Hedging a sold Knock-Out (KO) option with a Risk Reversal (RR) in layman’s terms…

Bank's Position:

• The bank has sold a knock-out (KO) call option to the trader.
• Underlying Asset: Stock XYZ
• Option Details: KO call option with a strike price of $100 and a barrier level of $110.
• Expiry: 3 months

Trader's Position:

• The trader has bought the KO call option from the bank.
• The trader is bullish on Stock XYZ and believes the price will rise up to the barrier, ready to forgo upside potential beyond it for a lower premium cost.

Hedging Strategies by the Bank:

Step 1: Identifying Risks

The bank identifies that while it does not have a payout obligation if the KO option is knocked out due to the barrier being reached, it is exposed to the risk of potential losses associated with the sold KO call option.

Step 2: Managing Vega, Skew, and Vanna Risk

The bank is concerned about vega risk (sensitivity to changes in implied volatility) and exposure to the volatility skew due to the combination of different strike levels in the options (strike and barrier).

The bank also inherits a risk known as vanna, which refers to the sensitivity of the option’s delta to changes in implied volatility. To manage vega risk and skew exposure, the bank employs a risk reversal strategy. The sold put helps mitigate vanna risk arising from selling KO options. As the stock price moves, the vanna risk in the sold put is offset by the vanna in the bought call, leading to a more stable overall position.

Step 3: Risk Reversal Strategy

• Buying Call Options: The bank buys call options on Stock XYZ with a similar expiry and strike price as the sold KO call option. This introduces a long vega position, helping offset potential losses from the sold KO call option due to increased implied volatility.
• Selling Put Options: The bank sells put options on Stock XYZ with the same expiry and a strike price of $100. This introduces a short vega position and helps manage skew exposure from the combination of different strike levels.

Step 4: Potential Outcomes at Barrier Reach

Barrier Not Reached:

• If the price of Stock XYZ does not reach the barrier level of $110, the KO call option remains active until its expiry.
• The bank monitors its vega and skew risk exposures through the combined positions of the sold KO call option, the bought call options, and the sold put options from the risk reversal strategy.

Barrier Reached:

• If the price of Stock XYZ rises to or beyond the barrier level of $110, the KO call option is knocked out, and the trader loses the right to exercise it.
• The bank does not have a payout obligation in this scenario but still faces the risk of potential losses associated with the sold option.
• The risk reversal strategy helps the bank manage vega risk and skew exposure.

Through careful hedging, the bank reduces its exposure to volatility risks while maintaining a stable position in the dynamic market created by the sold KO call option.