Volatility Forecasting and Interpolation

Forecasting volatility is important to financial asset pricing because a more accurate forecast will allow for a more accurate model to price financial assets. Currently the VIX is used as a measure of volatility in the market as a whole, but a major issue with this is that it is calculated based on manually traded options on the S&P 500. Another method of forecasting volatility is that of solving for volatility from the Black-Scholes model in option pricing, but this method is not consistent across prices; for different strike prices, a different volatility will be found, creating what is known as a volatility smile. I will develop a method which calculates a similar measure of volatility to the Black-Scholes method and the VIX, but using electronically traded options on the SPY ETF which tracks the S&P 500. I will also be incorporating the mathematical model developed by Britten-Jones and Neuberger in their 2000 paper, which is another variation from the method in which the VIX is calculated. The method developed will provide a smoother and more accurate forecast of volatility over any given time frame, with a 30 day forecast being the industry norm. The method will also have the ability to forecast volatilities for individual assets, not simply the whole market. Volatility Forecasting and Interpolation 2 Volatility plays an important role in financial markets, as an indicator of returns and the magnitude of possible changes in prices for securities. Volatility is defined in this paper as an annualized standard deviation of the percentage change in prices; the standard deviation must be annualized in order to compare the different methods. It is inaccurate to compare the standard deviation of daily percentage price changes to the standard deviation of monthly percentage price changes. One can see historical examples where fast, large changes in volatility were accompanied by a financial crisis; a recent example is the crisis of 2008, where the VIX peaked at well above average at 50. Even though volatility is a basic parameter of instruments, such as expected return (the amount of return seen by the security on average over a given time frame) and the beta associated with a stock (a measure of risk when running a regression of returns against a market index), there is still quite a bit of work involved in forecasting and predicting volatilities in the performance of financial securities. One of the methods used is to evaluate the variance in the percentage changes of prices over a previous period of time, but there is no guarantee that future periods will abide by this same parameter estimate. Another method, which is preferable to the backwards looking estimate from changes in the price is to use an ARCH or GARCH model to forecast future volatility. A major detraction from this method is that the future is outside of our data set, one would need to extrapolate into new periods, which can be dangerous under the best of circumstances. One of the major uses of volatility forecasting is the ability to accurately and precisely value an option. An option is a financial derivative which provides the holder the right to purchase or sell the underlying at a specific price. For example, if an investor were to purchase a call option on stock XYZ with a strike price of $15, then he could purchase the stock for $15 Volatility Forecasting and Interpolation 3 even if the stock price is above that value at let’s say $25; in the event the price of the stock falls below $15, the holder of the call option is under no obligation to pay $15 if the stock is below this value, at let’s say $10. The process is similar for a put option, except that with a put option, the right of the holder is to sell, not to buy at a given price. In essence, a call option is a bet on the security going up beyond a certain threshold, while a put option is a bet on the price of the security falling below a threshold. A major difference between simply going long a stock or short a stock and purchasing a call or put is that options have a time frame before they expire and become worthless. Due to this, pricing an option can be mathematically difficult and the price determined by various formulas and methods can be extremely sensitive to the projected volatility of the underlying security over the life span of the option. With the introduction of the Black-Scholes model, one can look at the price of an option on a security and use computer software to solve for the implied volatility of the option given all of its other parameters which are known. In traditional Black-Scholes option pricing, the volatility is considered a known, and the final price is calculated using their famous model. Using numerical methods, one can work from the price of an option, which is known from financial exchange data and is readily available, backwards to the volatility which traders are placing on the underlying security. This is called the implied volatility of the underlying security, which is often used as an indicator of future volatility moving forward throughout the option’s life span. This method has a few issues from a forecasting perspective. First, the volatility is not known or calculated and then incorporated into the Black-Scholes model to arrive at an option price, but rather numerical methods are used to arrive at what the volatility would need to be in order to arrive at the current price of the option. Second, an instrument will have a Volatility Forecasting and Interpolation 4 single volatility over a given time frame, but volatility smiles do exist, such as in the table below for options of the S&P 500 ETF. A volatility smile is when options at different strike prices will yield differing implied volatilities of the underlying security over the same time period, which is impossible. In the example detailed above, the value of $10 is the strike price for our option. Date of Expiration Strike Price Implied Volatility May 6, 2016 $180.00 44.95% May 6, 2016 $190.00 21.53% May 6, 2016 $200.00 16.36% May 6, 2016 $204.00 14.34% May 6, 2016 $204.50 14.07% May 6, 2016 $214.00 10.16% May 6, 2016 $225.00 13.58% All data taken from Yahoo Finance on Apr. 5, 2016 Graphically, this looks like the following: From our sample of seven strike prices from a single date of expiration, we can already see a large disparity between some of these volatilities. Intuition tells us this is a major problem; arguments have been made that one should use those options close to the stock price, such as using the implied volatility of an option with strike price of $50 when the underlying stock is trading at $50.23, but that does not utilize all the information provided by the option chain. Another issue with using the implied volatility is that forecasting implies a change in the volatility, but the Black-Scholes model assumes constant volatility in its formula. Volatility Forecasting and Interpolation 5 This issue is addressed by the 2000 paper “Option Prices, Implied Price Processes, and Stochastic Volatility” by Mark Britten-Jones and Anthony Neuberger in The Journal of Finance. In their paper, they introduce a model which does not use any previous information regarding volatility and is consistent with all available option prices. Their first major result is the following equation, but before that some variables will be defined. St is the price of a stock price at a time t, where time t is the time until expiration of an option on that stock. dSt is the incremental change in the random stock price from time t1 to time t2, which is required to complete the integral on the left hand side. K is the strike price of an option, such as $10 or $50 from the examples above. C is the price of a call option, put options will not be considered in this paper; the price C of a call option is a function of the time to expiration and the strike price at which it is issued. dK is the incremental change in the value of the strike price from K=0 to K=∞, which completes the integral on the right hand side. What this says, is that the expected value of the volatility between two points in time t1 and t2 is defined by the integral of the left hand side. A simplified model when looking at volatility from current time 0 towards a future time t2 is the equation following this paragraph. The simplification can be seen clearly by knowing that C(0,K) = max(S0 – K,0). This equality is a result of option pricing which says that when there is no time remaining in the life span of the option, the price of the call option is either 0 because the stock price is below the strike price, or the price of the call option is the difference between the current stock price and the strike price due to the stock price being above the strike price. Volatility Forecasting and Interpolation 6 A nontrivial task is seeing what role ( dSt St ) is playing in the above equation, in other words: seeing how ( dSt St ) works out to be a volatility measure. The algebraic steps are as follows: dSt St = μ dt + σ dB This first step is an equality coming from Ito’s Lemma and its application in the Black-Scholes paper “The Pricing of Options and Corporate Liabilities” in 1973. Here, μ is the expected return of the stock St over time, this is sometimes referred to as the drift and σ is the standard deviation of this movement. ( dSt St ) 2 = (μ dt + σ dB) ( dSt St ) 2 = μdt + 2μσ dtdB + σdB dt = 0 and dtdB = 0 and dB = dt The previous step is another result from stochastic calculus where dt is the change in the determined variable and dB is the random variation. A determined variable is a standard variable, where it changes with a known pattern and any value can be found at any moment. For a variable to follow random changes, such as dB above, the changes are wholly unknown and cannot be found for any given moment not in the past. Simplifying with the zeroes inputted, we achieve: Volatility Forecasting and Interpolation 7