Empirically Calculating an Optimal Hedging Method. Stephen Arthur Bradley Level 6 project 20cp Deadline: Tuesday 3rd May 2016

Transcription

1 Empirically Calculating an Optimal Hedging Method Stephen Arthur Bradley Level 6 project 2cp Deadline: Tuesday 3rd May 216 1

2 Acknowledgment of Sources For all ideas taken from other sources (books, articles, internet), the source of the ideas is mentioned in the main text and fully referenced at the end of the report. All material which is quoted essentially word-for-word from other sources is given in quotation marks and referenced. Pictures and diagrams copied from the internet or other sources are labelled with a reference to the web page or book, article etc. Signed Date

3 Contents 1 Introduction Put Call Parity Volatility Historical Volatility Implied Volatility Volatility Smile Comparison of Implied and Historical Volatility The Greeks Delta Delta Hedging Gamma Vega Black Scholes Model Assumptions Black Scholes Equation Option Pricing Delta Calculation Delta Hedging with Historical Volatility Hedging All Options Delta Hedging with Implied Volatility Comparison of the Performance of Implied and Historical models Heston Model The Feller Condition Euler-Maruyama Method Fast Fourier Transform Model Calibration Hedging with the Fast Fourier Transform Hedging with Euler-Maruyama Simulation GARCH model Model Features Information Criterion Model Calibration Hedging with GARCH Performance Comparison Parameter Fitting Performance Statistics

4 Abstract Hedging a portfolio describes any method which minimizes the risk of losses and there are many methods of doing so. The methods we shall examine will focus on delta hedging. This will require certain assumptions, for example, to define the behaviour and the evolution of the unobserved volatility process. Further, we will have to assume a model for the data and it is not clear which model will perform the best in practice. Exploring the application of some different models we calculate that the simple historical volatility model is one of the most impressive models for reducing risk but the GARCH model clearly out performs others we have considered, especially when time constraints are a limiting factor which they are in most real world applications. 1 Introduction When a financial option is bought or sold there is always risk affecting the potential profit or loss. The value of the underlying stock will change through time and the risk (with respect to a purchased option) is that this will cause the value of the option to fall or even become worthless. The risk with respect to a written (sold) option, is that an extreme movement in the underlying stock price could leave the buyer with an unbounded profit and the writer with the corresponding, unbounded loss. Hedging refers to any method which attempts to eliminate/minimise the risk associated with a portfolio. Calculating the hedge of an option can be a very complicated procedure. Thus, certain assumptions can be made to simplify calculations. One such assumption is that the volatility of the underlying stock can be modelled by a constant function; this assumption is made under the Black Scholes model. However, this particular assumption is very strong and often assumed to be an over simplification of reality which result in inadequate and unreliable forecasts and conclusions. Further, a constant volatility of the stock implies a constant variance in the price process. However, the trading price of the stock reflects the traders belief of the value of the stock and this can be effected by many things. These beliefs are often very similar among traders because they are all viewing the same available news about the company/stock. Thus, we tend to experience some periods of extreme deviation in price and some periods of gradual change; this would seem to be a contradiction and supports the notion of a time varying volatility. In general, a model in which the volatility is modelled by a time-dependent function will be more flexible in an attempt to better explain the data. Thus, it seems reasonable to assume that models with a non-constant volatility will outperform their counterparts in forecasting accuracy and will be more relevant and justified. Through this project we shall determine the benefit (if any) from using the more complicated model. The aim of this project is to decide which model assumptions perform the best in a practical sense. We will consider the hedging method and model assumptions as optimal if they generate the greatest profit with the minimum risk. All calculations and methods will assume the existence of a fair market in which there is no arbitrage (guaranteed profit). In the following sections we shall first explain different methods for modelling and estimating the volatility; historical (assuming constant), implied or stochastic. Then we shall discuss some hedging techniques and elaborate on delta hedging specifically. We will then explore the assumptions, features and equations of the models which we are implementing, and finally, calculate and compare the resulting profits under these different models and assumptions. Comparing the mean of the profits will show how successful the portfolio is under each method and comparing the variance of the profits will give an indication of the residual risk. We shall reach a conclusion of optimality by considering both of these statistics. More specifically, the profit mean divided by its standard deviation 4

