Welcome to the Sharpe Two Option Quiz!

Every Friday, we will challenge you with a mix of fundamental theory and real-world questions to test your options trading knowledge and get you one step closer to mastery.

Ideal for those keen to hone their skills on the art of options or prepping for a job interview, our quiz offers a concise yet insightful way to test and expand your understanding. You will find three sections:

**Theoretical Value:**Sharpen your grasp of options theory and its foundational principles.**Capital Concepts:**Apply theoretical knowledge through the lens of a retail trader navigating real market events.**7-Figures Bonus:**Step into the shoes of a professional trader and tackle complex, high-stakes scenarios on the trading desk.

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## Theoretical Value

Describe how the passage of time impacts an option's volatility and what sort of careful consideration a retail trader with a long view on volatility should take.

Explain how Delta and Vega interact in a portfolio consisting of long options positions with about three months before expiry. What are the implications for a trader with a long 20-delta strangle during periods of high market volatility and already declining from their peaks? How would it change if it was three weeks before expiry?

## Capital Concepts

### Question 1: PnL Analysis for an ATM Straddle Post-Economic Announcement

**Context:** Analyze the PnL dynamics for a trader who was long an ATM (At-The-Money) straddle on the SPY, expiring in 35 days, right before a major economic announcement.

**Scenario:** Following the announcement, the market experienced a significant downturn, but surprisingly, the implied volatility did not increase further. Discuss the key factors contributing to the biggest gain in the trader's PnL under these circumstances.

**Focus:** Emphasize the role of Delta and Gamma in this scenario, considering the ATM position of the straddle and the effect of a sharp market move without an accompanying rise in implied volatility.

### Question 2: Managing a Short-Term Straddle in a Falling Market

**Context:** Explore the dynamics of selling short-term straddles (5 days to expiration, or DTE) in a market scenario with a substantial downturn, but signs of stabilization are emerging.

**Scenario:** After the market takes another downturn, testing the trader's short-term straddle limits, their PnL remains positive. Explain the reasons behind this seemingly counterintuitive outcome.

**Focus:** Consider the impact of factors such as Theta (time decay), Delta and how the rapid approach of the expiration date influences the performance of the short-term straddle, especially in a volatile market environment.

## 7-Figures bonus

Define and describe a trading position that can simultaneously achieve the rare combination of being both long Gamma and long Theta. Explain the circumstances under which this might be possible and the specific characteristics of the options involved.

## Theoretical Value answers

### Question 1

**Implied Volatility and Time to Expiration**:Implied volatility (IV) represents the market's forecast of a likely movement in a security's price. Generally, IV tends to be higher for options with a longer time to expiration. This is because there's more uncertainty about the underlying asset's price movement over a longer period.

As the expiration date approaches, the IV can change significantly. If the market anticipates an event or significant movement in the underlying asset, IV might increase as the event nears and decrease rapidly after the event (known as "volatility crush").

**Theta (Time Decay)**:Theta measures the rate at which an option's value decreases over time, assuming all other factors remain constant. For a trader holding long volatility positions (like long calls or puts), time decay works against the position. The value of the option erodes as it gets closer to expiration.

The effect of Theta becomes more pronounced as the expiration date nears. Theta's impact is initially less significant for long-term options but accelerates as expiration approaches.

### Question 2

**Delta and Vega Interaction in Long Options Positions**

**Understanding Delta and Vega**:**Delta**measures the rate of change in an option's price per one-point move in the underlying asset. Delta can be positive (calls) or negative (puts) for long options. A 20-delta strangle each option has a delta of ±0.20, implying a moderate sensitivity to price movements in the underlying asset.**Vega**measures sensitivity to changes in the implied volatility of the underlying asset. Long options positions generally have positive Vega, meaning they gain value as implied volatility increases.

**Long Options Three Months Before Expiry**:In this timeframe, Vega plays a more dominant role than Delta. With three months to expiry, the options are less impacted by Theta (time decay), allowing Vega (volatility exposure) to be a more significant factor.

