Relationship between the implied volatility surfaces of the S&P 500 and the VIX

Transcription

1 Relationship between the implied volatility surfaces of the S&P 500 and the VIX Faculty of Economics and Business, University of Amsterdam Financial Econometrics MSc. Thesis By: Olivier Go, Supervisor: Prof. Peter Boswijk May 18, 2016 Abstract This thesis investigates the relationship between the volatility surfaces of the VIX and the S&P500. Historical data on the VIX June 2015 expiry is used to compare market data to expected analytical results. Furthermore, the Heston model is calibrated and fitted to the S&P 500 options data. Using the fitted parameters, the Heston model is simulated to see how well this model performs in pricing S&P and VIX options. It turns out that there is a strong correlation between changes in the S&P Skew and changes in the VIX implied volatility. The Heston model appears to be better at adjusting prices to these Skew changes. A new approximation for the VIX ATM volatility is tested. The relationship between S&P 500 skew and VIX ATM volatility is not significant in this new approximation. There is however a significant relationship between the S&P 500 ATM volatility and this VIX proxy.

2 Contents 1 Introduction 1 2 Theoretical explanation Basics of Option Theory Black-Scholes Formula Characteristics of Volatility VIX index Derivation of VIX pricing formula Model Heston Model Closed form solution Approach Data Option Data Index Data Results Analysis of Market Data Calculating skew VIX response to the S&P Monte Carlo Simulation with Calibration Response to changes in the Skew Comparing simulated data to market data Conclusion 43 A Appendix A 48

3 1 Introduction Ever since Black and Scholes (1973) published their paper on option pricing, their model has been the leading model for the valuation of options. The reasons for the popularity of this model are not hard to find. They managed to transform a few simplifying assumptions into a convenient pricing formula. Assuming that a stock price follows a geometric Brownian motion in a world of constant interest rates, constant dividend yields and no transaction costs or arbitrage opportunities, they came up with a formula to calculate the price of any standard option. Their model has two important implications. First of all, the value of a derivative depends only on the interest rate, dividend yield, stock price, strike price, time to maturity and volatility, which is the standard deviation of the returns. Given these input variables, the value of an option is determined based on the assumption that stock returns are lognormally distributed. This follows automatically from the assumption that the stock price follows a geometric Brownian motion. Taking a closer look at the variables of the Black-Scholes (BS) formula, the volatility is a key variable. In the BS formula, there are 6 variables determining the value of an option. The time to maturity and the strike price are known for any option. In this thesis, only European options will be considered, which means that the options cannot be exercised before their expiration date. Also, we will only look at monthly index options. These options expire on the third Friday of a month. This means that the time to maturity is fixed for these options. The current value of the index, dividend yield and the interest rate can vary a lot, but at any given time, these are known input variables as well. So in valuing options, these can be considered constants as well. This leaves only the volatility as an unspecified variable of the price of an option. Since there is no observed measure for volatility, a few different types of volatility have been proposed. The most straightforward is the historical volatility. The standard deviation of historical returns defines the historical volatility. Another way to calculate volatility is implied volatility. Calculating 1

4 implied volatility can be seen as inverting the Black-Scholes formula. Instead of using the volatility to calculate an option price, we take the current option price in the market and calculate which volatility corresponds to this option price. This is a nice way of calculating the volatility, since using the midpoint between the bid and ask price in the market, we have a value for the volatility that is accepted by the market. The obvious downside of this approach is that it is very model dependent. Using different models could lead to a different volatility. Thirdly, one could calculate the spot volatility. This is the square root of the spot variance. This will be defined more thoroughly in Section (3.1). A fourth possibility is to use a stochastic volatility model. That way, we may assume the volatility follows a stochastic process as well. The most common stochastic volatility model is the Heston model (Heston, 1993) The fact that there is no consensus on the best way to calculate volatility led Whaley (1993) to suggest the creation of volatility derivatives. And not long after that, the Chicago Board Options Exchange (CBOE) introduced the Volatility index in 1993 for two reasons. The first was to offer a benchmark of short term volatility. Secondly, the purpose of the VIX was to offer an index on which volatility options and futures could be written. The VIX index is a measure of the 30-day implied volatility of the S&P 500. VIX futures started trading in 2004, followed by options in 2006 (Whaley, 2008). The current S&P option prices are used to estimate the expected implied volatility over the next 30 days. In order to properly trade options on the S&P 500 and the VIX, it is very useful to know how the volatility of these products behave. Since the VIX index is a measure of the implied volatility of the S&P 500 options, these products are very much related. It is however interesting to see how the VIX options are affected by changes in the S&P 500. This thesis aims to find an answer to this problem by trying to answer the question: What is the relationship between the implied volatility surfaces of the S&P 500 and the VIX?. If we know the relationship between these two volatility surfaces, we have a better overview of how the volatility in these products affect each other 2

