The Black Scholes Model With Analysis of Volatilityecon1.altervista.org/econ/edu/cup/reports/2009/options.pdfmathematical understanding of the options pricing model and coined the

1 Yankai Shao—Bachelor Thesis in Economics, Spring 2009

Bachelor Thesis in Economics Department of Economics School of Sustainable Development of Society and Technology Mälardalen University Supervisor: Dr. Johan Lindén

The BlackScholes Model

With Analysis of Volatility

By

Yankai Shao

Spring, 2009

Västerås, Sweden


I. Abstract

Recently, while the financial crisis is spreading global wide, quite a few of renowned firms, even some governments, are struggling to avoid bankruptcy. Without any doubt, this catastrophe will strongly hit our human beings’ economy. Meanwhile, some experts come out to propose some useful policy both for the troubled firms and governments, but most of which is highly ideal and unrealistic. On the other side, as an investor, we should save our money in the interest-less bank account or fight to gain profit. Obviously, we are seeking for profitably practical method instead of empirical theory. Therefore, in our work, we re-derive the classical mathematical finance model—Black-Scholes model within a self-finance portfolio, and then build up hedge fund using the Black-Scholes parameters. We then examine and compare implied volatilities with historical volatilities to find a better measure. Our result indicates that the implied volatility is a better measure in predicting the future realized volatilities as well as the stock market returns.


II. Acknowledgement

I would like to express my deep-felt gratitude and appreciation to my supervisor Dr. Johan Lindén, who guided and advised me with kindness and fruitful discussion. His willingness to help me expand my knowledge is deeply appreciated. I also would like to thank my parents for their guidance, care and love. Moreover, many thanks to all my teachers for passing on precious knowledge to me and all my friends for supporting me all the time, I will always be grateful to have the honor to meet you at Mälardalen University.


Content I. Abstract .......................................................................................................................................... 2 II. Acknowledgement ........................................................................................................................ 3 1. Introduction .................................................................................................................................. 5 2. The Black‐Scholes‐Merton Model ................................................................................................. 6

2.1 Generalized Wiener Process ............................................................................................... 6 2.2 The process for a stock price ............................................................................................... 6 2.3 Itô’s Lemma ......................................................................................................................... 7 2.4 Volatility .............................................................................................................................. 8 2.5 Assumptions for Black‐Scholes formula .............................................................................. 9 2.6 Derivation of the Black‐Scholes‐Merton differential equation ........................................... 9 2.7 The solution of the Black‐Scholes‐Merton partial differential equation ........................... 10

3. Hedge .......................................................................................................................................... 12 3.1 The hedge parameters ...................................................................................................... 12

3.1.1 Delta ....................................................................................................................... 13 3.1.2 Gamma ................................................................................................................... 13 3.1.3 Theta ...................................................................................................................... 14 3.1.4 Vega ........................................................................................................................ 14

3.2 Numerical illustrations using hedge parameters ............................................................... 14 3.2.1 Delta hedging ......................................................................................................... 15 3.2.1 Delta‐Gamma hedging ........................................................................................... 15

4. Analysis of Volatilities .................................................................................................................. 16 4.1 Description of data and methodology. .............................................................................. 17 4.2 Implied volatility vs Historical volatility ............................................................................. 18 4.3 Efficiency of the implied volatility ..................................................................................... 21

5. Conclusion ................................................................................................................................... 25 6. Reference .................................................................................................................................... 26 7. Appendix ..................................................................................................................................... 27

7.1 Black‐Scholes Calculator using JAVA Applet ...................................................................... 27 7.2 Practical trading strategies. ............................................................................................... 37

7.2.1 Bearish/Decreasing Market .................................................................................... 37 7.2.2 Bullish/Increasing market ....................................................................................... 38 7.2.3 Neutral Market ....................................................................................................... 38 7.2.4 Volatile Market ....................................................................................................... 38

7.3 The step‐by‐step Derivation of the Black‐Scholes differential equation ........................... 39 7.4 The step‐by‐step solution of the Black‐Scholes partial differential equation ................... 40


1. Introduction

In finance, an option is a contract whose owner has the right to buy or sell an asset at a fixed price, the strike price or exercise price, on or before a given date, called the maturity date or expiration date. There are two basic types of options. A call option gives the holder of the option the right to buy an asset by the maturity date for the strike price. A put option gives the holder of the option the right to sell an asset by the maturity date for the strike price. In general, options can be either American or European, a distinction that has nothing to do with geographical location. American options can be exercised at any time up to the maturity, whereas European options can only be exercised at the maturity. Most of the options that are traded on markets, e.g. OMX Nordic, are American. Nevertheless, European options are generally easier to analyze than American options, and some of the properties of an American option are usually derived from those of its European counterpart. In 1973, Fischer Black and Myron Scholes first proposed a formula to price for the stock options, which made a major breakthrough in the field of mathematical finance. Later on, Robert Merton, student of Fisher Black, published a paper expanding the mathematical understanding of the options pricing model and coined the term "Black-Scholes" options pricing model. The model has had a huge influence on the way that trader’s price and hedge options. It has also been pivotal to the growth and success of financial engineering in the last 30 years. Finally, Merton and Scholes received the 1997 Nobel Prize in Economics for this and related work. Though ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish academy. In our work, we first recall the theoretical derivation of the Black-Scholes formula based on a self-finance portfolio consisting of a position in the option and a position in stock, which originally written in the work of Black, Scholes and Merton. Then, we introduce hedge parameters, or the “Greeks”, which are derived from the Black-Scholes model and used for hedge or neutral purpose. As the volatility plays a key role in calculating the option price as well as predicting the future market returns, we carefully analyze two volatility measures, implied volatility and t historical volatility, in a sampling period from February 2001 to April 2009 with 99 monthly volatility data of the S&P 500 (OEX) and NASDAQ 100 (NDX). We finally tested whether the implied volatility is a better measure in predicting the future realized volatilities and stock market returns. Besides, if there is a negative relationship between the implied volatility and stock market returns. In the appendix, we build a Java Applet to calculate the option price with the Black-Scholes model and propose some practical trading strategies. We believe that with a more accurate volatility measure and pricing model, the investors could construct their portfolio based on the trading strategies to make a low-risk and high-return profit.


