1. What is the price of a call as sigma -> infinity?
Ans:
It approaches S. The lognormal distribution is negatively skewed. As sigma -> infinity, although the probability of obtaining a very large S increases, a large portion of the probability mass is pushed towards the origin, making the option more likely to be out of the money (http://en.wikipedia.org/wiki/Log-normal_distribution).
2. Consider a product with maturity T=1, S_0=100, r=0. The product has a "one-hit" payoff, namely it pays \$1 when the underlying hits 120 for the first time, at which point the product terminates. What is the price of such product and how do you hedge it?
Ans:
It is worth 1*100/120 = \$0.8333. The replicating portfolio is simply to buy 0.008333 unit of stock at the inception and sell it off to collect 0.008333*120 = \$1 when the underlying hits 120 for the first time.
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For 1, "making the option more likely to be out of the money", do you mean in the money? Otherwise, the option would be worthless.
ReplyDeleteI meant "out of the money." Let me clarify: the increased probablity of ending up at very high spot price is counter-balanced by the "pushing towards zero" effect.
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