Greexotics – A First Step in the Land of Exotic Derivatives Greeks – Part 3

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Introduction

Building on our earlier analysis of Binary Options and Asian Options in the first two parts of this series, this third installment introduces Basket Options and Barrier Options, offering an in-depth look at their pricing, sensitivities, and practical implementation through Monte Carlo simulations and Python.

Basket Options, a type of multi-asset derivative, have an underlying composed of a portfolio or “basket” of securities rather than a single stock or index. In this article, we outline their key features, scope, and technical details. We examine the Delta, Vega, and Gamma of several basic multi-asset options, as well as advanced concepts like Cross-Gamma and sensitivity to correlation between basket components.

We also turn our focus to Barrier Options, a class of exotic derivatives. This section explores their pricing methodologies and includes Python code to illustrate practical computation techniques.

Basket options

Basket options are a fascinating subset of exotic options, offering tailored risk management and investment strategies for portfolios consisting of multiple assets. Unlike standard options, whose payoff depends on the performance of a single asset, basket options derive their payoff from the collective performance of a group of underlying assets. This unique feature makes them both versatile and complex to manage, especially given the intricate dependencies among the underlying assets.

Their pricing involves not only the individual characteristics of the underlying assets but also their interdependencies through correlation. Understanding advanced sensitivities like cross-Gamma and correlation risk is essential for traders and risk managers handling these instruments. By leveraging their ability to hedge or speculate on multiple assets simultaneously, basket options offer a cost-effective and efficient way to navigate the intricacies of multi-asset portfolios.

Definitions

A basket option is a derivative whose payoff depends on the weighted performance of a group (or basket) of underlying assets. These assets can include equities, indices, currencies, or other financial instruments. The payoff formula for a basket call option is often expressed as:

Where is the return of asset, is its weight in the basket, is the strike price (commonly set to zero) and is the notional value of the option (size of the contract).

Basket options are generally cheaper than purchasing individual options for each underlying asset because the diversification effect reduces the overall portfolio volatility.

Consider a basket option based on the average performance of two stocks: Stock A and Stock B, weighted equally. If these indices return +15% and -10% respectively over year, the payoff would be calculated as:

vs

The two separate options pay 150, while the basket option pays only 25.

Unlike basket options, where the payoff is based on the aggregate performance of a weighted basket of assets, best options and worst options focus on the extremes within the basket: the best-performing or worst-performing asset.

For example, the best call option pays based on the highest return among all assets, rather than the aggregate performance of the basket. This type of option is useful when the investor expects at least one asset to perform exceptionally well.

For instance, the worst put option pays based on the poorest return in the basket. This is particularly relevant in risk management when there is concern about the underperformance of a specific asset.

Greeks

While basket options share similarities with vanilla options, where the underlying is a single asset, the basket option treats the basket itself as the “single underlying.” This approximation allows for a comparable analysis of the Greeks associated with vanilla options, albeit with additional complexities arising from the interactions between the basket’s components.

Delta of a basket option

A basket call option provides the buyer with long delta exposure on the basket. This means the buyer benefits when the value of the basket rises, which indirectly implies a long position on each basket constituent. For example, in a basket comprising Index A and Index B, an increase in either index contributes to the overall basket value, generating a profit for the option holder.

Gamma of a basket option

Basket options also exhibit long gamma exposure. As the basket’s value rises, its delta increases, and when it falls, the delta decreases. Interestingly, changes in the price of one constituent affect the delta of all other constituents due to the interdependence of the basket. For instance, if Index A’s price increases, this raises not only its own delta but also the delta of Index B. This phenomenon, called Cross-Gamma, represents the sensitivity of one constituent’s delta to changes in the price of another constituent. It is generally positive, meaning a rise in one asset tends to amplify the deltas of others within the basket.

Vega of a basket option

Basket options have long Vega exposure, as they benefit from increases in the volatility of the basket. This mirrors the behavior of options on individual assets: higher volatility raises both the replication cost of the option’s payoff and the potential payoff itself. The basket’s volatility depends on the volatilities of its individual constituents and the correlation between them. Greater correlation leads to higher basket volatility, which, in turn, increases the option’s price.

The volatility of a basket is influenced by the variances of its constituents and their pairwise correlations. For a basket with two assets, A and B, and weights and (where ), the basket variance formula is:

Where is the correlation between A and B. Higher correlation amplifies basket variance and therefore its volatility , increasing the basket option’s price.

