface that is missing important structure such as skew for OTM put options and is therefore inconsistent with the real world. It may be tempting to think of η as having an independent existence from the ERP, however we would argue that this is unnecessary because it ought to be the same as μ for ATM/ITM puts as we will see in an empirical study below. Finally, the exact shape of a real implied volatility surface, such as that for the S&P 500, fluctuates through time and is anything but static. This is no different than saying that the market s estimate of the ERP is in constant flux, as is its estimate of future realized volatility. Forecasting these variables is not made any easier using the framework presented here. Indeed, this in no way makes modeling the dynamics of the volatility surface more tractable. For that, stochastic volatility models such as that of Heston [6] or the many variants on the jump-diffusion model are required. For a discussion of these and other techniques see Gatheral s excellent book on the subject [7]. Our results do, however, explain where skew and term structure come from in the first place: investors demand equal rates of return for taking equivalent risks. Once one accepts this basic premise, then the existence of the volatility surface is guaranteed by the fact that equity investors seem to demand a non-zero excess return for the risks that they are taking. The Equity Risk Premium and the Variance Risk Premium We will now consider a special case of the analysis above, and in doing so we can connect with real world option trading strategies to see whetherour findings are consistent with empirical results from the S&P 500 option market. We are going to examine a strategy that systematically sells fully collateralized one-month ATM put options as well as a strategy that systematically

11 buys one-month ATM call options on the S&P 500 index. Because a seller of an ATM put option bears all of the downside risk of the equity market, we expect that such an investor would receive the ERP for his efforts. Likewise, an ATM call buyer bears no risk and should receive no excess return. We start by returning to our decomposition of the put-call parity relationship. Restricting to the case K = S(t), our call and put equations become < ρ C (K, T t, g) >= 0 < ρ P (K, T t, g) > S(t)(μ r f )(T t) Here we have assumed a short time horizon in the second equation. This suggests that the long ATM call option makes no contribution to the index return while the fully collateralized short put option generates a return on invested capital given by μ r f = ERP. Thus, a prediction of our claim that the ERP is compensation for bearing downside market risk is that a fully collateralized ATM put seller should receive the full ERP while ATM call buyers should receive nothing. We can, in fact, test whether this has been borne out in reality. Using data on the listed options market for the S&P 500 we can perform the attribution analysis suggested by the put-call parity relation for ATM options. Our analysis covers the time period March 1983 through October 2012 and uses data for listed options on the S&P Our trading strategy is as follows: two days prior to each monthly option expiration date we close out our existing positions and we form a new portfolio that is long a one-month ATM call and short a one-month ATM put. In addition, we hold cash, which is assumed to earn three-month USD LIBOR, and is sufficient to fully collateralize the put option. The questions we are trying to answer are how much of the S&P 500 return can be attributed to the collateralized short put option and how much to the long call option? Figure 3 shows the cumulative results of this study, both including and excluding transaction costs. All returns are presented as returns in excess of three-month USD LIBOR. The first thing to notice is that when transaction costs 3 are excluded put-call parity holds to a very good approximation, even over 30 years of compounding returns. When we include transaction costs this, of course, is not the case. The second, and more interesting, observation is that since 1983 all of the return of the S&P 500 has come from the collateralized short put options. In other words, an investor would have been able to realize the full return of the market (i.e., the ERP) by only taking on exposure to the downside. Meanwhile, a systematic 2 For the period 1983 through 1995 we use daily data for options on S&P 500 futures sourced from the CME. Beginning in 1996 our primary data source switches to the Option Metrics Ivy database, which has daily data for S&P 500 (SPX) index options. 3 Transaction costs comprise both crossing the full bid/ask spread, which we have in our data set beginning in 1996, and our estimates of actual commissions paid to a broker for executing each trade.