The Delta of the options is less reactive to the underlying asset's immediate price movements than options closer to expiration. However, it still plays a role in the P&L as the market moves.

**Implications for a Long 20-Delta Strangle in High Volatility**

**During Periods of High Market Volatility**:As volatility peaks and starts declining, the long strangle (being long Vega) might lose some value since the options are more sensitive to changes in volatility.

The Delta of the options would cause fluctuations in the P&L in response to movements in the underlying asset, but these would be more pronounced in a high-volatility environment.

**With Three Weeks Before Expiry**:As the expiry date gets closer, the impact of Delta becomes more pronounced. The options become more sensitive to price movements in the underlying asset. The 20-delta characteristic of the strangle will now result in larger P&L swings in response to price changes.

Vega's impact is still significant, but starts to compete with Theta. The nearer to expiration, the more the options’ value is affected by time decay, potentially offsetting some gains from Vega in a volatile market.

In a declining volatility environment, the negative impact on the strangle's value due to decreasing Vega could be more pronounced as expiry nears.

## Capital Concepts answers

### Question 1

**Impact of the Market Downturn on the Straddle**

**Delta Dynamics**:Initially, the Delta of an ATM straddle is near zero, as the negative Delta of the put offsets the positive Delta of the call. However, as the market experiences a downturn, the Delta of the put increases in absolute value (becoming more negative) while the Delta of the call decreases.

This change in Delta leads to the straddle gaining value as the put option becomes more valuable with the declining market. In contrast, the loss in the call option is relatively less significant due to its lower Delta.

**Gamma Impact**:Gamma measures Delta's rate of change with respect to changes in the price of the underlying asset. Gamma is at its highest for ATM options, meaning any change in the underlying price will lead to a larger change in Delta.

As the market moves sharply downward, the ATM straddle's high Gamma amplifies the Delta change, especially for the put option. This results in a rapid increase in the value of the put, contributing significantly to the PnL.

**Lack of Increase in Implied Volatility**

In many scenarios, a sharp market movement, especially a downturn, is accompanied by a rise in implied volatility, which would typically benefit a long straddle due to its positive Vega. However, in this scenario, the implied volatility does not increase.

The lack of an increase in implied volatility means the straddle's value does not get the additional boost typically provided by rising Vega. Therefore, the primary contributors to PnL gains in this scenario are the changes in Delta and the effect of Gamma.

**Key Factors Contributing to PnL Gain**

**Increased Value of the Put Option**: The market downturn significantly increases the value of the put component of the straddle. This is the primary driver of PnL gains in this scenario, given the Delta and Gamma dynamics.**Gamma Amplification**: The high Gamma of the ATM straddle accelerates the Delta changes, especially for the put option, enhancing the PnL gain from the market downturn.

**Conclusion**

In this scenario, the biggest gain in the trader's PnL for the long ATM straddle on SPY post-economic announcement, without a rise in implied volatility, is primarily driven by the increased value of the put option due to the negative Delta's magnification and the high Gamma of the position. These factors outweigh Vega's lack of additional gains, as the implied volatility remains unchanged. The trader benefits from the directional move of the market, mainly due to the characteristics of Delta and Gamma in ATM options.

### Question 2

**Impact of Market Downturn on the Short Straddle**

**Delta Dynamics**:Initially, the Delta of a short ATM straddle is close to zero, as the negative Delta of the short put offsets the positive Delta of the short call.

In a falling market, the Delta of the put increases (becomes more negative) while the Delta of the call decreases. However, since the options are short, this change in Delta can be beneficial. The value of the short put increases (more negative), which is good for the seller, while the short call loses value (less positive), which is also beneficial for the position.

**Theta (Time Decay)**:Theta represents the rate at which the option's value decreases over time, assuming all other factors remain constant. For a short straddle, Theta favors the trader as the value of the options they've sold decreases over time.

With only five days to expiration, the effect of Theta is pronounced. The rapid time decay means the options lose value quickly, which benefits the seller.

