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**

**Options and Time Decay:**Detail how the Theta of an option behaves as it nears its expiration, and what effects do dividends and interest rates have on this Greek?**On put-cal parity:**Imagine you are analyzing the options market for ABC Corp and notice a significant discrepancy in the implied volatility between at-the-money call and put options with the same expiration date. The call option's implied volatility is 30%, while the put option's is 35%, despite similar liquidity and market conditions.Explain why this discrepancy in implied volatilities might be an arbitrage opportunity.

Describe a specific arbitrage strategy to exploit this discrepancy, including the positions you would take.

Discuss how this strategy is expected to yield a profit and what risks or market conditions you must consider.

## Capital Concepts

**On the shift in market sentiment**

Consider a retail trader with a specific options position on HYG (iShares iBoxx $ High Yield Corporate Bond ETF): you are short an out-of-the-money (OTM) 33 delta put option with seven days to expiration (DTE) and long an OTM 40 delta call option with 21 DTE. Unexpectedly, the central bank cuts rates. At first, the market reacts positively, and the rates of bonds are pushed lower and their prices higher while the volatility decreases:

Discuss the impact of that decision on the portfolio's overall Greeks and the likely pnl at the end of the day. Discuss what the retail trader should do to have a similar Greek exposure than at the beginning of the day.

Euphoric after this day of gain, the retail trader decided to do nothing. However, the rate cut is now perceived as negative for the economy. HYG tracking high-yield corporate bonds is severely impacted, and the volatility has increased significantly. Discuss the new Greek profile of the position and the impact of gamma on the pnl of the position. What urgent action should the trader take as we get closer and closer to the expiry date while volatility keeps increasing?

**7-Figures Bonus Question**

**On Spot-Vol Correlation**

As a professional trader, you have sold a structured product that resembles a skewed straddle with a slightly negative delta, reflecting your negative view on a basket of non-blue chip equities. This basket traditionally benefits from a negative spot-vol correlation, aligning with your risk management guidelines that define the acceptable limits for delta and vega exposure in your position. The structured product is due to expire at the end of the year.

As the year-end approaches, significant market flows trigger an unexpected rally in these equities. Contrary to the usual pattern, this rally is accompanied by an increase in volatility, resulting in a sudden shift to a positive spot-vol correlation.

Describe the impact of this sudden shift to a positive spot-vol correlation on your Greek risk limits, mainly focusing on delta and vega. Give insights into how your P&L might evolve as the rally continues, considering that your basket comprises non-blue chip stocks with less-than-ideal liquidity.

Given the change in market dynamics and the liquidity constraints of your equity basket, how might you adjust your trading strategy? What considerations should you consider if this change in correlation appears to be more than just a short-term anomaly and could persist until the end of the year?

## Theoretical Value

**Question 1: Options and Time Decay**

**Answer:** Theta represents the rate at which an option's price declines as the expiration date approaches, assuming all other factors remain constant. This is often referred to as "time decay." As an option nears expiration, its Theta typically increases, indicating an acceleration in the rate of time decay, especially for at-the-money options.

Dividends and interest rates affect Theta in different ways:

**Dividends:**When a stock pays dividends, the value of call options tends to decrease as the expected dividends are priced into the option. This is because the stock price will likely drop by the dividend amount on the ex-dividend date. Therefore, the Theta of call options might significantly decline as the ex-dividend date approaches. Conversely, put options might see a decrease in time decay if they are near the money, as the expected drop in the stock price due to the dividend could make them more valuable.**Interest Rates:**Higher interest rates increase the cost of carrying a position, which can lead to an increase in call option prices and a decrease in put option prices. Consequently, the Theta of calls may decrease slightly since the option's time value is higher. Conversely, the Theta of puts may increase as the time value is lower.

**Question 2: On call-put parity**

**1. Arbitrage Opportunity Due to Implied Volatility Discrepancy:**

Despite similar liquidity and market conditions, the notable difference in implied volatilities between the at-the-money call and put options for ABC Corp suggests a deviation from the call-put parity principle. This principle states that for options with the same strike price and expiration, the difference in price between a call and a put should align with the current market price and the strike price discounted to present value. A discrepancy in implied volatilities, with the call option at 30% and the put option at 35%, indicates potential mispricing and an arbitrage opportunity, particularly as it suggests a violation of the volatility arbitrage principle.

**2. Specific Arbitrage Strategy with Delta Hedging:**

The proposed strategy is to create a delta-neutral position to exploit this discrepancy. This involves buying the undervalued call option (with a lower implied volatility of 30%) and selling the overvalued put option (with a higher implied volatility of 35%), both at the money. The goal is to capitalize on the volatility arbitrage without assuming significant directional risk in the underlying asset.

Achieving delta neutrality is essential for mitigating directional market risk. This is done by dynamically hedging the net delta of the combined options position through purchasing or selling ABC Corp's shares and adjusting the hedge in response to market movements and changes in delta values.

**3. Strategy Profit Expectations and Risk Considerations:**

The profitability of this strategy hinges on the normalization of the implied volatilities of the call and put options, banking on the market to correct the initial mispricing. Profits are anticipated from the change in the volatility premium of the options rather than the movement of the underlying stock.

Risks to consider include the necessity of frequent rebalancing of the delta hedge, which can incur substantial transaction costs. Moreover, the underlying asset’s price volatility and shifts in overall market conditions can unpredictably affect the options' deltas and implied volatilities, potentially widening rather than narrowing the volatility discrepancy.