Best-of and Worst-of options are significantly influenced by factors such as dispersion, correlation, and volatility. Dispersion reflects the variability in returns among the basket’s underlying assets. High dispersion results in more diverse outcomes, with one asset potentially outperforming significantly while another underperforms. Conversely, low dispersion indicates similar returns across the assets.

Low correlation between assets increases dispersion by making returns more varied. Similarly, high volatility amplifies deviations in asset returns from the average, further increasing dispersion.

Best-of call options provide a payout equal to the highest-performing asset in the basket. As a result:

Delta of a best-of call

The buyer benefits from positive performance in any underlying, with delta shifting toward the best-performing asset as expiration approaches.

Gamma of a best-of call

Gamma is concentrated on the top-performing asset, with its delta increasing as its price rises. However, this reduces the delta of other assets due to relative performance adjustments.

Vega of a best-of call

Higher volatility increases the cost and potential payoff, with Vega converging toward the top-performing asset’s vanilla option as maturity nears.

Buyers are short correlation, as lower correlation (higher dispersion) increases the likelihood of higher payouts.

Worst-of put options provide a payout based on the worst-performing asset, making them:

Delta of a worst-of put

Profitable if at least one underlying underperforms, with delta concentrating on the worst-performing asset closer to expiration.

Gamma of a worst-of put

Gamma reflects sensitivity to price declines in the worst-performing asset, with cross-gamma dynamics like Best-of options.

Vega of a worst-of put

Higher volatility benefits the option holder, with Vega focusing on the worst-performing asset’s vanilla option.

Buyers are also short correlation, as lower correlation increases dispersion and the likelihood of significant underperformance by at least one asset.

Cross-Gamma and sensitivity to correlation

Cross-Gamma is a second-order sensitivity metric that measures how the delta of one asset in the basket changes in response to price movements in another asset. This interdependence arises because basket options are structured on the combined performance of multiple assets.

When the assets within a basket are positively correlated, the effect of Cross-Gamma is magnified. If one asset (e.g., Index A) rises, it increases not only its own delta but also the delta of other basket constituents (e.g., Index B). Positive correlation means the price of Index B is also likely to increase, compounding the benefits for the option holder. This synergy enhances the potential payoff. Conversely, if Index A’s price falls, it reduces the delta of Index B. Positive correlation means the price of Index B is also likely to decline, but the lower delta mitigates the negative impact on the option’s value.

Thus, positive Cross-Gamma and positive correlation work together to amplify gains and cushion losses, making basket options attractive for buyers seeking exposure to multiple assets’ interconnected dynamics.

Correlation significantly influences the volatility of a basket. For a basket with n assets, higher correlations among the constituents increase the basket’s overall volatility, as the assets are more likely to move in tandem, creating larger swings in value. Conversely, lower correlation reduces volatility, as the assets’ movements offset each other to some extent.

This has direct implications for pricing:

– A buyer of a basket option benefits from higher correlations because increased volatility raises the potential payoff. Buyers are said to be long correlation.

– A seller of a basket option, on the other hand, prefers lower correlations to reduce their exposure to extreme outcomes, making them short correlation.

However, correlation risk is inherently challenging to hedge. For example, a trader selling a basket call on the DAX and CAC40 might find it difficult to replicate or offset this risk using other derivatives due to uncertainties in implied correlations. This often results in wider bid-ask spreads for basket options, reflecting the difficulty in estimating and managing this risk effectively.

As the number of assets in a basket increases, the option’s behavior changes. In an equally weighted portfolio with many assets, diversification reduces the overall volatility of the basket. This dampened volatility can make the basket option less valuable because the likelihood of extreme payoffs diminishes. For instance, in a highly diversified basket with a high number of assets, the risk-reward profile becomes like that of a less volatile portfolio, which may lead to the basket option’s value approaching zero.

Consider a worst-of call option on two indices, A and B. When the price of Index A rises, the delta of Index B increases, as it is more likely to underperform. If A and B are positively correlated, the rise in A’s price may also drive up B’s price, amplifying the benefit to the option holder since B’s delta is now higher. The same applies if Index B’s price increases, boosting A’s delta and potentially its price due to positive correlation.

This dynamic is advantageous for the option buyer in positively correlated markets, as rising prices enhance deltas and overall value. Conversely, if prices fall, the impact is softened because lower deltas reduce losses from declining correlated assets.