5 will be a strong indication of the performance of the model because a large value would indicate a positive profit and a relatively insignificant risk. For large portfolios however, it will also be important to consider the running time of the hedging algorithm. 1.1 Put Call Parity Put call parity is a useful relation between the price of a call option and the price of its corresponding put option; if we know either price, the formula allows us to calculate the other. C t + K = P t + S t t A simple proof is as follows. If we possess the call option and an amount of cash equal to the strike price, then at time T this combination will be worth: (S T K) + + K = max(s T, K) Whereas, if we possess the corresponding put option and a share of the stock, our combination is worth: (K S T ) + + S T = max(s T, K) Since both combinations are equal in value at time T, they must also be equal at all times before T, to avoid an arbitrage opportunity [6]. This can be proved by contradiction as follows. Assume that portfolio A and portfolio B have an identical value at time T, i.e. A T = B T, and there exists a time t < T where A has a greater value than B (without loss of generality), i.e. A t > B t. Then we could buy portfolio B and take a short position on A (sell), then, at time T we can sell B to buy A and fulfil our short position. The overall profit is given by: B t + A t + B T A T = A t B t > This is guaranteed profit, so it contradicts the assumption of a fair market. Hence, we must have that A and B are always equal in value. 2 Volatility Volatility is a measure of how much a stock s price is expected to deviate over a short period of time. In general, many parameters in a financial model are known or readily available, for example: the time until expiration, the strike price and the current stock price. Volatility is the only unknown variable which contributes to the price of an option. Consider an at-the-money option, an option with strike price equal to the current stock price, then the price of the option will be an increasing function of the volatility. Thus, if the volatility is an underestimate, the option will be under-priced and if we purchase this option then the expected profit will be strictly positive. Similarly, if the volatility is an overestimate and the option is overpriced then shorting the option will generate a strictly positive expected profit. In either case, there is effectively an arbitrage opportunity. From this, it is easy to see that effective volatility estimation is crucial to the profit or loss of a portfolio. 2.1 Historical Volatility Historical volatility is an estimate which models the volatility as a constant process and is estimated solely from historical data of the evolving stock price. Historical volatility can also be called realized or statistical volatility. One method of estimating this value is under the assumption that the stock s volatility is the standard deviation of the log returns of the 5

6 stock, where the return at time t is given by: S t /S t 1. Let R t denote the returns, then we have: ( ) St R t := log S t 1 The sample variance of these log returns is then given by: ˆσ 2 = 1 n 1 n (R t R) 2 where the divisor is (n 1) to give an unbiased estimate and R is given by R := 1 n 1 n t=1 R t t=1 = 1 n 1 = 1 n 1 n ( ) St log S t 1 n [log(s t ) log(s t 1 )] t=1 t=1 = log(s n) log(s ) n 1 Thus, we have the standard deviation as; ˆσ = 1 n (R t n 1 R) 2 The final step here is to annualize this volatility. This ˆσ gives the 1-day standard deviation so we shall annualize this value in order to use it in the formulae and also for comparison with other estimates of the volatility. Under the assumption that the variance will be the same each day, the annual variance will be the daily variance multiplied by 252; this is standard convention since there are 252 trading days in a year. Equivalently, the annualized standard deviation (annualized volatility) is equal to the 1-day standard deviation, ˆσ, multiplied by 252. This gives the estimate of the historical volatility as: ˆσ annualized = 252 n (R t n 1 R) 2 Note that even on the weekends and holidays there are changes in the market due to news or events and this will cause larger differences between Friday and Monday values than one would expect from a typical one day jump. Thus, it would also be reasonable to use 365 here as opposed to Implied Volatility Implied volatility is estimated from the market price of the option using the Black Scholes equation (discussed in section 4). The Black Scholes equation gives the fair price of an option as a function of many parameters, including the volatility. Thus, if we know the current price of the option then the only value which is unknown is the volatility. We can then set our estimate of the volatility, ˆσ t, as the value of σ for which the Black Scholes equation gives the observed price. This is where Implied volatility gets it s name, i.e. it is the value of the volatility as implied by the option price under the Black Scholes equation. t=1 t=1 6

7 Implied Volatility Volatility Smile What is interesting about the implied volatility is how it changes with the strike price, i.e. how the implied volatility changes depending on how in-the-money or out-of-the-money the option is. An option is called in-the-money if exercising it at the current time would generate a profit and out-of-the-money if exercising the option would generate a loss. This relationship is shown in Figure 1. 1 Volatility Smile for selected call options Exercise Price Figure 1: Volatility smile for a section of the data; i.e. the strike price against the implied volatility for all options which become available on the 24th of April 214 and which expire on the 3th of May 214. Where the vertical line represents the current stock price (the price of the stock on 24th of April) This Figure shows that there is a clear relationship; the volatility seems to be an almost symmetrical, gradually increasing function of the difference between the strike price and the current stock price. There is a subtle problem with this pattern because the current price is not in the centre of the curve. However, the current stock price process we have access to was recorded at the end of each trading day, and the closing hours of a trading day are the most busy. This means that, the changing price process on each day of busy trading, has been approximated by a single value. Thus, although the process will be roughly representative of the true price process, we can expect small deviations from the most reasonable answers. In short, the current price line in Figure 1 indicates the last recorded value of the stock at the end of the day, but perhaps a more representative value for that day would have been in the centre of the curve. 2.3 Comparison of Implied and Historical Volatility The No-Arbitrage principle assures us that the volatility of an option will give the fair price, i.e. the price for which the expected overall profit will be zero. In reality however, the seller/writer of the option will be intending to make a profit, thus, the price will be a little higher than the fair market price. This means that the implied volatility and the realized (historical) volatility will take different values by design. Note that the price of an option is always an increasing function of the volatility as illustrated by Figure 2. Since the volatility is the only unobserved quantity, this figure 7