5 and we might be able to go forward from this research to devise a model to ensure the VIX and S&P 500 can be properly traded. The analysis will be carried out in two ways. First of all, an analytical approach will be applied on high-frequency historical market data. We can use the definition of the VIX and some characteristics of volatility to calculate how these volatility surfaces are theoretically connected. Aside from this, a Heston model will be simulated to calculate VIX option prices. The model will be calibrated on S&P 500 option prices to minimize the error between the market price and the price according to the Heston model. This thesis is organised as follows. Section (2) will have a closer look at the appropriate theory. Section (3) will take a closer look at stochastic volatility models. Section (4) focuses on the data available. In the next section, the empirical analysis is carried out. Finally, Section (6) summarizes and concludes. 2 Theoretical explanation The first part of this section will consist of some elementary option terminology. This part can be skipped by people who are familiar with derivatives. Section (2.2) has a closer look at the Black-Scholes formula. After that, some characteristics of volatility will be explained. In the last subsection, the VIX index will be more thoroughly explored. 2.1 Basics of Option Theory There are two main types of options, put and call options. A call option gives the owner the right to buy a unit of the underlying (in the cases considered here, the underlying will be an index) at a specific time for a specific price. For example, the AEX September 490 call option gives the owner the right to buy the AEX index on the third Friday of September at a value of 490, regardless of the actual value of the AEX. It is clear that the payoff of a call option (disregarding time value and transaction costs) is equal to max {S T K, 0}, 3

6 where S T is the value of the index at time T and K is the strike price, which was 490 in our example. If the strike price is below the current price, exercising a call option will lead to a profit. A put option gives the owner the right to sell a unit of the underlying at a specific time and strike price. Therefore, a profit is made if the value of the underlying is below the strike price. (Hull, 2006) Puts and calls can be either American or European. A European option can only be exercised at the expiration date. American options have the additional benefit of allowing the opportunity to exercise early. Since most index options are European style, the scope of this thesis will be on European options. An option can be either in-the-money (ITM), at-the-money (ATM) or outof-the-money (OTM). When an option is in-the-money, the intrinsic value is greater than 0. For example, a call option with strike price 500 is in the money if the value of the underlying is greater than 500. The option is at-the-money if the underlying is valued at 500. The call option is out-of-the-money if the strike price is higher than the underlying. This thesis will focus mainly on the S&P 500; an index on the 500 largest companies in the United States and one of the most widely used indicators of the American economy. This index is the sum of the total float-adjusted market capitalization of the largest 500 U.S. based firms. To calculate the market capitalization, the number of outstanding shares of every company is multiplied by its share price. 2.2 Black-Scholes Formula As discussed earlier, the foundation of option pricing is the Black-Scholes formula. In its simplest form, this model depends on 6 input parameters and the assumption that stock returns are normally distributed. The Black- Scholes model further assumes that volatility, dividend yield and interest rate are constant. Because the focus of this thesis is on S&P 500 options and VIX options, the assumption of a constant or zero dividend yield is acceptable. Since index options are usually European, we do not need to adjust the model 4

7 for our data. We use the 1 month LIBOR rate as a measure of the interest rate. This is done mainly because this way, the LIBOR expiration is around the same date as the VIX options expiration. Since the interest rate did not move a lot in the investigated period, the assumption of a constant interest rate is a reasonable simplification. The next variable is the volatility. The assumption of Black and Scholes is that volatility is a constant. As we will show in the next section, this assumption is clearly violated, since the very existence of a volatility smile contradicts this assumption. Furthermore, the time variation in the volatility indicates that the volatility can not be modelled as a constant. However, under the given assumptions, the option prices can be calculated as follows d 1 = ln( S σ2 ) + (r q + )T K 2 σ, d 2 = d 1 σ T (1) T call =N(d 1 )Se qt N(d 2 )Ke rt, put =N( d 2 )Ke rt N( d 1 )Se qt (2) where K=Strike price, S = Current Stock price or index level, r= risk-free interest rate, σ = volatility, q=dividend yield, T = Time to maturity. N( ) is the CDF of the standard Normal distribution. When the value of an underlying, S, goes up, we can see that the value of an option will change. When S goes up, a call is more likely to end up in the money and therefore, the value of a call option increases. Similarly, the value of a put will go down. The change of the value of an option with respect to a change in the underlying is called the Delta. Since the value of a call option increases when S increases, call options have a positive Delta. The Delta will not be investigated in much detail. The most important thing is that under the BS assumptions, the Delta of a call option is equal to e qt N(d 1 ), which is between 0 and 1 since we have positive yield q and positive time to expiration T. ATM calls have a Delta of 0.5, which is more commonly referred to as Delta 50. Both notations 5

