Arbitrage in Contingent Claims

Recall that arbitrage opportunities occur if the law of one price does not hold. The no-arbitrage conditions in options are based on the payoff at maturity.

Unlike forward commitments with symmetric profiles (as presented earlier), contingent claims have asymmetric payoff profiles. That is:

$$\begin{align*}c_T&=\text{max}(0,S_T-X)\\ p_T&=\text{max}(0,X-S_T)\end{align*}$$

Moreover, in contrast to forward commitments with an initial value of zero at initiation, the option buyer pays the seller a premium \(c_0)\) for call options and \(p_0\) for a put options. Profits at maturity are:

$$\begin{align*}π_\text{call}&=\text{max}(0,S_T-X)-c_0\\ π_\text{put}&=\text{max}(0,X-S_T)-p_0\end{align*}$$

An option is only exercised when it is in the money. As such, this condition calls for upper and lower no-arbitrage price bounds at any time \(t\).

Upper and Lower Arbitrage Bounds

Call Option

A call option is exercisable if the underlying price exceeds the exercise price. That is \(S_t>X\). As such, the lower bound of a call price is the underlying price minus the present value of the exercise price or zero, whichever is greater.

$$\text{Lower bound}=\text{max}(0,S_t-X(1+r)^{-(T-t)})$$

A call buyer will not pay more than the underlying price for the right to buy the underlying.  As such, the upper bound is the current underlying price.

$$\text{Upper bound}=\text{S}_{\text{t}}$$

Generally, the no-arbitrage bounds of a call option are stated as follows:

$$\text{max}(0,S_t-X(1+r)^{-(T-t)}<c_t\leq\text{S}_t)$$

Put Options

A call option buyer exercises a put option only if \(S_T<X\). As such, the upper bound on the put value is thus the exercise price.

$$\text{Upper bound}=X$$

The lower bound is the present value of the exercise price minus the spot price or zero, whichever is greater:

$$\text{Lower bound}=\text{max}(0,X(1+r)^{-(T-t)}-S_t)$$

Generally, the no-arbitrage bounds of a put option are stated as follows:

$$\text{max}(0,X(1+r)^{-(T-t)}-S_t)<p_t\leq\text{X}$$

Example: No-arbitrage Bounds of a Call Option

Consider a 3-year call option with an exercise price of USD 100 and a risk-free rate of 1.5%. If, after six months, the spot price of the underlying is USD 105, the no-arbitrage upper and lower bounds are closest to:

Solution

For a call option,

$$\begin{align*}\text{Lower bound}&=\text{max}(0,S_t-X(1+r)^{-(T-t)})\\&=\text{max}(0,105-100(1.015)^{-2.5})\\&=\text{USD 8.65}\\ \text{Upper bound}&= S_t=\text{USD 105}\end{align*}$$

Replication in Contingent Claims

Note that replication refers to a strategy in which a derivative’s cash flow stream may be recreated using a combination of long or short positions in an underlying asset and borrowing or lending cash.

Replication mirrors or offsets a derivative position, given that the law of one price holds and arbitrage does not exist. A trader can take opposing positions in a derivative and the underlying, creating a default risk-free hedge portfolio and replicating the payoff to a risk-free asset.

Replicating Call Options

Replication of a call option at the contract initiation involves borrowing at a risk-free rate, \(r\), and then utilizing the proceeds to buy the underlying asset at a price of \(S_0\).

At the expiration date \(t=T\), there exist two replication outcomes:

• If \(S_T<X\), exercise the option: Sell the underlying at \(S_T\)  and use the proceeds to repay the risk-free loan.
• If \(S_T<X\), no exercise: No settlement is needed.

If the exercise of the option is certain, we will borrow \(X(1+r)^{-T}\) just like in forwards. However, the exercisability of the option is not certain. As such, a proportion of \(X(1+r)^{-T}\) is borrowed depending on the likelihood of exercise at time \(T\) and linked to the moneyness of an option.

The non-linear nature of option payoff requires replicating transactions to be adjusted based on the likelihood of exercise.

Replicating Put Options

Replication of a put option at the contract initiation involves selling the underlying short at a price of \(S_0\) and lending the proceeds at the risk-free rate, \(r\).

At the expiration date \((t=T)\), there exist two replication outcomes:

• If \(S_T<X\), exercise the option: Buy the underlying at \(S_T\) from the proceeds of the risk-free loan.
• If \(S_T<X\), no exercise: No settlement is needed.

As with call options, a proportion of \(X(1+r)^{-T}\) is borrowed depending on the likelihood of exercise at time \(T\) and linked to the moneyness of an option.

Question

A 6-month put option on an underlying stock with no associated costs or benefits has an exercise price of $50. The underlying price at the contract inception is $47, and the risk-free rate is 1.5%. After three months, the underlying stock price is $45.75.

The lower bound of the put option price is closest to:

A. $4.06.

B. $50.

C. $45.75.

Solution

The correct answer is A.

The lower bound of a put option is given by:

$$\begin{align*}\text{Lower bound}&=\text{max}(0,X(1+r)^{-(T-t)}-S_t)\\&=\text{max}(0,50(1.015)^{-(0.5-0.25)}-45.75)\\&=\text{max}(0,4.064)\\&=$4.064\end{align*}$$