Why Delta-Neutral Call Backspreads Can Open for a Credit

# Why should delta-neutral backspread always result in credit?


# Why should delta-neutral backspread always result in credit?












Natenberg mentions in chapter titled "Volatility Spreads" :

> under the assumptions of a traditional theoretical pricing model, a delta-neutral ratio spread where more options are purchased than sold should always result in a credit.

Is there an intuitive logic behind this?

## Answer by dm63 (score 3, accepted)

https://quant.stackexchange.com/a/85414

With the stock currently at S, you sell one call at $K_1$ and buy two calls at $K_2$ where $S<K_1<K_2$. Using standard Black Scholes the value of this position is $$- S.N(d_1) + K_1. N(d_2) + 2S.N(e_1) -2 K_2.N(e_2)$$ where the d’s refer to the $K_1$ option and the e’s refer to the $K_2$ option. Given that the delta of the $K_2$ option is half that of the $K_1$ option we have that $N(d_1)=2N(e_1)$ (exactly) and $N(d_2)=2N(e_2)$ (approximately ) and the value of the position reduces to $$(K_1-K_2) N(d_2)$$. Since this quantity is negative , you must receive a credit to enter it. [I left out all the discount factors $e^{-rT}$ for clarity].

## Answer by Dr. Michael J. Stefano (score 1)

https://quant.stackexchange.com/a/85330

yes, it is intuitive in the sense that it means you paid less for the spread, your short call more than paid for your long calls, lowering your max loss potential, and, you have a straddle of sorts that has profit potential from a good move either way.

A call ratio backspread strategy is best used in a low implied volatility (IV) environment when you expect a significant increase in both the stock price and future volatility, which tend to go opposite each other (one reason it is not my cup of tea) The position is "long vega," meaning it benefits from an increase in IV after the trade is placed.

low IV environments are typically skewed lower to the upside strikes allowing you to get that spread at a net credit and benefit more from an IV increase.

Benefit from IV Expansion: The core principle is to buy low and sell high. Entering the position when implied volatility is low allows you to capitalize on a potential IV expansion, which increases the value of your net long options position. this is not really significant at all however, if the IV increase is accompanied by a price drop. it may help improve the effect of negative theta decay if price stalls though.

Cheaper Entry: You can initiate the trade for a lower cost (or even a net credit) when IV is low, making it more cost-effective.

Vega Sensitivity: A call ratio backspread has positive Vega, meaning its value increases as implied volatility rises. This makes it an ideal strategy for capturing an anticipated surge in market expectations for movement.

In comparison, a simple long call has no downside potential.

Neither does a call debit spread (bull call spread), although the debit spread has interesting adjustment possibilities for potential downside gain, and while it requires a net debit to open, its max loss is comparably/proportionately less than the call backspread.

I would do an AI search there rather than trying to figure out the call ratio backspread.