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No arbitrage principle and derivatives valuation

The no arbitrage principle is at the core of derivative pricing models. Derivatives are priced such that no arbitrage opportunity exists. This is why it is called arbitrage-free pricing. An arbitrage opportunity exists when an asset with the same future cash flows is sold at different prices such that a party can buy and sell it simultaneously to extract the difference as a riskless profit. They are rare because whenever they arise, investors trade such that the prices converge, and the mispricing is removed. However, this process is slow because such arbitrage opportunities can be exploited only if the profit exceeds the transaction costs.

For exploitation of arbitrage opportunities, we need to transact in two assets that produce the same payoffs, but such assets are very rare. For example, if we buy a share of Apple on one exchange and sell it at another. If they sell in different currencies, we also need to take care that the correct exchange rate is reflected. Similarly, bonds issued by the same issuer with different maturities can be arbitraged.

However, derivatives enable arbitrage of virtually any asset. For example, since the movement in the price of a derivative and its underlying are linked together, we can buy and sell them simultaneously to lock an arbitrage that must pay a return equal to the risk-free rate. If not, arbitrageurs should engage in this strategy by borrowing at the risk-free rate and earning a higher rate. This would force the strategy to earn only the risk-free rate and gravitate the market to a single price for the derivative. This is exactly the arbitrage principle which is used to value a derivative such that no above risk-free return should be possible in a perfect hedge.

Arbitrage is possible because investors can use a combination of underlying and the derivative to synthetically create different position through a process called replication. For example, if an investor buys an asset and sell a derivative, he should earn the risk-free rate on the overall portfolio. Similarly, if he buys a derivative on an equity index (is long the equity futures) and finance it with risk-free loan (is long the risk-free rate), he would effectively be long the equity index. Replication can potentially have lower transaction costs and it enables investors to exploit any pricing differentials.


While in pricing an underlying, we assume that investors are risk-averse and we use a discount rate which reflects the associated risk premium, in pricing a derivative, we assume that investors are risk-neutral because a derivative has no risk (in addition to the risk already reflected in the underlying). The fact that risk-aversion of investors does not affect a derivative’s price is called risk-neutrality and such a pricing mechanism is called risk-neutral pricing.

Factors that limit arbitrage

However, there are factors which limit market participants ability to exploit arbitrage opportunities:

  • Transactions costs may higher than the total price differentials making the arbitrage unprofitable.
  • Not everyone can borrow unlimited amounts at the risk-free rate.
  • Any gains from offsetting positions may not be liquid, for example, a transaction might be a hedge on paper, but a gain on one asset may be converted to cash only later, required the arbitrageur to plus the mismatch with his additional capital or borrowing.
  • Not all arbitrage transactions may be risk-free, for example, initial estimate of volatility may not be correct.
  • There may be restrictions on short selling limiting an arbitrageur’s ability to take a short position necessary to execute the arbitrage.
  • Removal of the mispricing depends on the realization by other investors of its existence.

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