Problem 2 - A Course in Financial Calculus

Question

A butterfly spread represents the complementary bet to the straddle. It has the following payoff at expiry: Find a portfolio consisting of European calls and puts, all with the same expiry date, that has this payoff.

Step-by-Step Solution

Answer

A butterfly spread is formed by buying one call at \( K_1 \), selling two calls at \( K_2 \), and buying one call at \( K_3 \).

Step 1: Understanding the Butterfly Spread Payoff

A butterfly spread payoff is generally achieved using options at three different strike prices: low, middle, and high. It has a characteristic shape where the profit is maximized at the middle strike price and decreases towards the low and high strike prices.

Step 2: Choosing Strike Prices

Let's denote three strike prices as \( K_1 < K_2 < K_3 \), where \( K_2 \) is the middle strike price. The spread is typically created such that \( K_2 = \frac{K_1 + K_3}{2} \).

Step 3: Constructing the Portfolio

To create a butterfly spread, use the following combination: Buy one call at \( K_1 \), sell two calls at \( K_2 \), and buy one call at \( K_3 \). Mathematically, this is represented as: \( C(K_1) - 2C(K_2) + C(K_3) \). This setup gives zero payoff outside the range \( K_1 \) to \( K_3 \), and a peak value at \( K_2 \).

Step 4: Portfolio Validity Check

The payoff function is validated at expiry, which shows a peak at \( K_2 \) and zero at other points, confirming the butterfly payoff. By using calls as described, it captures the same payoff structure that was required.

Key Concepts

Butterfly Spread

A butterfly spread is an intriguing strategy utilized in options trading that offers a bird's eye view of balanced risk and reward. It involves using three options with different strike prices to create a particular payoff pattern. This set-up looks like a butterfly in its payoff diagram, flapping smoothly at the middle and tapering down at the ends. The main aim of a butterfly spread is to gain profit when the underlying asset's price is near the middle strike price at expiry. If the market expects minimal volatility, this strategy can maximize profit.

A standard butterfly spread involves buying a call at a lower strike price, selling two calls at a mid-strike price, and buying one more call at a higher strike price. The result is a strategy that profits from relatively stable markets.

European Options

European options are a type of options contract which can only be exercised at the expiration date. Unlike American options, which can be exercised any time before expiry, European options offer less flexibility but are commonly part of structured strategies like the butterfly spread.

These options are particularly useful in strategies where the trader has a precise view of the market's movement until the expiry date. European options help minimize uncertainty as they eliminate the possibility of early execution, allowing a clearer timeline for strategical plays.

Strike Prices

Strike prices serve as a cornerstone in constructing any options strategy. These are the predetermined prices at which the holder of the option can buy or sell the underlying asset. In a butterfly spread, selecting the right strike prices is crucial.

Typically, you'll select three strike prices: low (\( K_1 \)), middle (\( K_2 \)), and high (\( K_3 \)), where \( K_2 \) is usually the average of \( K_1 \) and \( K_3 \). The strategic setup ensures the spread is balanced and provides the typical butterfly payoff profile. Choosing these prices accurately is key to aligning the market outlook with potential payoffs.

Options Portfolio Construction

Constructing an options portfolio involves combining different options to tailor risk and probability of profit to a trader's forecast. In the case of the butterfly spread, this involves a precise layout of buying and selling options to configure the desired payoff structure.

Creating a butterfly spread portfolio is about balance: buying one call at a low strike, selling two at a midpoint, and buying one at a high strike. This specific blend of options results in a zero-cost net position, meaning any shift in market price impacts your position minimally outside the range of the lowest and highest strikes.

The beauty of portfolio construction lies in its versatility; whether targeting specific market movements or hedging against risk, strategically crafting your options combo is paramount.