2. The Black‐Scholes‐Merton Model

We begin with a short recall of the derivation of the Black-Scholes formula within a self-finance portfolio using Black-Scholes partial differential equation approach based on Hull (2008).

2.1 Generalized Wiener Process

The drift rate is known as the mean change per unit time for a stochastic process and the variance rate is the variance per unit time. The basic Wiener process, dw, has a drift rate of zero and a variance rate of 1.0. The drift rate of zero means that the expected value of w at any future time is equal to its current value while the variance rate of 1.0 means that the variance of the change in w is a time interval of length T equals T. Then, a generalized Wiener process for variance x is defined in terms of dw as

where and are constants.

2.2 The process for a stock price

In our case, we assume the price process of a non-dividend-paying stock is a stochastic process which follows a generalized Wiener process with a constant expected drift rate and a constant variance rate. Obviously, the assumption of constant expected drift rate is inappropriate and needs to be replaced by the assumption that the expected return is constant. If S is the stock price at time t, then the expected drift rate in S should be assumed to be

for some constant parameter . This means that in a short interval of time, ∆ , the expected increase in S is ∆ . The parameter is the expected rate of return on the stock. In practice, we should also consider the volatility of a stock price. A reasonable assumption is that the variability of the percentage return in a short period of time, ∆ , is the same regardless of the stock price. This suggests that the standard deviation of the change in short period of time ∆ should be proportional to the stock price and leads to the model


where is the volatility of the stock price, is its expected rate of return, and is a generalized Wiener process.

2.3 Itô’s Lemma

Here we present the general expression of Itô’s Lemma, which was discovered by the mathematician K. Itô in 1951 and plays an unshakable position in financial analysis. To understand Itô's formula in its most simple form, we start with a Taylor expansion to the lowest orders for a function of two variables , :

,12

12

Where S is described by stochastic process given by

, ,

W is Wiener Process with a property

, , 2 , , , Substitute those properties into the equation we get

, , ,12

, ,12 ,

,12 , ,

Which is the Itô's formula. The generalized expression is

, …12

,


2.4 Volatility

The volatility is a measure of the uncertainty about the return provided by the stock. It can be defined as the standard deviation of the return provided by the stock in 1 year when the return is expressed using continuous compounding, In the case we are going to study later, we concentrate on finding the volatility of a stock by estimating volatility from historical data. Therefore, in order to estimate the volatility of a stock empirically, we should obverse the stock price at fixed intervals of time (e.g. daily, weekly, or monthly). Let’s define: n + 1: number of observations Si : stock price at the ith interval, with i = 0, 1, …, n t : length of time interval in years and let

ln

for i = 1, 2, …, n. the usual estimate, σ, of the standard derivation of the is given by

σ1

1

Or

σ1

11

1

Where is the mean of . It’s not easy to choose an appropriate value for n. Generally, more data lead to more accuracy, but the volatility does change over time and data that are too old may not be relevant for predicting the future volatility. A compromise that seems to work reasonably well is to use closing prices from daily data over the most recent 90 to 180 days. An often-used rule of thumb is to set n equal to the number of days to which the volatility is to be applied. Thus, if the volatility estimate is to be used to value a 2-year option, daily data for the last 2 years are used.


2.5 Assumptions for BlackScholes formula

Before turn to derive the formula for the value of an option in terms of price of the stock, we should first create an “ideal condition” in the market both for the stock and for the option. As written is Black & Scholes (1973). 1. The short-term interest rate is known and is constant through time. 2. The stock price follows a random walk in continuous time with a variance rate

proportional to the square of the stock price. Thus, the distribution of possible stock prices at the end of any finite interval is lognormal. The variance rate of the return on the stock is constant.

3. The stock pays no dividends or other distributions. 4. The option is “European”, that is, it can only be exercised at maturity. 5. There are no transaction costs in buying or selling the stock or the option. 6. It is possible to borrow any fraction of the price of a security to buy it or hold

it, at the short-term interest rate. 7. There are no penalties to short selling. A seller who does not own a security

will simply accept the price of the security from a buyer, and will agree to settle with the buyer on some future date by paying him an amount equal to the price of the security on that date.

2.6 Derivation of the BlackScholesMerton differential

equation

In this section, our study mainly lie on one market contains a risk-free bond B that pays an interest rate r and a stock S. The price of the stock follows a Wiener process (Geometrical Brownian Motion) with a constant drift and a stochastic term , where is the volatility. The two securities are given by

·0 1

· ·

0

The initial condition of the bond is 1 and the initial stock price is s. We now consider a portfolio h of the bond and the stock: , , where h holds the number of each security. After the comparison between relative self-finance portfolio and the derivation from Itô's Lemma (the step-by-step derivation is


presented in the Appendix), we finally get

∂∂

12

This is the Black-Scholes partial differential equation. Note that this equation is independent from . In a risk neutral word, we can explain the terms in the PDE as:

the change of value with respect to time t.

the change of value with respect to underlying price.

the change of value with respect to volatility.

the expected change of value of derivative security.

2.7 The solution of the BlackScholesMerton partial

differential equation

For one European option with strike price K, the price of this option FT at maturity T will be max , 0 , which is called the boundary condition of the Black-Scholes partial differential equation (PDE). Therefore, we have

12 0

max , 0

All the notations here adapted to the abbreviate notation in the previous section. As usual, we assume that , is a solution to the PDE above, where

· ·

W is a Wiener process under natural probability measure P. After the Girsanov transformation, we get

· ·

Where r is the risk-free interest rate and V is a Wiener process under risk neutral


probability measure Q. Under risk neutral probability measure Q, the stochastic part vanishes and we finally get:

, , max , 0

Finally we can write this as

, · · √ · ·

· · · Where

ln 2· √

· √

this is the final expression of the Black-Scholes formula (the step-by-step derivation is presented in the Appendix). Here, s is the initial underlying price, K is the strike price of the option, is the volatility on the market, r is the current interest rate on the market, t is the initial time and T is the maturity. When can simply substitute the data into the formula and find the corresponding option price under Black-Scholes model.


