Why a Down-and-In Barrier Call Can Have a Hump-Shaped Value Curve
Summary
The document explains the shape of a down-and-in call's value as spot moves around a lower barrier. At high spot levels, the chance of touching the barrier is small, so the option is worth little. As spot approaches the barrier, the probability of activation rises, which can increase the option's value even while the underlying call itself loses value. Once spot reaches the barrier, the option becomes active; further declines then reduce the call's value as it moves farther out of the money. A second answer emphasizes that the plotted curve only appears symmetric and points to standard barrier-option formulas as a way to examine its shape. The explanation is qualitative and depends on the option and market parameters; the cited example mentions a strike and barrier, but does not provide a complete numerical analysis. The document also contrasts the down-and-in behavior with a down-and-out option, whose value is zero at or below its barrier under the described setup.
Key ideas
- A down-and-in call gains activation probability as spot approaches its barrier from above.
- The option can rise in value near the barrier even as the underlying call loses value.
- After activation, further spot declines reduce the value as the call moves out of the money.
- The apparent symmetry of the plotted curve is misleading and depends on parameters.
Tags
Full text
# Graph of a down-and-in barrier option # Graph of a down-and-in barrier option 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. ## Answer by ir7 (score 4, accepted) https://quant.stackexchange.com/a/55396 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). ## Answer by stackoverblown (score 3) https://quant.stackexchange.com/a/55397 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 ## Answer by Rex (score 0) https://quant.stackexchange.com/a/63970 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.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.