8 shows that increasing the price of the option is identical to increasing the assumed value of volatility. Hence, by increasing the price of an option above the fair market price, the writer of the option has increased their implied volatility. This suggests that implied volatility will always exceed the true volatility of the market. Since the historical volatility is an unbiased estimate of the true volatility, we would expect the seller of the option to make a profit if the implied volatility exceeds the historical volatility. This implies that the seller is expected to loose money when their implied volatility is lower than the current historical volatility. Equivalently, this situation yields an expected profit for the buyer and this will effectively be arbitrage if the variance is sufficiently small. As we are hedging these options, we are assuming the role of the seller (taking a short position) so it is important to check this feature. The comparison is shown in Figure 21. surface plot illustrating that the option price increases with volatility 1 Option Price Volatility 9 8 Stock Price Figure 2: Black Scholes price for a call option with changing volatility and strike price 4 Comparison of historical and implied volatility.5 Comparison of historical and implied volatility Implied Historical 3.5 Implied Historical.4 3 volatility volatility Time (days) Time (days) Figure 3: Historical and Implied volatility for the dataset 8 8 1

9 From Figure 3 we can see that very few options have an implied volatility below the estimates of realised volatility. From the above, this implies that the vast majority of these options will expect a positive return. This is an expected result and motivates the writing of options. 3 The Greeks Once we decide on a model for the volatility process and for the evolution of the stock price, we can then hedge the portfolio. However, there are various methods to choose from before in order to hedge an option. Each method depends on certain financial objects, known as Greeks. These objects are referred to as Greeks simply because most financial objects of this type are denoted by Greek letters, [8]. In this section we shall consider some of them and elaborate on some hedging methods. 3.1 Delta An option s delta is the change in the price of the option for every unit change in the price of the underlying stock, i.e. an indication of the sensitivity of the option to the stock price. For example, if an option has a delta of 1 2 and the price of the underlying stock increases by x, the price of the option will increase by 1 2 x. Consider the effect of an increasing stock price on call and put options. The pay off of a call option is given by (S T K) + so every increase in S t increases the expected pay off of the option. Under the assumptions of a fair market, the price of the option will then increase which indicates that call options have a positive delta. On the other hand, a put option has a pay off of (K S T ) + whose expected value will decrease as the stock price increases. Under the same assumptions, the price of the option will decrease. Thus, put options have a negative delta. Furthermore, consider an in-the-money call option. Then, as time, t, approaches T (the expiration date) there will be a pay off of (S T K) + S t K and clearly this is linear in S t so an increase of 1 in the stock price would increase the pay off by 1, and so, increase the option price by 1. This implies that the delta of a call option tends to 1 as t tends to T. Similarly, consider an in-the-money put option. The pay off of this option is (K S t ) + and will tend to K S t as t tends to T, thus, each unit increase in the stock price will result in a unit decrease in the option price. This shows that the delta of an in-the-money put option tends to -1 as t tends to T. 3.2 Delta Hedging As above, an option s delta is equal to the change in value of the option per unit change in the underlying stock price. Thus, if we write (sell) an option and calculate it s delta at present time to be δ, then we can hedge the option by shorting (selling) δ shares of the stock. After doing this, a unit increase in the stock will increase the option s price by δ and will cause us to loose δ by shorting the stock, thus, the value of the portfolio will change by δ-δ=. When this holds, we say that the portfolio is delta neutral. This shows that all risk has theoretically been eliminated and confirms the validity of the hedging method. However, the delta of an option is not constant, so the number of shares of the underlying asset will need to be continually updated. This is a little impractical and in reality there will be transaction fees when buying/selling shares of the stock. For simplicity, in our calculations, we shall not consider transaction fees but, so that the results and conclusions are useful in a practical sense, we shall only implement the hedging method at the end of each day, i.e. we shall update the number of shares held in the portfolio once per day. 9

