Option Portfolio Modeling


 Rolf McLaughlin
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1 Value of Option (Total=Intrinsic+Time Euro) Option Portfolio Modeling Harry van Breen Introduction The goal of this white paper is to provide technical background on the workings and philosophy behind long term option portfolio modeling used by It gives an overview of the tools, techniques and methods employed to obtain one of the most accurate methods to valuate an option portfolio. The base concept behind option portfolio management is risk. The price of an option consists of two components, intrinsic value and time value. The intrinsic value is the value between the current price and the expiration price. The time value is the difference between option price and intrinsic value. Market condition play an important part in the time value component Intrinsic Value Time Value Total Value Value of Stock (Euro) Figure Example of the total, intrinsic and time value of an option with an expiration price of 00 Euro. Since the time value of an option is influenced by market conditions, it is virtually impossible to predict its value over time. Only at the expiration date, the time value is known to be zero. There is no model, which predicts the exact price accurately. The value of an option depends on many known and some unknown parameters. The price and volatility of the underlying stock is of prime importance. In chapter the behavior of a stock index is investigated according to Geomatric Brownian Motion (GBM). In this white paper, the price of an option is modeled based on current market prices. Based on options under current market conditions a discrete value surface can be created. A continuous value surface is created by filling the gaps using the Black Scholes model. A complete description is given in 3. Chapter 4 combines the behavioral description of chapter with the pricing prediction
2 model of chapter 3 to valuate future value of an option portfolio. This paper ends with some conclusions. The future valuation of an index The aim of future valuation of an index is to predict the future. From a statistical point of view, the result can not be a single value, but a range of values, each with its own probability of being correct. The behavior of an index can be separated into four categories. Each with its own distinctive characteristics. The categories are: ) Ultra short (speculating on index changes within one day, daytrading) ) Short (speculating on index changes within 6 month) 3) Medium (speculating on index changes within 6 months to several years) 4) Long term (speculating on index changes over 0 years) The methods and techniques described in this paper are related to medium term (6 months  5 years) speculation. For the behavior of the index itself, the Geometric Brownian Motion model is used.. Geometric Brownian Motion The most used mathematical approximation of index behavior is the Geometric Brownian Motion (GBM) distribution. The base assumption behind this model is that the probability density function (PDF) of the profit and/or earnings of a company or market are normally distributed. Consequently, the equation describing the probability function f(v,t) for the index to be V over time t is a lognormal equation. The result is given in the following equation: f (ln( V ) ln( V0 ) t t) t ( V, t) e V t In this equation α is the drift, σ the standard deviation of the index, V 0 is the index value at time t=0 and t is the number of years ahead one is looking... The drift α The drift α is the logarithmic rate of return on the value of the index. Notice that it is the logarithmic rate of return and not the regularly used rate of return by banks and other financial institutions. The relationship is given in the following equation: ln( RR) In this equation RR is the expected rate of return over a year and α the logarithmic rate of return. One can also obtain the drift α over a period of t years with the value of the stock at period end V t by period start V 0, see the following equation. V t ln V 0 t In practice the difference between the logarithmic returns and rate of return is fairly small <0%.
3 .. The standard deviation σ The σ is defined as the standard deviation of the logarithmic returns over a year. This value is equal to the volatility of the index. At least if the volatility is given as the standard deviation of the logarithmic return of the index. This is by no means always the case. In practice the difference between the standard deviation from logarithmic and percentage change is fairly small <0%.. Accuracy of Geometric Brownian Motion To investigate the accuracy of the GBM approximation, the actual change of the AEXindex (Dutch stock market index) in the period is taken. Based on daily closing prices in this period, see the figure below, the values for this index were for drift α=0.064 and for standard deviation σ= AEXIndex Max Min Figure Closing prices of the AEXindex in the period and 95% confidence interval for Geometric Brownian Motion prediction from the st of January 990. Since the closing prices for each day are known, it is possible to compare the probability density function (PDF) calculated by the GBM and the actual PDF based on the historic data for the same period. The result is given in Figure 3.
