Read a book Connolly «Buying and selling volatility» [1] know what the delta-neutral hedge separate option or option position. This method of trade was bought or sold the option underlying asset hedge their risks in such a way that the total delta position is always equal to zero. However, in practice, many traders to hedge their risks are not options for the delta, and after a fixed price of underlying asset. Finally, many do not hedge their risks, and sell or buy a «naked» options. The article compares these strategies.
Random Simulation
Especially for the solution of the problem through a package Excel file was created, in which the simulation of random motion of the price of a stock within 10 days. For simplicity, its current price is set at $ 100. The user enters the annual volatility of stock and its expected annual revenue, and the program on the basis of this information builds a random minute price series length of 10 days. It is expected that the share is traded around the clock. Thus, the length of the created-minute price series of 14,400 values. To understand better the program schedule is displayed price movements. For simplicity, we assume that we sell 100 call options (ie one full lot), which expired after 10 days, and within 10 days hedzhiruem options by buying and selling shares. The method of hedging is determined by the user in the program - either for hedging the delta, either through a fixed price range, or the lack of hedging at all. After starting the program creates a random number and price hedging transactions being sold option in accordance with established rules. Automatically calculated financial results of operations, which is included in a special table. Then create another random price series, again carried out by the hedge and the financial result is calculated, which also entered into the table. The number of random simulations of possible share price may be as high as, say, 500. In each case, the program performs all the necessary hedging and calculates profit or loss. As a result, we obtain a table containing the results of each experiment. On this basis, calculated automatically generalize indicators - the average financial results and its standard deviation (ie this will be just below).
In a random simulation is a prerequisite, which is based on the most common model for evaluating the options - black Shoulza - and many other models in the field of investment. The premise is that the stock price is log-normal distribution. It should be noted that this is one of the most objective methods of mathematical modeling of price movements.
Analysis of results
So, turn to the most interesting - the analysis of results of different methods of hedging.
The testing was as follows. Sell 100 call options with страйком $ 105 and the implied volatility 50% who either do not hedge, or hedge by buying shares through the fixed price range, or the delta.
Within each of the three options were tested with varying volatility of stocks, namely: 30% (volatility, which is moving action below implied volatility of an option), 50% (volatility of stocks is exactly the implied volatility of an option) and 70% (volatility of stock exceeds the implied volatility of an option).
Thus, only 9 tests conducted in each of them made 500 simulations random traffic campaigns. Note that in all cases, the expected profitability of the price was set at 0%, which implies the existence of a lateral trend - namely, in those circumstances, traders seek to sell the options.
The lack of hedging
To start with the first series of tests, when the sold option does not hedge at all, that is, any transactions with no underlying asset. Test results are given in Table 1. It clearly shows that when the volatility of shares was equal to 50%, that is exactly coincides with the implied volatility of option sold, on average, produces a loss of $ 11.05, which is very close to zero. With the volatility of shares equal to 30%, the average profit was $ 98.58, with volatility of 70% of an average loss of $ 110.45. The value of standard deviation of profit / loss, we have considered not going, but this information is useful to us in the future.
Thus, one can conclude that if the action will move to greater volatility than the implied volatility of the sold option, on average, we get a loss, if less - profitable. But how exactly to define the value of the average earnings or average loss? More specifically: if the volatility of the sold call option is equal to 50%, and the action moves to the volatile x%, what is the expected result of the financial position?
This can be calculated mathematically. It is necessary to calculate the models for Black-Shoulza premium call option with volatility 50% (respectively, valid till the expiry of 10 days, страйком $ 105; stock price in this case is $ 100, while the interest rate we take for 0%) and subtracted from the calculation of similarly premium exactly the same option, but with volatile x%. Expected financial results will be exactly equal to the difference between the two prizes.
Check this on our example. Prize option volatility to 50% is $ 145.43, with volatility 30% - $ 43.81, with a volatility 70% - $ 264.84. According to the above reasoning, when the volatility of shares equal to 30%, average income must be equal to $ 145.43 - $ 43.81 = $ 101.62, with volatility of 70% to a loss of $ 145.43 - $ 264.84 = - $ 119.41.