10 3.3 Gamma It is clear that the delta of an option will not be the only important quantity. In general, the value of the option cannot be expected to change as a linear function of the stock price. We can generalizing this method to also include the second derivative of the stock price process. An option s gamma is a measure of the sensitivity of the delta to the stock price, the change in the value of delta per unit change in the stock price, which defines gamma as a second derivative risk [8]. If we consider delta as the speed the option price changes with respect to the stock price, then we can consider the gamma as the acceleration of the option price with respect to the stock price. Since gamma is a second derivative of an unobserved quantity, estimation of the value of gamma will be very difficult and more likely to be inaccurate - apart from in some simple cases. 3.4 Vega Similar to delta, an option s vega, denoted ν, is the change in the option price per unit change in volatility. This method deals with a different type of risk. As we have seen in Figure 2, the volatility can have a strong impact on the value of an option. This suggests that an increase in the market volatility will increase the value of the option by increasing the expected profit. More specifically, if we were to write an option when the volatility was low then we would have to sell it at a low price but if the volatility increased then our potential losses would also increase. To hedge this type of risk and calculate a vega neutral position, we will need to buy a certain number of options. The number of options to buy will be a very difficult quantity to measure because it will depend sensitively on the volatility process, which is unobserved and estimated in a number of ways. 1

11 4 Black Scholes Model 4.1 Assumptions The assumptions under the Black Scholes Model are as follows. Volatility of the stock, σ, is constant in time (this is the most important and controversial assumption) No dividends are paid to the stock owner The interest rate is a known constant, r No transaction fees, so buying and selling stocks is free Liquidity is such that any amount of any stock can be bought or sold at any time However, the model can be manipulated in order to be more useful in real life applications. For example, the constant volatility can simply be replaced by the implied volatility or by a stochastic volatility measure. 4.2 Black Scholes Equation The Black Scholes partial differential equation states the following. ds = S(µdt + σdw ) where S t denotes the price of a stock at time t, µ denotes the drift, σ denotes the stock s volatility and W t is a Brownian motion with distribution N(, t). A solution to this equation must take the form: S t = Ae µt+σwt where A is a constant. Setting t= we can see that A must equal S (the initial value of the stock), i.e. S t = S e µt+σwt Let S t denote the discounted pay off such that S t = S t e rt. Then we have: E(S t ) =e rt E(S t ) =e rt E(S e µt+σwt ) =S e (µ r)t E(e σwt ) =S e (µ r)t E(e N(,σ2 t) ) =S e (µ r)t e σ2 t 2 σ2 (µ r+ =S e 2 )t Hence, when µ = (r σ2 2 ), S t will be a Martingale. This is a desired property to ensure that the stock price can be expressed as follows. σ2 (r S t = S e 2 )t+σwt 11

12 4.3 Option Pricing Here, we consider only European Call options, in which the call can only be exercised on the expiration date and not before. A European Put option can be calculated by first finding the price of the corresponding call option and invoking Put-Call Parity. Let K denote the strike price for the option, S denote the current stock price, T the time until expiration, and let r denote the fixed interest rate. Then the option price is calculated as follows. d 1 := S σ2 log( K ) + T (r + 2 ) σ T The price of the call option is given by: d 2 := S σ2 log( K ) + T (r 2 ) σ T C(S, T ) = S Φ(d 1 ) Ke rt Φ(d 2 ) and the price of the corresponding put option is: 4.4 Delta Calculation P (S, T ) = Ke rt Φ( d 2 ) SΦ( d 1 ) In order to implement a delta hedging technique we will need the ability to calculate delta for European options. Under the Black Scholes model, the delta of a call option is given by: Φ(d 1 ) where Φ(-) represents the cumulative distribution function of a standard Normal random variable and d 1 is as above, i.e. ( log( S σ2 K ) + T (r + 2 δ = Φ ) ) σ T Similarly, for put options we have delta as Φ(d 1 ). This is obvious from the above expressions for C(S, T ) and P (S, T ) when we recall that the delta is simply the differential of the option price with respect to the stock price, S. Figure 4: Resulting profit from delta hedging the options using historical volatility and the Black Scholes model 4.5 Delta Hedging with Historical Volatility The first hedging method we shall implement is the Delta Hedging method under the Black Scholes model, with the volatility estimated by the historical volatility. From this point on, for simplicity, we shall assume that the interest rate, µ or r, is equal to zero. 12

