Volatility

Significance of volatility in Derivatives

Project Report On

SIGNIFICANCE OF VOLATILITIES IN DERIVATIVES

SUBMITTED BY

Priyanka JainMMS FINANCE

BATCH: 2007-09SUBMITTED TO

PROF. NAVEEN BHATIA

HEAD-Finance

UNIVERSITY OF MUMBAI

N. L. DALMIA INSTITUTE OF MANAGEMENT STUDIES & RESEARCH

Certificate

This is to certify that Miss Priyanka S. Jain, student of N.L. Dalmia Institute of Management Studies and Research, has successfully carried out the project titled SIGNIFICANCE OF VOLATILITIES IN DERIVATIVES, under my supervision and guidance as partial fulfilment of the requirements of MMS course, Mumbai University Batch 2007-2009Prof. Naveen Bhatia

Prof. P.L. Arya

Project Guide

Director

Date:

Place: MUMBAI

Acknowledgement

I take this opportunity to express my deep and sincere gratitude to Prof. P.L. Arya and Prof. Naveen Bhatia for his valuable guidance and encouragement in implementing the project. It is because of his efforts that I was able to cover the manifold features of the project.

I also acknowledge the infrastructural support provided by N. L. Dalmia Institute of Management Studies & Research.

Finally I am thankful to all the friends who have given their full support in collecting the required information and continuous help during the preparation of the project.

Priyanka S. JainMMS Finance

TABLE OF CONTENTSParticularsPage

Executive summary5

Introduction to volatility

Underlying price as mean of distribution Volatility as Standard deviation Lognormal distribution7

Types of volatility

Historical volatility Implied volatility Volatility skew Estimating Volatility16

Volatility Strategies

Long straddle and short straddle Long strangle and short strangle35

Derivatives Greeks and Volatility

Delta

Gamma

Theta

Vega46

Bibliography65

EXECUTIVE SUMMARY

While the very core of derivative products is to manage risk, it is important to appreciate that all derivatives are highly geared, or leveraged, transactions. Traders/investors are able to assume large positions - with similar sized risks - with very little up-front outlay and the risk to the investor is high. A thorough grasp of product technicalities is only one aspect of the knowledge and skills that traders require. Every trader has a view of the market and their end objective is, of course, profit from that view. And the most effective route to achieving this is to form a view that proves to be correct, having positioned one's self to obtain the maximum profit from it

By their very nature financial markets are volatile. Through the use of derivative products, it is possible to manage volatility and risks of faced by the financial agents. Given the different risk bearing capacity of them, with some of the agents being risk-averse and some risk-lover, derivatives emerged essentially to satisfy both of them.

Derivatives are financial contracts whose values are derived from the value of an underlying primary financial instrument, commodity or index, such as: interest rates, exchange rates, commodities, and equities.

Volatility plays a great role in derivatives, especially in options. Volatility is both the boon and bane of all traders you cant live with it and you cant really trade without it. If a trader estimation of volatility is right, than through different volatility strategy he could make a lot of profit. Now many companies gives more emphasis on volatility related activities, because one can make money if one have the knowledge about it, whether market moves up or down.

As part of my project Derivatives and all about volatilities, I cover almost all the concept related to volatilities.

My project covers

Studied volatility and its importance in derivatives, Types of volatility, how to calculate different types of volatility i.e. Historical volatility, Implied volatility and Estimating volatility through GARCH model.

Studied the Volatility Strategies to handle the market volatility effectively and encase on it.

Studied the Derivatives Greeks used in derivatives market such as Delta, Gamma, Theta, Vega and Rho. And also studied how hedging can be done using the derivatives Greeks.

Studied the importance of Greeks on volatility spread

Introduction to Volatility

Volatility is a statistical measure of the amount of fluctuation in a stocks price within a period of time. A stock with high volatility would have rapid up and down movements in its stock price. A stock with very little movement in its price would constitute low volatility.Volatility is both the boon and bane of all traders you cant live with it and you cant really trade without it. Most of us have an idea of what volatility is. We usually think of choppy markets and wide price swings when the topic of volatility arises. These basic concepts are accurate, but they also lack nuance.

Some more about volatility Volatility is simply a measure of the degree of price movement in a stock, futures contract or any other market instrument.

Volatility is a statistical measure of a market or a security's price movements over a period of time. Mathematically volatility is often expressed as standard deviation.

It is a measure of the speed of the market. Market which move slowly are low volatility markets; markets which move quickly are high-volatility markets.

Volatility is very important for option trader. Option trader, like a trader in the underlying instrument, is interested in the direction of the market. But unlike the trader in the underlying, an option trader is also extremely sensitive to the speed of the market. If the market for an underlying contract falls to move at a sufficient speed, option on that contract will have less value because of the reduced likelihood of the market going through an options exercise price.If we knew whether a market was likely to be relatively volatile, or relatively quiet, and could convey this information to a theoretical pricing model, any evaluation of options on that market would be more accurate than if we simply ignore volatility.

If we assume only that the price movement of an underlying contract follows a random walk (random walk means one cant predict the path), and nothing about the likely direction of movement, the curves might represent possible price distributions in a moderate volatility, low volatility, and high volatility market, respectively. In a low volatility market, price movement is severely restricted, and consequently options will command relatively low premiums. In a high volatility market the chances for extreme price movement is greatly increased, and options will command high premiums.Since the different price distribution is symmetrical, it may seem that increased volatility should have no effect on an options value. After all, increased volatility may increase the likelihood of large upward movement, but this should be offset by the equally greater likelihood of large downward movement. Here, however, there is an important distinction between an option position and an underlying position. Unlike an underlying contract, an options potential loss is limited. No matter how far the market drops, a call option can only go to zero. In our example, whether the market finishes at 80 or 108 at expiration, the 105 call is worthless. However, if we buy the underlying contract at 100, there is a tremendous difference between the market finishing at 80 or 104. With an underlying contract all outcomes are important, with an option, only those outcomes which result in the option finishing in the money are important. This leads to an important distinction between evaluation of an underlying contract and evaluation of an option. If we assume that prices are distributed along a normal distribution curve, the value of an underlying contract depends on where the peak of the curve is located, while the value of an option depends on how fast the curve spreads out.

A normal distribution curve can be fully described with two numbers, the mean and the standard deviation. Graphically, we can interpret the mean as the location of the peak of the curve, and the standard deviation as a measure of how fast the curve spread out. Curve which spread out quickly, have a high standard deviation; curves which spread out very slowly, have a very small standard deviation. What is important to an option trader is the interpretation of these numbers, in particular what a mean and standard deviation suggest in terms of likely price movement.

The standard deviation not only describes how fast the distribution spread out; it also tells us something about the likelihood of an event ending up in a specific range. In particular, the standard deviation tells us the probability of a ball ending up in a specific range which is a specific distance from mean. For example, we may want to know the likelihood of a ball falling down through the maze and ending up in a trough lower than 5 or higher than 10. We can answer the question by asking how many standard deviations the ball must move away from the mean, and then determine the probability associated with the number of standard deviations.For Option traders the following approximation is useful:

+1 Standard deviation takes in approximately 68.3 %( about 2/3) of all occurrences. +2 Standard deviation takes in approximately 95.4 %( about 19/20) of all occurrences.