3. Hedge

For every investor, any amount of loss is unacceptable. Therefore, before we acquire profits, our first job is to know how to hedge the potential risk. A neutral portfolio is much better than a decreasing one. To hedge is our first step to the paradise of profit.

3.1 The hedge parameters

The hedge parameters or the Greeks measure the sensitivity of the option prices with respect to the dependent variables. These describe the change in the option value if any of the variables , , , or is changing when all others remain constant. The hedge parameters are defined by the partial derivatives:

∆ , Γ , Θ , ν

Where . Before derive the expressions of the four hedge parameters, we should introduce one useful equation. ( represents the standard normal density function and represents the cumulative standard normal distribution function.)

Proof:

√1

√2

√ 1√2

√

1√2

· √ 1√2

· ·√· √

1√2

· · · ·

So the equation is proved. Then we can find the expressions of the four hedge parameters easily. (In the figures below, we show how the price and the Greeks vary in some situations. We use the following values K = S = 50, T = 96 days, r = 4 % and σ = 30%)


3.1.1 Delta

∆ Φ · · · ·

Φ · · · ·

Φ 0

3.1.2 Gamma

Γ∆ Φ

· ·1√

·1 /

√2

Figure 3.1: Delta for a call option as function of the underlying price

Figure 3.2: Delta for a call option as function of time to maturity

Figure 3.3: The value of Gamma as function of the time to maturity

Figure 3.4: The value of Gamma as function of the underlying price


3.1.3 Theta

Θ · · · · · · · Φ

· · · · · · · Φ

√

Φ

3.1.4 Vega

ν · · · ·

· · · · · ·

√

3.2 Numerical illustrations using hedge parameters

In a complete market, an investor takes only a long or short position in call/put options exposes for a risk with the changing of the underlying stock. In order to protect this exposed risk, we will therefore buy a reasonable number of stocks to hedge the options. In this section, we concentrate on some numerical illustrations for Delta-hedging and Delta-Gamma-hedging.

Figure 3.5: The value of Theta as function of the time to maturity

Figure 3.6: The value of Theta as function of the underlying price


3.2.1 Delta hedging

Suppose this is a European option on the market with a strike price SEK 21 and a maturity of half a year, its underlying stock price is SEK 21 with a volatility of 30%. The interest on the market is 1%. We therefore calculate the number of stocks we should buy as a short position in a European call option to hedge the options. , is the option value, a is the number of options and b is the number of stocks. The total portfolio is given by:

· , · With a delta hedge:

0 · · ∆ ·

1

√∆· ln

12 · ∆

10.3 √0.5

ln2121 0.01

12 0.3 0.5

0.129636 … This gives 0.5517, so we need 0.5517a of the total number of options to hedge.

3.2.1 Delta‐Gamma hedging

Now, under the same conditions in the previous section, we also want to be a Gamma neutral, therefore, we should use a second option in our hedge. Suppose there is a put option with a strike price of SEK 20. Then our portfolio is

· , · · , And

0

So we get: · ∆ · ∆

0 · Γ · Γ

The reason why we should introduce one more option is that we need to have Gamma neutral in order to remove the sensitivity in ∆ and for Gamma neutral, one more option should be introduced. We solve the function group above and get

ΓΓ

∆ ∆ ·ΓΓ


4. Analysis of Volatilities

In the previous sections, we introduce the hedge parameters which are very essential for a financial analyst to construct the hedge fund with options. However, in order to get the value of the Greeks, we need to acquire the data of the volatility. Generally, there are two popular approaches to calculate the volatility, implied volatility and historical volatility, both of which are regarded as good measures in some degrees. During this financial crisis period, most of corporations and banks are seeking for some relatively safer ways to guarantee their money. Therefore, they require a higher accuracy and efficiency of the volatility to forecast the future realized volatility to build a nearly risk-free hedge fund. In this section, we analyze and compare the time series behavior of historical and implied volatility of the S&P 500 (OEX) and the NASDAQ 100 (NDX) indices, study various trends about volatility, and conclude a better measure of the volatilities. Previous research finds that the implied volatility is both a biased and an inefficient predictor of future volatility and uncorrelated with the information in past realized volatility. However, according to Christensen & Prabhala (1998), implied volatility does predict future realized volatility in isolation as well as, even better sometimes, in conjunction with the history of past realized volatility. They used volatility data sampled over a long period from November 1983 to May 1995 with one-month options on S&P 100 (OEX) and sampled the implied and historical volatility series at a lower (monthly) frequency. They found that, over the sampling period, the implied volatility is generally higher than the historical volatility but less variance. Then, a test of relationship between implied and historical volatility had been made by conventional analysis. By estimating a regression of the form

, the results indicate that OEX implied volatility is neither unbiased nor efficient. Moreover, the implied volatility has more predictive power than past volatility and, at very least, contains information about future volatility beyond that contained in past volatility. Some further alternative specifications are introduced in their work in order to double-confirm their results. When comparing with previous literature with contradictable ideas, they examined the slope coefficient for implied and historical volatility. Finally, they concluded that in a lower (monthly) sampling frequency instead of a day-to-day or weekly basis, the implied volatility is, indeed, an efficient forecast of future realized volatility that outperforms historical volatility and contains incremental information in forecasting. The results suggest that the implied volatility of at-the-money call options is predictable using a parsimonious set of variables in the market information set.


