European Options
European options only allow for the exercise of options at the expiry date,... Read More
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 \(t\).
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:
$$\text{max}(0,S_t-X(1+r)^{-(T-t)}<c_t\leq\text{S}_t)$$
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:
$$\text{max}(0,X(1+r)^{-(T-t)}-S_t)<p_t\leq\text{X}$$
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:
For a call option,
$$\begin{align*}\text{Lower bound}&=\text{max}(0,S_t-X(1+r)^{-(T-t)})\\&=\text{max}(0,105-100(1.015)^{-2.5})\\&=\text{USD 8.65}\\ \text{Upper bound}&= S_t=\text{USD 105}\end{align*}$$
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.
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 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.
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:
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:
$$\begin{align*}\text{Lower bound}&=\text{max}(0,X(1+r)^{-(T-t)}-S_t)\\&=\text{max}(0,50(1.015)^{-(0.5-0.25)}-45.75)\\&=\text{max}(0,4.064)\\&=$4.064\end{align*}$$