8 will be used, but mainly, I will refer to Delta regions. out-of-the-money calls have Delta < 0.5, in-the-money calls have Delta > 0.5. The Delta of a put varies from 1 to 0. At the money puts have a Delta of 0.5, an in-the-money put has Delta [ 1, 0.5) and out-of-the-money puts have Delta ( 0.5, 0]. For further reading on Deltas and other Greeks, see Hull (2006). When the underlying follows a geometric Brownian motion, the returns are assumed to be lognormally distributed. A lot of research has been done on the normality assumption. For example, it turns out that stock returns have fatter tails than the normal distribution would predict (Officer, 1972). This is similar to the observation of excess kurtosis in market data. Also, market data is very often left skewed. Corrado and Su (1997) show this phenomenon in historical S&P 500 data. Using historical S&P 500 returns, Corrado and Su calculated mean, volatility, skewness and kurtosis. Normally distributed log returns are expected to have a skewness close to 0 and a kurtosis of 3. The researchers find very different numbers than the lognormal distribution would suggest; they observed a lot of excess kurtosis and the skewness was observed to be significantly different from 0. Jarrow and Rudd (1982) devised a way to make adjustments for these shortcomings using an Edgeworth expansion. In their method, they assume a normal distribution and use standard option pricing methods to calculate the mean. The variance, skewness and kurtosis are then estimated from the data. This yielded better results, yet it does not completely solve the problem of the incorrect distributional assumption. Since the focus of this thesis will be on the ATM options and Delta regions 40 and 60, the violation of the distributional assumption is not a very big issue. It is however noteworthy that a similar approach needs to be used when extending this analysis to other Delta regions. Another option is to add jump components to the model. Cox and Ross proposed a jump-model (Cox & Ross, 1976), which they extended to their Cox- Ross-Rubinstein option pricing model (Cox et al., 1979). Almost simultaneous, the jump-diffusion model was proposed by Merton (1976). Jump models try to capture the effects of sudden big events in the economy. In essence, jump 6

9 models try to tackle the issue of the excess kurtosis but they can also explain the observed skewness and existence of the volatility skew. However, jump models are outside the scope of this thesis, since the issue of excess kurtosis is not so relevant for ATM options. 2.3 Characteristics of Volatility Before we are able to look at models for implied volatility, it is necessary to explore a few characteristics of volatility. Fortunately, most of these are commonly addressed in academic literature. The first thing is the concept of volatility clustering. Mandelbrot (1963) was among the first academics to notice this phenomenon. In essence, this theory states that volatility is time dependent; it turns out that there is some amount of auto-correlation in historical volatility. In periods of high volatility, the next day s volatility is likely to be high as well. Since uncertainty on the exchanges usually doesn t change in a short period, this may seem very straightforward. However, one of the core assumptions of the Black-Scholes model, the assumption of a constant volatility, is clearly violated because of volatility clustering. To be able to cope with volatility clustering, a model must be fit to explain volatility movements. Engle (1982) and Bollerslev (1986) respectively proposed the ARCH and GARCH model. Engle stated that the volatility in a specific period is a function of previous squared errors in his Autoregressive Conditional Heteroscedasticity model. Bollerslev generalized this model to his so-called General Autoregressive Conditional Heteroscedasticity model in Today, GARCH-models are still among the most commonly used volatility models. However, both ARCH and GARCH are deterministic volatility models. This means that the variance in a specific time period is a function of previous returns and/or variances. In these models, there is no room for stochastic shocks in the variance process. This thesis will not focus on either of the above-mentioned models but instead the focus will be on stochastic volatility models, which will be more thoroughly explained in Section (3). Another issue with volatility is the so called leverage effect. This was 7

10 first discussed by Black (1976) and states that stock returns and the change in volatility are negatively correlated. This means that when stock returns are negative, the volatility increases. One way to look at this, is that when stock returns are negative, the value of a firm drops and therefore, the debtto-equity ratio increases, making a firm more leveraged and therefore riskier. This increased risk leads to a higher volatility. Another argument for this finding could be that there tends to be more panic when the markets go down. Therefore, people are willing to pay more for options, hence the (implied) volatility increases. This specific argument mainly fits when the market goes down and generally leads to people buying downside options. Therefore, one could argue this will mainly lead to an increase in the skew. As shown by Bouchaud et al. (2008), the leverage effect is stronger for stock indices than for individual stocks. Since this thesis concerns indices, the volatility models used in the remainder of this thesis will take the leverage effect into account. These two characteristics have important implications for this thesis. In the Black-Scholes model, the assumption of geometric Brownian motions leads to assuming volatility is constant over a range of options. This is a very strong assumption and turns out to be incorrect mainly for the deep in-the-money and deep out-of-the-money options. This has led to the introduction of the volatility smile. The volatility smile is a graph of the implied volatility of an option versus its strike price. It turns out that the deep in-the-money and the deep out-of-the-money options are typically priced with higher implied volatility, corresponding to the higher kurtosis in the data. Therefore, the volatility vs. strike graph looks a bit like a smile. Since calls have a positive Delta, the value of a call increases when the underlying goes up. This happens because a call is more likely to end up inthe-money when the underlying goes up. Therefore, it also makes sense that the risk of a call option decreases when an underlying goes up. So it makes sense that the volatility goes down when the value of the underlying goes up. 8

11 This is in contradiction with the assumption of a constant volatility. In fact, it turns out that volatility is not constant but takes the shape of a smile. This smile is the so called volatility curve. This can also be seen in Figure (1). This graph is taken from Quant.stackexchange. Figure 1: Volatility Smile from quant.stackexchange.com As expected, when the underlying index goes down, the volatility goes up. On the other hand, the volatility initially goes down when the underlying goes up. After this initial decrease, the volatility starts increasing rapidly. This is why we call this a volatility smile. Applying Ito s Lemma to the price of a call option shows the dynamics of a Call option when the underlying changes. ds = µsdt + σsɛ dt dc = ( C C µs + t σ2 S 2 2 C 2 ) dt + C σsɛ dt (3) where ɛ N(0, 1). Now using that C 9 is equal to the Delta of an option and