13 For each individual option in the available portfolio, the value of delta is calculated (under the Black Scholes assumptions) at the end of each day and the option is hedged accordingly. For a call option this corresponds to buying delta shares of the stock and for a put option this corresponds to selling (taken a short position) on delta shares of the stock at the current (end of day) price. In either case, the value of delta needs to be calculated in a different way. This hedging technique has been applied to all options which expire in at least 5 days and the options with a expiry of 4 or fewer days have not been hedged, this is again for practical purposes because we are assuming that it is not a profitable use of time to hedge an option which expires in only a few days. This is a reasonable and common approach. Figure 5: The time (in days) until expiry of those options with an absolute profit which is larger than 1.5 The histogram of resulting profit/loss, displayed in Figure 4, illustrates the consistency of the results and confirms that this method eliminates the vast majority of the risk involved in writing these options (as is the methods purpose). Most values are close to zero, as we would expect, and most are positive which is encouraging. This method is further supported by the histogram in Figure 5 which shows that almost all (1357/143) of the options resulting in an extreme profit/loss, expire within less than 5 days of purchase (where an absolute value greater than 1.5 is classified as extreme, since 1.5 is the average price of an option). This means that they have not been hedged; for a practical approach as previously stated. Thus, the vast majority of the variance is due to options which were not hedged and this is a significant result because the options which were not hedged make up only 12.7% (3349/26388) of the total number of options. Since almost all the variance is due to a lack of hedging, it may be more useful to explore the analysis of only the hedged options. The hedged options have a mean of.512 and a standard deviation of 83. The value.512 seems insignificant but it is close to zero by design (in minimising the risk we decrease the potential profits). Also, it is greater than zero and since the standard deviation is fairly small, this profit is consistent enough to effectively be called arbitrage. This mean is not as insignificant as it appears. The average profit is equal to 3.44% of the average option price which is an impressive degree of arbitrage. Thus, we must conclude that this method functions very well, in fact, this has shown a very promising result and suggests that we could enjoy significant improvement in our profits by relaxing our rule for practicality, i.e. hedging options of any expiry date. To decide if this would be best suited to our purposes, we shall now analyse the effects of the exhaustive approach more closely. 13

14 Profit 3 Historical volatility approach compared with no hedging method Option (time ordered) #1 4 Figure 6: Contrast of the profit of hedging all options when using Black-Scholes and historical volatility (yellow) as opposed to not applying a hedge (blue) Hedging All Options From Figure 7 we can see that abandoning our practical approach has significantly reduced the variance in the profits (from.4596 to.8) but has also significantly reduced the potential profit. It is interesting that the median does not seem to have significantly changed (.154 from.162) which suggests that there are roughly equal numbers of profits and losses in each case. This is expected because hedging methods are designed to eliminate risk and not to turn losses into profits or to maximize profits. An important consideration of hedging is that someone who is going to the trouble of hedging a portfolio, is likely to apply the method to a large number of options. This suggests that, assuming the standard deviation of profits is sufficiently low, the only important value is the mean profit. Here, we have that the mean profit has changed from.1229 to.478. This is an extremely important result because, the average profit has been reduced to a third of its original size. Note that we are not suggesting the only important quantity is the mean. If we were to sell these options with no hedging method, the resulting profit would have mean.61 but this does not imply that no hedging method at all is better than our historical volatility approach. This is clear from the graph in Figure 6 which clearly shows how the risk has been minimised. In practice, this is always an unwanted side effect when hedging a portfolio and can be seen as the cost of a safe, risk free portfolio. Figure 6 illustrates how hedging methods will 14

15 Profit control the risk and simultaneously limit the profits. The initial losses shown in the hedged options are likely to be largely caused by the temporarily high volatility in the underlying, shown by the extreme profits and losses of the unhedged options in blue. Since the short-term options generate a larger profit when they are not hedged, this idea of the importance of the mean suggests that the application of a hedging method is not the optimal decision when the option is only available for a small number of days. Also note that we have not included transaction fees in our model which would give further evidence to the practical approach. However, this approach, in a hedging framework, implies that the profit from these short-term options is not exposed to risk. This is a very bold assumption to make for any option since it contradicts the assumption of a free market; that there are no arbitrage opportunities. Nonetheless, we are searching for the most practical approach so we shall examine the evidence before discarding the method. Hedging with Black-Scholes and Historical volatility Practical Hedge All Figure 7: Box diagrams comparing the practical method to the method of hedging all options, when using Black-Scholes and historical volatility We could argue that these profits are unaffected by risk if they are consistently positive and are independent of the market. Figure 8 shows a close up of the histogram of profits (short-term options only) when we do not hedge. This suggests that there is still risk involved because a reasonable number are ending in a loss. However, the histogram is a little misleading because the bar between.2 and.4 consists of over 22 options. Thus, these options are very consistent but this level of consistency doesn t imply risk free. To decide whether the series depends on the market behaviour we would like to know if the price process is correlated with the profits from the short-term options. We can calculate the mean profit from these options, daily, and compute the cross correlation between the price process, and this sequence of mean profits. This cross correlation is plotted in 15

16 Frequency Histogram of unhedged, short-term options Profit with no hedge Figure 8: Histogram showing the profit from short-term options (under 5 days) when unhedged. Figure 9 using a rough indication of significance which relies on a null hypothesis that the true correlations are zero with a variance of 1/sqrt112, since 112 is the length of this time series. The Figure shows that the market behaviour (represented by the price process of the stock) appears to be weakly correlated with these average profits. Lag, in the Figure, is referring to the price process, e.g. at lag K, we are examining the expectation: E( [X t K X] [P t P ] ) E( [Xt X] 2 ) E([P t P ] 2 ) where X,P represent the price process and profit process respectively. The most important feature of the cross correlation, is that there is a significant negative correlation for negative lags. Although the correlation is only briefly significant, this is extremely important because it shows that the past of the market behaviour is correlated with the present profits. This implies that the profits from these options depend on market behaviour and so, cannot be considered as risk free arbitrage opportunities. In a hedging framework, minimizing risk is the most important goal, thus, the most sensible thing to do here is to be cautious and assume that the correlation is significant. To conclude this section, we shall surrender our practical approach of only hedging certain options. Instead, we shall exchange potential profits for security, and hedge all available options. 16