Fleming (1998) examined the performance of the S&P 100 implied volatility as a forecast of future stock market volatility. The results indicate that the implied volatility is an upward biased forecast, but also contains relevant information regarding future volatility. In his work, he first made a survey about some previous work, compared the works in several aspects, and concluded that traditional regression analysis is biased and perhaps spurious in small samples. In estimating the implied volatility series, an end-of-day window of option prices, an analysis of overlapping daily observations, spurious regression problem inherent in volatility time-series, and an estimator which accounts for the covariance non-stationarity of implied volatility forecast errors are involved. Fleming first interpreted the Black-Scholes implied volatility with its estimation, then introduced the methodology of the implied volatility hypothesis and spurious regressions, and, at last, applied unbiased tests and efficiency and orthogonality tests. After series of tests, he finally concluded that although the implied volatility is biased forecast, it may also stem from the misspecification of the volatility process in the option valuation model and/or the existence of early exercise opportunities; it is efficient with respect to its past forecast errors, and its forecast errors are orthogonal to parameters often linked to conditional volatility, including the historical volatility rate and parameters embedded in ARCH specifications. Furthermore, more recently, Christensen & Hansen (2002) found some new evidence on the implied-realized volatility relation. They extended Christensen & Prabhala (1998)’s work by constructing a trade weighted average of implied volatilities from both in-the-money and out-of-the-money options and both on calls and puts. In their work, the robustness of the unbiasedness and the efficiency of implied volatility is verified for the market of OEX options with a database of a period from April 1993 to February 1997 and same frequencies, maturities of the options, and approached to compute volatilities. After the empirical testing and tests of separate put and call options, they verified Christensen & Prabhala (1998)’s results and even concluded that implied put volatility on average is slightly greater than implied call volatility, possibly because buying put index options is a relatively cheap and convenient way of implementing portfolio insurance.

4.1 Description of data and methodology.

In our sample, we observe on the OEX and NDX from February 2001 to April 2009. We concentrate our research on this 99-month data of both historical volatility and implied volatility. For the monthly historical data, we take the daily closing prices on the OEX and


calculate the daily returns ln as well as monthly standard deviations

for these daily returns. We multiply the monthly standard deviations with the square root of the approximately 252 trading days per year and find the actual monthly volatilities for the OEX during the sampling period. The same approach applies to the NDX in finding the monthly historical volatilities. For the monthly implied the volatility, we take the daily closing price for the VIX (monthly implied volatility of the S&P 500), which are derived from Black-Scholes model, over the sampling period and calculate the average value of these data for each one-month period. The same approach applies to the NDX in finding the monthly implied volatilities VXN. For the monthly closing prices of the S&P 500 and NASDAQ 100, we simply calculate the average of daily closing prices within one month. All the data is taking from CBOE (Chicago Board Option Exchange).

4.2 Implied volatility vs Historical volatility

According to Black and Scholes that the implied volatility should be equal to the historical volatility because the Black-Scholes model is the most accurate approach to price the European options. However, in reality, we observe on the OEX and NDX from February 2001 to April 2009 based on the data from CBOE (Chicago Board Option Exchange) and have the following results.

Figure 4.1: Monthly levels of Implied Volatilities (VIX) and historical Volatilities (HVOEX) for S&P 500


Figure 4.2: Monthly levels of Implied Volatilities (VXN) and historical Volatilities (HVNDX) for NASDAQ 100

As we can observe from the charts above, the implied volatilities are closely related to but considerably higher than the historical volatilities on both markets. Therefore, we could immediately get one conclusion that the Black-Scholes model is just an approximation approach to price the options and the real market is far beyond its assumptions. To be more accurate for our data, we illustrate the numerical data from the two previous charts.

Table 4.1: the Summary Statistics

Table 1 summarized the statistics for monthly levels of the implied and historical volatilities for the S&P 500 and NASDAQ 100 over the same period. As shown in the table, the mean values of the implied volatilities on both markets are higher


than of the historical volatilities, which confirm our conclusion from Figures 1 and 2. The mean closing value for VIX of 21.6582 is higher than the HVOEX mean of 18.4996; and the closing value for VXN of 30.8792 is higher than the HVNDX mean of 24.1528. The historical volatilities are more volatile than the implied volatilities. Firstly, the range of the implied volatilities fluctuate in a smaller range (VIX 51.8200, VXN 51.3215) than the historical volatilities (HVOEX 75.7500, HVNDX 69.8300) on their own markets, which indicates that the implied volatilities only change in a smaller range; Secondly, the mean-to-standard deviation ratio of 1.4406 for HVOEX implies it is more volatile than VIX, which has a ratio of 2.0674. This applies to the HVNDX and VXN, with mean-to-standard deviation ratios of 1.7635 and 2.1329 respectively. Therefore, in this degree, the historical volatility is less reliable for predicting the future volatility than the implied volatility. In order to test our hypothesis, we carefully examine some further statistics about the percentage change in volatility over the same time period on both markets.

Figure 4.3: Monthly % change in VIX and HVOEX


Figure 4.4: Monthly % change in VXN and HVNDX

Figures 3 and 4 clearly indicate that the implied volatilities are less volatile than the historical volatilities. Approximately, VIX’s monthly percentage change ranges from -49.67% to 102.31% while HVOEX’s ranges from -51.95% to 166.34%; VXN’s monthly percentage change ranges from -18.16% to 97.3% while HVNDX’s ranges from -49.25% to 154.73%. As reported in table 1, over the entire sampling period, the mean percentage change for HVOEX is 7.1272% while only 1.3245% for VIX; the mean percentage change for HVNDX is 4.4674% while only 1.3173% for VXN. The comparison above further confirms our hypothesis which insists that the historical volatility is less reliable than implied volatility due to its large range of volatility.

