Lecture 12,13,14

Puts

Option contract to sell an asset

bet the underlying asset is going down

• If the market price of the asset falls below the strike price before the option expires, you can exercise the option to sell the asset at the higher strike price, even though it’s worth less in the market.

• This lets you effectively "sell high" (at the strike price) even when the market price has dropped, leading to a profit

Calls

Option Contract to Buy an Asset

bet the underlying asset is going up

Suppose you buy a call option for a stock with a strike price of $50 that expires in a month.

• You pay a premium of $5 for this option.

• If the stock’s price rises to $60 before expiration, your option now allows you to buy it for $50 (while the market price is $60), giving you a profit potential.

• Your profit would be the difference between the market price and strike price minus the premium paid (in this case, $60 - $50 - $5 = $5).

1. If the Price Drops:

   • If the asset’s price doesn’t go above the strike price before expiration, the option expires worthless, and your only loss is the premium you paid.

American Option

can be exercised any time

European Option

can only be exercised at expiration

Strike/Exercise Price

The price at which the owner can buy/sell the asset

In the Money

exercise of the option would be profitable

Call: strike price < future market price

Put: strike price > future market price

At the Money

exercise price =asset price

Out the Money

exercise of the option would not be profitable

Call: market price < exercise price

50$ 60

• You have the right to buy the asset at $50, but it’s currently worth $60 in the market.

• Action: You can buy the asset at the cheaper $50 price and immediately make a profit of $10 per unit by selling it at the current $60 price.

Put: market price > exercise price

$50 $40

• You have the right to sell the asset at $50, even though it’s only worth $40 in the market.

• Action: You can sell the asset at the higher $50 price (via your put option), which would allow you to make a profit of $10 per unit compared to the current $40 market price.

• you can sell at the higher strike price, then potentially buy it back cheaper in the market to lock in a profit

Option Premium

the income received by an investor who sells an option contract

•Investors who purchase options must pay an upfront cost for this option benefit

•The upfront cost of the option is called an option premium

•Option writers are like insurance companies selling you that protection

•Thus they are entitled to receive the premium

Profit

Option Payoff-Option Premium

Long straddle

Buy call and put with same exercise priceand expiration.

The goal of a long straddle is to profit from significant price movement in either direction. You don’t need to predict if the price will go up or down, just that it will move substantially in one direction.

• f the price rises sharply, the call option gains value, while the put option might lose value, but the gains on the call can outweigh the loss on the put.

• If the price drops significantly, the put option gains value, and any loss on the call option is offset by the put’s profit.

• The potential profit is theoretically unlimited on the upside and substantial on the downside, as one of the options will become very valuable if the stock price moves enough.

• Loss Potential:

  • If the stock price doesn’t move much and stays around the strike price, both options may expire with little to no value, and you lose the premiums paid. The maximum loss is limited to the total premium paid for the call and put.

For European options, options cannot be exercised till expiration•Thus, if both strategies generate same payoff, their costs must be the same!

The expiration-only rule for European options prevents arbitrage by ensuring that options with the same expiration payoff cannot be exploited for risk-free profit; if two strategies yield identical payoffs only at expiration, they must have the same cost initially. This restriction means there’s no opportunity to capitalize on timing differences, as the payoff happens at a single, fixed time

Arbitrage Opportunity

When a trader can exploit price differences in markets or assets to make a risk-free profit. This typically happens when the same asset or financial instrument is priced differently in two or more markets, allowing a trader to buy low in one market and sell high in another without any risk or capital loss.

YOU SELL AN OVERPRICED ASSET AND BUY UNDER PRICED ASSET

So for example you would sell an overpriced call and buy and underpriced put

Intrinsic value

-profit that could be made if the option was immediately exercised

The right to exercise an American call early is valueless

as longas the stock paysno dividendsuntil the option expires

American puts are

worth more than European puts, all else equal

Payoff

what is the value of the option by the time we’re looking at it

Payoff

Payoff-Cost for option

Pay off is max(0, value you get)

if it is below 0 just put 0 for payoff

Put call parity

American Put

There is value in early excercising them

American Call

There is no value in early exercising them, so you should always sell them if you would like to close the position

Early Excercise

Covered Call

Callable Bonds

why do callable bonds have negative convexity-the issuer has more incentive to call the bond at par when interest rates fall, causing the price to fall as well: 

• Explanation

  Convexity measures the rate of change in a bond's duration in response to interest rate fluctuations. A bond's duration is the length of time that interest rate changes affect its price.