Options

Introduction

Words

Combinations

Options comes in two general flavors. Calls and puts. A call option gives the buyer of the option the right to buy the underlying asset from the seller at a future date. A put option gives the buyer the right to sell the underlying asset to the selles at a future data.

Buy Call

When you buy a call option, you’re making a leveraged bet on the price of the underlying asset going up. The price of the call option is influenced not only by the price of the underlying asset but also by factors like volatility, time to expiration, and interest rates (the “Greeks”).

Sell Call

Buy Put

Sell Put

European Options

American Options

Options as Leveraged Bets

Leverage is using debt or borrowed capital to undertake an investment or project.

With options, you’re not borrowing money in the traditional sense (taking on debt), but you’re still controlling a much larger exposure to the underlying asset with a smaller amount of capital, similar to how borrowing works. Therefore options have “embedded leverage”.

An option is considered a “leveraged” bet because it allows you to control a larger exposure to the underlying asset with a smaller upfront cost compared to buying or selling the asset directly. What follows is a breakdown of how that works.

Initial Cost is Lower than Buying the Underlying

When you buy an option, you only pay the option’s premium, which is usually a fraction of the price of the underlying asset. In contrast, if you were to buy or sell the asset directly (say a stock), you would need to commit the full market value.

Example:

In this example, for a much smaller investment ($200 instead of $10,000), you still have the potential to benefit from the price movement of 100 shares. This is the leverage aspect: you get exposure to the same amount of stock for a smaller upfront cost.

Magnification of Returns and Losses

Because you pay only a small fraction of the stock’s value to get exposure to the same amount of the stock, any gains or losses on the underlying asset are magnified in percentage terms. This is the core of the leverage:

Example (Call Option):

The other side of leverage is that, while your gains can be amplified, your losses are also magnified, although they are limited to the premium you paid for the option. If the stock price doesn’t move in your favor by the time the option expires, the option becomes worthless, and you lose 100% of your initial investment (the premium). This is different from holding the stock, where the stock could fall but not necessarily to zero.

This Applies to Both Calls and Puts

The Use of Options for Levered Bets

Options are not suited for someone who wants to simply track the market average or hold positions long-term without specific timing in mind. The time decay inherent in options (the fact that they lose value as they approach expiration) makes them risky for long-term views unless you have a precise idea of the timing of price movements.

For someone who has a quantitative model or a strong conviction about short-term price movements, options offer a way to:

Option Pricing

Black-Scholes

Hedging

There are two types of hedging: model-independent and model-dependent.

Model-independent hedging is quite uncommon, but the put-call parity is an example. Here the relationship is independent of the underlying asset’s movements.

Model-dependent hedging is when the hedging depends on a model, e.g., the Black-Scholes model.

Delta Hedging (Portfolio of stocks and options)

The delta is the

Gamma Hedging

Vega Hedging

Crash (Platinum) Hedging

Put-Call Parity

Put-call parity is a fundamental concept in options pricing, expressing the relationship between the price of European call and put options with the same strike price and expiration date. The parity ensures that there is no arbitrage opportunity in the market, i.e., traders cannot make a risk-free profit by exploiting price discrepancies between put and call options. We can write this relationship as:

[ \text{Call price} - \text{Put price} = \text{Stock price} - \text{Strike price} ]

where the strike price is the present value from expiration.

Formally, the put-call parity relationship can be expressed as:

[ C - P = S - K e^{-rT} ]

Where:

Intuition Behind the Formula

Put-call parity arises from the ability to construct two portfolios that have the same payoff at expiration:

  1. Long a Call Option and Short the Underlying Asset: This portfolio gives the holder the right to buy the asset at the strike price ( K ) (through the call option) and an obligation to deliver the asset at its current price ( S ).

  2. Long a Put Option and Long the Underlying Asset: This portfolio gives the holder the right to sell the asset at the strike price ( K ) (through the put option) while also holding the asset itself.

Since both portfolios provide the same payoff at expiration, the principle of no-arbitrage implies that their current values must be the same. Any deviation from this relationship would allow traders to make risk-free profits through arbitrage.

Implications

Put-call parity serves as a cornerstone in financial derivatives theory and is vital for understanding how options are priced in the context of arbitrage-free markets.

Arbitrage: Violation of put-call parity

Imagine that:

Now, if we sell the stock and the put option, and we buy the bond and the call option, we will earn $100 + $6 - $97 - $8 = $1, independent of what happens to the stock price six months from now.

Arbitrage: Violation of Put-Call Parity for European Option

Imagine the following scenario:

Now, if we sell the stock and the put option, and we buy the bond and the call option, we will earn:

[ 100 + 6 - 97 - 8 = 1 ]

This is an arbitrage profit of $1, independent of what happens to the stock price six months from now. Let’s break down how this works.

1. Understanding Put-Call Parity

First, let’s recall the put-call parity formula for European options:

[ C - P = S - PV(K) ]

Where:

In our case:

2. Checking if Put-Call Parity Holds

Plug these values into the put-call parity formula:

[ C - P = S - PV(K) ]

[ 8 - 6 = 100 - 97 ]

[ 2 = 3 \quad (\text{This is not true}) ]

Since ( 2 \neq 3 ), there’s a violation of put-call parity, indicating an arbitrage opportunity.

