## Pricing Decision Trees for Structured Products with Equities and Interest Rates Underlying Assets.

#### Elisabeth Dillies

When pricing structured products or derivatives, there are many aspects to consider. Models and quantitative approaches that should be used vary according to the considered payoff: underlying asset class, maturity, path dependency, number of underlying assets, available market data, and others: all influence the method to be applied. We categorize these aspects along with the following criteria: payoff type, pricing objective, and pricing model. In the following step, we add market data to the mix. This blog should be a guide to help you through the pricing decision-making process based on LexiFi’s pricing models and methods.

This short read might also be of interest to structured investment professionals who would like to know more about product pricing models and possible methodologies.

### 1. Concept definition

#### Calibration

A model has typically many parameters.

A calibrated model is a model whose parameter values are chosen so that it becomes consistent with a given set of market observations. Calibration typically involves finding values for those parameters such that the model can reproduce, as close as possible, the prices of a set of (typically liquid) “calibration instruments” observed in the market.

Calibration in simple terms means taking the model on one side and market values on the other, then matching market prices with the theoretical model prices. That is, we are looking for model parameters that allow matching the observed market prices with the calculated model prices. To achieve this, we try to capture market information such as rates, dividends, option prices, implied volatility, and any other information that helps us estimate the occurrence probability of events involved in the valuation of the considered financial instrument.

#### Smile calibration

Several models allow accounting for the volatility smile more or less accurately. Local volatility models allow capturing all available volatility points, provided that the observed surface is arbitrage-free, thus fitting exactly the smile. Even if this is true theoretically, it may incur non-negligible numerical convergence problems in practice. On the other hand, Heston (Stochastic Volatility) models typically deal with diffused volatility surfaces more realistically, at the cost of a frequently less precise fitting of the initial observed smile. Stochastic Volatility models can be extended with a Local Volatility to fit the smile while keeping a realistic volatility diffusion.

#### Path dependency

The more the product is path-dependent (i.e., depends on multiple observations during its lifetime), the more we need a realistic smile dynamic behavior. This means that we want to use a model that correctly simulates the distortion of the smile while we move forward through the product lifetime, conditionally to future fixings.

### 2. Equities

#### Jumps or no jumps?

If a payoff suggests that realistic product valuation needs to consider extreme or very improbable events (such as a call with short maturity and a strike far out of the money), then we might want to use jump models. “Classical” diffusion models have indeed trouble generating short-term high variability and do not correctly capture such low probability events. Typical products for which jump models may often be suitable are Cliquet and Junk options.

#### Linear and non-linear payoffs

If the product has a linear payoff (futures, forwards and swaps) it would be less dependent on the smile. We could then use the Merton model as it fits the skew and requires one volatility point for each maturity.

If the product is non-linear then our valuation is more dependent on the smile dynamic. In this case, we choose the Bates model to fit both the skew and convexity for a good smile dynamic.

In general if we want to account for the volatility smile, we opt for the Bates model.

#### Long to medium maturity

We do not consider jumps for products that are reasonably close to the money, where extremely low probability events do not impact valuations. For these cases, we can either opt for fast or precise pricing.

Fast pricing, multi-underlying assets

Fast pricing is advised in the case of multi-underlying assets product, portfolio valuations, or risk analyses. If we are satisfied with using one value per maturity for the volatility (likely the ATM point), we should consider the classical Black-Scholes model. However, if ATM skewness should be considered, the Shifted Black model would be a more suitable choice. On a side note, we could also use the Adjusted Black-Scholes model as it provides a better smile handling than the classic Black-Scholes model while being almost as fast.

Precise pricing

For more precise pricing, we must calibrate the smile. For non-path-dependent products, an approximate smile dynamic would be sufficient using the Local Volatility model. The Heston Stochastic Local Volatility would allow on the contrary for a good smile dynamic at the cost of longer calculation times though.

For path-dependent products, we can opt for partial smile fitting with the (Rough) Heston Stochastic Volatility model or an exact smile fitting with the Heston Stochastic Local Volatility model: both offer a good smile dynamic. The advantage of the Rough Heston model is that it can usually simulate more realistic volatilities, considering observable phenomena like volatility clustering.

