I'm trying to come up with a metric to value and compare spreads. One way that I was doing this was to compute the Expected Value of the spread.

To calculate the expected value I used the following from omnieq

$$E = C \left(1 - \mathrm{N}\left(d_{2_{s}}\right)\right) - L \cdot \mathrm{N}\left(d_{2_{l}}\right) - \min{\left(\frac{\left|K_{s} - K_{l}\right|}{1 + \frac{\mathrm{N}\left(d_{2_{s}}\right)}{\mathrm{N}\left(d_{2_{l}}\right)}}, L \left(\mathrm{N}\left(d_{2_{s}}\right) - \mathrm{N}\left(d_{2_{l}}\right)\right)\right)}$$

Where

• $$S_0$$ is the current underlying stock price
• $$K$$ is the strike price (with $$K_s$$ being the strike of the short leg and $$K_l$$ of the long--this is true wherever $$s$$ and $${l}$$ appear in the subscript)
• $$\sigma$$ is the implied volatility
• $$r$$ is the risk-free rate
• $$T$$ is the time to maturity in years
• $$q$$ is the dividend yield
• $$\mathrm{N}\left(x\right)$$ is the cumulative distribution function
• $$d_1 = \ln\left(\frac{S_0}{K}\right) + \frac{T \left(r - q + \frac{\sigma^2}{2}\right)}{\sigma \sqrt{t}}$$
• $$d_2 = d_1 - \sigma \sqrt{T}$$
• $$C$$ is the credit received from selling the credit
• $$L = \left|K_s-K_l\right|-C$$, the max loss of the credit

So for example, take a current (3/22 AH) SPY credit call spread of

-1 SPY 04/23 0.271∆ $402.50c @$2.43
+1 SPY 04/23 0.215∆ $405.00c @$1.77

• The credit for this spread is

$$2.43 - 1.77 = 0.66$$

• The max loss is

$$\left| 402.50 - 405 \right| - 0.66 = 1.84$$

• The expected value is

$$E = 0.66 \cdot \left(1 - 0.271\right) - 1.84 \cdot 0.215 - \\ \min\left({\frac{\left|402.50 - 405.00\right|}{1+\frac{0.271}{0.215}}, 1.84 \cdot \left(0.271 - 0.215\right)}\right)$$

$$E = 0.48114 - 0.3956 - \min{\left(1.10596707819,0.10304\right)}$$

$$E=-0.0175$$

Using expected value as a metric to compare spreads is OK but not ideal. Namely because if we modify the spread by widening the strikes (in this example and in almost every other example I look at), the increase in max loss dominates the increase in credit and EV goes down.

However, when comparing two credit spreads that are equivalent besides the width of the strikes, the spread with the wider longer strike is more valuable because

1. The break-even point is further OTM, increasing the probability of profit,
2. The exposure to the short leg's greeks is greater, allowing greater theta / greater time decay, and
3. The credit received is greater

Then my question is, what metric can I use to evaluate a credit spread that's dependent on the same variables as the Expected Value as well as the value provided by widening the strikes?

For example, I thought perhaps Expected Value times the spread's net theta might be interesting but this metric still does not recommend widening strikes as I would expect. There must be some modification that can be made such that widening the strikes on the long-side is seen as more valuable than narrowing, up to a point where max loss eventually dominates. Further widening the strikes on the short-side should be seen as making the spread less valuable.

Thanks.

• I'm not sure why the latex equations are being broken-up onto newlines. If someone would make edits to fix that, I'd greatly appreciate it. Mar 23 at 4:10
• Probably an instance of meta.stackexchange.com/a/358356/266114 Mar 23 at 6:45
• Should be better now Mar 23 at 8:06
• Hi, two follow up questions: 1) When you talk about credit spreads you mean equity option trading strategies that combine long and short positions in puts or calls on the same underlying and the same time to maturity but different strike prices? 2) When you talk about expected value, on which measure do you want to base this expectation? Trivially, the expected payoff value under the risk neutral / pricing measure should equal the strategy's value, and hence the expected value should be zero, no? Mar 23 at 8:43
• In my last comment, it was supposed to say discounted expected payoff. Mar 23 at 10:18