12 buyer of ATM call options has received nothing over this lengthy period. These results are entirely consistent with our expectations based on the assumption that the ERP is compensation for bearing downside market risk. 16 Value of $1 Invested (Log Scale) S&P 500 Puts (T-Costs) Puts (No Costs) Calls (T-Costs) Calls (No Costs) Figure 3. Excess Returns: Short Puts, Long Calls, and the Market Note: The performance relating to Puts and Calls represents the performance that would have been obtained by selling puts or buying calls, respectively, solely on the S&P 500. All performance is presented ex-cash. For purposes of removing the cash component from performance, cash rates are assumed to be 3 month USD LIBOR. Those lines that represent included costs assume transactions costs and bid/ask spreads. Those lines that represent no costs exclude transactions costs and bid/ask spreads. Option strike is that closest listed option available to ATM on day of sale. Positions are rolled 2 days before expiration. Options are all 1-month options or durations close to 1 month. Benchmark is a total return index. All returns are in USD. Returning to our theoretical framework can help shed some further light on this result. It is shown in the appendix that the expected profit for the long calls and short puts can be approximated for short time horizons, such as one-month, as follows: μ(t t) < ρ C (K, T t, g) > S(t) g(t t) T t 2 2π μ(t t) < ρ P (K, T t, g) > S(t) + g(t t) T t 2 2π

13 Here we have used the notation g(t t) as shorthand for the ATM expected implied-realized gap, also known as the variance risk premium. First, examining the expression for the expected profit of the short put option, we see that the return comes from two sources. The first term is half of the expected total return of the index, as should be expected from an option position with a delta of 0.5. The second term comes from the expected implied-to-realized gap, or VRP. This second source of return is often thought of as an independent risk premium, however, because the expected profit of the call option is zero, we can infer from these expressions that the expected implied-to-realized gap is simply related to the ERP and the risk-free rate of return. This is a direct consequence of put-call parity and the result that fully collateralized put sellers receive, in expectation, the ERP. In other words, the so-called VRP does not seem to be an independent concept as has been suggested elsewhere, see for example [8]. One interpretation of these results is that the market has set the VRP so that an investor who takes 100% of the downside market risk gets compensated with the ERP while an investor who takes no downside risk gets nothing. Conclusion The results in this paper hinge on three relatively straightforward notions. First, investors in equities demand a return on the capital they are putting at risk when providing unsecured financing to corporations. Second, all investors demand the same rate of return on capital that is exposed to the downside of the equity market no matter how that exposure is structured. Third, put-call parity must hold at all times. The implied volatility surface is therefore the result of a market that, in aggregate, manages to price all equivalent risks consistently. An implication of these results is that the ERP and the VRP are probably not separate concepts. The VRP is the avatar of the ERP in the BSM view of the world: the VRP arises because exposure to the downside of the equity market attained via the options market needs to be compensated with the ERP. Finally, we hope that this work helps put to rest the notion that statistical measures of risk such as trailing volatility, beta, value-at-risk, etc., are what investors are compensated for bearing. After all, systematic buyers of ATM call options have endured plenty of volatility and beta, but they have received no return for their efforts. These investors are simply not taking any downside risk and are entitled to no compensation as a result.

14 Acknowledgments I would like to thank my many colleagues at GMO who have contributed much to this paper. Melanie Rudoy, David Cowan, Sam Wilderman, Doug Francis, Martin Tarlie, Ben Inker, James Montier, Tom Cooper, Justin Klosek, Edmund Bellord, Nick Nanda, and Kai Wu have all been instrumental in helping me think about the theory and, most importantly, the real world uses of options.