4.3 Efficiency of the implied volatility

Based on the result we conclude above, we now examine the relationship between the implied volatility and stock market returns in order to check the efficiency of the implied volatility in predicting the future market returns. Based on Whaley (2001), there is a negative relationship between implied volatility and stock market returns. We carefully studied the percentages in stock market returns and the implied volatilities, the results are shown in the following graphs. Rising volatility often confirms expectation of declining markets, and corresponds with


downward moving prices, while falling volatility often accompanied with rising markets and supports a bullish trend in the near future.

Figure 4.5: Monthly % change in OEX and VIX

Figure 4.6: Monthly % change in NDX and VXN

These two graphs illustrate how monthly percentage changes in VIX and VXN vary inversely with monthly percentage changes in the S&P 500 and NASDAQ 100, respectively, for the 99 months from February 2001 to April 2009.


Throughout the whole time period, every step of increment in implied volatility is corresponding with a decrement in the stock market in a short future. Moreover, the larger changes in implied volatilities, the larger moving in stock markets returns. The most notable instances of the figures occurred in September 2001 following the World Trade Center terrorist attack, and from late August 2008 following the financial crisis throughout the world. In September 2001, VIX experienced a raise of 59.22% making an -11.53% drop in OEX, while VXN experienced a raise of 38.43% making an -21.45% drop in NDX. In the recently financial crisis, especially in October 2008, VIX suffered a raise of 102.31% making a -20.39% drop in OEX, while VXN suffered a raise of 97.30% making a –21.55% drop in NDX (the largest drop in our sample). The negative relationship between stock market returns and changes in XIV was tested by Whaley (2001) using regression analysis, and confirmed by a negative slope coefficient of the regression. We represent this classical result with using the previous data:

Table 4.2: Regression data of T-Test for significance of relationship

In order to examine whether there is a significant relationship between percentage change of VIX and of OEX, we utilize a two-tailed hypothesis test for the slope of regression, where the null hypothesis is H0: β1= 0, and the alternative hypothesis is Ha: β1≠ 0. As shown in Table 4.2, t-statistic equals -12.3570 and is significantly smaller than the lower bound of 1% level of significance; there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. Therefore, we can conclude that a significant relationship exists between percentage change of VIX and of OEX. We repeat the test for percentage change of VXN and NDX and find the t-statistic equals -8.6656, which is significantly smaller than the lower bound of 1% level of significance; there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. Therefore, we can conclude that a significant relationship exists between percentage change of VXN and of NDX. Given that a significant relationship exists between the percentages change of implied volatility and market indexes, we proceed to examine the type of correlation between them.


Table 4.3: Regression data of T-Test for correlation

We utilize a lower-tailed hypothesis test for the correlation coefficient, where the null hypothesis is H0: ρ= 0 and the alternative hypothesis is Ha: ρ< 0. As shown in Table 4.3, t-statistic equals -12.4216 and is significantly smaller than the lower bound of 1% level of significance; there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. Therefore, we can conclude that a negative relationship exists between percentage change of VIX and of OEX. We repeat the test for percentage change of VXN and NDX and find the t-statistic equals -8.7106, which is significantly smaller than the lower bound of 1% level of significance; there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. Therefore, we can conclude that a negative relationship exists between percentage change of VXN and of NDX.


5. Conclusion

Nowadays, computing the option price using Black-Scholes formula is extremely easy and convenient with the help of computers. However, a comprehensive understanding of the theory of the Black-Scholes model is the fundamental knowledge for every financial engineer or analyst to understand and derive more financial derivatives. Moreover, the basic self-finance portfolio proposed in Black-Scholes model is almost the universal tool to analyze any type of options. Black and Scholes provided us a fast and convenient approach to find the option prices. However, it is just an approximation with a series of ideal assumptions. As the real market is stochastic and unpredictable, we should take the volatility as an indicator of the future market returns in order to make profit or hedge risk. Many previous researches found that the implied volatility is biased and inefficient in forecasting the future movement of both the market and the realized volatility. In contrast, we conclude that the implied volatility is unbiased, efficient, and a better measure in predicting the future realized volatilities as well as stock market return over a longer (monthly) sampling frequency and non-overlapping data. Furthermore, Christensen and Prabhala (1998) used S&P 100 index option data to study the relation between implied and historical volatility, and find that the implied volatility is superior to the historical volatility in predicting the future volatility, especially after the market crash of 1987. Their results are confirmed by Fleming (1998), who examined the quality of implied volatility in predicting market on S&P 100 index option and concluded that the S&P 100 may be used as simply as an index of market sentiment and an alternative method for evaluating asset pricing models. The predictive power of implied volatility has been tested on 35 futures markets by Szakmary et al (2002). Moreover, there is a negative relationship between the implied volatility and stock market returns, which is also examined by Whaley (2001) using regression analysis. At the end of our work, we provide a Java applet with all Java code as an aid for the investors as well as some practical trading strategies. We believe that with solid mathematical finance knowledge, comprehensive understanding of the Black-Scholes hedging parameters, better volatility measures, our markets will be healthier so that we could pass this miserable financial crisis period peacefully.


6. Reference

F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economics, 81:637–654, 1973, Chicago, USA Robert C. Merton, Theory of Rational Option Pricing, the Bell Journal of Economics and Management Science, Vol. 4, No. 1 , pp. 141-183, 1973, Cambridge, USA John C. Hull, Options, Futures, and Other Derivatives, 7th Edition, Prentice Hall: 237~251, 2008 J. Röman, Lecture Notes in Analytical Finance I, Mälardalen University, 2007 M. Brenner and M. Subrahmanyam, A simple approach to option valuation and hedging in the Black-Scholes model. Financial Analysts Journal 50, March/April (1994) 25-28.