12 that the deterministic part of equation (3) drops out of the variance, we see that var(dc) = 2 σ 2 S 2 dt. The two most important parts of the volatility curve are the ATM volatility and the skew/slope. The terms skew and slope are interchangeable and will both be used in this thesis. The ATM volatility is the volatility at Delta 50. The downside is defined as everything below Delta 50. The upside is everything above Delta 50. The slope is the gradient of the ATM implied volatility with respect to the ATM strike. In order to make a linear approximation, we could look at Delta regions above and below the Delta 50, which is the ATM strike and vol. A possible choice is comparing Delta regions 40 and 60, since linear approximations will be fairly accurate in these Delta regions. The focus will be on the ATM vol and the slope. The far downside ( (0, 0.05)) and far upside ( (0.95, 1)), are not the main priority, because we know that the Black-Scholes implied volatility will be poorly estimated in these regions. Another possibility is to use moneyness, which is the ratio of the value of the underlying to the Strike price. So if I have an index currently valued at 2200, the strike with 2000 has moneyness = 1.1 or 110%. However, because the difference between the strike and current underlying value are not very big around the ATM strike, I prefer working with Delta regions. For the S&P, we use strikes 1950 until Therefore, the Delta regions will be more distinctive than the moneyness. 2.4 VIX index The VIX index was mentioned in the introduction. This section will take a closer look at the dynamics of the VIX and the way it is calculated. The VIX is a so-called volatility index. This means that, unlike most indices, the VIX is not a weighted average of several companies but the implied volatility of the S&P 500 index. The VIX is defined as the expected 30-day volatility, which means that the monthly expiration is not necessarily on the third Friday of a month, unlike index options. The VIX is calculated using two different S&P 500 expirations. 10

13 All calculations below are carried out for both expiry s, using subscript j = 1,2. In the following part, the rather complex calculation of the VIX will be explained. The value of the variance strip of an expiry month is calculated as follows. (CBOE,2014) σ 2 j = 2 T j i K i e R jt j Q Ki 2 j (K i ) 1 ( ) 2 Fj 1, j = 1, 2 (4) T j K j,0 where T j is the time to expiration. F j is the forward index level, K j,0 is the first strike below F j. K j,i is the i-th strike price of the out-of-the-money option. For all strikes K i > K j,0, the price of the call option is used and for the strikes K i < K j,0 the price of the put option is used. K i = K i+1 K i 1 2. R is the risk free interest rate. Q j (K i ) is the mid point of the bid-ask spread for the options with strike K i. The time to expiration is calculated in calendar days, converted to minutes to obtain more precise numbers. Hence, T = M T oday + M Settlementday + M Otherdays. (5) Minutes in a year M T oday is the number of minutes until midnight of the current day. M Settlementday is the number of minutes from midnight until 8:30 a.m. The value for M Otherdays is the total number of minutes between the current day and the expiration day. The 1-month LIBOR yield is used as the value for R j. Now before we can calculate the VIX index, we need to know which strikes are included in the calculation. First we need to determine the value for F; the strike price that has the smallest difference between the put and call option price. F j = Strike Price + e R jt j (call Price - put Price), with j=1,2 and R and T defined as above. Once we have obtained a value for F, we can calculate K 0, which is the first out-of-the-money strike. We now include all put options on strikes K 0 < F until we have 2 consecutive strikes that have a bid price of 0. The same thing 11

14 is done for all call options on strikes K 0 > F until we have 2 consecutive strikes with bid price 0. Thirdly, select both the put and call for K 0 and take the average value of the 2 options. The derivation of this formula for pricing of the VIX volatility can be found in the next subsection. Now we have everything we need to apply formula (4) and calculate σ 1 and σ 2. Once we have obtained these values, we are ready to calculate the value of the VIX index. This is a weighted average of the near-term and next-term volatility. ( [ V IX = 100 T 1 σ1 2 NT2 N 30 N T2 N T1 ] [ + T 2 σ2 2 N30 N T1 N T2 N T1 ]) N365 N 30 (6) In this formula, N X is defined as the total number of minutes until event X. So N T1 is the total number of minutes until the front month expiration. N T2 is the total number of minutes until the second expiration. N 30 and N 365 are the total number of minutes in 30 days and 365 days, respectively. Since the VIX is essentially not very different from any other index, we can use the VIX as an underlying. On this underlying, we can create put and call options as well. This is hardly different from the way we look at, for example, S&P options. A VIX call option has payoff max {V IX T K, 0}. Pricing these options can be done using Black-Scholes but will lead to the same issues as we had when pricing S&P 500 options (excess kurtosis, no constant volatility and more). Just like in Figure (1), Figure (2) shows that the VIX volatility surface is far from constant and that we must find a model to fit the smile. This will be the Heston model, described in Section (3) 12

15 VIX volatility surface : : : vol Strike Figure 2: Volatility Smile for the VIX Jun expiry on three different dates. ATM strike was, chronologically, 13.06,13.74, Derivation of VIX pricing formula As a starting point, we will assume zero interest rates and dividends. This will merely simplify the formulas and will not influence the result. In their 1978 paper, Breeden and Litzenberger show that the conditional probability density function (PDF) of a stock price at time T can be written as a the second derivative of an option price. p(s t, T ; S t, t) = 2 C(St, K, t, T ) K 2 = 2 P (St, K, t, T ) K=ST K 2 (7) K=ST The above notation is slightly different than that in Breeden and Litzenberger and is taken from Gatheral (2006). Now that we have a formula for the PDF, we can calculate the value of a claim of a certain payoff g(s t ). 13

