Why No-Arbitrage Is Needed for Binomial Option Pricing
Summary
The document examines whether a one-step binomial option model can be used when the risk-free return does not lie between the asset’s down and up returns. It explains that this condition is tied to the absence of arbitrage and to obtaining a valid risk-neutral probability for pricing payoffs.
One answer illustrates the arbitrage: if the risk-free return equals the up return, selling a call and investing the proceeds at the risk-free rate can produce a nonnegative payoff with a possible gain and no initial investment. Another answer distinguishes replication cost from risk-neutral valuation, while noting that in an arbitrage market prices may not be uniquely determined. The discussion is conceptual and limited to a simple one-period setting; it does not address transaction costs or more general market structures.
Key ideas
- A valid risk-neutral probability in the binomial model requires the risk-free return to lie between the down and up returns.
- When that condition fails, an arbitrage strategy can arise in the one-step example.
- Without no-arbitrage, replication-based prices may not be uniquely determined.
- The explanation focuses on a simple binomial setting.
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Full text
# Does the Binomial Pricing Model require a no-arbitrage assumption?
# Does the Binomial Pricing Model require a no-arbitrage assumption?
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?
## Answer by Raskolnikov (score 1)
https://quant.stackexchange.com/a/36463
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.)
## Answer by Wiktor Madejski (score 0)
https://quant.stackexchange.com/a/39799
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.