# Option volatility pricing advanced trading strategy and technique

If implied volatility is too low, vertical spreads should focus on purchasing the at-the-money optlon. If implied volatility is too high, vertical spreads should focus on selling the at-the-money options.

A trader is not required to execute any vertical spread by first buying or selling the at-the-money option. Such spreads always involve two options, and a trader can choose to either execute the complete spread in one transaction, or leg into the spread by trading one option at a time.

Regardless of how the spread is executed, the trader should focus on the at-the-money option, either buying it when implied volatility is too low, or selling it when implied volatility is too high. The choice of the at-the-money option is slightly different when we move to stock options. If we define the at-the-money option as the one whose de1ta is closest to 50, then we may find at the at-the-money option is not always the one whose exercise price is closest current price of the underlying contract.

In stock options, the forward price is the current price of stock, plus carrying costs on the stock, less expected dividends. Of course,greater risk also means greater reward. By contrast,the vertical spreader's profits are limited,but he will also be much less bloodied if the market makes an unexpected move in the wrong direction.

Rearranging the components of a synthetic underling position, we can create four other synthetic relationships:. In the absence of any interest or dividend considerations, the value of the synthetic market can be expressed as:.

The three-sided relationship between a call, a put, and its underlying contract means that we can always express the value of any one of these contracts in terms of the other two:. No matter what happens in the underlying market, the underlying position will do exactly. The foregoing position, where the purchase of an underlying contract is offset by the sale of a synthetic position, is known as a conversion.

As before, we assume that the call and the put have the same exercise price and expiration date. Typically, an arbitrageur will attempt to simultaneously buy and sell the same items in different markets to take advantage of price discrepancies between the two markets. If the synthetic is overpriced, all traders will want to execute a conversion buy the underlying, sell the call, buy the put.

If the synthetic is underpriced, all traders will want to execute a reversal sell the underlying, buy the call, sell the put. Such activity, where everyone is attempting to do the same thing, will quickly force the synthetic market back to equilibrium.

If the cash flow resulting from an option trade and a trade in the underlying instrument is identical, the synthetic relationship is simply:. This will be true if interest rates are zero, or in futures markets where both the underlying contract and options on that contract are subject to futures-type settlement. Assuming that all options are European no early exercise permitted , we can now express the synthetic relationship in futures markets where the options are settle in cash as follows:.

Taking into consideration the interest rate component, we can express the synthetic relationship as:. Anytime a strategy is executed one leg at a time, there is always the risk of an adverse change in prices before the strategy can be completed. The practical solution is to avoid carrying conversions and reversals to expiration when there is a real possibility of expiration right at the exercise price.

If al1 contracts are subject to futures-type settlement, any credit or debit resulting from changes in the price of the underlying futures contract wil1 be offset by an equal but opposite cash flow from changes in prices of the option contracts. This type of position, where the underlying instrument in a conversion or reversal is replaced with a deeply in-the-money option, is known as a three-way. Since a box eliminates the risk associated with carrying a position in the underlying contract, boxes are even less risky than conversions and reversals, which are themselves low-risk strategies.

Another method of eliminating a position in the underlying contract is to take a synthetic position in a different expiration month, rather than at a different exercise price as with a box. These combined long and short synthetic positions taken at the same exercise prices but in different expiration months is known as a jelly roll or simplya roll. The value of the roll is the cost of holding the stock for the three-month period from June to September.

Regardless of the exact theoretical value, there ought to be a uniform progression of both individual option prices and spread prices in the marketplace. If this uniform progression is violated, a trader can take advantage of the situation by purchasing the option or spread which is relatively cheap and selling the option or spread which is relatively expensive.

The trader can start with conversions and reversals, then look at vertical spreads and butterflies, and finally consider straddles and time spreads. A trader who exercises a futures option early does so to capture the interest on the option's intrinsic value.

This intrinsic value will be credited to his account only if the option is subject to stock-type settlement. Since the only reason a trader would ever consider exercising a stock option call early is to receive the dividend, if a stock pays no dividend there is no reason to exercise a call early.

If the stock does paya dividend, the only time a trader ought to consider early exercise is the day before the stock goes ex-dividend. At no other time in its life is a stock option call an early exercise candidate. Whereas a stock option call can only be an early exercise candidate on the day prior to the stock's ex-dividend date, a stock option put can become an early exercise candidate anytime the interest which can be earned through the sale of the stock at the exercise price is sufficiently large.

The importance of early exercise is greatest when the underlying contract is a stock or physical commodity. This difference will especially affect the difference between European and Am erican put values, since early exercise wil1 allow the trader to earn interest on the proceeds from the sale at the exercise price. An option trader in either the stock or physical commodity market will find that the additional accuracy offered by an American model, such as the Cox-Ross-Rubenstein or Whaley models,will indeed be worthwhile.

