###### Arbitrage, Replication, and the Cost o ...

Arbitrage refers to buying an asset in a cheaper market and simultaneously selling... **Read More**

Call and put options can be used to manage risk for holders of the underlying risk. Two common strategies are to reduce exposure by using a covered call (selling a call option) or to use a protective put (buying a put option).

A covered call is a relatively conservative strategy in which the underlying asset is owned, and a call option on the underlying is sold. The value of the position at the expiration of the call option is the value of the underlying plus the value of the short call.

- V
_{T}= S_{T}– max{0, S_{T}– X}- V
_{T}= S_{T}if S_{T}≤ X - V
_{T}= S_{T}– (S_{T}– X) = X if S_{T}> X

- V

The profit for the covered call can be computed by comparing the value at expiration (V_{T}) with the value at inception (V_{0}), where V_{0} is the initial value of the underlying less the call premium (short call). So V_{0} = S_{0} – c_{0}. The profit is therefore as follows:

- Π = S
_{T}– max{0, S_{T}– X} – (S_{0}– c_{0})- Π = S
_{T}– S_{0}+ c_{0}if S_{T}≤ X - Π = S
_{T}– S_{0}– (S_{T}– X) + c_{0}= X – S_{0}+ c_{0}if S_{T}> X

- Π = S
- Maximum profit = X – S
_{0}+ c_{0} - Maximum loss = S
_{0}– c_{0} - Breakeven: S
_{T}* = S_{0}– c_{0}

The option part (short call) of a covered call strategy in isolation would appear very risky. Selling a call without owning the stock exposes the investor to unlimited losses. However, selling a covered call in which the stock is owned reduces the overall risk.

However, selling a covered call to reduce risk also has the effect of limiting potential gains. The call option writer could miss out on gains in a strong bull market, but the compensation for giving up the potential upside gains is that the premium in a bear market will cushion the loss on the underlying.

The higher the exercise price of the call option, the less premium is received but, the greater the potential upside gain. Determining the exercise price has to factor in future expectations for the underlying and depends on the investor’s risk preferences.

Buying a put option to add to an underlying long position provides downside protection while still retaining upside potential. The value at the expiration of the put option is the value of the underlying plus the value of the long put.

- V
_{T}= S_{T}+ max{0, X – S_{T}}- V
_{T}= S_{T}+ (X – S_{T}) = X if S_{T}≤ X - V
_{T}= S_{T}if S_{T}> X

- V

The initial value of the position is the initial price of the underlying, S_{0}, plus the premium on the put p_{0}. Thus, the profit being the difference between V_{T} and V_{0} is as follows:

- Π = S
_{T}+ max{0, X – S_{T}} – (S_{0}+ p_{0})- Π = X – (S
_{0}+ p_{0}) if S_{T}≤ X - Π = S
_{T}– (S_{0}+ p_{0}) if S_{T}> X

- Π = X – (S
- Maximum profit = ∞
- Maximum loss = S
_{0}+ p_{0}– X - Breakeven: S
_{T}* = S_{0}– p_{0}

A protection put can appear to be a great transaction with no drawbacks. It is often viewed as a classic example of insurance where the premium is paid to the insurance seller, the put writer. The higher the exercise price, the less risk is assumed by the holder of the underlying and the more risk assumed by the seller. Like traditional insurance, coverage is provided for a certain period of time. After expiration, the underlying holder must decide whether to renew the insurance by purchasing another put option.

QuestionIf an underlying asset is priced at $200 (S

_{0}) and a put option with a premium of $10 (p_{0}) carries a $180 (X) exercise price, what is the breakeven of a protective put strategy? If at expiry, the underlying asset is $150 (S_{T}), what is the profit generated from the protective put strategy?A. Breakeven = $190; Profit at expiry = -$30

B. Breakeven = $210; Profit at expiry = $30

C. Breakeven = $200; Profit at expiry = $10

SolutionThe correct answer is A.

The breakeven S

_{T}* = S_{0}– p_{0}= $200 – $10 = $190 and the profit at expiry when S_{T}= $150 and X = $180 (S_{T}≤ X) is as follows: Π = X – (S_{0}+ p_{0}) = $180 – ($200 + $10) = -$30. Note that in this instance, the loss on the value of the underlying alone ($150 – $200 = -$50) is cushioned by the addition of the protection put.

*Reading 59 LOS 59b:*

*Determine the value at expiration, profit, maximum profit, maximum loss, breakeven underlying price at expiration, and payoff graph of a covered call strategy and a protective put strategy, and explain the risk management application of each strategy*