It is evident that these figures are very close to the results obtained during testing. A slight deviation is due to the fact that we did just 500 tests. If you do not 500, as, say, 10 thousand simulations, the calculated on the basis of that sample numbers would be closer to the results calculated by mathematical.
In a fixed price range
Consider now how our results change if we sold the lot of options to hedge equity base through a fixed interval.
Specific ways of hedging via fixed-price range, there are many. In our case, at the level of $ 101, $ 102, $ 103, $ 104 and $ 105, we will buy 20 shares. That is, once the price reaches $ 101, we buy 20 shares. If the price rises to $ 102, we will buy another 20 shares. And so up to $ 105 - it страйк our option.
By the time will be purchased for a total of 100 shares which, when further growth of prices will be fully sold to hedge the option. If, however, after growth, for example, up to $ 101 price will revert back, we are at $ 100 sell previously purchased 20 shares and Fix loss. Unfortunately, any hedging of underlying asset implies a loss - this will not деться.
So, what results were obtained when hedging the option, through a fixed interval? The table shows that when the volatility of shares 30%, on average, had a profit of $ 101.56, with volatility of 50% - profit of $ 3.79, with a 70% loss of $ 111.28. Just shows that the average financial results very close to those obtained with a simple sale of the call option without hedging.
In fact, if the computer power and time are not allowed to make 500, but, say, 1 million of testing, we have found that the average financial results in the two cases are to each other even closer. Although it may seem strange, but in the general case of selling the call option with the transaction of its hedging and selling similar option, followed by hedging with the same expected financial results. Does this mean that all the efforts of hedge option does not have any meaning? No.
Let us analyze the second important indicator that we have not yet been considered. This is the standard deviation of the financial result. It is a measure of how far away from the value of its average value. The larger the standard deviation, the greater dispersion of values around the mean value, the smaller - the smaller the variance.
In the area of investment, the standard deviation of return on investments is a common measure of risk. The high standard deviation indicates that it is very difficult to estimate in advance what will be return on investment. If the attachment has a small standard deviation of returns, we can much more accurately predict in advance what profit the investor will receive (it is located far from its expected value).
So, back to the financial results. The table shows that in the first case, when we sold the call option, and not to hedge it, the standard deviation of the financial result was higher than in the second case, when the sold option hedging. Thus, when the volatility of shares equal to 30%, standard deviation for the case without hedging was $ 146.50, and for the case of hedging - $ 100.55.
It is easy to see that the same pattern seen in two other cases. You can make an important conclusion that the option hedging leads to lower standard deviations of the profit / loss strategy. Moreover, any hedging has this and only this goal. Hedging does not affect the amount of expected profit, it only reduces the risk of the strategy (standard deviation).
Hedging on the delta
Finally, consider what we will, if the option to hedge in the manner described in the book, Connolly "Buying and selling volatility" [1], namely - on the delta. This method of hedging a trader buys and sells shares in a manner that the total position delta has always been zero. In the program we have produced almost continuously hedge, buying and selling shares every minute. What does this lead? As expected, the average value of the financial strategy of the result received very close to the results obtained in the first two cases. However, the standard deviation of the financial results were significantly lower than in the first two cases, and ranged from $ 1 to $ 3!
In fact, if the hedging on the delta was carried out not every minute, and indeed continually, the standard deviation would be zero! This means that irrespective of which way she went to stock price, the strategy of selling the call option with hedging of delta would the same financial result.
For example, if the action moving with a constant volatility 30%, we would have a profit of $ 101.62 - regardless of the dynamics of the share price. The last statement may seem surprising, but it is actually true.
Interestingly, the continuous delta hedging an option on the profit or loss depends only on two parameters: from the implied volatility of the sold option and the volatility of equities. Selling an option, call implied volatility of 50% and hedzhiruya him to the delta, we have, according to Connolly, sell volatility, as we will always make a profit, if the action will move with volatility less than 50%, and loss - with the volatility of over 50% .
Moreover, we can advance to say what the gain or loss shall receive at a given volatility of equities. So, what exactly will share the road - will it grow, fall or stand on the spot - we are not interested. Our financial results will be determined solely by volatility.