Since each strategy combines an underlying position with an option position, it follows from Chapter 11 that the resulting protected position is a synthetic option:.

As with the purchase of a protective optlon, a covered write consists of a position in the under ng and an option. It can therefore be expressed as a synthetic position:.

A popular strategy, known as a fence, is to simultaneously combine the purchase of a protective option with the sale ofa covered option. For example, with an underlying contract at 50, a hedger with a long position might choose to simultaneously sell a 55 call and purchase a 45 put.

Fences are popular hedging tools because they offer known protection at alow cost, or even a credit. Fences go by a variety of names: Conversely, in a low implied volatility market a hedger should buy as many options as possible and sell as few options as possible.

A hedger who constructs a position with unlimited risk in either direction is presumably taking a volatility position. There is nothing wrong with this, since volatility trading can be highly profitable.

But a true hedger ought not lose sight of what his ultimate goal is: When he does that, he will have a position theoretically equivalent to owning a call. This process ofcontinuously rehedging an underlying position to replicate an option position is often referred to as portfolio insurance. If the mix of securities in a portfolio approximates an index, and futures contracts are available on that index, the manager can approximate the results of portfolio insurance by purchaslng or selling futures contracts to increase or decrease the holdings in his portfolio.

Even if options are available on an underlying asset, a hedger may still choose to effect a portfolio insurance strategy himself rather then purchasing the option in the marketplace.

For one thing, he may consider the option too expensive. If he believes the option is theoretically overpriced, in the long run it will be cheaper to continuously rehedge the portfo1io. Or he may find insufficient liquidity in the option market to absorb the number of option contracts he needs to hedge his position.

Finally, the expiration of options which are available may not exactly correspond to the period over which he wants to protect his position.

If an option is available, but expires earlier than desired, the hedger might still choose to purchase options in marketplace, and then pursue a portfolio insurance strategy over the period following the option's expiration. Moreover,the volatility of the underlying contract appears to be mean reverting. When volatility rises above the mean, one can be fairly certain that it will eventually fall back to its mean; when volatility fal1 s below the mean, one can be fairly certain that it will eventual1y rise to its mean.

Rather than asking what the correct volati1ity is, a trader might instead aSk, given the current volatiUty climate, what' right strategy? Rather than trying to forecast an exact volatility,a trader will try to pick a strategy that best fits the volatility conditions in the marketplace.

To do this, a trader will want to consider several factors:. Market participants are making the logical assumption that what has happened in the past is a good indicator of what will happen in the future. When the historical volatility declined, the implied volatility rarely dec1ined by an equal amount. And when historical volatility increased, the implied volatility rarely increased byan equal amount. Because volatility tends to be mean reverting, when historical volati1ity is above its mean there is a greater likelihood that it will dec1ine, and when historical volatility is below its mean there is a greater likelihood that it will increase.

When a disparity does exist,a trader can execute an arbitrage by hedging the mispriced index against either other stock indices or against a basket of stocks. There are several different methods of calculating stock index values, but the most common methods entail weighting the stocks either by price or by capitalization. The purchase of a futures contract offers one important advantage over the purchase of the component stocks: Consequently, there is an interest rate savings equal to the cost of borrowing sufficient cash to purchase all the stocks in the index.

This type of trading strategy, where one buys or sells a mispriced stock index futures contract and takes an opposing position in the underlying stocks, is one type of index arbitrage.

Since computers can often be programmed to calculate the fair value of a futures contract, and to execute the arbitrage when the futures contract is mispriced, such astrategy is also commonly referred to as program trading. A buy prograrn consists of buying the stocks and selling the futures contract,and a sell program consists of selling the stocks and buying the futures contract.

There are real1y two types of stock index options, those where the underlying is a stock index futures contract,and those where the underlying is the index itself. Although the ultirnate decision about the underlying price is trader's, in a stock index futures option rnarket a trader should be very careful about using an underlying futures price different from the quoted price. If he 1s wrong about the price at which the index is actually trading because the individual stock prices do not reflect the true rnarket,his theoretical evaluation of the futures contract wil1 be incorrect.

It may seem odd, but in fact it doesn't matter whether the index opens the next morning at a higher price, lower price, or unchanged. What matters is that the marketplace believes that the market will change, and that all contracts are priced accordingly. In such a case, the trader rnust exercise those options which, given the perceived change in the underlying price,now have a value less than parity,and replace them with other contracts which are not limited by parity constraints.

Refiners who purchase crude oil and refine it into gasoline and heating oil are often sensitive to the value of crack spreads ,the spread between the price of crude oil and its derivative products. Using, as before,a grid where the horizontal axis x-axis represents movement in the underlying contract and the vertical axis y-axis represents profit or loss, we can interpret the theoretical edge, delta, and gamma as follows:.