4 F(%) F(%) F(%) F(%) PDF for % change in Day PDF for % change in Month Data 0,5 Data 0,8 GBM 0, GBM 0,6 0,5 0,4 0, 0, 0,05 05% 0% 5% 0% 5% 0% 5% 040% 0% 0% 0% 40% % Change % Change PDF for % change in 6 months PDF for % change in months 0,07 0,06 0,05 0,04 0,03 0,0 0,0 060% 40% 0% 0% 0% 40% 60% Data GBM 0,06 0,05 0,04 0,03 0,0 0,0 000% 50% 0% 50% 00% Data GBM % Change % Change Figure 3 Probability density function comparison of GBM and actual data from 8 years of AEXindex closing prices. On the horizontal axis the change in index value is given over the period of time. On the vertical axis the probability is shown based on actual data and GBM. Notice that there is a distinct difference between the GBM model and the actual changes in index value. Changes of a number of percents in a day is not uncommon,while the GBM model doesn t allow such large changes in such a short period of time. However, over longer periods the behavior of the index becomes more and more like a GBM. In short, the GBM model is primarily intended to predict medium and long term (> 6 months) changes in stock index value and not short term. For short term valuation (<6 months), numerical historic data is more accurate. Short term valuation falls outside the scope of this paper. 3 Future value of an option The value of an option depends on many factors. A very small change in stock value means a large change in option values. This large and uncertain leverage makes investments in options more uncertain. One might state that there is no model, which comes close to market behavior. On the other hand, there are some parameters, which have a distinctive influence on the value of an option:  value of the stock  volatility of stock  strike price  time until expiration  dividend payout  interest rates The method presented in this paper is based on current option prices and extrapolates these into the future. The assumptions and base method, which we dubbed value surface is described in paragraph 3.. This value surface is a discrete approximation of market behavior and based on what
5 Option Value (Euro) is known as the volatility smile. Chapter 3. describes how the Black Scholes model is used to fill the gaps between fixed points to create a continuous model. 3. Creating a value surface The value surface concept is based on the assumption that the price of an option is defined by: ) value of the stock ) strike price 3) time until expiration 4) market conditions By market condition all variables are meant, which define the value of an option except the three other given. Market conditions can be perceived as a black box, which include parameters such as dividend, interest rate, volatility, etc. etc. By assuming that this black box is an unknown parameter, it is possible to reverse engineer the value of the black box from current market data. The values and parameters inside the black box determine the time value of the option, see. The value surface is nothing more or less than the value of all the Put or Call options from a stock combined into a single 3D graph. The xaxis on this graph is the strike value in percentage of current underlying stock or index value, the yaxis the time until maturity and the zaxis the value of each individual option. An example is given in Figure Expiration time (years) Strike Value (% stock value) Figure 4 Value Surface: Option values in relation to time to maturity and strike value In the example figure, each dot is an actual option taken from current market prices. As can be seen, there are gaps between the values taken from market. To create a continuous model, the gaps between the options need to be filled with an accurate approximation. This can be done using the Black Scholes model.
6 Implied Volatility (%) 3. Current valuation and filling the gaps using BS model The gaps between individual options in the discrete value surface, Figure 4, is filled using the Black Scholes equation. The BlackScholes equation is a common way to valuate European style options (such as index options) and is based on Geometric Brownian Motion of the index value. Taking into account dividend the BlackScholes equations become: C( V, t) e P( V, t) Ee d d s V ln r q E t V ln r q E t N ( x) qt V N rt x N e rt d Ee N d d V N d k dk t t C(S,t) P(S,t) V E N(x) r t σ q Price of Call option Price of Put option Value of underlying stock Expiration price of option CDF of standard normal function Interest rate of risk free state bond Time until maturity Volatility of index (logarithmic version) Dividend yield of underlying stock Options, with expiration and strike prices in the gaps of the discrete volatility surface can now be approximated. The BlackScholes equation requires values for: interest rate, volatility, value of underlying stock, expiration price, time until maturity and dividend yield. All variables within the Black Scholes model are constant throughout the value surface. Therefore, instead of using the (historic) volatility of the index, the implied volatility is used. Implied volatility is based on the market price of an option and then calculated inverse using the Black Scholes equation. Since there is no inverse Black Scholes equation, the inverse calculation has to be done numerically. For example, using the Newton Raphson method. Replacing the price of the options in the value surface with their implied volatility provides a new 3D graph where the zaxis is the implied volatility. This transformation is given in the figure below Expiration time (years) Strike Value (% stock value) Figure 5 Result of implied volatility graph from figure 4.