T. Coleman, Y. Kim, Y. Li, and A. Verma, Dynamic Hedging With a Deterministic Local Volatility Function Model, Cornell University, 2000 D. Emery, W. Guo, and T. Su, A closer look at Black–Scholes option thetas, Springer Science & Business Media, LLC 2007 John C. Cox, Stephen A. Ross, and Mark Rubinstein, Option Pricing: A Simplified Approach, Journal of Financial Economics (September 1979) G. Bakshi, C. Cao, and Z. Chen, Empirical Performance of Alternative Option Pricing Models, Journal of Finance, V. 52, Issue 5 (Dec. 1997), 2003-2049. B.J. Christensen and N.R. Prabhala, The relation between implied and realized volatility, Journal of Financial Economics 50 (1998) 125-150 J. Fleming, The quality of market volatility forecasts implied by S&P 100 index option prices, Journal of Empirical Finance 5 (1998).317–345 C. Chiu, Analysis of Historical and Implied Volatility of the S&P 100 and NASDAQ 100 Indices, Stern School of Business, New York University, May 2002 A. Szakmary, E. Ors, J Kim and W. Davidson, The predictive power of implied

volatility: Evidence from 35 futures markets, Journal of Banking & Finance 27 (2003) 2151-2175

B. Christensen and C. Hansen, New evidence on the implied-realized volatility relation, The European Journal of Finance 8, 187-205 (2002)


7. Appendix

7.1 BlackScholes Calculator using JAVA Applet

We implement our theory using JAVA Applet to help you find out the price of European options and the values of the Greeks within the Black-Scholes model. The interface of our program is:

Figure 7.1: the Interface of the Black-Scholes model in JAVA

You can choose the style of the option and fill in the fields in the input part, then press

button, can get:

Figure 7.2: the results of the Black-Scholes model in JAVA

These results are accurate and will be very useful for you to construct your portfolio using the proposed strategies in our work. If you are a student or a scholar who is interested in financial engineering, we also attach our JAVA code in the following part:


/**

* Black-Scholes model

* Auther: Yankai Shao

* Mälardalen University */

import java.awt.*;

import java.awt.event.*;

import javax.swing.*;

import javax.swing.event.*;

import javax.swing.table.*;

import java.text.*;

public class BlackScholes extends JApplet

implements FocusListener,

ActionListener{

// class variables

// panels

private JPanel inputPanel = new JPanel();

private JPanel outputPanel = new JPanel();

private JPanel controlPanel = new JPanel();

private JPanel dataPanel = new JPanel();

// texts

private JTextField maturityField = new JTextField();

private JTextField volatilityField = new JTextField();

private JTextField interestField = new JTextField();

private JTextField strikeField = new JTextField();

private JTextField initialField = new JTextField();

private JTextField optionPriceField = new JTextField();

private JTextField deltaField = new JTextField();

private JTextField gammaField = new JTextField();

private JTextField thetaField = new JTextField();

private JTextField vegaField = new JTextField();

// String constants

private final String INPUT_PANEL = "Input";

private final String OUTPUT_PANEL = "Output";

private final String MATURITY = "Maturity (year)";

private final String VOLATILITY = "Volatility";

private final String INTEREST = "Interest Rate";

private final String STRIKE = "Strike Price";

private final String INITIAL = "Initial Price";


private final String OPTIONPRICE = "Option Price";

private final String DELTA = "Delta";

private final String GAMMA = "Gamma";

private final String THETA = "Theta";

private final String VEGA = "Vega";

private final String RADIO1 = "European Call Option";

private final String RADIO2 = "European Put Option";

private final String CALCULATE = "Calculate";

private final String RESET = "Reset";

private final String BLANK1 = "";

private final String BLANK2 = "";

// Error messages

private final String NOT_A_NUMBER = " Enter a number";

private final String NON_POSITIVE = " Enter a positive number";

private final String NON_INTEGER = "Enter an integer number";

private final String M_BIGGER_P = " maturity should be equal or

greater than protection period";

// numerical constants

private final double MATURITY0 = 1.0;

private final double VOLATILITY0 = 0.25;

private final double INTEREST0 = 0.03;

private final double STRIKE0 = 100;

private final double INITIAL0 = 95;

// numerical variables

private double maturity = MATURITY0;

private double volatility = VOLATILITY0;

private double interest = INTEREST0;

private double strike = STRIKE0;

private double initial = INITIAL0;

private int currentRadioButton = 0;

double P,delta,gamma,theta,vega;

// labels

private JLabel maturityLabel = new JLabel(MATURITY);

private JLabel interestLabel = new JLabel(INTEREST);

private JLabel volatilityLabel = new JLabel(VOLATILITY);

private JLabel strikeLabel = new JLabel(STRIKE);

private JLabel initialLabel = new JLabel(INITIAL);

private JLabel optionPriceLabel = new JLabel(OPTIONPRICE);

private JLabel deltaLabel = new JLabel(DELTA);


private JLabel gammaLabel = new JLabel(GAMMA);

private JLabel thetaLabel = new JLabel(THETA);

private JLabel vegaLabel = new JLabel(VEGA);

private JLabel blank1Label = new JLabel(BLANK1);

private JLabel blank2Label = new JLabel(BLANK2);

// Button

private JButton calculateButton = new JButton(CALCULATE);

private JButton resetButton = new JButton(RESET);

// Radio buton group

ButtonGroup group = new ButtonGroup();

// Radio Button

private JRadioButton callButton = new JRadioButton(RADIO1);

private JRadioButton putButton = new JRadioButton(RADIO2);

// Number formatter

private DecimalFormat numberFormatter = new DecimalFormat();

// class methods

public void init() {

// Initialise decimal formatter

DecimalFormatSymbols symbols = new DecimalFormatSymbols();

symbols.setDecimalSeparator('.');

numberFormatter = new DecimalFormat("###.####",symbols);

// get content pane

Container contentPane = getContentPane();

// set titles

inputPanel.setBorder(BorderFactory.createTitledBorder(INPUT_PANEL)

);

outputPanel.setBorder(BorderFactory.createTitledBorder(OUTPUT_PANE

L));