Therefore, continuing a strategy of hedging options in the delta can be truly called a trade volatility. It should be noted that many sources of the term is used more broadly to refer to any strategy aimed at changing the volatility (eg, buying and selling straddle), which is actually not quite correct.
To calculate the delta requires knowledge of the volatility. Usually used by the current implied volatility of an option. But in fact want to use the true volatility of equities. It may not match the implied volatility of option.
We sell the call option implied volatility of 50%. However, delta hedging calculated based on the volatility of shares 30%, 50% and 70%. If we used the delta, based on implied volatility of option (50%), we would not have achieved this result, and the standard deviation of profit / loss is not equal to $ 0.
In reality, the problem is that to estimate the true volatility, which is moving action at this time, it is very difficult. The historical volatility, which represents the average volatility for some historical period of time, an inaccurate assessment of the current volatility.
Therefore, to calculate the delta of traders, in most cases use the implied volatility of an option. More experienced players use "empirical delta, which is based on" empirical volatility - that is, their own assessment of the volatility of equities. In any case, you have to say that for the delta hedge - even if it is not quite accurate - has meant that the standard deviation of the financial result of the strategy is less than with alternative methods of hedging. Consequently, the delta-neutral hedge is still the preferred way to hedge stock options, reducing the risk to the minimum possible level.
Conclusion
It should be noted that when testing does not take into account the commission for the transactions, which can be great, especially when using a continuous (or near-continuous) on the delta hedging. It is clear that this is adversely affect the final result.
Also worth mentioning that in the ho-de tests, we assumed that the expected return on equities is equal to 0%. This premise is quite natural because traders usually sell options in terms of lateral trend. However, with a positive expected return on equities in the case of bovine market, or negative in the case of Bear - the results will be different. In this article, this is not taken into account.
Finally, despite the fact that the article dealt with the sale and hedging the call option, the findings are true also for the purchase of call option, as the buyer's financial results are always exactly equal to the financial result of the seller, taken as negative. Also, all the conclusions are valid for the put option - short or long.
The reader can check all of the findings in this article, as well as to test any other ways to hedge by using the CD-ROM magazine, a description which can also be found on the disk.
In conclusion would like to add that, perhaps, someone will be able to open up new ways to hedge or to find other parties to this very interesting question.
Random Simulation
Especially for the solution of the problem through a package Excel file was created, in which the simulation of random motion of the price of a stock within 10 days. For simplicity, its current price is set at $ 100. The user enters the annual volatility of stock and its expected annual revenue, and the program on the basis of this information builds a random minute price series length of 10 days. It is expected that the share is traded around the clock. Thus, the length of the created-minute price series of 14,400 values. To understand better the program schedule is displayed price movements. For simplicity, we assume that we sell 100 call options (ie one full lot), which expired after 10 days, and within 10 days hedzhiruem options by buying and selling shares. The method of hedging is determined by the user in the program - either for hedging the delta, either through a fixed price range, or the lack of hedging at all. After starting the program creates a random number and price hedging transactions being sold option in accordance with established rules. Automatically calculated financial results of operations, which is included in a special table. Then create another random price series, again carried out by the hedge and the financial result is calculated, which also entered into the table. The number of random simulations of possible share price may be as high as, say, 500. In each case, the program performs all the necessary hedging and calculates profit or loss. As a result, we obtain a table containing the results of each experiment. On this basis, calculated automatically generalize indicators - the average financial results and its standard deviation (ie this will be just below).
In a random simulation is a prerequisite, which is based on the most common model for evaluating the options - black Shoulza - and many other models in the field of investment. The premise is that the stock price is log-normal distribution. It should be noted that this is one of the most objective methods of mathematical modeling of price movements.
Analysis of results
So, turn to the most interesting - the analysis of results of different methods of hedging.
The testing was as follows. Sell 100 call options with страйком $ 105 and the implied volatility 50% who either do not hedge, or hedge by buying shares through the fixed price range, or the delta.
Within each of the three options were tested with varying volatility of stocks, namely: 30% (volatility, which is moving action below implied volatility of an option), 50% (volatility of stocks is exactly the implied volatility of an option) and 70% (volatility of stock exceeds the implied volatility of an option).