7 Option Value (Euro) Notice that the implied volatility for options, which are just about to expire is much higher compared to options with longer expiration times. For options with a long expiration time, the implied volatility becomes near constant. Using the implied volatility and the Black Scholes equation it is possible to fill the gaps in the value surface, thus creating a continuous value surface. The endresult is given in Figure Expiration time (years) Strike Value (% stock value) Figure 6 Continuous Value Surface: Option values in relation to expiration time and strike value 3.3 Predicting future option prices using the Value Surface Using the value surface graph it is relatively easy to extrapolate the value of an individual option into the future. By changing the expiration time and strike value according to future values, the best estimate for the future price can be obtained. This is best illustrated by means of an example. Example: Question: What is the expected price of an option one year from now when the index increases 0% in that year. The option of interest has a strike value of 480 and expires over three years. At present, the index is 43. Evaluation: The present value of the option can be taken from the value surface (Figure 4): Strike value 480/43=% Option value 45,68 Euro Expiration 3 years When One year predicting from now, future the options index prices is assumed an option to have might increased fall outside 0% the and value thus becomes surface boundaries. If that 43+0%=475,. is the case, the The best expiration guess is time taken reduces by expansion one year of and the the value value surface can be outside taken its from current the value boundaries. surface: Expansion is achieved by taking the implied volatility of the option nearest to the requested Strike option value price 480/475,5=0% and using the BS equation accordingly. Option value 5,8 Euro Expiration years Answer: Expected value is 5,8 Euro.
8 3.4 Obtaining values for risk free interest rate and dividend yield There are two constants within the value surface, the risk free interest rate r and the dividend yield d. Both can be obtained from the market. For the risk free interest rate, the interest on a 0 year state bond is taken (TBond). In general one can state that state bonds are risk free. The 0 year bond is chosen because it provides the best match. The dividend yield of a stock can be obtained with putcall parity. Developed in medieval England, the putcall parity equation derives from the lemma that a portfolio consisting of cash + call option equals underlying stock and put option. Both options have the same strike price. Assuming a continuous dividend paying underlying stock, as usually done for an index, this all translates into the following equation. C(S,t) P(S,t) V E r t q Price of Call option Price of Put option Value of underlying stock Expiration price of option Interest rate of risk free bond Time until maturity Dividend yield of underlying stock In this equation, all variables are known except for the dividend yield q. Using the least squares equation applied to all options on the market for a certain stock/index it is fairly easy to determine the current dividend estimation. Using the putcall parity lemma is an accurate and fast way to determine the market estimation of future dividend payouts on an index. 3.5 Accuracy of value surface approach The value surface approach is by no means a 00% accurate way to determine future option values. However, it is a well founded best guess. The values estimated are based on current market sentiment and conditions, which can fluctuate widely on a daily basis. Like the Geometric Brownian Motion estimation of market movements, the value surface approach works best for long term options. An example of an accuracy test is given in Figure 7.
9 Figure 7 Result of an accuracy test of the surface model. The prediction of current option prices is compared to the actual current option prices. For example, the call option at strike value 500 and expiration date 7dec00 was estimated to cost 55,4 Euro using market condition on jan08. This is 6% lower than the actual market price of 58,75 Euro. The accuracy of options outside the value surface deteriorates and it is ill advised to found ones option strategy on options far removed from the boundaries of the value surface. The value surface approach is very robust to changes in implied volatility, market interest rates, dividend payouts and rate of returns. The reason is that these values apply to the gaps in the value surface and are only used to finetune the prediction of values within the gaps. 4 Valuation of a Portfolio An option portfolio is a combinations of options and cash forming the value of the portfolio. Valuation of individual options is done using the value surface. In this chapter value calculation of the entire portfolio is calculated under the assumption that the cash is invested in (state) bonds. In this chapter the future value of an option portfolio is investigated. This is done by two representation graphs based on two variables (time and index prices).