### 3. Interest Rates

First, let’s introduce the difference between what is called “short rate” models and “market models.” Short rate models simulate non-observable instantaneous interest rates. In contrast, market models directly model observable market rates (Libor), making them easily market consistent.

Former models typically need a costly volatility calibration phase while the latter can be easily fed with observed Libor volatilities, at the cost of often non-intuitive behavior and difficult-to-interpret results.

Let’s also note that the short rate models used today in the industry perfectly match the initial yield curve “by construction”.

Short rate models are also known in the finance literature under other - but mathematically equivalent - terminologies, such as yield curve models or alike. They all describe a no-arbitrage consistent stochastic evolution of an initially observed yield curve.

#### Pricing with near ATM options

If we do not need to calibrate the smile, we have to choose between using a short rate model or a “market model”.

Fast pricing, multi-underlying assets

In the case of a fast-pricing objective, we turn to short rate models. These allow typically generating positive and negative rates. We can either simulate the yield curve evolution by considering the evolution of the short-term part only or simulate a correlated evolution of both the short-term and long-term part of the yield curve.

To simulate the short-term part of the yield curve only, the Hull-White 1 factor and the One-factor Quasi Gaussian Linear Local volatility models can both be used. The Hull-White 1 factor assumes deterministic constant volatility, while the One-factor Quasi Gaussian Linear Local volatility model accounts for the skewness. Both models allow for a full forward fitting.

The Hull-White 2 factors model allows for simulating both the short and long-term part of the yield curve with a constant correlation, assuming deterministic volatility. It also allows for a full forward fitting. The Hull-White 2 factors model is for instance adequate for Steepener product pricing.

Market consistent, few underlying assets

When dealing with simple products with few underlying assets with the objective of market-consistent pricing, we can opt for a Libor model. These models allow simulating simultaneously all Libor forward rates with a parameterized correlation (from 1 to 3 factors), assuming a deterministic volatility. If we need to generate positive and negative rates and take the skewness into account, then we must use the Shifted Forward-Libor model. If positive rates and constant volatility are enough, we can use the Forward-Libor model, although it is a less stable and tractable model.

#### Smile calibration for precise pricing

Non-path-dependent

For a pricing operation with a smile calibration on a non-path-dependent product, we also have to choose between short rate and market models.

When diffusing all Libor forward rates, if we need to generate positive and negative rates, we must choose the Shifted Heston Lognormal Forward-LIBOR model. It allows for a full smile fitting, yet approximate smile dynamic. If not, we can use the Heston Lognormal Forward-LIBOR model, which allows for a partial smile fitting and an approximate smile dynamic.

To diffuse only the short rate, the Cheyette Local Volatility or Dupire Volatility models can be used. These models allow for a full smile fitting, an approximate smile dynamic, and a full forward fitting.

Path-dependent

If the product is path-dependent, we should diffuse the short rate through the One-Factor Quasi Gaussian Stochastic Local Volatility model (Pieterbarg) for partial smile fitting or the Cheyette Stochastic Local Volatility model for full smile fitting. Both models allow for a good smile dynamic and a full forward fitting.

#### Market data

Two kinds of instruments can be used to obtain volatility points and calibrate the smile in the interest rate case: Caps/Floors and Swaptions. Caps/Floors are used for products with Libor underlyings, while Swaptions are used for products with CMS underlying assets. When using Swaptions, we should pick the right tenors according to the CMS tenors.

### 4. Quantitative engine: LexiFi’s experts and expertise

LexiFi’s technology offering specializes in derivatives and structured product valuation and management. Our dedicated quantitative team has been working to extend, refine and enhance LexiFi’s pricing engine, benefiting from over 20 years of LexiFi’s expertise and client feedback. The software offers users an automated model selection from a long list of suitable pricing models taken from both market and proprietary models - find out more about the pricing process following this page!

More about pricing models available in LexiFi Apropos can be found on the Pricing Models page on our website. The Pricing methods page gives a good overview of our methodologies, market data manipulation, and implementation techniques.