Other factors, such as changes in interest rates, dividends on the underlying stock, and liquidity risk, must be considered. The trader needs to be vigilant in monitoring these market dynamics and be prepared for timely adjustments in the hedging strategy.

This strategy, while theoretically sound, requires careful execution and continuous monitoring to manage the various risks and to ensure profitability from the volatility arbitrage opportunity.

## Capital Concepts

**Part 1: Initial Impact of Rate Cut**

#### Impact on Greeks and P&L

**Delta Impact****Short OTM Put (33 Delta, 7 DTE):**The rise in bond prices makes it less likely that HYG will fall below the put strike, decreasing the put's delta (making it less negative). Vanna (sensitivity to volatility) and Charm (time decay effect on delta) also contribute to this decrease, especially with diminishing volatility and the approaching expiration.**Long OTM Call (40 Delta, 21 DTE):**The increase in bond prices raises the probability of exceeding the call strike, increasing its delta. However, the Vanna and charm effects on the call are less pronounced due to the longer DTE and the rate cut's impact on underlying prices.

**Vega Impact****Both Options:**The decrease in volatility diminishes the value of both the short put and the long call, but the put's closer expiration makes it more sensitive to this change.

**P&L at the End of the Day**The portfolio's delta becomes slightly more positive, and the vega exposure decreases. P&L is likely positively impacted by the rate cut, but the extent depends on the specific price movements and volatility changes.

#### Adjusting for Similar Greek Exposure

**Short Put Position:**The trader could close or roll the short put to a further expiration to manage the reduced delta and vega exposure. The decision should be made based on an assessment of the volatility. We were initially short, and volatility dropped. Enough to get out of the trade?**Delta Management:**If the goal is to maintain a similar delta profile, the trader might need to adjust their equity exposure or consider other options and strategies to balance the delta change.**Vega Adjustment:**Given the decreased volatility, reassessing the overall vega exposure is essential. This might involve adjusting the strike prices or expiration dates of the options.

**Part 2: Market Sentiment Shift**

#### Impact on Greeks and P&L

**Delta and Gamma****Short Put (33 Delta, 7 DTE):**The price decline increases the put's negative delta and gamma (the rate of delta change), heightening directional risk and sensitivity to price movements.**Long Call (40 Delta, 21 DTE):**The delta likely decreases due to the price drop, and gamma increases, but less so than the put due to the longer DTE.

**Vega**The rise in volatility increases the value of both options, but the short put is more affected due to its nearer expiration.

**P&L at End of Day**The increased volatility and delta/gamma changes could significantly negatively impact the short put, potentially leading to losses and eroding previous gains.

#### Urgent Actions

**Manage the Short Put:**The trader should consider closing, rolling, or hedging the short put to address the increased negative delta, gamma, and vega.**Reevaluate the Long Call:**Determine the feasibility of maintaining the long call in the context of the changed market conditions and increased volatility.**Overall Strategy Reassessment:**The drastic shift in market outlook necessitates a thorough trading strategy review. Exiting positions initially designed to benefit from lower volatility might be more prudent, especially considering the escalated risks from gamma as expiration approaches.

## 7-Figures Bonus

**Delta Impact**The initial slight negative delta indicates a position favoring a decrease in equity prices. However, the unexpected rally in the equities pushes this negative delta towards or beyond the risk limits, especially since increasing prices typically devalue short positions.

The shift to a positive spot-vol correlation exacerbates this issue. Typically, negative delta positions benefit from declining volatility in a downturn, but the rally coupled with increased volatility presents an unusual challenge.

**Vega Impact**With the original position being short on vega, the expectation was to benefit from a decline in volatility. However, the rally's unexpected association with increased volatility (a positive spot-vol correlation) significantly alters the risk profile, pushing the vega exposure towards or past established risk limits.

**P&L Evolution as the Rally Continues**

The unusual positive spot-vol correlation in this scenario reverses the typical dynamics expected from a negative spot-vol correlation. Instead of volatility decreasing with the rally, it increases, potentially amplifying losses on the short vega position.

The P&L may be further stressed due to the liquidity constraints of the non-blue chip stocks. Adjusting the position effectively in a less liquid market can be challenging, potentially leading to suboptimal execution and additional costs.

The skew of the structured straddle plays a critical role here. It may offer some mitigation if the skew is adjusted to account for the rally. However, this effectiveness depends heavily on the specifics of the skew relative to market movements.

**Adjusting the Trading Strategy**

**Delta Management**Considering the negative delta, purchasing equities or utilizing delta-hedging strategies with other derivatives becomes essential to mitigate the increased risk due to the market rally.

**Vega Management**The short vega position, which is now disadvantageous due to the rising volatility, necessitates adjustments. This could involve buying options to balance the vega exposure or restructuring the straddle's skew to better align with the current market dynamics.

**Liquidity and Market Impact:**Adjustments must be mindful of the liquidity constraints inherent in non-blue chip stocks. Gradual and strategic trades, possibly using instruments with lower market impact, are advisable to avoid exacerbating price movements.

**Long-Term Considerations:**If this shift to a positive spot-vol correlation is not just a temporary anomaly but a longer-term trend, it might require a significant overhaul of the trading strategy. This could involve diversifying into different asset classes or redefining the risk parameters to accommodate the new market environment.

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