+3 Standard deviation takes in approximately 99.7 %( about 369/370) of all occurrences.

Each number of standard deviations is preceded by a plus or minus sign. Because normal distribution is symmetrical, the likelihood of an up movement and down movement is identical.

The characteristics of normal distribution have been so closely studied that formulas have been developed which facilitate the computation of both probability associate with every point along a normal distribution curve, as well as the area under various portions of the curve. If we assume that price of an underlying instrument is normally distributed, these formulas represent a unique set of tools with which we can solve for an options theoretical value. This is one of the reasons Black and Scholes adopted the normal distribution assumption as part of the model.UNDERLYING PRICE AS THE MEAN OF A DISTRIBUTIONWhen we enter the present price of an underlying instrument we are actually entering the mean of a normal distribution curve. An important assumption in the B & S model is that, in the long run, a trade in the underlying instrument will just break even. It will neither make money nor lose money. The mean of the normal distribution curve assumed in the model must be the price at which a trade in the underlying instrument, either a purchase or sale, would just break even.VOLATILITY AS A STANDARD DEVIATION

In addition to the mean, we also need a standard deviation to fully describe a normal distribution curve. This is entered in the form of volatility. With some slight modification, we can define the volatility number associated with an underlying instrument as a one standard deviation price change, in percent, at the end of a one year period.

For example, suppose that an underlying futures contract is currently trading at 100 and has a volatility of 20%. Since this represents a one standard deviation price change, one year from now we expect the same futures contract to be trading between 80 and 120 approximately 68% of the time, between 60 and 140 approximately 95% of the time, and between 40 and 160 approximately 99.7%.

If the underlying contract is a stock currently trading at Rs. 100, then the 20% volatility will have to be based on the forward price of the stock at the end of one year. If interest rate is 8% and the stock pays no dividends, the one-year forward price will be 108. Now a one standard deviation price is 20%*108=21.60. So one year from now we would expect the same stock to be trading between RS. 86.40 and Rs. 129.60 approximately 68% of the time, between Rs.64.80 and Rs.151.30 approximately 95% of the time, and between Rs.43.20 and Rs.172.90 approximately99.7% of the time. LOGNORMAL DISTRIBUTIONSIs it reasonable to assume that those prices of an underlying instrument are normally distributed?The normal distribution assumption has one serious flaw. A normal distribution curve is symmetrical. Under a normal distribution assumption for every possible upward move in the price of an underlying instrument there must be the possibility of a downward move of equal magnitude. If we allow for the possibility of a Rs. 50 instrument rising Rs. 75 to Rs. 125, we must also allow for the possibility of the instrument dropping Rs.75 to a price of Rs.25. Since it is impossible for traditional stocks and commodities to take on negative prices, the normal distribution assumption is clearly flawed.As the interest can be compounded at different intervals, volatility can also be compounded at different intervals. For purposes of theoretical pricing of options, volatility is assumed to compound continuously, just as if the prices changes in the underlying instrument, either up or down, were taking place continuously but at an annual rate corresponding to the volatility number associated with the underlying instrument.

What would happen if at every moment in time the price of an underlying could go up or down a given percent, and that these up and down movements were normally distributed? When price changes are assumed to be normally distributed, the continuous compounding of these price changes will cause the prices at maturity to be lognormally distributed. Such a distribution is skewed toward the upside because upside prices resulting from a positive rate of return will be greater, in absolute terms, than downside prices resulting from a negative rate of return.The Black-Scholes Model is a continuous time model. It assumes that the volatility of an underlying instrument is constant over the life of the option, but that this volatility of an underlying instrument is constant over the life of the option, but that this volatility is continuously compounded. These two assumptions mean that the possible prices of the underlying instrument at expiration of the option are lognormally distributed. It also explains why options with higher exercise prices carry more value than options with lower exercise prices, where both exercise prices appear to be an identical amount away from the price of the underlying instrument. For example, suppose a certain underlying contract is trading exactly at 100. If there are no interest considerations and we assume a normal distribution of possible prices, then the 110 call and the 90 put, both being 10% out-of-the-money, ought to have identical theoretical values. But under the lognormal assumption in the black-Scholes model, The 110 call will always have a greater value than the 90 put. In absolute terms, the lognormal distribution assumption allows for greater upside price movement than downside price movement. Consequently, the 110 call will have a greater possibility of price appreciation than the 90 put.So, the lognormal assumption build into the black-Scholes model overcomes the logical problem we initially posed. If we were to allow for the possibility of unlimited upside price movement of an underlying instrument, a normal distribution assumption would force us to allow for unlimited downside movement. This would require us to accept the possibility of negative prices for the underlying instrument, clearly not a possibility for most optionable instruments. A lognormal distribution, however, does allow for open ended upside prices while bounding downside prices by zero. This is a more realistic representation of how prices are actually distributed in the real world.So, the most important assumptions governing price movement in the black-Scholes Model are:

Changes in the price of an underlying instrument are random and cannot be artificially manipulated, nor is it possible to predict beforehand the direction in which price will move.

The percent changes in the price of an underlying instrument are normally distributed.

Because the percent changes in the price of the underlying instrument are assumed to be continuously compounded, the price of the underlying instrument at expiration will be lognormally distributed.

The mean of the lognormal distribution will be located at the forward price of the underlying contract.

The first of these assumptions may meet with resistance from some traders. Technical analysts believe that by looking at past price activity it is possible to predict the future direction of prices. The important point here is that the Black-Scholes Model makes the assumption that price changes are random and their direction cannot be predicted. This does not mean that there is no predictive requirement in using the black-Scholes Model. However, price prediction will focus on the magnitude of the price changes, rather than on the direction of changes.There is also a good reason to question the third assumption, that prices are lognormally distributed at expiration. This may be a reasonable assumption for some markets, but a very poor assumption for other markets. Again, the important point here is for the trader who uses a theoretical pricing model to understand the assumptions on which the theoretical values are based. He can then make his own decision, based on his knowledge of a particular market, as to whether these assumptions, and hence the theoretical values generated by the model, are likely to be accurate.Finding Daily and Weekly Standard Deviation

An important characteristic of volatility is that it is proportional to the square root of time. As a result of this, we can approximate volatility over some period of time shorter than a year by dividing the annual volatility by the square root of the number of trading period in a year.

Suppose someone is interested in a daily volatility. While it would take a logarithmic calculation to give an exact daily volatility, if we ignore the relatively minor effect of continuous compounding over such a short period of time, it is possible to make an estimate of daily volatility. First we must determine the number of daily trading periods in a year. This is, if we look at prices at the end of every day, how much time a year can price changes? If we restrict ourselves to exchange traded options, even though there are 365 days in a year, prices cannot really change on weekends or holidays. This leaves us with about 256 trading days during the year. Since the square root of 256 is 16, to approximate a daily volatility we can divide the annual volatility by 16. We can do the same type of calculation for a weekly standard deviation.Types of Volatilities

Future volatilityFuture volatility is what every trader would like to know; the volatility which best describes the future distribution of prices for an underlying contract. In theory it is this number to which we are referring when we speak of the volatility input into a theoretical pricing model. If a trader knows the future volatility, he knows the right odds. When he feeds this number into a theoretical pricing model, he can generate accurate theoretical values because he has the right possibilities

Of course, traders rarely talk about the future volatility since it is impossible to know what the future holds.

