# Level 1 CFA® Exam:

Minimum & Maximum Values of Options

### Minimum Value

No matter if we are dealing with an American or a European option or whether it’s a call or a put, the option value will never be lower than 0. This is so because the option premium consists of the option's intrinsic value and time value. Even if the time value of an option is zero, the option will be worth as much as its intrinsic value. And by definition intrinsic value is always equal to or greater than zero. What follows is that the value of an option is also either equal to or greater than zero.

### Maximum Value

Following the intuitive approach, the maximum value of a call option at time t is the price of the underlying at time t.

If the price of a call at time t was higher than the value of the underlying at time t, it would be more effective to buy the underlying rather than only a right to buy it. This rule is true for both American and European-style calls.

But in the case of put options, we need to make a distinction between American and European options.

The maximum value of an American put is set at a level that does not exceed the exercise price. In the case of a European put, which cannot be exercised prior to its maturity date, the maximum value is the present value of the exercise price (we use a risk-free interest rate for discounting).

For a specific option, we are not only able to say that its value is equal to or greater than zero, but we may also establish the actual lower bound of its value.

To establish the lower bounds for European options, we can use put–call parity:

\(c_{0}+\frac{X}{(1+r)^{T}}=p_{0}+S_{0}\)

- \(c_{0}\) - current price of a call option
- \(X\) - exercise price
- \(r\) - risk-free interest rate
- \(T\) - time to expiration (number of days to expiration divided by 365)
- \(p_{0}\) - current price of a put option
- \(S_{0}\) - current price of the underlying asset less the PV of any future benefits & plus the PV of any future costs expected to be earned or incurred on the underlying before the expiration of the option

Assuming that the value of a put is always equal to or greater than zero, the value of a European call option is greater than or equal to the price of stock minus discounted exercise price:

\(c_{0}\geq{S_{0}-\frac{X}{(1+r)^{T}}}\)

However, we should also remember that an option’s value cannot be less than zero. So, the formula for minimum value of an European call option is:

\(c_{0}\geq{Max(S_{0}-\frac{X}{(1+r)^{T}};0)}\)

(...)

We can determine the lower bound of a European put option also using put-call parity. Assuming that the value of a call is always equal to or greater than zero, we end up with the following relationship:

\(p_{0}\geq{\frac{X}{(1+r)^{T}}- S_{0}}\)

However, we should also remember that an option’s value cannot be less than zero. So, the formula for minimum value of an European put option is:

\(p_{0}\geq{Max(\frac{X}{(1+r)^{T}}- S_{0};0)}\)

For an American-style put, since it is possible for an investor to exercise the option at any moment, the minimum value of the option cannot be lower than its intrinsic value:

\(P_{0}\geq{Max(X- S_{0};0)}\)

We have a 3-month American put and a 3-month European put. The exercise prices of these options are the same and are equal to USD 50. Their underlying is currently trading at USD 45. What are the minimum values of these options if the risk-free interest rate is 4.5%?

European-style put:

\(Max(\frac{X}{(1+r)^{T}}- S_{0};0)=Max(\frac{50}{1.045^{0.25}}-45;0)=\\=Max(4.45;0)=4.45\)

American-style put:

\(Max(X- S_{0};0)=Max(50-45;0)=5\)

- No matter if we are dealing with an American or a European option or whether it’s a call or a put, the option value will never be lower than 0.
- The maximum value of a call option at time t is the price of the underlying at time t. It holds true for both European-style and American-style call options.
- The maximum value of an American put is set at a level that does not exceed the exercise price. In the case of a European put, the maximum value is the present value of the exercise price.
- To establish the lower bounds for European options, we can use put–call parity.