3. Constructing the Arbitrage Portfolio

To exploit this arbitrage, we construct a portfolio where we:

The net cash flow today is:

[ 100 + 6 - 97 - 8 = 1 ]

We receive $1 upfront.

4. Analyzing Payoffs at Maturity

Now, let’s examine the payoff of this portfolio at maturity based on the stock price ( S_T ).

Case 1: ( S_T \geq 100 ) (Stock price is at or above the strike price)

In this case, the bond pays $100, and we use that $100 to buy back the stock via the call. The net value is zero.

Case 2: ( S_T < 100 ) (Stock price is below the strike price)

In this case, the bond’s payout ($100) exactly covers the obligation from the put. The net value is zero.

5. Conclusion

Regardless of whether the stock price is above or below $100 at expiration, the net payoff from the portfolio is zero, and we have already received $1 upfront. Therefore, this is a risk-free arbitrage profit of $1.

Shorting the Stock: An Alternative Approach

If you do not own the stock initially, you can still execute the arbitrage strategy by shorting the stock. However, the cost of borrowing the stock must be taken into account. Here’s how it works:

  1. Short Sell the Stock:
    • Borrow the stock from another party and sell it at the current market price of $100.
    • This gives you $100 in cash upfront, but you may incur a borrowing cost (e.g., fees or interest).
  2. At Maturity:
    • If the stock price ( S_T \geq 100 ):
      • Exercise the call option you bought. You can purchase the stock at $100 and use it to close your short position by returning the stock to the lender.
    • If ( S_T < 100 ):
      • The put option you sold will be exercised, requiring you to buy the stock at $100. You then use this stock to close your short position.
  3. Closing the Short Position:
    • In both cases, the bond you purchased matures and pays you $100, which you use to fulfill your obligation of buying and returning the stock.
  4. Profit Consideration:
    • For this arbitrage to be profitable, the total cost of borrowing the stock must be less than $1, as this is the profit margin provided by the strategy. If the borrowing cost is higher, the arbitrage opportunity would not yield a net profit.

This approach allows you to execute the arbitrage strategy without needing to own the stock initially, as long as the borrowing costs are minimal enough to maintain the profitability of the trade.

Summary

The scenario demonstrates how arbitrage occurs when put-call parity is violated. By creating a portfolio that balances the positions (selling the stock and put, and buying the bond and call), you lock in a guaranteed profit.

Volatility Option Strategies

You can benefit from volatility with options, e.g., with long straddles and long strangles, which are strategies that capitalize on large price movements in the underlying asset, regardless of the direction (up or down).

Volatility is a key component in option pricing. The higher the expected volatility of the underlying asset, the more expensive the option premium becomes. This is because volatility increases the probability that the option will move into the money (for calls: above the strike price; for puts: below the strike price).

When you expect the price of an asset to move significantly but are unsure of the direction, you can use certain options strategies to benefit from volatility itself, rather than making a directional bet (i.e., just betting on whether the price will go up or down).

Long Straddle

A long straddle involves buying both a call option and a put option at the same strike price and expiration date for the same underlying asset. This strategy benefits from large movements in either direction (up or down), but it requires the asset price to move significantly to overcome the cost of both options (i.e., the combined premium paid).

How It Works:

The key is that if the price of the underlying asset moves far enough in either direction, one of the options will become profitable (and the other will expire worthless). The goal is for the profitable option to make enough to cover the cost of both the call and the put premiums.

Example:

If the stock price moves up to $120:

Similarly, if the stock price drops to $80:

If the stock price stays near $100 (no significant movement):

Thus, you benefit from a large price move in either direction, and you lose only if the price stays stagnant.

Long Strangle

A long strangle is similar to a straddle, but with one key difference: the strike prices of the call and put options are different. The call option is purchased with a higher strike price, and the put option is purchased with a lower strike price. This strategy can be cheaper than a straddle because the options are further out-of-the-money, but it also requires a larger price movement in the underlying asset to be profitable.

How It Works:

Example:

If the stock rises to $120:

If the stock falls to $80:

If the stock stays between $90 and $110, both options expire worthless, and you lose the $6 premium.

Again, you benefit from large price movements in either direction, but because you bought out-of-the-money options, you need a more significant move to make a profit compared to the straddle.

When to Use These Strategies:

Risks:

So What Can We Use Options For?

Option theory on the sell-side is primarily used for pricing, managing risk, and profiting from transaction volume and spreads. On the buy-side, however, option theory serves as a strategic tool to enhance returns and manage risks when capitalizing on identified opportunities. Instead of simply buying an asset (Y) based on a quant model that predicts its movement (e.g., when X rises, Y follows), options can offer leverage, timing flexibility, and risk-adjusted strategies. By using options creatively, whether through calls, puts, spreads, or volatility plays, investors can amplify profits, limit downside, or capture additional dimensions of market dynamics, like volatility, making the overall strategy more efficient and targeted.

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