Historical VolatilityHistorical volatility is a measure of actual price changes during a specific time period in the past. It is the annualized standard deviation of daily returns during a specific period. Historical volatility is the measure of a stocks price movement based on historical prices. It measures how active a stock price typically is over a certain period of time. Usually, historical volatility is measured by taking the daily (close-to-close) percentage price changes in a stock and calculating the average over a given time period. This average is then expressed as an Annualized percentage. Historical volatility is often referred to as actual volatility or realized volatility. Short-term or more active traders tend to use shorter time periods for measuring historical volatility, the most common being five-day, 10-day, 20-day and 30-day. Intermediate-term and long-term investors tend to use longer time periods, most commonly 60-day, 180-day and 360-day.

There are varieties of ways to calculate historical volatility, but most methods depend on choosing two parameters, the historical figure over which the volatility is to be calculated, and the time interval between the successive price changes. The historical period may be ten days, six months, five years, or any period trader chooses. Longer period tend to yield an average or characteristic volatility, while shorter periods may reveal unusual extremes in volatility.

Hence historical volatility is an indication of past volatility of the stock in the market. But, in practice, traders usually work with what is known as implied volatility.

Calculation of Historical Volatility

Steps to calculate Historical Volatility

Step 1:Measure the day-to-day price changes in the market:

Calculate the natural log of the ratio (RT) of a stocks price (S) from the current day (t) to the previous day (t-1):

The result corresponds closely to the daily percentage price change of the stock.Step 2:Calculate the average day-to-day price changes over a certain period:

Calculate the Average daily price change (Rm) for a period of time (n)

Step3:

Find out how far prices vary from the average calculated in Step 2:

The historical volatility (HV) is the average variance from the mean (the standard deviation), this is the average deviation of the values from the mean, and is estimated as:

Sometimes historical volatility is estimated by ditching the mean and using the 2nd formula mentioned. The latter formula for HV is statistically called a non-centered approach. Traders commonly use it because it is closer to what would actually affect their profits and losses. It also performs better when n is small or when there is a strong trend in the particular stock.

Step 4:

Express volatility as an annual percentage:To annualize the historical volatility, the above result is multiplied by the square root of the average number of trading days in a year (252 days). The resulting HV of approximately 13% & 12% suggest the Index will likely fluctuate this far from its current price if this level of Volatility remains constant.

In the above graph the red line shows Nifty closing while the blue line indicates the historical volatility for the relevant period.

DateNifty Price (T)T/(T-1)LognormalStdevAnnual Hist Vol

2-Jan-074007.41.010340.010280.0166026.56699

3-Jan-074024.051.004150.004150.0166026.56607

4-Jan-073988.80.99124-0.008800.0166226.58464

5-Jan-073983.40.99865-0.001350.0165926.54252

8-Jan-073933.40.98745-0.012630.0166026.56703

9-Jan-073911.40.99441-0.005610.0166126.57597

10-Jan-073850.30.98438-0.015740.0166426.62190

11-Jan-073942.251.023880.023600.0167026.71825

12-Jan-074052.451.027950.027570.0167726.83813

15-Jan-074078.41.006400.006380.0167326.76270

16-Jan-074080.51.000510.000510.0167226.74682

17-Jan-074076.450.99901-0.000990.0167126.73777

18-Jan-074109.051.008000.007970.0167126.73696

19-Jan-074090.150.99540-0.004610.0167026.72540

22-Jan-074102.451.003010.003000.0166826.69274

23-Jan-074066.10.99114-0.008900.0166926.70973

24-Jan-074089.91.005850.005840.0166926.70237

25-Jan-074147.71.014130.014030.0166926.70983

29-Jan-074124.450.99439-0.005620.0167026.71782

31-Jan-074082.70.98988-0.010170.0167026.72245

1-Feb-074137.21.013350.013260.0166826.68089

2-Feb-074183.51.011190.011130.0166826.69449

5-Feb-074215.351.007610.007580.0166926.69728

6-Feb-074195.90.99539-0.004620.0166926.69957

7-Feb-074224.251.006760.006730.0166926.70337

8-Feb-074223.40.99980-0.000200.0166826.68759

9-Feb-074187.40.99148-0.008560.0166926.70665

12-Feb-074058.30.96917-0.031320.0168226.91263

13-Feb-074044.550.99661-0.003390.0168026.87561

14-Feb-074047.11.000630.000630.0167926.86630

15-Feb-074146.21.024490.024190.0168526.95450

19-Feb-074164.551.004430.004420.0168526.95367

20-Feb-074106.950.98617-0.013930.0168726.99674

21-Feb-074096.20.99738-0.002620.0168726.99447

22-Feb-0740400.98628-0.013820.0169027.03360

23-Feb-073938.950.97499-0.025330.0169827.16719

26-Feb-0739421.000770.000770.0169527.12573

27-Feb-073893.90.98780-0.012280.0169727.14689

28-Feb-073745.30.96184-0.038910.0171527.44728

1-Mar-073811.21.017600.017440.0171727.46905

2-Mar-073726.750.97784-0.022410.0172327.56821

5-Mar-073576.50.95968-0.041150.0173827.80894

6-Mar-073655.651.022130.021890.0174327.89210

7-Mar-073626.850.99212-0.007910.0174127.85250

Implied VolatilityImplied Volatility can be defined as the volatility of an instrument as implied by the prices of an option on that instrument, calculated using an options pricing model. An options value consists of several components The strike price Expiration date, The current stock price, Dividends paid by the stock (if any), The implied volatility of the stock and Interest rates. Instead of substituting a volatility parameter into an option model (e.g. Black-Scholes) to determine an option's fair value, the calculation can be turned round, where the actual current option price is input and the volatility is output. Therefore implied volatility is that level of volatility that will calculate a fair value actually equal to the current trading option price. This calculation can be very useful when comparing different options on the same underlying & different strike prices. The implied volatility can be regarded as a measure of an option's "expensiveness" in the market, and issued by traders setting up combination strategies, where they have to identify relatively cheap and expensive option contracts. Rising implied volatility causes option prices to rise while falling implied volatility results in lower option premiums.As there are many options on a stock, with different strike prices and expiration dates, each option can, and typically will, have a different implied volatility. Even within the same expiration, options with different strike prices will have different implied volatilities.

Implied volatility represents the markets expectation of a stocks future price moves. High-implied volatility means the market expects the stock to continue to be volatile i.e., make large moves, either in the same direction or up and down. Conversely, low implied volatility means the market believes the stocks price moves will be rather conservative. Because implied volatility is a surrogate for option value, a change in implied volatility means there is a change in the option value. Many times, there will be significant changes in the implied volatility of the calls vs. the puts in a stock. This signals that there may be a shift in the bias of the market.

