*In Part 1, Jeff Augen helps options traders to understand more about implied volatility. To finish his discussion, he takes a closer look at various near-expiration options for Goldman Sachs before and after the market collapse and examines what is known as a volatility skew.*

The environment of the steeper skew is highly beneficial to options traders because it allows complex positions to be structured across many strikes. It also increases liquidity. Referring to table 3, only four put strikes are likely to be tradable in the reduced volatility example on the right side of the table, while the actual situation facilitated trading across all nine strikes listed on the left side ($50 to $90). This result is surprising because only 19 days remained before expiration. A conservative investor could have structured a ratio trade consisting of 10 long $55 puts and 20 short $50 puts with absolutely no risk unless the stock fell from $86.85 to below $45.00 in the remaining two weeks. The trade would have yielded a maximum return of $4.80 with the stock expiring at the short strike ($50). In most cases an investor would expect such a trade to expire with both sides out-of-the-money. These dynamics hint at a more aggressive solution that involves 30 short $50 puts and 10 long $55 puts. Assuming that both sides expire worthless, the trade would generate $640 of profit.

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30 short $50 puts x $0.44 = $1,320 10 long $55 puts x $0.68 = $680 net short = $640

Although this return might seem small, it dramatically exceeds any interest generating investment in the post crash environment where the Fed Funds target rate is set at 0.00% to 0.25%. More precisely, the return on this trade would be equal to the profit ($640) divided by the total cost for the long side ($680 for 10 contracts) plus the margin requirement for the 20 naked short puts ($18,010).

$640 / ($680 + $18,010) = .034

Considering that the trade was only open for 19 days, the return in percentage terms (3.4%) is surprisingly large. Moreover, had the stock fallen to a point between the long and short strikes, the trade would have generated a much larger return. Optimally, with the stock trading at $50 on expiration day, the position would be worth $5.00 ($5,000 for 10 long $55 puts that are $5 in-the-money), and the profit would be $5,640 calculated as the final value of the trade plus $640 obtained from the original net short position. Total profit would rise to 30%, albeit an unlikely scenario.

We can extend these calculations using 30 day prices to project a monthly return. A variety of positions can be structured using the three lowest strikes on our list.

$50 put = $1.17 @ 145% implied volatility $55 put = $1.60 @ 136% implied volatility $60 put = $2.07 @ 126% implied volatility

We could, for example, structure a 2:1 ratio composed of 20 short $50 puts and 10 long $55 puts. This position would be fully protected down to an underlying price of $45. The more aggressive trade outlined above could be duplicated using 30 short $50 puts and 10 long $55 puts. Although not as well protected in a deep crash of the stock, this trade delivers a significantly larger return. The best answer is most likely a hybrid of the two choices. We can construct such a trade using a 3:1 ratio that is short $50 puts and long $60 puts. The position, therefore, would be net short $1.44 as follows:

30 short $50 puts at $1.17 = $3,510 10 long $60 puts at $2.07 = $2,070 net short = $1,440

If the stock closes expiration above both strikes, the trade will generate a minimum return of 7.2% during the 30 days remaining before expiration.

$1,440 / ($2,070 + $18,010) = .072

Monthly compounding of this conservative trade would, therefore, generate an annual return of 144%. As before, each monthly version of the trade would be far more profitable if the stock fell far enough to close expiration between the long and short strikes. At the short strike, the position would generate an $11,440 profit calculated as $10,000 for the long $60 put plus the original short sale of $1,440 – a total monthly return of 57%. A more likely scenario, however, would be for the stock to settle in the $10 space between the two strikes. If, for example, the stock closed expiration at $55, the long $60 put would be worth $5.00 and the profit would be $6,440 calculated as $5,000 plus the value of the original net short position ($1,440).

$6,440 / ($2,070 + $18,010) = .321

This return is reasonable because the stock is very likely to land in this large region between the strikes and the required movement is not particularly large. Calculated in standard deviations using at-the-money implied volatility of 95%, a 1 month 1 standard deviation price change would be $23.82.