Thus, only 9 tests conducted in each of them made 500 simulations random traffic campaigns. Note that in all cases, the expected profitability of the price was set at 0%, which implies the existence of a lateral trend - namely, in those circumstances, traders seek to sell the options.
The lack of hedging
To start with the first series of tests, when the sold option does not hedge at all, that is, any transactions with no underlying asset. Test results are given in Table 1. It clearly shows that when the volatility of shares was equal to 50%, that is exactly coincides with the implied volatility of option sold, on average, produces a loss of $ 11.05, which is very close to zero. With the volatility of shares equal to 30%, the average profit was $ 98.58, with volatility of 70% of an average loss of $ 110.45. The value of standard deviation of profit / loss, we have considered not going, but this information is useful to us in the future.
Thus, one can conclude that if the action will move to greater volatility than the implied volatility of the sold option, on average, we get a loss, if less - profitable. But how exactly to define the value of the average earnings or average loss? More specifically: if the volatility of the sold call option is equal to 50%, and the action moves to the volatile x%, what is the expected result of the financial position?This can be calculated mathematically. It is necessary to calculate the models for Black-Shoulza premium call option with volatility 50% (respectively, valid till the expiry of 10 days, страйком $ 105; stock price in this case is $ 100, while the interest rate we take for 0%) and subtracted from the calculation of similarly premium exactly the same option, but with volatile x%. Expected financial results will be exactly equal to the difference between the two prizes.
Check this on our example. Prize option volatility to 50% is $ 145.43, with volatility 30% - $ 43.81, with a volatility 70% - $ 264.84. According to the above reasoning, when the volatility of shares equal to 30%, average income must be equal to $ 145.43 - $ 43.81 = $ 101.62, with volatility of 70% to a loss of $ 145.43 - $ 264.84 = - $ 119.41.
It is evident that these figures are very close to the results obtained during testing. A slight deviation is due to the fact that we did just 500 tests. If you do not 500, as, say, 10 thousand simulations, the calculated on the basis of that sample numbers would be closer to the results calculated by mathematical.
In a fixed price range
Consider now how our results change if we sold the lot of options to hedge equity base through a fixed interval.
Specific ways of hedging via fixed-price range, there are many. In our case, at the level of $ 101, $ 102, $ 103, $ 104 and $ 105, we will buy 20 shares. That is, once the price reaches $ 101, we buy 20 shares. If the price rises to $ 102, we will buy another 20 shares. And so up to $ 105 - it страйк our option.
By the time will be purchased for a total of 100 shares which, when further growth of prices will be fully sold to hedge the option. If, however, after growth, for example, up to $ 101 price will revert back, we are at $ 100 sell previously purchased 20 shares and Fix loss. Unfortunately, any hedging of underlying asset implies a loss - this will not деться.
So, what results were obtained when hedging the option, through a fixed interval? The table shows that when the volatility of shares 30%, on average, had a profit of $ 101.56, with volatility of 50% - profit of $ 3.79, with a 70% loss of $ 111.28. Just shows that the average financial results very close to those obtained with a simple sale of the call option without hedging.
In fact, if the computer power and time are not allowed to make 500, but, say, 1 million of testing, we have found that the average financial results in the two cases are to each other even closer. Although it may seem strange, but in the general case of selling the call option with the transaction of its hedging and selling similar option, followed by hedging with the same expected financial results. Does this mean that all the efforts of hedge option does not have any meaning? No.
Let us analyze the second important indicator that we have not yet been considered. This is the standard deviation of the financial result. It is a measure of how far away from the value of its average value. The larger the standard deviation, the greater dispersion of values around the mean value, the smaller - the smaller the variance.
In the area of investment, the standard deviation of return on investments is a common measure of risk. The high standard deviation indicates that it is very difficult to estimate in advance what will be return on investment. If the attachment has a small standard deviation of returns, we can much more accurately predict in advance what profit the investor will receive (it is located far from its expected value).
So, back to the financial results. The table shows that in the first case, when we sold the call option, and not to hedge it, the standard deviation of the financial result was higher than in the second case, when the sold option hedging. Thus, when the volatility of shares equal to 30%, standard deviation for the case without hedging was $ 146.50, and for the case of hedging - $ 100.55.
