5.09.2005

Q&A: What's the no-arbitrage value of the call option...

in this Q&A?

A

  • construct a portfolio of stocks and calls in which final payoff (at expiration) is up/down state-independent: S- - c- * n = S+ - c+ * n
    • Buying stocks and selling calls ((ie. covered call) so that regardless of which way the underlying moves, the portfolio value should be the same (perfectly hedged).
    • Calculate the hedge ratio (shares per call) : n = ($5 - $0) / ($90 - $75) = 0.3333.
  • To form a perfectly hedged portfolio, an investor needs to buy 1/3 of a share of stock for each call that written (sold), or buy 1 share of stock and sell (write) 3 calls. i.e. consider the position of the portfolio at the expiration of the call if the investor writes 1 call:
Value S+ = $90 S- = $75
1/3 share Stock $30 $25
Call -$5 $0
Expiration $25 $25
    • Guaranteed outcome is $25 for this portfolio: PV = $25 / 1.06^0.5 = $24.28
  • Investor would have to buy 1/3 share of the stock and sell 1 call option: $24.28 = S/3 - c
    • c = $80/3 - $24.28 = $2.38

Category: C++ Quant > Derivatives > Valuation

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