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## Wednesday, May 16, 2012

### PnL curve pushed towards hockey stick - 2 "pushers"

The vanilla European Call or Put's valuation graph (against spot price) is a _smooth_ curve Above the kinky hockey stick. I like to remind myself the curve depicts a "range of possibilities". The "original" curve is a Snapshot of a bunch of what-if possibilities such as "If underlier becomes \$x my put's theoretical value is \$y".

The curve drops towards the hockey stick as expiration approaches. This sentence is important but too complex too abstract. Let's be concrete and say we are on the 2nd last day of an IBM put. Let's say IBM is around \$100 now.

To keep things simple, let's say our option is ATM -- Scenario 1. The range of possibilities remains the same because IBM can still become \$1 or \$1000 or any value in between. Theoretical value of our put in that range of scenarios would be different than the earlier days of the option. IBM = \$50 -> put = \$51... IBM=\$100->put=\$11... IBM=\$200->put=\$1.50. Note, in reality IBM will fluctuate around the spot rather than long-jumping, so we can erase all but the middle section of this new curve. Still a smooth curve. If you compare this smooth curve to the original smooth curve, new curve is physically closer to the hockey stick.

What if our 2nd put is deep OTM with K = \$50 (Scenario 2)? Well the hockey stick is now a different hockey stick. (Actually it's shifted to the left). But again as expiration approaches, the "smooth curve" moves down to the hockey stick.

This downward shift is known as option decay. There are rare exceptions, but vast majority of options lose value over time.

During the decay, the smooth curve presumably becomes more convex, as it morphs into the kinky hockey stick. This means gamma increses????

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Now, there's a 2nd way the smooth curve moves towards (or away from) the hockey stick. When IBM implied vol surface experiences a uniform drop, the smooth curve drops in a similar fashion. More specifically, if the implied vol smile curve for our maturity drops, the smooth curve drops towards the hockey stick. This is still too vague too abstract. Let's be concrete and say smile curve drops to 0 for our maturity. Our put becomes identical to a short position in IBM if current price is below our strike, or worthless if above strike. That means, our PnL graph against IBM price is the hockey stick itself.

Therefore, drop in sigma_i has a similar effect as option decay. A surge in sigma_i will lift the smooth curve away from the hockey stick. Note all of the graphs we mentioned are plotted by BS -- so-called theoretical values.

It's all common sense after you internalize it.