Note at time and volatility have similar effects on an option position. But unlike time, which can only move in one direction, volatility can either rise or fall.

The accuracy of values generated bya theoretical pricing model rests on two points: The underlying contract cannot a1ways be freely bought or sold; there are sometimes tax consequences; a trader cannot always borrow and lend money freely, nor at the same rate; there are always transaction costs. The most serious flaw in the frictionless markets hypothesis is the assumption at there are no transaction costs. While a strategy might or might not be affected by tax or interest rate considerations, there are always transaction costs.

While a changing interest rate will cause the value of a trader's option position to change, interest rates usually do not change in a way which will have a significant impact on an option's value,at least in the short run. Since the effect of changing interest rates is a function of time to expiration,and si ce most listed options have terms of less than ni months, interest rates would have to change violently to have an impact on any but the most deeply in-the-money options.

With the introduction of long-term equity options, known as LEAPs, the consequences of changing interest rates may well become more of a concern. We have only considered one alternative volatility scenario, where volatility is either increasing or decreasing.

But there are an infinite number ofpaths which volatility might follow over life of an option. Atrader might even assume that volatility is itself random, and at predicting volatility with any degree of accuracy is not possible. Models which assume stochastic volatility do exist and might, under some conditions, be more suitable than a traditional pricing model.

At the same time, such models add another dimension of complexity to a trader's life, and for this reason are not widely used. A diffusion process is a convenient, but clearly inexact, approximation of how prices change in the real world. Exchange-traded contracts cannot follow a pure diffusion process because exchanges are not open 24 hours per day. Theoreticians tend to agree that underlying contracts in most markets follow a combination ofboth a diffusion process and a jump process.

Avariation of the Black-Scholes model which assumes that the underlying contract follows a jump-diffusion process has in fact been developed.

Unfortunately, the model is considerably more complex mathematically than the traditional Black-Scholes model. Moreover, in addition to the five customary inputs, the model also requires two new inputs: In other words, the volatility ofa market is not independent of the price of the underlying contract. On the contrary, the volatility over time seems to be dependent on the direction of movement in the underlying contract. In some cases a trader expects the market to become more volatile if the movement is downward and less volatile if the movement is upward; in other cases a trader expects the market to become more volatile if the movement is upward and less volatile if the movement is downward.

Because volatility in some markets does seem to be dependent on the price of the underlying contract, a further variation of the Black-Scholes model has been proposed. The constant-elastictty ofvariance,or CEV, mode1 8 is based on an assumed relationship between volatility and the price level of the underlying contract. This relationship determines the probability of price moves of various magnitudes at each moment in time.

Price changes are stil1 random under a CEV assumption,but the randomness varies with the price of the underlying contract. A perfectly normal distribution can be fully described by its mean and standard deviation. But two other numbers, the skewness and kurtosis, are often used to describe the extent of the difference between an actual frequency distributlon and a true normal distribution.

If a distribution is positively skewed, the right-hand tail is longer an the left-hand tail. If the distribution is negatively skewed,the left-hand tail is longer than the right-hand tail. A true normal distribution has askewness of zero. Adistribution with a positive kurtosis has a tal1, pointed peak leptokurtic , while a distribution with a negative kurtosis has a low, flat peak platykurtlc.

Aperfectly normal distribution has a kurtosis of zero mesokurtlc. The language of options. Contract specifications Two types: The difference between an option and a futures contract: A futures contract requires delivery at a fixed price. The seller must make delivery and the buyer must take delivery of the asset. The buyer of an option can choose to take delivery a call or make delivery a put.

Therefore, we can calculate the proflt or 10ss at each exercise price involved and simply connect these points with straight lines. Ab ove highest exercise price all calls will go into-the-money, so the entire position will act like an underlying position which is either long or short underlying contracts equal to the number of net long or short calls. Below the lowest exercise price all puts will go into-the-money, so the entire position will act like an underlying position which is either long or short underlying contracts equal to the number of net long or short puts.

Introduction to Theoretical Pricing Models If he purchases options, not only must he be right about market direction, he must also be right about market speed. The minimum factors you must consider: The price of the underlying contract. The amount of time remaining to expiration. The direction in which he expects the underlying market move. The speed at which he expects the underlying market to move.

Propose a series of possible prices at expiration for the underlying contract. Assign an appropriate probability to each possible price. Maintain an arbitrage-free underlying market. From the prices and probabi1ities in steps 1, 2, and 3, calculate the expected return for the option. From the option's expected return, deduct the carrying cost. The option's exercise price. The current price of the underlying contract. The risk-free interest rate over the life of the option. The volatility of the underlying contract.

