Here is a graph of Price vs Spot from Joshi's Quant Interviews book,
The first line is a down-and-out barrier option and the other one is a down-and-in barrier option. The strike is 100 and the barrier is at 95.
Why does the down-and-in option look like a hump? I would have thought there would be someone asymmetry due to the barrier causing the option to spring into life.
Intuitively, underlying call keeps losing value as the spot goes down, but the barrier option value (which starts at almost nothing for high spot) keeps growing as the spot approaches the barrier level (the chance to get something, even if it's an out-of-money call, is growing). When the spot hits the barrier level, the value of the call is still ok (unless barrier level is very low) and, more importantly, is becoming real for the first time. After that, if the spot continues to slide, the call keeps losing value (at this point it has replaced the barrier option).
If you put some numbers into down-in/out barrier call option formulae that can be found in many books, you will see that the down-in curve is not symmetric. It just looks like it in that plot.
Below the barrier, the prices are obviously just Black-Scholes values, as the spot price goes higher the chance of it going below the barrier is obvious becoming smaller and smaller so the price is vanishing with higher spot. For the down-out barrier call prices, the prices are zero at and below the barrier obviously.
For risk free rate of 5%, volatility of 20%, expiry in 1 year, other parameters as you stated, I got
An easier way to understand is to look at the spot price line from right to left (i.e. from 120 to 80). That should make the understanding more intuitive.
