# Does a Call Spread always need to be symmetric?

I have a plot of a Call Spread Option at time $t ={0}$ but the graph of the call spread is not completely symmetric. My question is: does it have to be? Here is the plot I'm referring to: I'm just wondering because at maturity time the Call Spread becomes symmetric, so it anyone can provide a bit of information on this I'd really appreciate it. Thanks!

• What do you mean symmetric? A call spread is not symmetric. You can plot the payoff to observe that – Gordon May 22 '16 at 23:46

## 1 Answer

I assume this is a plot of option value versus price of the underlying. The only case where it ought to be symmetric is if the pdf of the underlying is symmetric eg normally distributed. I'm guessing your chart assumes a lognormal underlying, which is a non symmetric pdf, so the graph is non symmetric.

• +1 Indeed. Intuitively, recall that in the limit of an infinitesimally small wedge size $K_2-K_1$, a (normalised) call spead converges to a digital option, i.e. its price is intrinsically tied to the probability of $S_T$ finishing in the money, i.e. to the cumulative distribution function of the random variable of interest. When the distribution function of your random symmetric, the cdf is symmetric as well. – Quantuple May 23 '16 at 8:07