Options in Practice: Strategies, Risk, and Arbitrage

Options differ in when they can be exercised, which is a crucial factor in trading decisions. There are three main exercise types:

• European options: Can only be exercised at expiration.
• American options: Can be exercised any time up to and including expiration.
• Bermudan options: Can be exercised early, but only on specific dates (we’ll cover these in more detail in an advanced course).

For now, let's focus on European versus American options. Which is more flexible? The American option, because it allows early exercise.

Early exercise can be especially valuable for put options. Imagine you hold a put with a strike price of $50, and the stock plummets near zero. A European put would force you to wait until expiration, risking the stock rebounding and the option expiring worthless. With an American put, you can exercise immediately, locking in the payoff of $50 per share and reinvesting those funds for additional return.

The key takeaway: American options offer an extra layer of optionality that can be strategically advantageous. While we will cover precise pricing and optimal exercise in a later course, keep in mind that exercise style is an essential decision when selecting an option.

Beyond exercise style, trading options involves several choices. Here are the main considerations:

1. Strike price - determines the price at which the option can be exercised.
2. Direction - choose between calls (betting the stock rises) or puts (betting it falls).
3. Timing - select the option’s maturity, ranging from 1 month to over a year.
4. Size - the number of options or underlying shares involved.

Strike price is often the most challenging choice. Essentially, it reflects the magnitude of the expected price movement. For example:

• You buy a one-month call with a strike of $110 when the stock is $100.
• After one month, the stock rises to $109.
• Calls struck at $102.50, $105, and $107.50 are all profitable, but the $110 call expires worthless.

There's no partial credit: getting the direction and timing right isn’t enough if the strike is off.

Options can also be classified based on their relation to the current stock price:

Calls:

• In-the-money (ITM): Strike < stock price
• At-the-money (ATM): Strike ≈ stock price
• Out-of-the-money (OTM): Strike > stock price

Puts:

• ITM: Strike > stock price
• ATM: Strike ≈ stock price
• OTM: Strike < stock price

ITM options are more expensive because they have intrinsic value, ATM options are moderately priced, and OTM options are cheaper but carry higher risk. Only options that expire ITM provide a positive payoff.

At the strike price, the option’s payoff slope changes sharply: for a call, the value rises above the strike; for a put, the value falls below it. This often creates a “tug-of-war” as traders on opposite sides hope the option ends up in or out of the money.

Traders approach options differently depending on their goals:

• Hedgers: Use options to protect existing positions.
• Speculators: Take directional bets on price movements.
• Arbitrageurs: Exploit price discrepancies across markets.

Understanding exercise style, strike selection, and payoff behavior is the foundation for all these strategies. In the next lesson, we’ll explore specific option strategies and how to combine these elements effectively.

Options are not just for speculation – they can also be powerful tools to manage risk. Think of risk in terms of volatility, or the uncertainty of outcomes. The goal of hedging is to reduce this volatility, much like buying insurance.

Example: Imagine you're a farmer who needs to buy seed for this year's harvest. You're willing to pay $50 per bushel. Paying more could make it impossible to sell your crops profitably. You want to lock in a price to protect yourself.

One option is a futures contract, which might cost nothing upfront. A futures contract would allow you to lock in $50 for seed in three months. But if the market price falls to$30, you’re stuck paying $50, which hurts your competitiveness.

Now consider using a call option instead. Yes, options require an upfront premium, but they never force you to transact at an unfavorable price. Here’s how it works:

• Scenario 1: Seed prices rise above $50. The call option lets you buy at$50, saving money and increasing your profit margin when selling crops.
• Scenario 2: Seed prices fall below $50. You simply don't exercise the option and buy at the lower market price. The option acted like insurance against rising costs.

The key difference between futures and options is flexibility vs. obligation:

Options can also hedge the output price. For instance, the farmer could buy a put option to sell corn at $90 per bushel. If market prices drop, the put guarantees the minimum sale price, while still allowing higher profits if prices rise.

In short, options serve as insurance: they limit potential losses while retaining upside potential.