16 E[g(S T ) S t ] = = 0 F 0 p(k, T ; S t, t)g(k)dk 2 P K g(k)dk + 2 C 2 F K g(k)dk 2 (8) The rest of this section will mainly follow Gatheral (2006) in his derivations of variance swaps and volatility swaps. We can break (8) down like this because of the result in (7). The second derivatives with respect to the strike price of a call and a put option are equal. From this second form, we can begin rewriting the expected value of our claim using partial integration and the put-call parity, which states that the value of a call option and the current value of the strike price should be equal to the value of a put on the same strike price and the current value of the underlying. Rewriting (8) results in the following formula. E[g(S T ) S t ] = P K g(k) F F 0 F 0 C K g (K)dK F P K g (K)dK + C K g(k) =g(f ) P 0 K g (K)dK C F K g (K)dK =g(f ) P F F (K)g (K) + P (K)g (K)dK 0 0 C(K)g (K) + C(K)g (K)dK =g(f ) + F 0 F F P (K)g (K)dK + F F C(K)g (K)dK (9) This shows that twice differentiable payoff can be replicated with a portfolio of options with strikes ranging from 0 to with weights equal to the second derivative of the payoff of the option at that strike. Furthermore, in a zero interest and dividend world, the forward F will be equal to S 0. Therefore, we can write the return of a stock as follows. 14

17 log S T F = log S T T T = log S t = S ds t S t T 0 σs 2 t dt (10) 2 This last result is quite significant. The last term is half of the variance of the stock. The other term in the last part is the payoff of a strategy where you continuously hedge a stock by maintaining a constant amount of money in the stock. Using risk neutrality, we see that [ T ] [ E σs 2 t dt = 2E log S ] T 0 F [ 0 = 2 p(k)dk + 0 ] (11) c(k)dk This shows that we can indeed replicate the variance of a stock using a strip of European options. Now we combine the expression in (9) and (4) to obtain the following result. Using the logarithmic transformation for g( ), we obtain that g (K) = K 2. Plugging this in (9) V IX 2 T 2 = = = = F 0 K P (K) dk + K 2 P (K) K 2 dk + Q(K) K 2 dk + C(K) dk F K 2 K 0 F Q(K) K 2 dk + 1 K 2 0 Q(K) K 2 dk 1 K 2 0 K 0 K 2 F C(K) K 2 dk + K F dk F K 0 (K F )dk K 0 (K 0 F ) 2 2 P (K) C(K) dk K 2 (12) The first term is an integral of all midpoints of strikes divided by K 2. Since there are not infinitely many strikes and all strikes are actually discrete numbers, we see that this first term is the continuous equivalent of the summation in the VIX formula. The second term is a matter of rearranging and these two 15

18 terms give us the desired result we found in (4) 3 Model Aside from an analysis using historical data, the VIX will be analysed using a stochastic volatility model. 3.1 Heston Model Before the model can be explained, there is one thing that needs to be more thoroughly explained. We have talked about implied volatility, but the term spot volatility is very important in the Heston model. The spot volatility is the square root of the spot variance and is defined as σ t = lim h 0 1 h E [( ) 2 ] St+h S t S t (13) The Heston model, first derived by Heston in 1993, states that the price S t of an underlying follows the stochastic differential equation ds t = µs t dt + ν t S t dw S t (14) with µ equal to the expected return minus the annual dividend yield and ν equal to the spot variance of the asset returns. So ν t is equal to the spot volatility of the asset. The variance satisfies the following stochastic differential equation, dν t = κ(θ ν t )dt + ξ ν t dw ν t (15) Both Wt S andwt ν are so called Wiener processes, or Brownian motions, with correlation dw S t dw ν t = ρdt (16) A more thorough explanation of Brownian motions can be found in Etheridge 16

19 (2002). The parameter θ is the long run variance. Sometimes the historic variance is taken as a value for θ. The first part of equation (15) indicates that when the current variance is larger than the long term average variance, or ν t > θ, the next variance will be downwards corrected. The value of κ is the mean-reversion coefficient, which tells us how much the variance will be corrected downwards (upwards) if it overstates (understates) the long run variance. The last parameter to explain is ξ, which is the volatility of ν t, so it is the volatility of the volatility, or more commonly referred to as the vol of vol. This parameter is the most interesting parameter for this thesis. Since the aim of this thesis is to determine the relationship between the volatility surface of the S&P 500 volatility surface and the volatility surface of the VIX, we are mainly looking at the relationship between the volatility of the S&P ATM options and the volatility of the ATM VIX options, which corresponds to the expected forward volatility of the S&P volatility. This is where the parameter ξ becomes interesting. Because the VIX is a measure of the implied volatility, the parameter ξ for the S&P 500 should be proportional to the VIX volatility (Jablecki et al., 2014). They use Ito s Lemma and results from Gatheral (2006) to find an approximation for the Black-Scholes variance, with x t equal to the logarithm of the moneyness and β(v 0 ) can be any function of v 0, the initial variance. In the Heston Model, β(v 0 ) is equal to 1. σ 2 BS 1 2 ρξβ(v 0)x T (17) The derivative of (17), which can be considered the slope of the Black- Scholes volatility, is therefore a function of the moneyness and the vol of vol. So translating this result to this thesis, the Skew of the S&P 500 is proportional to the parameter ξ, which in turn is proportional to the VIX. (Jablecki et al., 2014) We need to keep in mind that the VIX is an index on the expected volatility of the S&P 500, and not on the current spot volatility. However, 17