Summarize the most irnportant assurnptions governing price movement int the Black-Scholes Model: Changes in the price of an underlying instrurnent are randorn and cannot be artificially manipulated, nor is it possible to predict beforehand direction in which prices will move. The percent changes in the price of an underlying instrurnent are norrnally distributed. Because the percent changes in the price of the underlying instrurnent are assumed to be continuously cornpounded, the prices 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. Future volatility is what every trader would like to know, the volatility at best describes the future distribution of prices for an underlying contract. It is volatility being implied to the underlying contract through the pricing of the option in the marketplace.

Even though the term premium real1y refers to an option's price, it is common among traders to refer to the implied volati1ity as the premium or premium level. If the current implied volatility is high by historical standards, or high relative to the recent historical volatility of the underlying contract, a trader might say that premium levels are high; if implied volatility is unusuallylow,he might say that premium levels are low.

Using an Option's Theoretical Value The purchase or sale of a theoretically mispriced option requires us to establish a hedge by taking an opposing positlon in the underlying contract. The delta of a call option is always somewhere between 0 and 1.

The delta of an option can change as market conditions change. An underlying contract always has a delta of 1. The steps we have thus far taken illustrate the correct procedure in using an option theoretical value: Purchase sell undervalued overvalued options. Establish a delta neutraI hedge against the underlying contract. Adjust the hedge at regular intervals to remain delta neutral.

At that time we plan to close out the position by: Letting any out-of-the-money options expire worthless. Selling any in-the-money options at parity intrinsic value or, equiva1ently, exercising them and offsetting them against the underlying futures contract. Liquidating any outstanding futures contracts at the market price. Traders can freely buy or sell the underlying contract without restriction AlI traders can borrow and lend money at the same rate. Transaction costs are zero.

There are no tax considerations. Option Values and Changing Market Conditions. The de1ta is a measure of how an optio 's value changes with respect to a change in the price of the underlying contract. Theoretical or Equivalent Underlying Position. THE GAMMA The gamma sometimes referred to as the curvature of an option, is the rate at which an option's delta changes as the price of the underlying changes.

Deltas range from zero for far out-of. At-the-money calls have deltas of approximately 50, and at-the-money puts approximately At-the-money options have greater gammas than either in- or out-of-the-money options with otherwise identical contract specifications. At-the-money options have greater vegas than either in- or out-ofthe-money options With otherwise identical contract specifications.

Out-of-the-money options have the greatest vega as apercent of theoretical value. Introduction to Spreading Spreading is simply a way of enabling an optlon trader to take advantage of theoretlcally mispriced options, while at the same time reducing the effects of short-term changes in market conditions so that he can safely hold an optlon positlon to maturity.

Volatility Spreads Regardless ofwhich method we choose, each spread will have certain features in common: Eachspread will be approximately delta neutral.

Each spread will be sensitive to changes in the price of the underlying instrument. Each spread will be sensitive to changes in implied volatility. Each spread wil1 be sensitive to the passage of time.

If no movement occurs, a backspread is likely to be a losing strategy. Designate the opposite of a backspread as a ratio vertical spread. STRADDLE A straddle consists of either a long call and a long put, or a short call and a short put, where both options have the same exercise price and expire at the same time. STRANGLE Like a straddle, a strangle consists of a long call and a long put, or a short call and a short put, where both options expire at the same time.

BUTTERFLY A butterfly consists of options at three equally spaced exercise prices, where all options are of the same type either all calls or all puts and expire at the same time. TIME SPREAD calendar spread or horizontal spread Time spreads, sometimes referred to as calendar spreads or horizontal spreads, consist of opposing positions whlch expire in different months.

The most common type of time spread consists o The most common type of time spread consists of opposing positions in two options of the same type either both calls or both puts where both options have the same exercise price. An increase decrease in dividends lowers raises the forward price of stock. The effect of changing interest rates and dividends on stock option time spreads is shown below: Adjust at regular intervals — In theory, the adjustment process is assumed to be continuous because volatility is assumed to be a continuous measure of the speed of the market.

Adjust when the positlon becomes a predetermlned number 01 deltas or short. Delta DirectionaI Risk-The risk that the underlying market will move in one direction rather than another. When we create a position which is delta neutral, we are trying to ensure that initially the position has no particular preference as to the direction in which the underlying instrument will move.

A delta neutral position does not necessarily eliminate all directional risk, but it usually leaves us immune to directional risks within a limited range. Gamma Curvature Risk -The risk of a large move in the underlying contract, regardless of direction.

The gamma position is a measure of how sensitive a position is to such large moves. AbeBooks is an online marketplace for books. Millions of books offered from thousands of booksellers around the world are for sale through the AbeBooks website.

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