$86.85 x 0.95 / SQRT(12) = $23.82

The distance to $55, the midpoint between the strikes, is $31.85 or 1.3 standard deviations. Option pricing theory, built on the normal distribution, sets the probability of this downward price change at 9.7%. Taken out of context, this particular value is not helpful because it represents the probability of a precise change from the starting price to exactly $55. A more reasonable approach would be to calculate the probability of the stock closing expiration in various trading ranges with distinct profit characteristics. The following list includes the four most significant ranges for this trade which are further quantified in table 4.

1. Above both strikes ($60) where the initial $1.44 of excess premium is retained as profit 2. Between the upper ($60) and lower ($50) strikes – the so called “profit zone” where the trade returns more than the initial net short amount of $1.44 3. Between the lower strike ($50) and the breakeven point ($45.70) where profit shrinks below the initial net short amount of $1.44 4. Beyond the breakeven point ($45.70) where the trade loses money

A determination of the exact breakeven point referenced in items 3 and 4 must include the excess $1.44 of premium realized from the initial position structure – that is, the final breakeven point must be adjusted to remove this initial gain.

**Table 4:** Risk assessment for the 3:1 ratio trade described above (30 short $50 puts and 10 long $60 puts for Goldman Sachs with 30 days remaining before the March 2009 expiration). Calculations are based on the normal distribution.

Description |
Result |
Probability |

stock closes expiration above $60 strike price |
Trade profit = $1,440 (7.2%) |
87.0% |

stock closes expiration in the profit zone between the long ($60) and short ($50) strikes |
Trade profit between $1,440 (7.2%) and $11,440 (57%) |
6.9% |

stock closes expiration between the short strike ($50) and the breakeven point ($45.70) |
Trade profit shrinks to a value between $0.00 and $1,440 |
1.9% |

stock closes expiration below the breakeven point ($45.70) |
Trade loses money at the rate of $2,000 for each additional dollar of underlying price decline |
4.2% |

Based on the characteristics outlined in table 4, the trade has an excellent risk: return profile. Most of the advantage accrues from the steep implied volatility smile that allows us to sell 145% volatility while purchasing only 126%.

The smile also allows us to create a deep out-of-the-money short position with options that would be illiquid and nearly worthless if priced using the much lower at-the-money implied volatility of 95%. These dynamics make it difficult to structure a similar trade with call options because the volatility skew disproportionately increases the price of the long strike over the short strike. Bear market volatility skews generally favor trades that are structured with short components at lower strikes.

Summarizing the data outlined in table 4, we can conclude that the trade will most likely deliver a significant profit (94% probability of returning more than 7.2% on invested capital). Only 4.2% of trades are likely to lose money. Furthermore, because the trade is structured with deep out-of-the-money options that have significant amounts of remaining time premium, it is very likely that outsized downward spikes can be handled through adjustments to the original position.

Finally, the steep volatility skew that characterizes today’s market is likely to persist for an extended period of time – possibly several years. Unlike other distortions, the skew has memory because it is built into the hedging models of thousands of institutional and private investors who lost money during the last crash. Moreover, until the world financial situation completely stabilizes, institutional investors are unlikely to regain the previous level of complacency that allowed them to confidently sell out-of-the-money naked put options.

* Jeff Augen is currently a private investor and writer,and has spent over a decade building a unique intellectual property portfolio of databases, algorithms, and associated software for technical analysis of derivatives prices. This work has been the subject of three books from Pearson Education (Financial Times Press):* The Volatility Edge in Options Trading, The Option Trader’s Workbook,

*and*Trading Options at Expiration.

*Click here to sign up for a free, online presentation by Larry Connors, CEO and founder of TradingMarkets, as he introduces The Machine, the first and only financial software that allows traders and investors to design and build quantified portfolios.*