17 Correlation.2 Cross correlation between the price process and the average daily profit from short-term options Lag Figure 9: Cross correlation of the observed price process with the series of daily mean profits from short-term options. 4.6 Delta Hedging with Implied Volatility This method has been calculated in the same way as the previous method but with an implied volatility instead of the historical volatility. It seems reasonable to expect that this method will perform better than the previous method purely due to the increased flexibility of the implied volatility. From our previous conclusion, we shall now consider hedging all options in the portfolio; in order to make a fair comparison. Since we have shown that a simple histogram can be misleading, we shall consider a comparison similar to that of Figure 6. This comparison is shown in Figure Comparison of the Performance of Implied and Historical models Figure 1 shows a very similar structure to that of Figure 6 but contains some subtle differences. Firstly, the initial options generate a significant loss under historical volatility, due to the previously mentioned reasons, but under implied volatility we do not incur such losses. The difference here is entirely due to the independence structure of the volatility process. We calculate each value of implied volatility from the market data at that time as opposed to accumulating a large amount of data to sufficiently estimate its value under a historical model. Another difference is that many of the profits are greater than those under the historical volatility model. This is not obvious from Figure 1 so we shall use a different visualization. Collecting all options which are available for an identical number of days, we can much more easily observe and compare the effects of historical and implied volatility under the Black-Scholes model. This plot is shown in Figure 11. This plot shows that under implied volatility, the generated profit is almost always greater. 17

18 Profit 3 Implied volatility approach compared with no hedging method Option (time ordered) #1 4 Figure 1: Contrast of the profit of hedging all options when using Black-Scholes and implied volatility (yellow) as opposed to not applying a hedge (blue). To explore whether this method has out performed the historical volatility model we shall examine the statistics. model variance mean Historical Implied Firstly, the mean is greater under the implied volatility model, as expected from Figure 11. More interestingly however, the variance of the generated profits has more than doubled. We could argue, as before, that both variances are very low so the only meaningful comparison is that of the mean but this would be a minimalist approach. Comparing these outputs using box diagrams, in Figure 12, indicates the distribution of profits. It is obvious that the increased variance is caused by an increase in the number of positive outliers, i.e. values exceeding the 75 th percentile of the data. This is clearly visible from the diagram because the outliers (red crosses) in the positive y-axis, referring to positive profit, have become much more dense. This increased number of larger profits will be responsible for the increased mean. However, there has also been an increase in the number of negative outliers which is indicated by the density of the negative outliers. This feature of rough symmetry implies that the profits gained have been at the expense of risk. In our calculations, the implied volatility of each option has been independent of all previous data. This feature is largely responsible for the models success but could also be considered as a problem in some cases. 18

19 Profits 9 Profit per length of availability under Historical and Implied volatility Number of days from availability until expiry Figure 11: Comparison of the profits under implied and historical volatility using bins of identical lengths (in days) of availability When fitting a model, sometimes we observe a strange, unexpected change in the parameters, e.g. in the fitted value of volatility, which seems like an anomaly when compared to previous patterns. Under a historical volatility model, we assume that the volatility is constant, which implies that all data are equally important and so, estimates which seem extreme will borrow strength from all previous data, i.e. more extreme values would be stabilized via a running mean with all previous values. When we have a sufficient amount of data, this would effectively bury the anomaly without the need for extra effort. The problem is that these extreme values cannot be corrected or buried under the implied volatility model. The flexibility of a model to allow for a time-dependent volatility is very attractive and has proved very powerful. Thus, we shall maintain the assumption that the volatility is time-dependent. However, in order to include this idea in a model, we shall consider a consistent probability structure for the volatility. This can be done under the Heston model. 19

20 Profit 1.2 Profits under historical and implied volatility models Historical Implied Figure 12: Box diagrams comparing the resulting profits when using the Black-Scholes model with historical volatility and with implied volatility. 5 Heston Model This model takes a different approach to volatility. Previously, we have allowed volatility to be non-constant via considering the implied volatility to allow more flexibility in the model. Volatilities calculated in this way will not depend on one-another, and so, the volatility process could change sharply at every step. Although volatility represents randomness in the price process, the volatility process itself should admit some form of structure and a totally random, independent structure does not seem very justifiable. Here we shall allow for a volatility structure which evolves slowly through time. This model is not as widely implemented as Black-Scholes because it is more complex; this is due to the added complication of the stochastic volatility as opposed to the very simple idea of a constant one. The model is defined by the differential equations shown below. ds t = µs t dt + σ t S t dw (1) t (1) dv t = γ(v t θ)dt + κ v t dw (2) t (2) dw (2) t = pdw (1) t + 1 p 2 dw (3) t (3) Where the parameters are defined as follows; S represents the stock price process µ is the drift parameter, representing the interest rate 2