Options are also used by speculators who bet on price movements. They leverage derivatives to amplify potential gains – but at the risk of losing the entire premium. Here are a few strategies:

1. Buying calls:

• Pay the premium upfront.
• Profit if the stock rises above the strike price before expiration.
• Success depends on selecting the right strike and expiration.

2. Selling puts:

• Receive cash upfront.
• Risk exposure if the stock price drops.
• Profitable if the stock stays above the strike.

3. Bull spread (buy low, sell high):

• Buy a call at a lower strike (e.g., 100) and sell a call at a higher strike (e.g., 110).
• Reduces net cost because selling the second call partially offsets the first call’s premium.
• Upside is capped at the difference between the strikes ($10 in this example).
• Suitable if you expect a moderate increase rather than unlimited upside.

Comparison of strategies:

• Buying a call offers unlimited upside but higher upfront cost.
• Selling a put generates income but exposes you to downside risk.
• A bull spread reduces cost but limits maximum profit.

Key point: Options can expire worthless. For hedgers, a worthless option is like paying an insurance premium: it provides peace of mind. For speculators, a worthless option is a total loss of capital. Unlike equities, ETFs, or crypto, options are inherently leveraged and non-linear, meaning small movements in the underlying can lead to large gains or losses.

Summary:

• Hedgers: use options to protect against adverse price movements; the cost of the option is like paying for insurance.
• Speculators: use options to amplify potential gains; losing the option premium means total capital loss.

Rule of thumb: Always know your strike and risk exposure. The difference between profit and loss often comes down to just one price level – the strike.

Let's finish with a fascinating type of trader: the arbitrageur. Arbitrageurs look for price discrepancies across markets to earn risk-free profits. How? By exploiting relationships between stocks, bonds, calls, and puts that should, in theory, be equivalent.

Equity options can be replicated using a combination of stocks and bonds. If the market prices of the replicating portfolio and the actual option differ, an arbitrageur can:

1. Sell the “expensive” portfolio.
2. Buy the “cheap” portfolio.
3. Lock in a guaranteed profit, because the cash flows offset each other.

Opportunities like these are rare and fleeting, often lasting only fractions of a second, but they are fundamental to understanding the pricing of derivatives.

Arbitrage Example

Consider two portfolios:

Portfolio A: Long a call, short a put; both options have the same strike ($K$) and expiration ($T$).

Portfolio B: Long a stock, short a bond with notional value ($K$), earning the risk-free rate ($r$) maturing at time ($T$).

Let's examine the payoff at expiration:

Notice that Portfolio A and Portfolio B have identical payoffs.

Finding Arbitrage

Suppose the market prices are:

• Portfolio A: $8
• Portfolio B: $5

An arbitrageur could:

1. Sell Portfolio A for $8.
2. Buy Portfolio B for $5.
3. Pocket the difference: $3 in risk-free profit.

This works because the portfolios’ cash flows offset each other perfectly, leaving no net risk.

Key Takeaways

• Arbitrage opportunities are usually short-lived, requiring fast execution across multiple markets.
• Understanding arbitrage helps explain why derivative prices are tied closely to their underlying assets.
• While this is an advanced concept, it introduces an important principle: mispriced derivatives can create a guaranteed profit – but only if executed correctly and instantly.

So far, the securities we’ve studied were spot instruments, meaning their value is directly determined by their current market price. Derivatives, in contrast, are instruments derived from another asset, called the underlying. Because their value depends on the underlying asset, derivatives provide tools for risk management, speculation, and strategic investment.

Formally, a derivative is any contract whose value is linked to another measurable variable, often the price of an asset. Through financial modeling, stochastic mathematics, and hedging strategies, it is possible to estimate a derivative’s price based on the behavior of its underlying. At any point in time – whether at creation or expiration – a derivative’s value is determined either by the current market price of the underlying or by predefined payoff rules.

Derivatives generally fall into four categories:

• Futures - agreements to buy or sell an asset at a predetermined future date and price.
• Forwards - customized contracts similar to futures but traded over-the-counter (OTC).
• Swaps - contracts to exchange cash flows or other financial instruments; these will be covered in an advanced module.
• Options - contracts granting the right, but not the obligation, to buy or sell an underlying asset at a specific price within a certain period.