It is seen that the volatilities of both the calls & puts increase with the falling index levels & top out at a point when index is bottoming out & vice versa. So, it can be inferred quite conclusively that Call & Put IVs have an inverse correlation with the movement in the broader market. Although, the inference drawn is from the historical data and therefore doesnt guarantee to repeat itself in future, it, still, can be used in conjunction with other technical indicators to improve the decisiveness of the market direction predicted using those indicators.

Implied volatility is that level of volatility which is calculated from the current trading option price. This can help to gauge whether options are cheap or expensive. However the prices of deep ITM and deep OTM options are relatively insensitive to volatility

As there are many options on a stock, with different strike prices and expiration dates, each option can, and typically will, have a different implied volatility. Even within the same expiration, options with different strike prices will have different implied volatilities.

For example, suppose a certain futures contract is trading at 98.50 with interest rate at 8%. Suppose also that a 105 call with three months to expiration is available on this contract, and that our best guess about the volatility over the next three months is 16%. If we want to know the theoretical value of the 105 call we might feed all these inputs into a theoretical pricing model. Using the Black-Scholes model, we find that option has a theoretical value of .96. Having done this we might compare the options theoretical value to its price in the marketplace. To our surprise, we find that the option is trading for 1.34. The discrepancy between our value of .96 and the marketplaces value of 1.34 must be due to difference of opinion concerning one or more of the inputs into the model. Since all other inputs are expect volatility. So, the marketplace must be using volatility other than 16% to evaluate the 105 call.

To know the actual volatility in the marketplace, we can ask the following question; if we hold all other inputs except volatility what volatility must we feed into our theoretical pricing model to yield a theoretical value identical to the price of the option in the marketplace? In our example, clearly the volatility has to be higher than 16%. So, we start to raise the volatility and fitting it into black-Scholes model we find that at a volatility of 18.55, the 105 call has a theoretical value of 1.34. This volatility is known as implied volatility. When we solve for the implied volatility of an option we are assuming that the theoretical value (the options price) is known, but that the volatility is unknown. In effect, we are running the theoretical pricing model backwards to solve for this unknown.

The implied volatility in the marketplace is constantly changing because option prices, as well as other market conditions, are constantly changing. It is as if the marketplace were continuously polling all the participants to come up with consensus volatility for the underlying contract. This is not a poll in the true sense, since all traders do not huddle together and eventually vote on the correct volatility. However, as bids and offers are made, the trade price of an option will represent the equilibrium between supply and demand. This equilibrium can be translated into an implied volatility.

Assuming a trader had a reliable theoretical pricing model, if he determines the future volatility of an underlying contract he would be able to accurately evaluate options on that contract. He might then look at the difference between each options theoretical value and its price in the marketplace, selling any options which were overpriced relative to the theoretical value, and buying any options which were under priced. If given choice between selling one of two overprice options, he might simply sell the one which was most overpriced. However, a trader who has access to implied volatilities might use a different yardstick for comparison. He might compare the implied volatility of an option to either a volatility forecast, or to the implied volatility of other options on the same underlying contract. Going back to our example, with a theoretical value of .96 and a price of 1.34, the 105 call is .38 overpriced. But in volatility terms it is 2.5% overpriced since its theoretical value is based on a volatility of 16% while its price is based on a volatility of 18.5% (the implied volatility). Due to the unusual characteristics of an options, it is often more useful for the serious trader to consider an options price in terms of implied volatility rather in terms of its total price.

If a contract has a high value and a low price, then a trader will want to be a buyer. If a contract has a low value and a high price, then a trader will want to be seller. For an option trader this usually means comparing the future volatility with the implied volatility. If implied volatility is low with respect to the expected future volatility, a trader will prefer to buy the option; if implied volatility is high, a trader will prefer to sell options. Of course, future volatility is an unknown, so we tend to look at the historical and forecast volatilities to help us make an intelligent guess about the future. But in the final analysis, it is the future volatility which determines an options value.Generally, the implied volatilities of calls and puts show a distinct pattern, called the skew of implied volatility. Implied volatility tends to be higher for out-of-the-money (OTM) options compared to at-the-money (ATM) options. This is because OTM options present more risk on very large moves to compensate for this risk, they tend to be priced higher. But equally OTM calls and puts do not necessarily have the same implied volatility, and this difference represents the bias or skew of the market. The skew can be caused by a strong directional bias in the stock or the market, or by very large demand for either calls or puts, which pushes implied volatility higher. Implied volatility acts as a proxy for option value. It is the only parameter in option pricing that is not directly observable from the market, and cannot be hedged or offset with some other trading instrument. Because all other factors can be locked in, the price of the option becomes entirely dependent on the implied volatility. This is an important fact to consider when looking for relative value in options.Volatility Skew is U shaped, the bottom of the U being the at-the-money strike. From there, the skew rises on both sides. This formation is sometimes referred to as the smile curve. The farther an options strike is from the market price of the underlying instrument, the higher the options volatility. Volatility skews are present when two or more options on the underlying have a significant difference in implied volatility levels. The volatility skew gauges and accounts for the limitation that exists in most option pricing models and is used to give an edge in estimating an option's worth.There are two types of volatility skews that exist in the marketplace, Price skew and

Time skew Price skew occurs when the implied volatility differs across the strike prices of options, on the same stock and the same expiration months. The options trader has various strategies that are best utilized when a price skew has been identified. Strategies include the bull call, bull put, bear call, bear put spreads as well as the call ratio and put ratio back spreads. Just always remember when using either a price skew or a time skew that the primary goal of the options strategist is to be selling higher volatility and buying lower volatility. For example, assume an investor is bullish on a particular underlying and at the same time has identified a price skew where the implied volatility of the strike price they are selling is significantly higher than the implied volatility they are buying. The option strategist would implement a bull call spread as the best strategy.

A bull call spread is a strategy in which a trader buys a lower strike call and sells a higher strike call to create a trade with limited profit and limited risk.

The other type of skew is the time skew and it is identified by different levels of implied volatility on stock options with the same or sometimes different strike prices, but different expiration months. Again, just as with the price skew, there are particular strategies that work best and focus on selling higher volatility and buying lower volatility. When a significant time skew has indeed been identified then the strategist can utilize any number of calendars or time spread strategies. Time spread strategies include the call calendar, put calendar, diagonal call and diagonal put spreads. To illustrate the usage assume the trader possesses a bullish bias on the underlying and has discovered a time skew where the implied volatility for the strike price and month they are selling is much higher than the strike price and expiration month they are buying. In this case the option strategist could implement a call diagonal spread as a very suitable strategy. A diagonal call spread is constructed by purchasing a call with a distant expiration date and a lower strike price along with a short call with a closer expiration date and a higher strike price.

The important thing is that the investor needs to understand is what type of volatility skews they are dealing with. From that point depending on the particular bias the trader has is it bullish, bearish, or sideways then the most appropriate options strategy can be selected.

