In a binomial option model, if we take the uptick as 6%, downtick as 5% (assume equally probable), and RFR of 6% (continuous compounding), then we have a violation of $0 < d < 1 + r < u$. Does this mean we cannot proceed with the pricing model at all? Is no-arbitrage one of the required assumptions?
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$\begingroup$ In this case, your probabilities are no longer positive and you can not proceed. $\endgroup$– GordonOct 30, 2015 at 12:55
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4$\begingroup$ yes its a required assumption. see the beginning of 11.1 prenhall.com/behindthebook/0132242265/pdf/… $\endgroup$– nathanesauOct 30, 2015 at 13:43
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2 Answers
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Let's illustrate with a one step tree. Take a call option. Without even making a specific assumption about the payout of the option, except that it will be greater in case of an uptick than a downtick: $f_u>f_d$. The price at time 0 for the option will be $f=(1+r)^{-1}f_u$ by the risk-neutral valuation formula, since you assume $1+r=u$.
Sell the option at time 0, receive $(1+r)^{-1}f_u$ and put them on a risk-free bank account. You receive $f_u$ at time $T$. Buy back the option which is now either worth $f_u$ in which case you just have the money necessary, either worth $f_d$ in which case you gain $f_u-f_d$ while your initial investment was zero. An arbitrage opportunity. (You could equally well make the argument using the stock instead of the option, it doesn't require you to use the risk-neutral valuation formula.)
answered Oct 16, 2017 at 5:45
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If you consider the price as the cost of the payoff replicating portfolio then no need for no-arbitrage assumption!
With violated no-arbitrage assumption you can make money using arbitrage. But then again think what would be the price for any positive payoff? It could be 0, it could be anything. It is since no matter how much you invested - if the short/long limitless fractional transaction are allowed - you can generate any cash-flow.
Take away: assumption on $0<d<1+r<u$ lets us define proper market. It is market without arbitrage possibilities. Based on this assumption risk neutral measure is determined ie. $$ p^* =\frac{(1+r)-d}{u-d},$$ which can be used for pricing the payoffs.
If for mentioned pricing model you use that measure then yes its derivation is based on $0<d<1+r<u$ assumption.