20 if the options are properly priced, the parameter ξ should reflect the VIX volatility quite well. This can be tested using historical data and Monte Carlo Simulation. We can estimate the parameters of the Heston Model from the historical data and calculate what the expected value of ξ is. This value can be compared to the log changes (or the square of these changes) of the VIX to see whether the Heston Model gives a good explanation of stock movements and the volatility. The Heston model has some obvious advantages compared to more basic volatility models. First of all, the mean-reverting property is captured in parameter κ. Also, the model is robust to non-lognormal distributed returns. The leverage effect, described in Section (2.3), is also permitted in the Heston model. This is the case because the stock returns and volatility are allowed to be correlated in this model. Since this correlation is allowed to take on negative values, the leverage effect can be captured. (Mikhailov & Nögel, 2003) One of the downsides of every volatility model, including the Heston model, is that volatility is not observable in the market and therefore, the model is very input-dependent. This also makes it difficult to estimate the input parameters of the model properly. The solution to this problem is very often calibration. Calibrating a model is another way of saying that the model parameters are optimized to the best fit with the market data. However, in the case of optimization of 5 variables, it is very likely that we end in a local optimum instead of the global optimum, depending on the starting parameters. It is therefore important to either fix a few parameters or to test multiple starting points. Thirdly, the Heston model fails to create a short term skew as strong as the implied skew from the market. This finding might become a problem for this thesis, since the main focus is on estimating the effects of changes in the skew. (Mikhailov & Nögel, 2003) 3.2 Closed form solution In this subsection, the closed form solution for a call option of the Heston model will be derived and explained. All derivations are taken from Heston 18

21 (1993) and the notation from that paper will be used. This notation is slightly different from the one used in Section (3.1), ds t = µs t dt + ν t S t dz 1 (t) (18) dν t = κ(θ ν t )dt + σ ν t dz 2 (t) (19) This model is an extension of the model used by Stein and Stein (1991), which assumed that the two Wiener processes z 1 (t) and z 2 (t) are independent. Heston s model permits a correlation between the two processes. The last step towards (19) comes from Cox, Ross and Ingersol (1985), who used similar stochastic volatility models for interest rates. From these interest rates models, they derived that the process can be described by (19). The value of the asset U(S,ν,T) is determined by a few equations. Since this thesis concerns call options, the value of U(S,ν,T) = max{0, S K}. First of all, this value must satisfy the partial differential equation 1 2 U 2 νs2 + ρσνs 2 U 2 ν σ2 ν 2 U ν + ( κ(θ ν t ) λ(s, ν, T ) ) U ν U + rs 2 U ru + T = 0 (20) The term λ(s, ν, T ) is the volatility risk premium. We will assume this is 0. Now the value of a call option is assumed to be of a similar form as the Black-Scholes option price; C(S, ν, T ) = SP 1 Ke rt P 2 (21) plugging this formula into formula (20) and using some advanced mathematics, we obtain the following formula for the function of ln S t. For the detailed proof, see the appendix of Heston (1993). f(x, ν, T ; φ) = e C(T t;φ)+d(t t;φ)ν+iφx (22) 19

22 where, C(t; φ) = rφit + a [ ]) 1 ge dr ((b σ 2 j ρσφi + d)t 2 ln, 1 g D(t; φ) = b [ ] j ρσφi + d 1 g, σ 2 1 ge dr g = b j ρσφi + d b j ρσφi d d = (ρσφi b + j) 2 σ 2 (2u j φi φ 2 ) (23) u 1 = 1 2, u 2 = 1 2, a = κθ, b 1 = κ + λ ρσ, b 2 = κ + λ The probabilities P 1 and P 2 can be found from P j = (x, ν, T ; ln(k)) = π 3.3 Approach 0 [ ] e iφ ln(k) f j (x, ν, T ; φ) Re dφ (24) iφ The analysis carried out will be twofold. First, the market data will be used to make inference about the relationship between several variables. The variables of interest is the ATM volatility of the VIX. As potential explanatory variables, we have the ATM volatility of the front and back S&P expirations. Also, the effect of the S&P 500 on the VIX index will be calculated and used as a variable. The fourth variable of interest is the cross term between the front month ATM volatility and the VIX. The reason to include this variable is that since the front month volatility is more often changed in the S&P 500, this will not only have a direct effect on the VIX volatility but also an indirect effect through VIX. After this market data analysis, the model will be calibrated to obtain estimates for κ, ν 0, θ, ξ and ρ and with these parameters, the Heston Model will be programmed into Matlab and a Monte Carlo simulation will be done to calculate the volatility in the model. The downside of this approach is that this will make the model very dependent on the data we use. It is very plausible that the correlation we calculate from 1 months option data will not reflect the actual correlation very well. Similarly, we could be in a very volatile or very calm month, leading to an 20