Because there are many options on a stock, with different strike prices and expiration dates, each option can, and typically will, have a different implied volatility. Even within the same expiration, options with different strike prices will have different implied volatilities.

Calculation of implied volatilityO find out implied volatility, break down the Black & Scholes model into its components and then calculated the IVs through goal seeking analysis.

For each day,calculate the volatility for six readings 3 for call (Itm & Atm)and 3 (Itm & Atm) for put with the highest no of contracts.Estimating Volatilities

Most popular option pricing models, such as Black-Scholes, assume that the volatility of the underlying asset is constant. This assumption is far from perfect. In practice, the volatility of an asset, like the assets price is a stochastic variable. Unlike the asset price, it is not directly observable. So, tracking and estimating current level of volatility is very important for trader.

Two models are commonly used to keep track of the variations in the volatility or correlation through time.

1. Exponentially weighted moving average (EWMA)

2. Generalized autoregressive conditional heteroscedasticity (GARCH)

Estimating volatility

Define, as the volatility of a market Variable on day n, as estimated at the end of day n-1. The square of the volatility 2n on day n is the variance rate. Suppose that the value of the market variable at the end of day i is Si. The variable ui is defined as the continuously compounded return during day I (between the end of day i-1 and the end of day i):

ui = ln Si/Si-1An unbiased estimate of the variance rate per day, 2n, using the most recent m observation the ui is

2n = (1/(m-1)) (un-i - ubar)2Where u is the mean of the uisUbar = (1/m) Un-iFor the purposes of monitoring daily volatility, the formula in equation is usually changed in a number of ways:

1. ui is defined as the percentage change in the market variable between the end of day i-1 and the end of day i, so that :

Ui = (Si-Si-1)/Si-1

2. Ubar is assumed to be zero.

3. m-1 is replaced by m.

These three changes make very little difference to the estimates that are calculated, but they allow us to simplify the formula for the variance rate to

2n = (1/m) u2n-1Weighting Schemes

The above equation gives equal weight to u2n-1, u2n-2,.., u2n-m. Our objective is to estimate the current level of volatility, n. It therefore makes sense to give more weight to recent data. A model that does this is

2n = iu2n-iThe variable i is the amount of weight given to the observation i days ago. The s are positive. If we choose them so that i< j when i>j. less weight is given to older observations. The weights must sum to unity, so we have

i =1

An extension of the idea in above equation is to assume that there is a long-run average variance rate and that this should be given some weight. This leads to the model that takes the form

2n = VL iu2n-iwhere VL is the long-run variance rate and is the weight assigned to VL. Because the weights must sum to unity, we have

i = 1This is known as an ARCH (m) model. It was first suggested by Engle. The estimate of the variance is based on a long-run average variance and m observations. The older an observation, the less weight it is given. Defining = VL, the model in above equation can be written as

2n = + iu2n-i

Let us now discuss two important approaches to monitoring volatility using the above equations.The Exponentially weighted moving average model (EWMA)

The exponentially weighted moving average model is a particular case of the model in equation where the weights i decreases exponentially as we move back through time. Specially, i+1= i, where is a constant between 0 and 1.

This weighting scheme leads to a particular simple formula for updating volatility estimates. The formula is

2n = 2n-1 + (1- )u2n-1

The EWMA approach has the attractive feature that relatively little data need to be stored. At any given time, we need to remember only the current estimate of the variance rate and most recent observation on the value of the market variable. When we get a new observation on the value of the market variable, we calculate a new daily % change and equation to update our estimate of the variance rate. The old estimate of the variance rate and the old value of the market variable can then be discarded.

The EWMA approach is designed to track changes in the volatility. Suppose there is a big move in the market variable on day n 1, so that U2n 1 is large. From equation this causes our estimate of the current volatility to move upward. The value of (a value close to 1) produces estimate of the daily volatility that respond relatively slowly to new information provided b the daily percentage change.

The Risk Metrics database, which was originally created by j. p. Morgan and made publicly available in1994, uses the EWMA model with =0.94.For updating daily volatility estimates in its Risk Metrics database. The company found that, across a range of different market variables, this value of gives forecasts of the variance rate that comes closest to the realized variance rate. The realized variance rate on a particular day was calculated as an equally weighted average of the u2i on the subsequent 25 days.

GARCH MODEL

The GARCH model, is proposed by Bollerslev in 1986. The difference between the GARCH model and EWMA model is analogous to the difference equation and equation . In GARCH variance rate is calculated from a long run average variance rate, VL, as well as from n-1 and. un-1. The equation For GARCH is

2n = VL + u2n-1 + 2n-1

where is the weight assigned to VL, is the weight assigned to u2n-1, and is the weight assigned to 2n-1. Because the weights must sum to one, we have

+ + = 1Mean reversion

The GARCH model recognizes that over time the variance tends to get pulled back to a long-run average level of VL. The amount of weight assigned to VL is . The GARCH is equivalent to a model where the variance V follows the stochastic process.

Choosing between the models

In practice, variance rates do tend to be mean reverting. The GARCH model incorporates mean reversion, whereas the EWMA model does not. GARCH Model is therefore theoretically more appealing than EWMA model.

Now the question is how best fit parameters in GARCH Model can be estimated. When the parameter is zero, the GARCH Model reduces to EWMA. In circumstances where the best-fit vale of w turns out to be negative, the GARCH model is not stable and it makes sense to switch to the EWMA model.

In the EWMA and GARCH models, the weight assigned to observations decrease exponentially as the observations become older. The GARCH model differs from the EWMA model in that some weight is also assigned to the long run average variance rate. Both the EWMA and GARCH models have structures that enable forecasts of the future level of variance rate to be produced relatively easily.

Maximum likelihood methods are usually used to estimate parameters from historical data in GARCH and similar models. These methods involve using an iterative procedure to determine the parameter values that maximize the chance or likelihood that historical data will occur. Once its parameters have been determined, a model can be judged by how well it removes autocorrelation from the u2i.

Volatility Strategies

Long straddleLONG STRADDLE - Bullish & Bearish trade, forecasting explosive movement in price either way

COMBINATIONS: Combinations represents strategies which involve taking positions in both calls and puts on the same stock.

It involves buying a call and a put option with same exercise price and date of expiration.

Involves initial cost of investment.

Also known as bottom straddle or straddle purchase. While most straddle are executed with one-to-one ratio (one call for each put), this is not the requirement. A straddle can also be rationed, so that it consists of unequal numbers of calls and puts. Any spread where the number of long market contracts (long calls or short puts) and short market contracts (short calls or long puts) are unequal is considered a ratio spread Example:

Buy Call

Buy Put

Strike Price: Rs. 70.0

Strike Price: Rs. 70.0

Premium : Rs. 5.0 Premium : Rs. 4.0 Initial Investment: Rs. 9.0

When stock price is Rs. 70 - Maximum Loss of Rs. 9. Hence loss is limited up to the initial investment.

Investor can earn profit if the stock price moves in either direction significantly. Hence, suitable if volatility is expected. But our view should be different from general view.