This module focuses on futures, forwards, and options, with swaps reserved for a later discussion.

In the previous lesson, we covered the fundamentals of options – calls and puts, how premiums work, and the potential outcomes if an option expires in or out of the money. We also discussed the factors that affect an option’s value: the underlying stock price, strike price, time to expiration, risk-free rate, dividends, and volatility. In this lesson, we explore two key characteristics that make options unique: leverage and nonlinearity.

Leverage allows investors to control a larger position than they could with cash alone. Traditionally, leverage involves borrowing capital to amplify potential returns, with the expectation that the investment will earn more than the borrowing cost. In the context of options, leverage works differently: you don’t borrow money, but your potential return is magnified because each option represents multiple shares of the underlying stock.

For most stock options, one contract represents 100 shares. Buying a single call option gives the holder the right – but not the obligation – to buy 100 shares at the strike price. Since the cost of one option is far less than buying 100 shares outright, you achieve leverage naturally: a small initial outlay controls a much larger position.

Let's explore leverage by comparing three ways to invest in a bullish stock scenario where the expected return is 10%.

a. Buying Stock with Cash

Suppose a stock costs $500 per share and you want 100 shares. You need$50,000 upfront. A 10% return on this investment yields $5,000. No leverage is involved here – the return is proportional to your cash invested.

b. Buying Stock Using Financing

Now imagine you only have $50,000 in cash but borrow another$50,000 to buy $100,000 worth of stock. If the stock rises 10$10,000. Since only $50,000 was your money, the actual return on your cash is 20% – double the unleveraged return.

Leverage allows you to magnify returns, but it also magnifies losses. If the stock instead fell 10%, you would lose $10,000, or 20% of your cash invested.

c. Using Options for Leverage

With a call option, you could potentially control the same 100 shares by paying a fraction of the stock’s cost. For example, if one option costs $5,000 instead of$50,000, a 10% increase in the stock could produce the same $10,000 gain – a 200% return on your initial option investment.

Options allow you to achieve high leverage without borrowing money, but the tradeoff is that you can lose your entire premium if the option expires worthless.

Leverage amplifies outcomes in both directions. Small positive moves can turn into large gains, while negative moves can produce large losses. Excessive leverage can even wipe out your investment or leave you owing more than you put in when borrowing is involved.

Regulations exist to limit leverage, ensuring investors don’t expose themselves to catastrophic losses. The key takeaway: leverage is a multiplier. It doesn’t predict returns; it simply amplifies whatever happens. Use it with caution.

Options are unique because they inherently provide leverage, but they must be purchased fully with cash – you cannot borrow funds to buy them. This is because leverage in options comes from the conversion factor, not from borrowing. For standard stock options, one contract represents 100 shares of the underlying stock.

For example, if a stock rises by $1, the option's value may increase by roughly$100 because it represents 100 shares. The actual pricing is more complex, involving stock price, strike, volatility, time to expiration, interest rates, and dividends. Despite this complexity, options can easily generate 100% or more returns, giving them higher leverage than buying stocks on margin. The leverage is built into the contract itself.

If you are bullish on a stock, you could invest by:

1. Buying the stock outright with cash.
2. Buying the stock with a mix of cash and borrowed funds.
3. Buying a call option with a specific strike and expiration.

Timing matters with options. Stocks do not expire, so even if your prediction takes slightly longer to materialize, you still earn a return. Options, however, expire. If the stock doesn’t move enough before expiration, the option becomes worthless. This can result in a 100% loss of the premium.

For example, imagine a stock at $200. You could buy options with strikes spaced at$2.50 intervals above and below the stock price, and expirations ranging from this month to several months out. The sheer number of combinations – calls vs. puts, strike prices, and expiration dates – means option traders must get three things right to profit:

1. Direction: Calls if the stock rises, puts if it falls.
2. Timing: The price must move before expiration.
3. Strike: The price must surpass the strike for a positive payoff.

Getting only two out of three right still results in an out-of-the-money option, which expires worthless.

Sometimes, predicting the stock's direction is difficult. Instead, traders can bet on volatility. A common strategy is the straddle, which involves buying both a call and a put at the same strike:

• Profit occurs if the stock moves significantly in either direction.
• The total cost is the sum of the premiums for both options, so the stock must move enough to cover these costs.
• This approach focuses on magnitude of price movement rather than direction.