23 unrepresentative value for θ. However, it is worthwhile to see whether calibration leads to results that are close to the values observed in the market. If these numbers are not close, the Heston Model might be a bad model to use, since it will have very little explanatory power. For computational efficiency and to get results that actually make sense, it is necessary to set a few conditions on the values of the parameters. As we noted in Section (2.3), index movements are subject to the leverage effect. The leverage effect can be modelled by limiting ρ to the interval ( 1, 0]. Also, it is necessary that the volatility is always positive. This means that θ and ν t have to be positive numbers. However, since ν t is also driven by a Brownian motion, we need to make sure that this entire process is positive. Therefore, we normally could impose the Feller condition 2κθ > ξ 2. It is unfortunately not always possible to obtain reasonable outcomes of the model that also satisfy the Feller condition. Therefore, we will initially calibrate the model without this condition. If it turns out that this leads to major issues, this restriction will be imposed. One of the major advantages of the Heston model is that it has an analytical (closed-form) solution. This closed-form will be used to price options and these values will be compared with the market price and the Black-Scholes price. After this initial analysis, we will use the calibrated Heston parameters to simulate movements of the S&P. These simulations will be used to price options on both expirations of the S&P. After pricing these options, the same calculations and regressions as for the market data are carried out. This way, we can see whether the Heston model manages to produce the same options prices as we observe in the market. From these prices, the implied volatility is calculated and the same regressions are performed. This approach has one major problem. To calculate the VIX, we need data on all strikes in the S&P 500. However, not all data is available. Since we also don t have a proper pricing model for far out-of-the-money options, these prices can not be accurately estimated. Therefore, for this last analysis, the VIX ATM volatility has to be approximated. 21

24 4 Data 4.1 Option Data High-frequency data of the VIX and S&P 500 implied volatility surface is obtained through Bloomberg. In order to calculate the volatility surface, the bid and ask price in the market is obtained for a range of strikes. The midpoint between these two will be considered the market price of an option. The investigation only concerns call options. This data is taken every 10 minutes from May 21st 09:00 until June 12th 22: Even more frequent data could have been observed as well, but it was possible that even more frequent data could lead to more additional noise than valuable extra data in the data set. Especially in less liquid times and options, this could lead to a massive increase in data points with only very little gain in information. The sampling period is obviously not randomly chosen. First of all, data gathering starts on the first day when this expiry is the front month. Since the VIX May expiry was on May 20th, the June expiry options were not as liquid as the May expiry. However, the day after the May expiry, these options became the front month and therefore the most liquid VIX options. Another issue with options is that as we get closer to expiration, the Delta of all outof-the-money options goes to 0 and the Delta of all in-the-money options go to 1. Since most of my analysis will be based on Delta regions, calculations will become less reliable and less informative as we approach expiration. Therefore, the last 3 trading days before the June expiration are taken out of the sample and sampling stopped at June 12th. This date is also fine for the S&P June expiry, which is on June 19th, 2 days after the VIX expiry. This was also convenient because the VIX is priced using the next 2 S&P expiries. Since the S&P May expiry was on May 15th and the June expiry was on June 19th, the VIX future was based on the same 2 S&P expiries throughout the entire sampling period. In this time interval, there was no data recorded for some specific times. I am not sure whether this was an error in obtaining the data or if this happened 22

25 because there were no orders in the market. I have decided to exclude these times. Also, trading in these products is allowed from 09:00 until 22:00. I have decided to only look at the data from 15:30 until 22:00 (GMT+1 hour), on every day because these are the American trading hours, which means that liquidity was greatest in these periods. All together, this gives 572 values for the bid and ask per option. The VIX June expiry data consists of options with strikes ranging from 10 to 22. Since the VIX index stayed in the interval [11.88, 15.21], these values definitely suffice to analyse Delta 40, 50 and 60 of the VIX. The S&P June expiry data has bid and ask prices of options with strike prices ranging from 2000 to In the observed period, the S&P moved from to So again, this range is clearly sufficient for obtaining Delta 60, 50 and 40. Because the S&P July expiry was further away, the Delta of the out-of-the-money strikes are different. This makes sense since the Delta can be seen, as follows from risk-neutrality in the Black-Scholes model, as the probability of an option ending in or out-of-the-money. With more time to go until expiration, there is more time for the underlying to move, which will lower the Delta of the in-the-money options and increase the Delta of out-ofthe money options. To overcome this potential issue, data on more strikes were gathered for the July expiry. For this expiry, the 1950 to 2240 strikes were added to the data set. From the bid and ask prices, the mid market price of an option can be obtained. Given these market prices, it is possible to invert the Black-Scholes model to calculate the implied volatility from this option. Also, given this implied volatility, the corresponding Delta ( ) can be obtained. Inverting the model can only be done by numerical methods. Once we have the implied volatility and the implied Delta of the entire option surface, we can calculate the exact strike that corresponds to the 60 and 40 regions. This will be done by interpolation between the first strike above and below a specific Delta region. More information on the calculations that will be carried out with the data is in Section (5.1.1). 23