10 DOW FTSE 00 NIKKEI 5 BRENT SPOT CRB INDEX GOLD $ US M3 TB JPM US BOND INDEX JPM GLOBAL BOND INDEX 4. Why state bonds? Investing the cash component into (short term) state bonds is the best guaranty on a fixed return rate (interest rate on state bond),e.g.: TBills. Since the guaranty on state bond is 00%, the return rate is the lowest in the market available. Therefore, a more lucrative alternative could be to invest in company bonds, gold, currency, stock, stock index, or other investment opportunities. When investing the capital into an alternative to state bonds it is important to realize that risk increases rapidly with the correlation between the index used for the putcall options and the sector one is investing into, see Figure 8. DOW FTSE NIKKEI BRENT SPOT CRB INDEX GOLD $ US M3 TB JPM US BOND INDEX JPM GLOBAL BOND INDEX Figure 8 Correlation matrix between indices, bonds, oil and gold. Correlation is based on data The correlation value is if there is a 00% correlation and 0 if there is no correlation what so ever. The value is negative if the correlation is negative, e.g.: the JPM US Bond index goes up as the US M3 TB goes down. The following guidelines are often used for evaluation of correlation values: is the absolute value below 0., no correlation is measured. Is the correlation above 0.5, there seems to be a large correlation. When designing an option portfolio and one doesn t want to take too much a risk, try to make sure there is no correlation between the value of the options (its underlying index) and capital. 4. Capital component growth in an option portfolio For the option portfolio valuation calculation, it is assumed that the capital is invested into state bonds with a fixed return rate over a given period. In addition, it is assumed that any interest payouts are directly reinvested into the same state bond. Any changes in the cash position due to expiration of options will directly result in selling and/or buying of extra bonds. All this combined results in the following equation for the capital component of the option portfolio.
11 Cash(t) r t Cash 0 N t i EV i (t) Cash at time t Interest rate of state bond time in years Cash at time 0 Number of options in portfolio Time until expiration of option i Expiration value option i at time t This equation defines the value of cash at time t. All cash flow generated by expiring options within the portfolio are taken into account. 4.3 Value of an option portfolio over time The value of the option portfolio is the sum of its capital value and each individual option. The capital value is defined by the equation given in chapter 4. and the value of each individual option is given in chapter 3. Assuming that interest rates, dividend yields and market sentiment are constant, the only unknown factor is the value of the index in the future. Since it is not possible to determine the future value, it is imperative that the future value of an option portfolio is expressed as a function of time and/or index value. Roughly there are two ways to graphically represent the value of the option portfolio:. Estimate the value with time fixed and underlying index variable. Estimate the value with time variable and underlying index fixed In addition, the most obvious alternatives are plotted into the same graphs. Since one of the main objectives of option portfolio s is to outperform the underlying index, the underlying index is an obvious choice. An example of the first representation type is given in Figure 9.
12 Figure 9 Option portfolio containing three Dutch AEXindex options and starting capital 30kEuro. The Brown line signifies the starting capital (30 keuro), the red line is the value of the portfolio if invested in the index itself, the black line represets the value on a savingsaccount and the green line represents the value of the option portfolio itself at different times in the future (,,3 and 4 years from now). The value (vertical axis) depends on the future value of the AEXindex itself (horizontal axis). In this figure, the value of the option portfolio is always above an investment into the index itself. Notice that the option portfolio is not so lucrative anymore in 0. The reason for this is that options with an earlier expiry date are sold/bought, but no alternative strategy is taken into account. In the example case one option expired in dec00, but no options are to be bought/sold at expected at that time. In reality, this is probably not the case. Therefore, one has to take expiration dates into account when evaluating the portfolio. An alternative representation to that of evaluation of the portfolio value at a fixed time into the future is to fix the index value and plot the value over time. En example of this second representation is given in Figure 0.
13 Figure 0 This figure represents the value of the option portfolio and compares it with the value it would have been if an investment was made into the underlying index. On the horizontal axis the future time in years is given and on the vertical axis the value of the option portfolio is given. A worst/base/best case is plotted, where the thick lines are the option portfolio value and the thin lines the alternative underlying index investement. In this example, the obvious thing to do is to close the portfolio over,4 years. At that time the value of the option portfolio is always higher compared to an investment into the index. At that time the improvement is 35 keuro absolute or 7%3% relative. The example was taken from actual market data on sep Conclusions In this white paper a comprehensive description of option portfolio modeling is given. The main conclusions can be summed as follows: The geometric Brownian motion is a fairly good approximation of future stock value behavior. The longer away the prediction is made for, the more accurate it is. The value surface approach to option valuation is one of the best approximation of predicating future option values. It is not based on a model, but uses current option valuations to predict future valuation under different market conditions. It is possible to create a option portfolio consisting of state bonds and index options, which outperforms the market index.
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