It is easy to see that the same pattern seen in two other cases. You can make an important conclusion that the option hedging leads to lower standard deviations of the profit / loss strategy. Moreover, any hedging has this and only this goal. Hedging does not affect the amount of expected profit, it only reduces the risk of the strategy (standard deviation).
Hedging on the delta
Finally, consider what we will, if the option to hedge in the manner described in the book, Connolly "Buying and selling volatility" [1], namely - on the delta. This method of hedging a trader buys and sells shares in a manner that the total position delta has always been zero. In the program we have produced almost continuously hedge, buying and selling shares every minute. What does this lead? As expected, the average value of the financial strategy of the result received very close to the results obtained in the first two cases. However, the standard deviation of the financial results were significantly lower than in the first two cases, and ranged from $ 1 to $ 3!
In fact, if the hedging on the delta was carried out not every minute, and indeed continually, the standard deviation would be zero! This means that irrespective of which way she went to stock price, the strategy of selling the call option with hedging of delta would the same financial result.
For example, if the action moving with a constant volatility 30%, we would have a profit of $ 101.62 - regardless of the dynamics of the share price. The last statement may seem surprising, but it is actually true.
Interestingly, the continuous delta hedging an option on the profit or loss depends only on two parameters: from the implied volatility of the sold option and the volatility of equities. Selling an option, call implied volatility of 50% and hedzhiruya him to the delta, we have, according to Connolly, sell volatility, as we will always make a profit, if the action will move with volatility less than 50%, and loss - with the volatility of over 50% .
Moreover, we can advance to say what the gain or loss shall receive at a given volatility of equities. So, what exactly will share the road - will it grow, fall or stand on the spot - we are not interested. Our financial results will be determined solely by volatility.
Therefore, continuing a strategy of hedging options in the delta can be truly called a trade volatility. It should be noted that many sources of the term is used more broadly to refer to any strategy aimed at changing the volatility (eg, buying and selling straddle), which is actually not quite correct.
To calculate the delta requires knowledge of the volatility. Usually used by the current implied volatility of an option. But in fact want to use the true volatility of equities. It may not match the implied volatility of option.
We sell the call option implied volatility of 50%. However, delta hedging calculated based on the volatility of shares 30%, 50% and 70%. If we used the delta, based on implied volatility of option (50%), we would not have achieved this result, and the standard deviation of profit / loss is not equal to $ 0.
In reality, the problem is that to estimate the true volatility, which is moving action at this time, it is very difficult. The historical volatility, which represents the average volatility for some historical period of time, an inaccurate assessment of the current volatility.
Therefore, to calculate the delta of traders, in most cases use the implied volatility of an option. More experienced players use "empirical delta, which is based on" empirical volatility - that is, their own assessment of the volatility of equities. In any case, you have to say that for the delta hedge - even if it is not quite accurate - has meant that the standard deviation of the financial result of the strategy is less than with alternative methods of hedging. Consequently, the delta-neutral hedge is still the preferred way to hedge stock options, reducing the risk to the minimum possible level.
Conclusion
It should be noted that when testing does not take into account the commission for the transactions, which can be great, especially when using a continuous (or near-continuous) on the delta hedging. It is clear that this is adversely affect the final result.
Also worth mentioning that in the ho-de tests, we assumed that the expected return on equities is equal to 0%. This premise is quite natural because traders usually sell options in terms of lateral trend. However, with a positive expected return on equities in the case of bovine market, or negative in the case of Bear - the results will be different. In this article, this is not taken into account.
Finally, despite the fact that the article dealt with the sale and hedging the call option, the findings are true also for the purchase of call option, as the buyer's financial results are always exactly equal to the financial result of the seller, taken as negative. Also, all the conclusions are valid for the put option - short or long.
The reader can check all of the findings in this article, as well as to test any other ways to hedge by using the CD-ROM magazine, a description which can also be found on the disk.
In conclusion would like to add that, perhaps, someone will be able to open up new ways to hedge or to find other parties to this very interesting question.
Michael Glukhov
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