**Reasons Behind Positive P&L in a Falling Market**

**Rapid Time Decay**:The primary reason behind the positive P&L, despite the market downturn, is the accelerated time decay of the options as the expiration date approaches. This decay leads to a rapid decrease in the options' premium, which is advantageous for the seller of the straddle.

**Delta Neutral Position Initially**:The straddle was initially Delta-neutral, meaning the position's value wasn't highly sensitive to small moves in the underlying asset's price. Even though the market downturn would have caused some changes in Delta, the short time frame to expiration means that these changes have less impact on the overall position.

**Market Stabilization Signs**:The emerging signs of market stabilization suggest that the price of the underlying asset might not move dramatically in the remaining days to expiration. This lack of significant price movement is ideal for a short straddle strategy, as it maximizes the benefit from Theta decay.

**Conclusion**

In conclusion, the positive P&L of a trader selling a short-term straddle in a falling market, especially when signs of stabilization are emerging, can be primarily attributed to the significant impact of Theta decay in the last few days before expiration. The initial Delta-neutral nature of the position also contributes to this outcome, as it reduces the position's sensitivity to immediate market movements. The market stabilization further supports this by limiting significant price movements that could adversely affect the straddle. Therefore, despite the market downturn, the specific dynamics of a short-term straddle under these conditions can lead to a positive P&L outcome for the trader.

### 7-Figures Bonus answer

A trading position that is long Gamma and long Theta is a rare combination in options trading, as these two Greeks usually have an inverse relationship. However, it's possible to construct a strategy that achieves this under certain circumstances. Let's define these terms and then explore how such a position might be structured.

**Definitions**

**Gamma**measures the rate of change of an option's Delta relative to a one-point move in the underlying asset. Being long Gamma means benefiting from large movements in the underlying asset, as it causes the Delta of the option to change more rapidly, potentially increasing the position's value.**Theta**represents the rate of time decay of an option's value. Being long Theta means benefiting from the passage of time, as it causes the value of the options to decrease, which is beneficial when selling options.

**Achieving Long Gamma and Long Theta**

Typically, being long in options (buying calls or puts) gives you long Gamma but short Theta, as the value of your options decreases over time. Conversely, selling options gives you short Gamma but long Theta, as you benefit from the time decay of the options you've sold. However, specific strategies can allow a trader to be long both Gamma and Theta.

**The Strategy**

**Calendar Spread (Time Spread)**:One of the most common ways to achieve a long Gamma and long Theta position is through a calendar spread. This involves selling a short-term option and buying a longer-term one with the same strike price.

The short-term option has a higher Theta but lower Gamma, and the long-term option has a lower Theta but higher Gamma.

As time progresses, the short-term option (which the trader is short) loses value rapidly due to its higher Theta, benefiting the trader. Meanwhile, the long-term option (which the trader is long) has less time decay and maintains its value better due to higher Gamma.

**Specific Market Conditions**:This strategy works best in a market where the underlying asset is not expected to make significant moves in the short term but has the potential for larger moves in the longer term.

The key is that the short-term option decays faster than the long-term option gains value, which can happen in low to moderate-volatility environments.

**Characteristics of the Options**

**Different Expirations**: The crucial characteristic is the difference in expiration dates between the short-term and long-term options.**Same Strike Price**: Typically, both options will have the same strike price, aligning their price movement with the underlying asset.**Implied Volatility Considerations**: The implied volatility environment can significantly impact the effectiveness of this strategy, as it affects both Gamma and Theta.

**Risks and Considerations**

While a calendar spread can achieve long Gamma and long Theta, it's not without risks. The primary risk is a significant move in the underlying asset in the short term, which could lead to losses on the short-term option that outweigh the benefits from Theta decay.

The trader must also manage the position actively, potentially adjusting the strike prices or rolling the options forward to different expirations based on market movements and volatility changes.

**Conclusion**

In summary, achieving a long Gamma and long Theta position is challenging but possible, typically through strategies like calendar spreads. This strategy requires careful consideration of market conditions, option characteristics, and active position management. While it offers the dual benefit of profiting from time decay and potential large moves in the underlying asset, it also carries unique risks that must be diligently managed.

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