Conversely, if a trader expects low volatility, they can sell a straddle – selling both a call and a put at the same strike. Profit occurs if the stock stays near the strike price, but large movements in either direction can result in significant losses.

Options differ from most financial instruments because of their nonlinear payoff. While stocks and futures have linear payoffs (profit/loss changes proportionally with the price), options have a “hockey stick” shape:

• Flat region: When the stock price is below the strike, a call option has no intrinsic value.
• Rising region: Once the stock exceeds the strike, the call becomes in the money, and profit increases linearly with the stock price.

This combination of flat and rising segments makes the overall payoff nonlinear. The strike price is a “line in the sand”: if crossed, profits start to accumulate; if not, the option can expire worthless.

The intrinsic value measures what the option would be worth if exercised immediately. A call with no intrinsic value is out of the money and would yield zero if expiration were immediate.

Nonlinear payoffs aren't common in other markets, but they do appear in situations like housing investments, which we will explore next.

Imagine you want to buy a home but lack sufficient cash to cover the full price. You turn to the mortgage market. While mortgages come in many forms, all share a fundamental property: leverage. Borrowing money to acquire an asset – whether a house or stocks – is a form of leverage, as you pledge the asset itself as collateral. The principle is simple: you control a valuable asset without paying the entire cost upfront.

Unlike stock investments, homeownership is often non-speculative, particularly if the property is your primary residence. Secondary homes may be treated differently, often carrying stricter terms. For instance, U.S. regulations limit stock purchases on margin to 50% of the purchase price, whereas mortgages commonly allow 80% financing – requiring only a 20% down payment. This leverage amplifies gains: if a home appreciates 10%, the return on your down payment is magnified.

However, owning a home comes with unique carrying costs: mortgage payments, property taxes, insurance, utilities, and maintenance. Unlike stocks or bonds, real estate requires ongoing upkeep. Additionally, housing values fluctuate, meaning the property could lose value, just like financial assets. Nevertheless, a home provides tangible benefits beyond financial returns – a place to live.

To illustrate nonlinearity, we turn to credit risk theory, specifically the Merton model. Robert Merton, co-creator of the Black-Scholes-Merton option pricing framework, showed that equity in a firm can be modeled as a call option on the firm’s assets.

Consider a company with total assets worth $V$ and debt $D$ due at time $T$. The stockholders’ equity behaves like a call option:

• If $V$ > $D$, shareholders receive $V - D$.
• If $V$ = $D$, shareholders receive nothing (limited downside).

The key insight: equity is nonlinear. Gains are unlimited if asset values rise, but losses are capped at zero if assets fall below debt.

We can apply the same idea to housing. Let $H$ be the market value of your home and $M$ your mortgage. Your home equity behaves like a call option with strike price $M$:

• If $H$ > $M$, selling the home covers the mortgage, leaving you with profit ($H - M$).
• If $H$ = $M$, selling only pays off the mortgage; equity is zero.
• If $H$ < $M$, the home is underwater, but with a non-recourse mortgage, you can walk away without paying the difference.

This framework highlights how homeownership combines leverage and nonlinearity, similar to options.

Non-recourse loans amplify optionality. With zero or minimal down payments, homeowners essentially receive a free call option on the house:

• If home prices rise, sell for profit.
• If prices fall, walk away with minimal financial loss.

While this seems attractive to buyers, it introduces systemic risk. Speculators can exploit these conditions, repeatedly flipping homes to profit from price increases. When prices fall, lenders bear the losses. This behavior contributed significantly to the U.S. housing crisis during the Great Financial Crisis, creating widespread defaults and repossessions.

From a financial perspective, the mortgage positions the lender as being short a put option:

• When the home value exceeds the mortgage, the option is out of the money, favorable for the lender.
• When the home value falls below the mortgage, the lender absorbs the loss, analogous to a put being in the money.

In short, providing mortgages with little or no down payment transfers option-like benefits to homeowners, incentivizing speculation and increasing the risk of housing bubbles.