Risk: Limited, but should not really be viewed as a low risk strategy because you are paying out for two options which are both wasting assets

Reward: Unlimited

WHEN TO USE: You believe that the stock/index is about to make a large move in either direction. A good time to utilize straddles is where there has been a prolonged period of extreme quietness (in prices) and implied volatility is around multiyear lows. If this is the case look to do longer dated months rather than the shorter ones

VOLATILITY EXPECTATION: Very bullish. Volatility increases improve the position substantially. Volatility should therefore be monitored closely.

PROFIT: Unlimited for an increase or decrease in the underlying

LOSS: Limited to the premium paid in establishing the position. Loss will be greatest if the underlying is at the initiated strike at expiry.

BREAKEVEN: Reached if the underlying rises or falls from option strikes by the same amount as the premium cost of establishing the position.

TIME DECAY: Hurts a lot, remember you have double time erosion because of the two options bought. Decay depends a lot on volatility if volatility increases time decay will decrease etc.

LONG STRADDLE IDEAS Work best on stocks/indexes that are likely to experience explosive moves.

Consider legging into them - buying the calls today and buying the puts on a rally or vice-versa

Always best to use some sort of time stop because of the time decay

If youre expecting a very large breakout then better to trade strangles Are there any clues on the chart to consider liquidating the opposing option? You buy a straddle; market breaks out to the upside, and then possibly looks to sell the put

Very hard trade to make money on if you buy the options when volatility is high

The new option trader often finds long straddle attractive because strategies with limited risk and unlimited profit potential offer great appeal, especially when the profit is unlimited in both directions. However, if the hoped for movement fails to materialize, he soon find that losing money little by little hurt a lot at the end, because you have paid for two option. So, one have to very careful while choosing this strategy.

SHORT STRADDLEForecasting little movement or a good contraction in movement

For traders who are generally neutral about a stocks potential. The stock may rise or fall but is generally in a range bound pattern. The trade works because both the option premiums fall.

Can also be created by selling a call and a put option with same exercise price and date of expiration.

Generates initial positive cash flow for the investor as he is taking short position in both call and put options.

Also known as top straddle or straddle writes. Example: Sell Call

Sell Put

Strike Price: Rs. 70.0

Strike Price: Rs. 70.0

Premium : Rs. 5.0 Premium : Rs. 4.0

Initial Inflow: Rs. 9.0

When stock price is Rs. 70 - Maximum profit of Rs. 9. Hence profit is limited up to the initial investment.

Loss can be unlimited if the stock price moves in either direction significantly. Hence, suitable if volatility is not expected

Risk: Unlimited

Reward: Limited

WHEN TO USE: For aggressive investors who don't expect much short-term volatility, the short straddle can be a risky, but profitable strategy. If you only expect a moderately sideways market consider selling strangles instead

VOLATILITY EXPECTATION: Bearish, volatility increases wreck the position. Straddles are not as susceptible to volatility increases as strangles. Keep an eye on volatility throughout the position

PROFIT: Limited to the premium received, highest profit when the market settles at the sold strike

LOSS: Unlimited for either an increase or decrease in the underlying.

BREAKEVEN: Reached if the underlying rises or falls from sold strike by the same amount as the premium received from establishing the position.

TIME DECAY: Helps, especially when the trade is initiated in periods of high volatility

LONG STRANGLE Bullish & Bearish trade, forecasting explosive movement either way, even more so than with a Long Straddle

Also known as bottom vertical combination.

Created by buying a call and a put option of same stock and expiration period but of different strike prices.

Strike price of put option is less than strike price of call option.

Initial investment is needed.

Example:

Buy a Call

Buy a Put

Strike Price: Rs. 70.0

Strike Price: Rs. 65.0

Premium : Rs. 5.0

Premium: Rs. 4.0

Initial Investment: Rs. 9.0

When stock price is in the range of both strike prices i.e. Rs.70 Rs. 65

Maximum Loss of Rs. 9.0. Hence loss is limited to the initial investment.

Investor can earn profit if the stock price moves in either direction significantly. Hence, suitable if volatility is expected.

Risk: Limited

Reward: Unlimited

The Trade: Buying out-of-the-money calls and puts

WHEN TO USE: You believe the stock/index will have an explosive move either up or down. This strategy is similar to the buy straddle but the premium paid is less but then a larger move is needed to show a profit.

VOLATILITY EXPECTATION: Very bullish, increases in volatility work marvels for the position

PROFIT: The profit potential is unlimited although a substantial directional movement is necessary to yield a profit for both a rise and fall in the underlying.

LOSS: Occurs if the market is static; limited to the premium paid in establishing the position

BREAKEVEN: Occurs if the market rises above the higher strike price at B by an amount equal to the cost of establishing the position, or if the market falls below the lower strike price at A by the amount equal to the cost of establishing the position.

TIME DECAY: This position is a big wasting asset. As time passes, value of position erodes toward expiration value. If volatility increases, erosion slows, if volatility decreases, erosion speeds up.

LONG STRANGLE IDEASCan mix the strikes up depending on whether you lean towards the bull or bear tract but are still overall neutral - Perhaps you feel the odds slightly favor a bull move. If stock is at 5.00 instead of buying the 4.50P and 5.50C you could buy the 4.50P and 6.00 call

Use some sort of time stop because time erosion is your enemy

If you expect a mega move then better strategy than straddles because strangles are cheaper to buy therefore can buy more with the same amount of capital

Look to trade when market has been quite for a long time and volatility is low, if this is the case look to trade the longer dated months

Look at the charts for any clues, if market is breaking out then maybe ditch the wrong option

Short Strangle

Strangle can also be created selling call and put options.

This is known as short strangle or top vertical combination.

Created by selling a call and a put option of same stock and expiration period but of different strike prices.

Strike price of put option is less than strike price of call option.

Generates positive inflow for the investor.

Example:

Sell a Call

Sell a Put

Strike Price: Rs. 70.0

Strike Price: Rs. 65.0

Premium : Rs. 5.0

Premium : Rs. 4.0

Initial Inflow: Rs. 9.0

When stock price is in between the strike prices, investor can earn a limited profit.

If the stock price moves significantly, than the investor can make loss which can be limited. Hence, suitable if volatility is not expected.

DERIVATIVE GREEKS AND VOLATILITY

DELTA Change in option price per unit with respect to change in the price of the underlying.

Delta for call varies from 0 to 100 as we go from OTM to ITM and DELTA for a put varies from 0 to 100 as we go from OTM (Put) to ITM Put.

From put call parity we know that C-P= F and therefore it follows that Delta of put = 1- DELTA OF CALL.

Delta is 50 around the forward price (Not the spot price)

The exact formula is X= S* EXP (r +^2/2) tDELTA HEDGING:

Long units of underlying + Short one call = Market neutral

The market neutrality is there in very narrowly defined conditions.ISSUES IN DELTA HEDGING

The log-normal assumption may not be valid.

The volatility estimate may not be correct.

The hedge may not be done frequently enough to prevent losses due to hedge slippage or Gamma Risk.DELTAS SENSTIVITY TO VOLATILITY

As volatility increases all options become more like ATM with their Deltas Approaching 50.

