Concepts of arbitrage, replication, and risk neutrality in derivatives pricing

The pricing of derivatives is based on the no-arbitrage principle. There are mainly four types of underlying assets on which derivatives are based: equities, fixed-income securities, currencies, and commodities. The underlying for derivatives can be interest rate as well, but that is not an asset.

We need to take the present value of future cash flows to price financial assets. The value of asset equals the expected future price plus any interim payments such as dividends or coupons discounted at an appropriate discount rate depending on the riskiness of the asset. Thus the investor arrives at a fundamental value of the asset and compares that with the market value and trade accordingly.

The expectation plays a role in the valuation. The investor forecasts the value of the asset over his holding period. Suppose the asset has no interim cash flows and E(ST) be the expected price of the asset in the future. Now, this estimated price is based on the probability distribution. For example, take a non-dividend paying stock whose price you want to estimate and your holding period is one year. You expect the price the stock after one year to be following:

As you can see from the table that the expected spot price after one year comes with a risk (volatility), as the price can range from $50 to $170. Note we are just assuming simple expectation. The probability distribution can have a much wider range of less probability at each price level. The probability distribution can be normal as well where the tails are very long. So, we would require a higher return on the investment because of the inherent risk. Assuming λ to be risk premium and r to be risk-free rate, the spot price today can be written as S₀ = E(S_T)/(1+r+ λ)^T i.e. discounting the expected future cash flow at the appropriate discount rate.

Benefits and costs of holding an asset: Few assets have interim cash flows like dividends and coupon payments. These benefits are called the monetary benefits. We can have nonmonetary benefits with some assets as well especially with commodities. Those benefits are generally referred to as convenience yield. The nonmonetary benefits are generally opaque and difficult to measure. Financial assets have nonexistent or extremely limited convenience yield. Commodities have convenience yield. If a commodity is in short supply or it is extremely difficult to short the commodity, then it is likely to have a higher convenience yield because by owning the asset, you have the flexibility to sell it when the price rises. Because of the convenience yield, the spot price of a commodity could be even higher than the expected future spot price.

The costs for holding the asset primarily include storage cost and the opportunity cost of the money invested. The commodities like gold, oil, and wheat incur storage cost. There are other costs as well such as insurance cost to protect commodities from theft and destruction.

When we include costs and benefits of holding an asset, then we need to include that in the pricing as well. We incorporate the effects of these costs and benefits by determining their value at the end of the holding period with the exception of the opportunity cost. We assume that these costs and benefits are certain unlike the price of the asset and discount these costs and benefits with the risk-free rate to get the present value. The costs reduce the current price, and the benefits increase the current price. The net of the costs and benefits is referred to as the term carry or cost of carry. Denoting θ as the present value of the costs and γ as the present value of the benefits, we can calculate the spot price of an asset using the formula: S₀ = E(S_T)/(1+r++ λ)^T - θ + γ.

The principle of Arbitrage: Arbitrage occurs when two assets or portfolios produce identical results but sell at different prices. So, a trader can buy at lower price and sell at a higher price to earn a risk-less profit. The arbitrage opportunities occur very infrequently and are exploited very quickly by the traders.

It is difficult to find two assets that would have identical cash flows at a future date and sell at different prices today. So, generally, arbitrage opportunities are found when the shares of a company traders at different exchanges and after converting the shares in one currency, they have different values. The same asset can remain at different prices if the price different is very small, and the transaction costs are higher than the price differential. So, the arbitrage transactions do not occur for such price discrepancies.

However, with the help of derivatives, one can create a hedged portfolio that would earn risk-free rate. This hedged portfolio eliminates the arbitrage opportunities and thus helps in derivatives pricing. The derivatives and risk-free rate can be combined to replicate the returns of an asset. Note that in the below formulas, the position in derivative means the position that is opposite to the underlying position. So, if we are long in the asset, then the derivative position will be short.

Asset + Derivative = Risk-free asset
Asset - Risk-free asset = -Derivative
Derivative - Risk-free asset = -Asset

Replication generally has more costs than simply taking a position in the asset. So, if all the assets are priced correctly, the replication seems useless. However, the ability to replicate something with something else can be valuable to investors as it helps them in making arbitrage profits. Sometimes, replication can even have lower transaction costs. For example, replicating return of an index fund has lower transaction costs with derivative and risk-free rate rather than purchasing every security in the index fund.

Risk aversion, risk neutrality, and arbitrage-free pricing: Most of the investors are risk-averse, i.e. they do not accept the risk without expecting some additional return for that risk. A risk neutral investor is one who does not care for the risk and does not require a risk premium to invest in a risky asset. A risk-seeking investor is one who does who is ready to take a negative risk premium and prefers risky securities than riskless securities with the same return.

A derivative can be combined with an asset to produce a risk-free position. The derivative price is the price that guarantees the risk-free combination of the derivative and the underlying to produce a risk-free rate of return. Thus, the derivative price can be inferred from the characteristics of the underlying, the characteristics of the derivative, and the risk-free rate. The investor's risk-aversion is not a factor in determining the derivative price. So, the derivative price is calculated assuming that the investor is risk-neutral. That's why the derivatives pricing is sometimes called risk-neutral pricing. That means the expected payoff of the derivative can be discounted at the risk-free rate rather than the risk-free rate plus a risk premium.

If the calculated price of the derivative is different than the market price, then there is an arbitrage opportunity. The arbitrage transactions will bring back the price to where it should be, and that price will eliminate the arbitrage opportunities. This process of pricing derivatives by arbitrage and risk neutrality is called arbitrage-free pricing.

Limits to arbitrage: Transaction costs, inability to short-sell, inability to borrow unlimited money at the risk-free rate, and money put into margin accounts (initial margin and maintenance margin) are limits to arbitrage. When the short selling is difficult, then the arbitrage might exist in only one direction but not in the opposite direction. The modeling risk is also a hindrance in arbitrage as the pricing of some complex derivatives is subject to model risk.

Check your concepts:

(58.1) What is the impact of the risk aversion in the spot price of an asset?

(a) Higher is the risk aversion; higher is the spot price
(b) Higher is the risk aversion; lower is the spot price
(c) Risk aversion has no impact on the spot price of an asset

(58.2) How can a synthetic long position in an asset be replicated using derivative and risk-free asset?

(a) Buying risk-free asset and selling derivative
(b) Buying risk-free asset and buying derivative
(c) Selling risk-free asset and buying derivative

(58.3) The derivative pricing is based on the principle that the investor requires:

(a) Positive premium for assuming the risk
(b) Negative premium for assuming the risk
(c) No premium for assuming the risk

(58.4) The arbitrage can be best described as:

(a) Earning risk-free rate without taking any risk
(b) Earning risk-adjusted profit for taking the risk
(c) Earning profit at no risk and no capital investment

Solutions:

(58.1) Correct Answer is B: The spot price of an asset is calculated by taking the present value of the expected future spot rate. The discount rate increases with the risk-averseness of the investor. The higher the risk averseness, the higher is the discount rate, and the lower is the spot price.

(58.2) Correct Answer is B: A synthetic position in an asset can be replicated by taking a long position in the risk-free asset and a long position in the derivatives.

(58.3) Correct Answer is C: The derivative pricing is based on no arbitrage and risk neutrality. An investor is considered as risk neutral if he requires no premium for assuming the risk.

(58.4) Correct Answer is C: A risk-free rate can be earned by investing in a risk-free asset. One is expected to earn risk-adjusted risk for taking the risk. The arbitrage is earning a profit at no risk without any capital investment.

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