5.09.2005

Q&A: What's the no-arbitrage value of the put 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- + p- * n = S+ + p+ * n
    • Buying stocks and calls (ie. protected put) 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 = ($10 - $0) / ($90 - $75) = 0.66.
  • To form a perfectly hedged portfolio, an investor needs to buy 2/3 of a share of stock for each put, or buy 1 share of stock and 3/2 puts. 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
2/3 share Stock $60 $50
Put $0 $10
Expiration $60 $60
    • Guaranteed outcome is $60 for this portfolio: PV = $60 / 1.06^0.5 = $58.28
  • Investor would have to buy 2/3 share of the stock and 1 put option: $58.28 = 2*S/3 + p
    • p = $58.28 - $2*80/3 = $4.9

Category: C++ Quant > Derivatives > Valuation

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