Bull Spreads

Bull spreads are the most popular vertical spread strategies and correspond to a bullish view on the market. Investors use bull spreads when they believe an underlying asset value is going to increase above a specific level K₁ but will not be able to reach a level K₂ (with K₂ > K₁). In this case, investors are willing to capture the positive performance of the underlying asset and pay a smaller premium for this option.

A bull spread strategy can be constructed using call options (bullish call spread) or put options (bullish put spread).

Holding a bull call spread requires an initial investment equal to Call (K₁, T) − Call (K₂, T).

A bull spread strategy can also be realized by combining a short position in a put struck at K₂ and a long position in a put struck at K₁. The strategy is said to be bullish since the idea is to gain profit from selling a first put struck at K₂ and expecting the stock price to increase. However, it is important to note that the payoff of a bull put spread is always negative, whereas the payoff of a bull call spread is always positive.

This is due to the fact that an investor holding a bull put spread strategy is in fact selling a product and receives a global premium equal to Put (K₂, T) − Put (K₁, T).

Delta

The bell shape of the curve reminds us of the shape of the Gamma of vanilla options.

The Delta of a bull call spread is always positive, which proves that the holder of the bull spread strategy is always long the stock price.

Gamma and Vega

The Gamma and Vega of bull call spreads are more difficult to manage since their sign changes on the basis of movement in the underlying’s price.

The point at which the Vega of a call spread changes sign is around (K₁ + K₂)/2, assuming zero dividends and rates (flat forward).

If the stock price is below this breakeven point, the Vega of a call spread is positive and it becomes negative for stock prices above this point.

In the general case, the Vega changes sign around the forward, which we note could be several percent away from the (K₁ + K₂)/2 point.

Intuitively, you can think of being long vega between K₁and (K₁ + K₂)/2 because you have more to win than to lose and being short vega between (K₁ + K₂)/2 and K₂because you have more to loose than to win.

Skew

Buying a bull call spread implies buying a call with a lower strike K₁ and selling a call with a higher strike K₂.

Therefore, the buyer of the spread is long skew.