Conversely as decrease time to maturity or volatility, Delta moves away from 50 levels.

From ITM calls delta move towards 100 and for OTM towards 0 as volatility decreases.

GAMMA

The rate of change of Delta with that of the underlying is known as Gamma.

Delta is not constant and changes from 0 to 100 for calls as the underlying price moves up and from -100 to 0 for puts as the underlying price moves up.

For any long option position whether call or put Gamma is always positive. The frequency with which delta hedging needs to be done is a function of level of Gamma Risk. A trader is often required to make a trade-off between frequency of hedging and trading cost.

GAMMA RISK:

GAMMA for Put and call =

{1/S0**SQRT (T)}*N (d1)

N (d1) = 1/2* exp (-d12/2)MOST IMPORTANT THING TO UNDERSTAND:

The holder of an option who is short Delta units of the underlying (long Call) will achieve a positive cash flow. If subsequently the price movement is sufficiently large and negative cash flow if price movement is sufficiently small. It is easy to understand this in terms of convexity (Gamma) and time decay (Theta) which are two main features of any option.

We have already learnt from B & S that gamma and Theta are of the same magnitude (opposite in sign)

Change in option price for a large change in underlying = Delta X ( Change in underlying) + O.5* (Gamma)^2* (Change in underlying) Now for a Delta hedged portfolio the first term in the above equation will be offset by the change in underlying and the second term for a long option position will be positive and proportional to change in the price of the underlying. Though Theta and Gamma are equal and opposite, Theta is more or less fixed in terms of loss per day where as gamma is directly proportional to change in the price of the underlying.

When there is a big move in the price of the underlying gamma change is more than Theta and therefore for a delta hedged portfolio with long potion and short underlying (positive gamma) position it returns in a positive cash flow.

A numerical example in the next slide would further clarify this

So = 100; X =100; r = 10% PA VOL= 25% T =1M

Delta= 56; Therefore, 100 Calls+ 56 Stock = Delta neutral Gamma= .0536 AND Theta = .0625

Lets assume that in a days time the price moves to 105 & the new delta is .7936. Change in options price = (New Delta + Old Delta)/2 * Change in underlying less time decay

This = (.7936- .5592) /2 *(5) - .0625 = 3.3

Change in option portfolio = 3.32*100= 332 (+)

Change in underlying = 56*5= 280 (-ve)

Net Change = 332-280 = 52SENSTIVITY OF GAMMA: The magnitude of gamma is consistent with the uncertainty whether the option will expire in or out of money. It is maximum for ATM options (uncertainty is maximum) The gamma for ITM and OTM options increase with the increase in volatility and the time to maturity (Uncertainty increases) Gamma for ATM option falls with the increase in volatility and increase in time to maturity (Is the uncertainty decreasing?)

THETA The rate of change of option price with that of time is known as Theta.

SENSTIVITY OF THETA:

A long option position will always have a negative Theta and a short option position a positive Theta.

A Gamma and Theta position will be opposite in sign and size will also correlate. This follows form B&S partial differential equation when applied in the context of a delta hedged portfolio.

Theta = -0.5 * (VOL)^2 * (S)^2 * Gamma The owner of a Gamma (who is net long options) is subject to time decay of spot doesnt move much, but benefits from price movements. As expiration approaches Gamma of an ATM option becomes increasingly large and the same is also true about Theta.

Every option position is a trade-off between market movement and time decay. The Theta of an ATM option increases as expiration approaches. This implies that a short term option will decay more quickly than a long term option.

As we increase volatility the theta of all options increases. Higher volatility means that there is a greater risk premium (Option premium (ST- PV (K)) with the option and each days decay will also be greater when no movement occurs. Theta (Call) = -SoN'(d1) /2T- rXexp (-RT) N (d2) Theta (Put) = -SoN'(d1) /2T+ rXexp (-RT) N (-d2)

VEGA

Vega is also known as kappa or lambda. Vega is the rate of change of the value of the portfolio with respect to the change in volatility.

A Vega of 0.12 indicates that an options value will increase/decrease by R. 0.12 With every 1% increase/decrease in volatility.

Characteristics of Vega:

Vega of long positions (i.e. long calls and long puts) is positive. Vega of short positions (i.e. short calls and short puts) is negative. The Vega of underlying security is zero.

Vega of at-the-money options is generally the largest.

Vega decreases as an option goes further in or out-of-the-money.

Long-term options have higher Vega than short-term options.

The Vega of both the call and put options is equal and positive reflecting the fact that an option price is directly related to volatility. B&S model is actually inconsistent with changing volatility. The model assumes that volatility will remain constant for the full life of the option Vega increases with the increase in time to maturity. ATM options are most sensitive to total points changes in volatility, where as OTM in percentage terms.

Vega of an ATM is constant at various volatility levels and Vega of an ITM or OTM option increases with the increase in volatility.Vega of European Call and Put Options:

According to the Black-Scholes model Vega of a European call or put option on a non-dividend paying stock is as follows:

The figure below shows the variation of Vega of options of different time to expiry with the stock price.

The figure below shows the variation of Vega of options of different time to expiry with the stock price.

VOLATILITY SPREAD AND GREEKSVOLATILITY SPREADS In volatility spread, magnitude of movement important but not direction. All volatility spreads are delta neutral.

All spreads which are helped by a movement in underlying have a positive Gamma.

All spreads which are hurt by a movement in underlying have a negative Gamma.

POSITIVE GAMMA NEGATIVE GAMMALONG STRADDLE SHORT STRADDLE

LONG STRANGLE SHORT STRANGLE

SHORT BUTTERFLY LONG BUTTERFLY

SHORT TIME SPREAD LONG TIME SPREAD A positive Gamma trader hopes for volatile conditions and a negative Gamma trader hopes for stable conditions.

Any spread that has a positive gamma has a negative Theta.

Spreads which are helped by a rise in volatility have a positive Vega. These include long straddles/strangles, long time spread and short butterfly. A long time spread always benefits from increased volatility. This is because a longer term option is more sensitive to changes in volatility than a shorter term option. When Implied volatility rises, time spread tend to widen When implied volatility falls, time spread tends to fall. We also know that time spreads loose their value as the options move Into the money Ideally a trader who is long on a time spread wants two contradictory things

He wants markets to sit still so that the time decay will have a beneficial impact on the spread. He wants the implied volatility to go up. 9Which can only happen in every

This actually happens when G-7 meets to discus exchange rates. When the meeting is initially announced, the IV increases without actual increase in the exchange rates. These two contradictory impacts of implied volatility and movement differentiate time spreads from other volatility spreads. A long time spread benefits from increase in interest rates and a short time spread from a decrease in interest rates.

Once a trader has taken a view, one would look for spreads which have a positive theoretical edge, i.e., purchasing options which are overpriced.

If implied volatility is generally low look for spreads with a positive Vega. Likewise if IV is high look for spreads with a negative Vega.

Straddles/Strangles (Long/Short) may be sensible strategies when option appears to be underpriced /overpriced.

Straddles/Strangles may have a large theoretical edge, but other risks remains.

Long time spreads are likely to be profitable when IV is low but expected to rise.