Monte Carlo simulations

To calculate the Greeks of a basket option, such as a best-of call option, using Monte Carlo (MC) simulations, we can simulate paths for the underlying asset prices and then compute the option’s price and its sensitivity to various parameters like delta, gamma, and vega.

Here’s a Python code that implements a Monte Carlo simulation to estimate the Greeks for a best-of call option:

Barrier options

Introduction

Barrier options are widely used in financial markets for their cost efficiency, risk management capabilities, and flexibility in crafting tailored investment strategies. They are typically less expensive than standard options because the embedded barrier condition reduces the likelihood of a payout, making them a cost-effective tool for both speculative and hedging purposes.

Investors use knock-in options, such as down-and-in puts or up-and-in calls, to gain exposure only if the underlying asset crosses a specific price level, thereby limiting upfront costs while aligning with market expectations. Knock-out options, like up-and-out calls or down-and-out puts, are commonly employed to hedge against moderate price movements while avoiding the need for higher premiums associated with standard options. Additionally, barrier options are integral components of structured financial products, allowing institutions to design custom payoff structures that align with specific client goals or market scenarios.

Definitions

An up-and-in barrier option is a type of option that gives its holder the right to buy (call) or sell (put) an underlying asset only if the price of the underlying asset matches or exceeds a predefined barrier level (the knock-in price, in this case) at least once during the option’s lifetime. If the barrier is not breached, the option does not become active and holds no value. A down-and-in barrier option is very similar, with the only difference being that the underlying price has to drop below the barrier for the option to get value.

An up-and-out barrier option is a type of option that gives its holder the right to buy (call) or sell (put) an underlying asset if the price of the underlying asset does not match or exceed a predefined barrier level (the knock-out price, in this case) over the option’s lifetime. If the barrier is breached, the option becomes worthless. A down-and-out barrier acts similarly, with the only difference being the direction of the underlining, this time requiring that the stock falls at least once below the barrier for the option to lose all its value.

Pricing

For pricing an American barrier option like up-and-out put we can use Black Scholes, and instead we need the binomial tree model because it can handle both the path dependency of the barrier condition and the flexibility of early exercise. Unlike the Black-Scholes model, which assumes European-style exercise and ignores price paths, the binomial tree evaluates the option’s value at each step, accounting for whether the barrier is breached and comparing the intrinsic value with the continuation value. This step-by-step approach makes the binomial tree ideal for accurately pricing American-style barrier options.

Example

Let’s look at an up-and-out American put option, with a barrier equal to 70, initial stock value S at 50, strike K at 52, risk-free rate equal to 5% and for simplicity each discrete jump happens once a month. We also take for now as given the risk neutral probability of an up move p=0.51 and that of a down move q=0.49.

Looking at the figure above, we need to do the following steps to calculate the option value at each node.

Node B:

Continuation value:

Immediate exercise:

It is therefore rational not to exercise the option at this node, and as the option is under the barriers the value is 1.95.

Node C:

Option value:

Immediate exercise:

It is therefore rational to exercise the option at this node, hence the value becomes 12.

Node A:

Option value:

Immediate exercise:

It is therefore rational not to exercise the option at this node, and the final option value is 2.

Python code

Lastly, before describing the code we should mention that the up and down size (u and d) elements used to calculate the risk neutral probability p are based on the Cox-Ross-Rubinstein (CRR) model, introduced in 1979, and the most widely used method for pricing options through a binomial tree framework.

Also, under the CRR model, the expected price of the asset after one time step, under the risk-neutral measure, should equal the current price grown at the risk-free rate, hence we have that:

And after rearranging to solve for p we get that:

The function below is split into 3 main sections which combine to lead to the final option price for either up-and-out puts or calls. First, we need to initialize the asset price at each point in time, followed by the payoff calculation at maturity and then the backward pass used to calculate the value of the option like in the previous example.

Running the above function with u and d parameters following CRR model and the same initial parameters like in the previous example we get a function value for our barrier put option of 2.06, which is close to our initial estimate from the simplified example.

References

1. “Pricing and Hedging Financial Derivatives: A Guide for Practitioners”, Irene Perdomo and Leonardo Marroni
2. “Numerical techniques for the valuation of basket options and their Greeks.”, Hager, Corinna & Hüeber, Stefan & Wohlmuth, Barbara. (2010). The Journal of Computational Finance.