21 σ is the time dependent volatility v is the variance, i.e. v t = σ 2 t for convenience, since a solution to equation (1) depends only on σ 2 t, and not σ t θ is the long run mean of v γ is the rate of relaxation to this mean κ is the variance noise W (1) and W (2) are correlated 1-dimensional Brownian Motions p is the degree of correlation between W (1) and W (2), as enforced by equation (3) in which W (3) is another standard Brownian Motion independent of W (1) and W (2) [1]. 5.1 The Feller Condition Here is a short and simple result worth mentioning. The Feller condition states that: 2γθ > κ the process v t is strictly positive From the formula alone it is clear that this is a necessary result because the square root of v t (i.e. σ t ) is used and appears in Formula (1). Also, from the concept of volatility (behaving similarly to a standard deviation) we understand that negative values would not be intuitively meaningful. This is an extremely important feature of the model and can act as a quick check for the validity of fitted/assumed parameters. Alternatively, this condition could be written as a constraint in a constrained optimisation problem, as we shall implement in our model calibration. 5.2 Euler-Maruyama Method The Heston model is not as commonly used as Black Scholes due to it s increased complexity and the computational cost of it s application, for example, there is no closed form expression for the price of an option under the Heston model. Thus, we will need a way to simulate the the evolution of the price process and estimate the price of the option. As mentioned above, we shall simplify this model by setting the interest rate, µ, equal to zero. This gives: ds t = v t S t dw (1) t (4) dv t = γ(v t θ)dt + κ v t dw (2) t (5) Thus, assuming these parameters are known, a simultaneous realisation of {S t } t> and {v t } t> can be simulated using the Euler-Maruyama Method [7]. The Euler-Maruyama Method is used to simulate stochastic processes from stochastic differential equations (SDE s). The application of this algorithm is explained below. 1. replace ds t and dv t with S (t + 1) S (t) and v (t + 1) v (t), respectively 2. replace dt with to annualize (there are 252 business days per year) 3. make S (t + 1) and v (t + 1) the subjects of the equations 4. generate the process iteratively from equations 4 and 5 This results in equations: S (t + 1) = S (t) + v (t)s (t) ɛ (1) t ( 1 v (t + 1) = v (t) γ [v (t) θ] 252 ) + κ ( v (t) pɛ (1) t + ) 1 p 2 ɛ (2) t 21

22 where S() refers to the current stock price, v() refers to the current volatility and ɛ (1) t, ɛ (2) t are independent, Gaussian noise processes with mean zero and variance ; since the length of the interval is in years (considering only trading days). Note that S() is known and v() can be estimated by the current market volatility (current estimate of historical volatility). Assuming that the parameters are known (or estimated/fitted), we can use the Euler- Maruyama method above to simulate the future of the process. Further, the hypothetical profit can be calculated for each simulation and an average should be roughly equal to the expectation of the resulting profit; by the weak law of large numbers under the assumption that the parameters are adequately estimated. That is to say, we can use this algorithm to give good estimates of for call and put options respectively. E(S T K) + and E(K S T ) + By the assumption of a fair market, this expectation will be equal to the fair price of the option at the present time. Then we could calculate a delta hedge for the options from the following formula: C(S t + ɛ) C(S t ) (S t ) = lim ɛ (6) ɛ and we can approximate this by taking a very small ɛ. Clearly this will be very sensitive to errors in the price of the option so we will have to decide on a reasonable number of iterations in order to balance this crucial issue of accuracy with computational efficiency. This issue is especially important because we are considering a large portfolio of over 26, options which means that the algorithm will have to run over 26, times. Ultimately, however, the focus will be on accuracy and the algorithm will be required to perform an extremely large number of calculations. Such a problem makes this method rather impractical. 5.3 Fast Fourier Transform The Fast Fourier Transform (FFT) is a powerful solution to this problem of efficiency. The FFT significantly reduces this computational cost and makes the model much more attractive and is likely to be the main cause for the popularity of the Heston model. Take f( ) as the density function of the log-return of the underlying price process, then (subject to the assumption that f is an integrable function) the Fourier transform of f is given by: F (φ) = f(x) = 1 2π + + e iφx f(x)dx e iφz F (φ)dφ where F is the characteristic function of f, [3]. The FFT method is very practical in calculating summations of of the form: n e i 2π n (h 1)(k 1) g(h) (7) h=1 for some function g( ), [3]. This is useful to us because the fair price of a call option is given by: 22

