Numerix Blog

Webinar Announcement: CCDS for Vanillas, Exotics & Portfolios - June 30

Please join us on June 30 for a one-hour webinar on CCDS for Vanillas, Exotics & Portfolios: Advanced Methods for Hedging Counterparty Risk.

When: Wednesday, June 30, 2010 – 9:30 a.m. EST

Description: The standardized contingent credit default swap (CCDS) gives traders a simple way to view credit valuation adjustment (CVA) for individual vanilla instruments. However, in order to effectively hedge counterparty risks for exotics or portfolios (or better yet—cross-asset portfolios containing exotics), you’ll need a way to structure customized CCDS, since these hedges are not addable.

In this webinar, we will discuss the standard CCDS contract and demonstrate how to use intuitive deal-scripting to rapidly structure custom CCDS, with examples for:

• Vanilla IR swap
• Cross-currency swap
• Commodity swap
• Exotics
• Portfolios

We will also discuss the Numerix hybrid model framework as it relates to deterministic vs. stochastic credit (wrong-way risk/exposure), Potential Future Exposure (PFE), and selecting the model and calibration parameters.

Speaker: Dan Li, Vice President – Financial Engineering, Numerix 

Registration: To receive dial-in instructions, click here to register.

Comparing Market Model Accuracy to the Volatility Smile

Following up from yesterday’s post on LIBOR Market Models in Numerix, I thought it would be valuable to show how the incorporation of shifts and stochastic volatility enables more accurate pricing for instruments with high market skew.

To measure the calibration fit of various models in use today, we tested the standard BGM model, a shifted BGM (SBGM) model and a LMM with stochastic volatility (LMM-SV) for a grid of 55 call swaptions at various expiry/maturity combinations (from 1Y/1Y to 25Y/5Y) and relative strikes (-2% to +2%). (Note: The plotted implied model vols are obtained from Monte Carlo simulation pricing of the swaptions.)

Ideally, a model should track the market swaption volatility (representing the market-quoted price of the swaption) as close as possible across expirations, maturities and relative strikes. The better the fit, the more accurate the calculations for pricing and risk analysis.

Findings:

• LMM-SV provided the best overall fit to the market volatility smile across all expiry/maturity combinations
• SBGM accuracy declined for shorter-dated maturities
• BGM was generally only effective for ATM options (0% relative strike)

LIBOR Market Models for Smile-Dependent Exotics

Continuing our series on the use of advanced models (see The Bates Model), today we address the issue of pricing complex trades that are heavily dependent on the volatility smile, such as CMS, LIBOR-linked structures and hybrids. For these exotics, the Numerix includes a suite of LIBOR Market Models (LMM), with support for stochastic volatility, which is required to calibrate to the entire volatility surface.

To understand why this model is necessary, let’s look at recent advances in market models for pricing interest-rate derivatives.

Tracing the Evolution of the LMM
In 1997, Brace, Gatarek and Musiela [1] introduced the BGM model—a standard formulation of the LIBOR market model using a lognormal diffusion of the underlying forward LIBORs. By modeling observable rates directly and enabling correlation between observable rates, BGM offered a major advantage over short-rate models that relied on unobservable instantaneous rates.

This multi-factor model allowed de-correlation of LIBOR/CMS rates and provided analytical formulas for European caplets/swaptions, which are necessary for effective calibration. However, lognormal modeling of the underlying LIBORs implies a flat volatility smile. This limitation made BGM ineffective for pricing smile-dependent exotics, such as range accrual notes.

The need for more-complex calibration algorithms led to two key extensions of the classic formulation. First came the shifted BGM, which introduced the ability to reproduce an implied volatility skew [2]. Further, a stochastic volatility process was added to the LIBORs’ evolution to calibrate both the skew and curvature of the smile [3].

Bates Model and Cliquet Pricing in Numerix

Recently, we added the Bates stochastic volatility jump-diffusion model to the Numerix CrossAsset model library. This is the market standard model for pricing exotic options that depend heavily on the forward skew, such as cliquets and other forward-starting trades.

For these types of options, the classic Black-Scholes model [1] is inadequate, because it implies constant volatility. The Heston model [2] extended the Black-Scholes model with a stochastic volatility process, providing close agreement with the implied volatility surface and realistic dynamics for the smile. However, no stochastic volatility model—including the widely used Heston model—can fit short-dated skew without incorporating the possibility of jumps in the underlying asset.

The Bates model [3] does just that, extending the Heston model by explicitly adding a jump-diffusion process. This is more consistent with empirical evidence, providing better calibration to the short-term skew and more realistic forward-skew dynamics.

Numerix Implementation
Numerix supports the classic Bates model, as well as an enhanced version that provides a better fit to market volatilities (more information is available by contacting Numerix sales). It includes support for:

• Monte Carlo simulation, as well as finite difference methods (used for weakly path-dependent instruments such as barrier options or American options)
• The option to draw jumps from either log-normal (Merton) or double-exponential (Kou) distribution
• Use of Bates as an equity component in the Numerix hybrid model framework for pricing structured products with underlyings from multiple asset classes

Applying the Bates Model to Cliquet Options
One of the uses of the Bates model is in pricing cliquets, or “ratchet options.” These are essentially a series of forward-start options, in which the strike price is reset to the current value of the underlying at each option expiration. The cliquet has become increasingly popular as way for sophisticated investors to trade on volatility assumptions. They’re also used in many retail structured products that give investors a lower-risk way to participate in equity and bond markets.

Video: Carr & Antonov on Quantitative Evolution

While attending the recent Quant Congress in New York, Drs. Peter Carr (Bloomberg) and Alexander Antonov (Numerix) took time to share insights on how quantitative methods are adapting to today's market environment.

In many cases, traditional techniques for pricing OTC derivatives simply aren't appropriate anymore and can lead to complete model failure. In this video, Carr and Antonov discuss changes in modeling, methods and calibration that are necessary to account for issues such high volatility, counterparty credit risk and liquidity risk.

Watch the video here:

Share your thoughts about this and other topics at the Numerix Forum — Quant Connection.