26 4.2 Index Data Aside from this data on the option surface, two different data sets are obtained on the index. First of all, for the same period as the option surface, high frequency data on the underlying is obtained for the VIX and S&P. This data is required to be able to invert the Black-Scholes model and to make any inferences regarding the options. Additionally, historical high, low, opening and close prices of the S&P 500 and VIX will be obtained. This data is obtained through Yahoo Finance and is freely available on the Internet. This is a set of closing prices of the S&P 500 and the VIX index from January 1999 until April This can be used to have a first glance at the correlation between these two indices. As we stated earlier, the VIX is a measure of the implied volatility of the S&P 500 index. We also claimed that if the underlying goes up, the volatility goes down. Therefore, the VIX and S&P 500 returns are expected to be negatively correlated. In fact, it is expected that on any day that the S&P 500 goes up (down), the VIX should go down (up). A straightforward analysis of these returns tell us that the correlation between the log change of the VIX and the log change of the S&P 500 is As expected, this number is negative. On 69% of the days in our sample, the S&P 500 and VIX moved in opposite directions. This is also in line with our expectations and is better observed in Figure (3). Only a rather small subset of the data set is plotted but we can already observe that many of the negative spikes of the S&P 500 correspond to a positive spike in the VIX and vice versa. We also note that the VIX returns are of a much higher order, which implies that the VIX volatility is much higher than the S&P 500 volatility. Even though the main analysis will be done using the high frequency data, it is still worthwhile to have a look at the long term data. This could give us a good idea about the long-term variance in the S&P and VIX. This can be used as input in the Heston model. 24

27 In Table 1, some statistics about the long term data set can be found. Index Min Max Mean daily Return Std.Error of Return Observations S&P , ,69 0,00574% 0, ( ) ( ) VIX 9,89 80,86-0,00673% 2, ( ) ( ) S&P , ,69 0,02094% 0, VIX 10,32 25,27 0,114% 0, Table 1: Data for the entire period is in the top panel. The bottom panel consists of data in the last 252 trading days (1 year). We can see that the S&P 500 has dropped to on March 9th 2009 and climbed up to on April 24th 2015, which means the value tripled in approximately 6 years. However, looking at the VIX shows how much more volatile this index is. The VIX closed at 9.89 on January 24th 2007 and peaked on November 20th 2008, in the midst of the crisis at a level 9 times the low, which was obtained less than 2 years before. Clearly the VIX moves much more than the S&P 500, again indicating a higher volatility. 25

28 0.01 Time Series Plot:S&P Return S&P Return Sep 27, 13 Jan 25, 14 May 25, 14 Sep 22, 14 Jan 20, 15 May 20, 15 Time (a) 0.1 Time Series Plot:VIX Return VIX Return Sep 27, 13 Jan 25, 14 May 25, 14 Sep 22, 14 Jan 20, 15 May 20, 15 Time (b) Figure 3: Plots of the S&P 500 and VIX Return in the period September 27th 2013 until May 10th Now that we have the data on the indices, we can have a look at some of the statements we made earlier. The main problem with the Black-Scholes Model 26

29 is the normality assumption. The data of both S&P and VIX have been fitted to the normal distribution to see if we indeed observe these shortcomings. We can see that the data is indeed skewed to the left. This means that the tail is larger on the left side. Calculating the skewness in the S&P returns gives S&P Return histogram Return NormPDF Return (a) 120 VIX Return histogram Return NormPDF Return (b) Figure 4: Plots of the S&P 500 and VIX Returns and a standard Normal distribution. 27

30 a value of A larger skewness on the left side means that more extreme outcomes are more likely on the left side than on the right. This matches the expectations we have regarding stock movements because negative shocks tend to be more extreme shocks than positive shocks, making outliers on the left side more likely. It might be surprising that the VIX return has a positive skewness. After all, many people agree that the negative shock argument holds. It is important to keep in mind how the VIX is constructed. The VIX usually moves opposite from the S&P 500. Therefore, if the S&P is more prone to negative shocks, you would expect the VIX to have more positive outliers. This is indeed the case, since the skewness in the VIX is However, aside from the skewness, we can also have a look at the fourth moment of the returns; the kurtosis. It is often argued that stock returns are not normally distributed but that they have excess kurtosis. This can also be clearly observed in the graphs above. The amount of S&P returns greater than 0.02 in absolute value clearly exceeds the amount we would expect from a normally distributed data set. A normally distributed variable has a kurtosis equal to 3. We observe a kurtosis in the VIX returns of and the S&P returns have kurtosis We can definitely state that there is excess kurtosis in these returns and that the normality assumption is violated. 5 Results As stated earlier, the aim of this investigation will be on the period right after the May VIX expiration, which lasted from May 20th 2015 until just before the June expiration, which was on June 17th Throughout the entire results section, the interest rate R will be considered constant at 0.19%. This is a reasonable assumption since the 1 month LIBOR rate was at this level throughout the entire investigated period and hardly changed in the past 6 months at all. In fact, the LIBOR moved from 0.15% in October 2014 to 0.19% in June This increase is so small on a 1-month basis, that the assumption of constant interest rates is valid in this time interval. 28