// set content pane

contentPane.setLayout(new BorderLayout());

contentPane.add(dataPanel, BorderLayout.CENTER);

contentPane.add(controlPanel, BorderLayout.SOUTH);


// set control panel

controlPanel.add(calculateButton);

calculateButton.addActionListener(this);

controlPanel.add(resetButton);

resetButton.addActionListener(this);

// set data panel

dataPanel.setLayout(new GridLayout(0,2));

dataPanel.add(inputPanel);

dataPanel.add(outputPanel);

// set input panel

inputPanel.setLayout(new GridLayout(7,2));

inputPanel.add(callButton);

inputPanel.add(blank1Label);

inputPanel.add(putButton);

inputPanel.add(blank2Label);

inputPanel.add(maturityLabel);

inputPanel.add(maturityField);

maturityField.setText(numberFormatter.format(MATURITY0));

maturityField.addFocusListener(this);

inputPanel.add(volatilityLabel);

inputPanel.add(volatilityField);

volatilityField.setText(numberFormatter.format(VOLATILITY0));

volatilityField.addFocusListener(this);

inputPanel.add(interestLabel);

inputPanel.add(interestField);

interestField.setText(numberFormatter.format(INTEREST0));

interestField.addFocusListener(this);

inputPanel.add(strikeLabel);

inputPanel.add(strikeField);

strikeField.setText(numberFormatter.format(STRIKE0));

strikeField.addFocusListener(this);

inputPanel.add(initialLabel);

inputPanel.add(initialField);

initialField.setText(numberFormatter.format(INITIAL0));

initialField.addFocusListener(this);


// set output panel

outputPanel.setLayout(new GridLayout(5,2));

outputPanel.add(optionPriceLabel);

outputPanel.add(optionPriceField);

outputPanel.add(deltaLabel);

outputPanel.add(deltaField);

outputPanel.add(gammaLabel);

outputPanel.add(gammaField);

outputPanel.add(thetaLabel);

outputPanel.add(thetaField);

outputPanel.add(vegaLabel);

outputPanel.add(vegaField);

// set radio buttons

callButton.addActionListener(this);

putButton.addActionListener(this);

group.add(callButton);

group.add(putButton);

callButton.setSelected(true);

}

public void focusGained(FocusEvent e) {

}

public void focusLost(FocusEvent e) {

// obtain source

Object source = e.getSource();

// if the source is maturity

if (source == maturityField) {

maturity = readPositive(maturityField,

maturity,

maturityField.getToolTipText());

return;

}

// if the source is volatility field

if (source == volatilityField) {

volatility = readPositive(volatilityField,


volatility,

volatilityField.getToolTipText());

return;

}

// if the source is interest field

if (source == interestField) {

interest = readPositive(interestField,

interest,

interestField.getToolTipText());

return;

}

// if the source is strike field

if (source == strikeField) {

strike = readPositive(strikeField,

strike,

strikeField.getToolTipText());

return;

}

// if the source is initial field

if (source == initialField) {

initial = readPositive(initialField,

initial,

initialField.getToolTipText());

return;

}

}

// Reads double numbers

private double readPositive(JTextField field,

double oldValue,

String title) {

boolean isOK = true;

double newValue = 1;

try { // test input

newValue = Double.parseDouble(field.getText());

}

catch (NumberFormatException e) { // ERROR message

JOptionPane.showMessageDialog(null,

NOT_A_NUMBER,

title,

JOptionPane.ERROR_MESSAGE);

isOK = false;


}

if (newValue <= 0) { // ERROR message

JOptionPane.showMessageDialog(null,

NON_POSITIVE,

title,

JOptionPane.ERROR_MESSAGE);

isOK = false;

}

if (isOK) {

return newValue;

}

else {

field.setText(Double.toString(oldValue));

return oldValue;

}

}

public void actionPerformed(ActionEvent e) {

Object source = e.getSource();

// if reset button

if (source == resetButton) {

// resetall TextFileds and variables to the initial values

callButton.setSelected(true);

maturity = MATURITY0;

maturityField.setText(numberFormatter.format(MATURITY0));

volatility = VOLATILITY0;

volatilityField.setText(numberFormatter.format(VOLATILITY0));

interest = INTEREST0;

interestField.setText(numberFormatter.format(INTEREST0));

strike = STRIKE0;

strikeField.setText(numberFormatter.format(STRIKE0));

initial = INITIAL0;

initialField.setText(numberFormatter.format(INITIAL0));

return;

}


// if calculate button

if (source == calculateButton){

double

d1=(Math.log(initial/strike)+interest*maturity)/(volatility*Math.sqrt

(maturity))+volatility*Math.sqrt(maturity)/2;

double d2=d1-volatility*Math.sqrt(maturity);

if(callButton.isSelected()){

P =

initial*Erf.cdf(d1)-strike*Math.exp(-interest*maturity)*Erf.cdf(d2);

delta = Erf.cdf(d1);

gamma =

Math.exp(-d1*d1/2)/(volatility*initial*Math.sqrt(2*Math.PI*maturity))

;

theta =

-initial*volatility*Math.exp(-d1*d1/2)/(2*Math.sqrt(2*Math.PI*maturit

y))

+initial*interest*Math.exp(-interest*maturity)*Erf.cdf(d2);

vega =

initial*Math.sqrt(maturity)*Math.exp(-d1*d1/2)/Math.sqrt(2*Math.PI);

}

if(putButton.isSelected()){

P =

-initial*Erf.cdf(-d1)+strike*Math.exp(-interest*maturity)*Erf.cdf(-d2

);

delta = Erf.cdf(d1)-1;

gamma =

Math.exp(-d1*d1/2)/(volatility*initial*Math.sqrt(2*Math.PI*maturity))

;

theta =

-initial*volatility*Math.exp(-d1*d1/2)/(2*Math.sqrt(2*Math.PI*maturit

y))

+initial*interest*Math.exp(-interest*maturity)*Erf.cdf(d2)