23 C(S t ) =E [ (S T K) +] = = = = + + log(k) + log(k) + log(k) (S T K) + f(log[s T ])d(log[s T ]) (S T K)f(log[S T ])d(log[s T ]) (e log[s T ] e log[k] )f(log[s T ])d(log[s T ]) (e x T e log[k] )f(x T )dx T which can be efficiently approximated by a sum in the same form as (7). After some extensive calculus, Moodley [3] gives us the following form of the characteristic function, under the Heston model. C T (k u ) e αku π N e 2π N (j 1)(u 1) e ibvj F CT (v j ) η 3 (3 + ( 1)j δ j 1 ) j=1 The value of delta will need to be estimated here and will be very sensitive to changes/errors in the price of the option; this approximation of the call price will clearly need to be very accurate. Although this method suggests that estimates will be calculated extremely quickly, we must question whether this method is appropriate here. FFT methods are very useful in signal processing because they will determine which frequencies are present in the time series and the amplitude of each frequency, [2]. This has many applications but in order to apply it here, we must assume that the daily stock prices can be decomposed into sinusoids of different frequencies. It is not clear if this assumption makes sense due to the unpredictable nature of the stock market. An attempt at a justification may be that the stock price is so unpredictable that we can treat it as a highly noisy process composed of many different frequencies of noise but this is not an intuitively sound justification. 23

24 Parameter value 5.4 Model Calibration In order to calculate a hedge as before, we must calculate the value of delta. To perform this calculation, we require the parameters of the Heston model. Thus, the parameters of the Heston model will need to be estimated from the data, i.e. calibrated to the data. For each day, we can use a simple method to fit the parameters on that day. Under our current assumptions, the parameters (i.e. γ, θ, κ, p) are constant in time so it seems contradictory to make separate estimates each day but this will help us to see if their structure is consistent and to test our assumptions. All parameters for a given day are calibrated to the options which become available on that day. A simple calibration method is to take the set of parameters (along with a value for the instantaneous volatility) for which the estimated option value is closest to the observed option prices. That is to say, on a day where n options become available, we will take the parameters, φ, that minimise: n [Observed option price FFT(φ)] 2 i=1 where FFT is a function to estimate the value under the Heston model. These fitted parameters are shown in Figure 13. Fitted Heston Parameters instantaneous volatility rate of relaxation (gamma) long run mean (theta) volatility of volatility (kappa) correlation (p) Days (2nd Jan - 5th June, 214) Figure 13: Calculated daily parameter values for the Heston model On the whole, the Figure seems to clarify that this fitting technique is sufficient because each curve has such a consistent structure. The only problem with this consistency is the sharp spikes, which appear in all model parameters (γ, θ, κ and p) in a synchronized way at about three different days; this is not a disaster because there is a total of 17 days in which options become available. These unexpected jumps in all model parameters do not appear in the estimate of the instantaneous volatility. This suggests that the spikes could not have been caused by some 24

25 news or event involving the underlying stock. To elaborate further, these discontinuities occur on the following dates: 28-Jan, 19-May and 5-Jun. Usually we would associate such jumps in volatility structure with the anticipation of an event. For example, in the lead up to an earnings announcement we can expect the (implied) volatility to rise because the writers of the options want to limit the potential risk in case the earnings announcement has a significant impact on the market. However, these announcements for the first and second quarter occurred on April 17th and July 18th. So we can expect some other news, maybe news which is indirectly related to this stock, to be the cause of this if it is not simply an error in our calibration. For this reason, we shall consider sharp changes as an error in model fitting; it is not unreasonable to expect some anomalous results here since our method was so minimal. Formally, we shall use the following condition. If the estimate of a parameter, on day m, has increases by 1% of its original value, on day m 1, then we shall say that the estimate for day m is an anomaly and we shall replace its value with the original estimate from day m 1. Removing each anomaly in this way results in the estimates to the left of Figure 14 which are arguably more reliable..9 Parameters with each anomaly removed.9 Smoothed Parameters gamma theta kappa p Days Days Figure 14: The graph to the left shows the daily estimates of the Heston parameters with each anomaly removed, and the graph to the right shows the smoothed parameters, i.e. a running mean of the estimates. Defining an anomaly in this way does not seem to have completely solved the problem for parameter theta since we still see some spikes, especially towards the end. However, the estimated value of theta does appear to gradually increase towards the end so it seems that the difference is systematic rather than an anomaly. An evolving structure would contradict our model assumptions but this gradual change does not seem significant enough to stop our model from being effective. Considering that the structure of each parameter is assumed to be constant, we can go a step further and smooth the daily estimates for the parameters using their running mean (mean of all previous estimates). This method results in the estimates on the right of Figure 14. These look convincing and we will use both sets of parameters to compare hedge performance. 25