15 Appendix: the expected return of put and call options In this appendix we will derive some of the formulae used in the main text. First, we will calculate the expected value of a put option that expires at time T under the assumption that the underlying index, with starting value S(t), has an expected continuously compounded rate of return given by μ and an expected realized volatility of σ. Following Figelman s derivation and notation [4] we have, < P(0, K) > = max(k S(t)e Y T, 0) f Y T ; μ σ2 2 (T t), σ2 (T t) dy T Here, f(x; m, v) is the Gaussian probability density function with mean m and variance v. In other words, we are calculating the expected value of the put option s payoff function under the assumption that the distribution of future prices for the index is log-normally distributed. Noting that the integrand vanishes when ln K S(t) < Y T, this formula can easily be rearranged to give the following expression for the expected value of the put: < P(0, K) > = KN( d 4 ) S(t)e μ(t t) N( d 3 ) log S(t) σ2 K + μ + 2 (T t) d 3 = σ T t d 4 = d 3 σ T t This is a specific instance of Rubinstein s formula for the expected value of a put option [5]. In these expressions N(x) is the standardized cumulative normal distribution function. Likewise, the expected value of a call option at expiration is given by, < C(0, K) > = S(t)e μ(t t) N(d 3 ) KN(d 4 )

16 The similarity of these expressions with the BSM formulae for the price of put and call options is not an accident. Indeed, we can write the market value of a call option in the standard BSM framework as, C M (T t, K) = C BSM S(t), K, σ I, r f, q, T t = S(t)e q(t t) N(d 1 ) Ke r f (T t) N(d 2 ) d 1 = log S(t) K + r f q + σ 2 I 2 (T t) σ I T t d 2 = d 1 σ I T t Where σ I is the appropriate implied volatility. We can obviously make a similar identification for the market value of a put option. By comparison with the above, we can see that the expected value of call and put options at expiration (t=t) is simply given by, < C(0, K) > = C BSM (S(t)e μt, K, σ, 0,0, T t) < P(0, K) > = P BSM (S(t)e μt, K, σ, 0,0, T t) Using these formulae, we can now derive the main results from the body of this paper. The expected profit from buying a call option and selling a put option are respectively: < C(0, K, g) > C BSM S(t), K, σ I, r f, q, T t = C BSM S(t)e μ(t t), K, σ R, 0,0, T t C BSM (S(t), K, σ R + g, r f, q, T t) < P(0, K, g) > + P BSM S(t), K, σ I, r f, q, T t = P BSM S(t)e μ(t t), K, σ R, 0,0, T t + P BSM (S(t), K, σ R + g, r f, q, T t) This is the result that was used to derive the volatility surface in the main sections of this paper. Note that we have made explicit the dependence of these expectation values on the implied-torealized volatility gap, g.

17 Finally, setting K= S(t) and expanding the right-hand side of each of these equations in a Taylor series to first order in (T-t) gives the following result: μ(t t) < C(0, K, g) > C BSM S(t), K, σ I, r f, q, T t 2 μ(t t) < P(0, K, g) > + P BSM S(t), K, σ I, r f, q, T t 2 g(t t) T t S(t) 2π + g(t t) T t S(t) 2π Here g(t t) = σ I (T t) σ R. This is the result used in the main section of the paper.

18 References [1] Hammond, P. B. (Jr.), Leibowitz, M.L., Siegel, L.B. (Eds.), Rethinking the Equity Risk Premium, CFA Institute Research Foundation, December [2] Black, Fischer; Myron Scholes (1973). "The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81 (3): ; Merton, Robert C. (1973). "Theory of Rational Option Pricing," Bell Journal of Economics and Management Science (The RAND Corporation) 4 (1): [3] Derman, E. When You Cannot Hedge Continuously, The Corrections of Black-Scholes (RISK, 12-1 January 1999, pp ). [4] Figelman, Ilya, Expected Return and Risk of Covered Call Strategies, Journal of Portfolio Management, Summer 2008: [5] Rubinstein, Mark, A Simple Formula for the Expected Rate of Return of an Option Over a Finite Holding Period, Journal of Finance, 39, December 1984: [6] Heston, S. A Closed Form Solution for Options with Stochastic Volatility, with Application to Bond and Currency Options, Review of Financial Studies 6, [7] Gatheral, Jim, The Volatility Surface: A Practitioners Guide, Wiley Finance, [8] Carr, Peter; Liuren Wu, Variance Risk Premia, Review of Financial Studies 22,