-strike*interest*Math.exp(-interest*maturity);

vega =

initial*Math.sqrt(maturity)*Math.exp(-d1*d1/2)/Math.sqrt(2*Math.PI);

}

optionPriceField.setText(numberFormatter.format(P));

deltaField.setText(numberFormatter.format(delta));

gammaField.setText(numberFormatter.format(gamma));

thetaField.setText(numberFormatter.format(theta));


vegaField.setText(numberFormatter.format(vega));

}

}

} And: /**

* cumulative normal distribution

* Auther: Yankai Shao

* Mälardalen University */ public class Erf {

static final int ncof = 28;

static final double[] cof ={-1.3026537197817094,

6.4196979235649026e-1,

1.9476473204185836e-2,

-9.561514786808631e-3,

-9.46595344482036e-4,

3.66839497852761e-4,

4.2523324806907e-5,

-2.0278578112534e-5,

-1.624290004647e-6,

1.303655835580e-6,

1.5626441722e-8,

-8.5238095915e-8,

6.529054439e-9,

5.059343495e-9,

-9.91364156e-10,

-2.27365122e-10,

9.6467911e-11,

2.394038e-12,

-6.886027e-12,

8.94487e-13,

3.13092e-13,

-1.12708e-13,

3.81e-16,

7.106e-15,

-1.523e-15,

-9.4e-17,

1.21e-16,

-2.8e-17};

public static double cdf(double x) {

// Return cumulative distribution function.


return 0.5*erfc(-0.707106781186547524*x);

}

private static double erfc(double x) {

// Return erfc(x) for any x.

if (x >= 0.) {

return erfccheb(x);

}

else {

return 2.0 - erfccheb(-x);

}

}

private static double erfccheb(double z) {

double t=2.0/(2.0+z);

double ty=4.0*t-2.0;

double d=0.0;

double dd=0.0;

double tmp=d;

for (int j=ncof-1; j>0; j--) {

tmp = d;

d = ty*d-dd+cof[j];

dd=tmp;

}

return t*Math.exp(-z*z+0.5*(cof[0]+ty*d)-dd);

}

7.2 Practical trading strategies.

7.2.1 Bearish/Decreasing Market

Negative price spread

1. Purchase one call option with strike price b and one put option with strike price a. (b>a)

2. Sell one call option with strike price b and one put option with strike price a. (b>a)

Ratio-spread with put options

1. Buy two of the higher strike price put options that are near the current price of the underlying stock.


2. Sell one put option at a lower strike price than those purchased.

7.2.2 Bullish/Increasing market

Positive price spread / Bull spread 1. Buy a call option (usually an at-the-money one) with a strike price of a

and a put option with a strike price of a. (b>a) 2. Sell a call option with a strike price of b and a put option with a strike

price of b. (b>a)

Diagonal spread/ Time spread

1. Purchase a far out-the-money call option that has at least several months to the maturity.

2. Sell a close out-the-money call option with a very near expiration date.

Positive back spread

1. Issue one call option. 2. Buy twice as many options with a higher strike and the same maturity.

7.2.3 Neutral Market

Short Straddle

1. Sell a put option and a call option with the same strike price and maturity. 2. Or, buy a underlying stock and twice as many call options.

Short Strangle

1. Sell a put option with strike price a. (b>a) 2. Sell a call option with strike price b. (b>a)

7.2.4 Volatile Market

Buy/Long Straddle

1. Buy one call option and one put option with the same strike price, usually but the at-the-money ones.

2. Or, buy the underlying stock and twice as many put options.


Short Butterfly

1. Sell a call option with strike price b. (b<a<c) 2. Buy two call options with strike price a. (b<a<c) 3. Sell a call option with strike price c. (b<a<c)

7.3 The stepbystep Derivation of the BlackScholes


Based on the previous assumptions and conditions, we have The value of the portfolio, V(t), is defined as:

· · The portfolio is said to be self-finance if:

· ·· · · · · ·· · · · · ·

We then define a relative portfolio , as:

·,

·, 1

The value process in terms of the relative portfolio is:

· · · · · Here, let , . We know from Itô's Lemma that

· · ·

· · · · · ·

To make this similar to dV(t) and use the abbreviate notation ,

, , , , , we then get


· · 12 · ·

·· ·

After comparison, we know that

·

So we get

12 · ·

· · · · ·

And 12 · ·

·

Since 1, we finally get

∂∂

12

7.4 The stepbystep solution of the BlackScholes partial


Based our previous work, we now take the derivate of F with Itô's formula:

∂∂

12

· ·12 ·

·12 · · ·

· · We integrate it and take the expected value, under risk neutral probability measure Q, the stochastic part vanishes and we finally get:


, , max , 0

Now we need to find the expression of ST, let’s assume ln , from Itô's formula we get:

12

1· ·

12

1·

12 ·

Integrate it:

12

ln12

Then we find:

· exp12 ·

The probability distribution of Z is therefore a lognormal distribution 1 2⁄ , . The probability density function g(S) is given by

12

exp ln /2

2

The price of the call option is then given by:

| , max , 0

First, we define the following variables:

12 , , √

Where z is the standard normal distribution. Then

· exp √ ·

And


1√2

expln

21

√2exp

2

By definition above, we have

· · √ ·

· √

And

1√2

exp 21√

For the call option, we have

· , 0

If · 0 ln , and the value of z is

√, which is

the lower limit. We now start to integrate to get the price of the call option from t to T.

· ·

· · √

·

max · · ·√ , 0

Where 1

√2/

Substitute the lower limit condition into the integration and get:


· · · ·√

· · · · ·√ · ·

· ·

· ·

√2· · ·√ /

√2·√ / · · √

Finally we can write this as

, · · √ · ·

· · · Where

ln 2· √

· √

Documents

The Black Scholes Model With Analysis of Volatilityecon1.altervista.org/econ/edu/cup/reports/2009/options.pdfmathematical understanding of the options pricing model and coined the