ADJUSTMENTS REQUIRED IN A VOLATILTIY SPREAD:

A volatility spread may be initially Delta neutral but Delta of the option is likely to change as the price of the underlying falls/rises. Similarly changes in volatility and time to expiration can change the Delta of a spread. A trader can decide the frequency of adjustments by: A pre specified Delta limit EG 500 Delta long/short daily/weekly adjustments. Adjust by market feel.For example, when the underlying at 50 Delta = 0, Gamma = 200, when the underlying moves by two point the position will be 400 delta long. If a trader feels that 48 is a good support level, he probably neednt adjust. When we purchase or sell the underlying to achieve Delta neutrality, we are not changing the other risk such as gamma, Theta and Vega.

When we use options to achieve Delta neutrality we are also altering the other risk characteristics.

For example, in a short straddle, if the underlying moves down, the delta of put becomes more negative and call reduces and since we are short on calls and puts our position becomes Delta positive.

To achieve delta neutrality, we can either sell contracts of the underlying, sell calls or purchase puts. By buying puts, he is reducing his existing Gamma, Theta and Vega levels.

By selling calls he is adding to existing gamma, Theta and Vega risk.

Buying and selling of option alter other risks, but if options are underpriced / overpriced they also improve the theoretical edge.

Also the original strangle position is now altered. For example the original position may be short (20 X 20; now it may be short (25 X 30).

To increase/reduce risk/theoretical edge is a choice which a trader has to make. A trader who had a positive gamma (long options) is always adjusting against the trend, i.e., if the market goes up his delta increases and he has to sell underlying.

A trader with a negative gamma is always adjusting with the trend. Take an example to look for the theoretical edge, which is usually in terms of our volatility estimate and implied volatility differences.

Lets consider the following three spreads (These are options on futures).

Spread 1Spread 2Spread3

SHORT STRADDLE CALL RATIO VERT SP LONG PUT BUTTERFLY

-10 MAY 50 CALLS +10 MAY 51 CALLS +10 MAY 49 PUTS

-8 MAY 50 PUTS -15 MAY 52 CALLS -20 MAY 50 CALLS

+10 MAY 51 PUTS

All the spread are negative gamma spreads. Lets see the theoretical edge for the spreads using our volatility estimate of 15% and interest spread of 8% P.A.

RISK CONSIDERATIONS IN VOLATILITY SPREADS

SPREAD 1TH PRICE M PRICE THE PRICEDELTA

-10 MAY 50 CALLS0.921.071.50-440 (44*-10)

-8 MAY 50 PUTS1.421.591.36+440 (-55*-8)

SPREAD 2TH PRICE M PRICE THE PRICEDELTA

+10 MAY 51 CALLS0.570.77-2.0+310 (31*10)

-15 MAY 52 CALLS0.330.53+3.0-315(21*-150)

SPREAD 3TH PRICE M PRICE THE PRICEDELTA

+10 MAY 49 PUTS 0.911.05-1.4-420 (-42*10)

-20 MAY 50 PUTS1.421.59+3.4+1100(-55*20)

+10 MAY 51 PUTS2.062.22-1.6-680(-68*10)

All the spreads are delta neutral.

Spread 1 has a theoretical edge of 2.86; spread 2 has 1 and spread 3 has 0.4. SPREAD 1SPREAD 2SPREAD 3

Theo. Edge2.861.000.40

Spread 1 has the highest theoretical edge as it involves selling both the overpriced options. If we make the spreads equivalent in terms of theoretical edge, we can do three times of spread two and seven times of spread three.

Now, if we are right about our volatility estimates and the option price adjust to our volatility estimate. All the spreads will give us the same theoretical edge. However, the risk is that we may be wrong about our volatility estimates and we need to evaluate the spread risks in terms of Vega, Gamma and Theta.

GAMMATHETAVEGA

Spread 1-10* 13.4 -10* -.0099 -10* .076

-8* 13.4 - 8* -.0098 -8 * .076

-241.2 +.1774 -1.368

GAMMATHETAVEGA

Spread 2+ 30*12.1 +30* -.0090 +30*.068

- 45* 9.8 -45* -.0073 -45* .055

-78.0.0585-0.435

GAMMATHETAVEGA

Spread 3+70* 13.3 +70*-.0098 +70*.075

-140* 13.4 -140* -.0098 -140*.076

+70* 12.1 +70* -.0087 +70* .068

-98.0+.0770-.630

GAMMATHETAVEGA

Spread 1-241.2 +.1774 -1.368

Spread 2-78.0 .0585 -0.435

Spread 3-98.0 +.0770 -0.630

Under the current defined narrow conditions spread 1 has the highest Gamma and Vega risk for a given level of theoretical edge.

All the spreads will loose the theoretical edge once the volatility happens to be greater than 15%. Can we determine the breakeven volatility for each spread? If one does that calculation it can be seen that IVs of spread 1 is 17%, spread 2 is 20% and spread 3 is 22%.

GAMMATHETAVEGA

Spread 1-241.2 +.1774 -1.368

Spread 2-78.0 .0585 .0585

Spread 3-98.0 +.0770 +.0770

BREAK- IV 17% 20%22%

The higher the break even volatility of a spread, higher is the error of our margin in our volatility estimate. If the volatility turned out to be less than 15 %, than spread 1 will also give the highest profit. ( Highest negative Vega).DELTA RISK: A volatility spread is delta neutral atleast within a limited range. GAMMA RISK:

All the spread have a negative gamma and if the underlying makes a large move, all the spread will be adversely affected due to a large movement in the underlying.

Here spread 1 is again the most riskiest and the spread 3 is the least riskiest on the upside move and spread 2 on the downside move.

Spread 3 appears to be the least risky but is difficult to execute.

Also spread 3 has the largest size and therefore the highest transaction costs. A quick method could be to use Vega/gamma in the numerator and divide this by the theoretical edge.

The quick method poses one problem, the Gamma, Vega measured at the initiation of spread hold their value only in a limited Range.

In other words the option sensitivities are defined only in very narrow market conditions.GENERAL RULE:

Straddle and strangle are riskiest of all the spreads.

They are high return-high risk strategy.

Bibliography Beckers, Stan. 1981. Standard Deviations Implied in Option Prices as Predictors of Future Stock Price Variability, Journal of Banking and Finance 5, pp. 363 382.

Black, Fischer and Myron Scholes. 1973. The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81:May/June, pp. 637 654.

Hull, John C. 2003. Option, Futures, and Other Derivatives. Pearson Education, Inc.

Anthony, J. (1988) 'The Interrelation of Stock and Options Market Trading - Volume Data' journal of Finance, Vol. 43, pp. 949-964.

Cornell, B (1981): The relationship between volume and price variability of futures markets, Journal of Futures Markets, Vol. 1, no 3, pp 303.16.

NCFM Derivatives Core Module

Web Bibliography www.nseindia.com www.bseindia.com www.ivolatility.com www.investopedioa.com www.riskglossary.com www.contentlinks.asiancerc.com/ www.indiaderivatives.com www.moneycontrol.com

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