Structured Certificate Pricing & Credit Risk

Overview

This project studied the pricing of an equity protection investment certificate through a replication-based derivatives framework.

The product was an equity-linked structured certificate on the FTSE MIB, with capital protection and an upside cap. The analysis compared the observed market price with a theoretical replication value obtained from option-pricing components and market inputs.

The project also considered how credit risk could be incorporated into the valuation, using the idea that part of the replicating portfolio behaves like issuer debt while the option-spread component remains non-negative.

Problem

Structured certificates combine bond-like and option-like components. Their market price may differ from the value implied by a simple replication portfolio, because of issuance margins, model assumptions, liquidity, dividends, volatility inputs and issuer credit risk.

The goal was to analyze whether the certificate's issue and market prices were consistent with a theoretical replication value, and to identify which modeling extensions were needed to account for credit risk.

Replication Logic

The payoff was represented through a portfolio of vanilla option components:

Long call with strike 0
+ long put at the protection level
- short call at the cap level

Equivalently, the payoff can be interpreted through put-call parity as:

Zero-coupon bond at the protection level
+ call spread between protection and cap

This second representation is useful for thinking about credit risk, because the bond-like component exposes the investor to issuer default risk.

Pricing Model

The base valuation used Black-Scholes-style option pricing with interest rate, dividend and volatility inputs.

Certificate value =
  call(S, K = 0)
  + put(S, K = protection)
  - call(S, K = cap)

The theoretical value was normalized by the initial underlying level to express the certificate price on a comparable scale.

Market Inputs

• Underlying: FTSE MIB historical levels.
• Volatility: historical volatility estimated from log returns before issuance.
• Dividend input: dividend yield converted into continuous intensity.
• Interest rates: spot rates obtained from ECB Nelson-Siegel-Svensson yield-curve parameters.
• Contract terms: protection level, cap level, issue date and maturity date.
• Observed prices: market prices used for theoretical-versus-observed comparison.

Yield Curve Modeling

The project used the Nelson-Siegel-Svensson parameterization to obtain maturity-matched spot rates from ECB yield-curve data.

r(tau) = NSS(tau; beta0, beta1, beta2, beta3, tau1, tau2)

For each valuation date, the remaining time to maturity was computed and the corresponding spot rate was extracted from the fitted term structure.

Credit-Risk Extension

The project considered extending the valuation beyond a default-free replication model by incorporating issuer credit risk.

The credit-risk intuition was based on the parity representation:

Protected bond-like component
+ non-negative option spread

This creates a structured exposure to issuer credit risk: simpler than a fully bilateral derivative exposure, but richer than a plain-vanilla bond because the certificate payoff depends on an equity-linked component.

Possible extensions included CDS-implied default intensities, reduced-form credit-risk adjustments and structural approaches to equity-linked credit exposure.

Implemented Elements

• Data preparation for underlying prices, dividend information and observed certificate prices.
• Construction of a business-day valuation calendar.
• Historical volatility estimation from FTSE MIB log returns.
• Residual time-to-maturity calculation for each valuation date.
• Nelson-Siegel-Svensson spot-rate extraction from yield-curve parameters.
• Black-Scholes valuation of call and put components.
• Replication-based certificate price calculation.
• Theoretical-versus-observed price comparison.
• Emission premium and mispricing analysis.
• Exploration of implied-volatility inversion and non-unique solutions.

Outputs

The analysis produced theoretical certificate values, observed-market comparisons and emission-premium diagnostics.

The model highlighted a discrepancy between observed prices and the replication-based theoretical value, motivating discussion of model assumptions, volatility inputs, issuer margin and credit-risk adjustments.

Evaluation Limits

The project was a derivatives-pricing prototype rather than a full production valuation engine.

• Volatility: historical volatility may not represent market-implied expectations.
• Dividends: dividend assumptions materially affect equity-linked certificate pricing.
• Rates: NSS spot-rate extraction depends on curve parameters and interpolation choices.
• Credit risk: issuer default risk requires additional calibration from CDS or credit spreads.
• Liquidity and margins: observed market prices can include bid-ask effects, fees and issuance margins.
• Model risk: Black-Scholes assumptions may be too restrictive for structured products.

Modern Extension

A modern version of the project would separate the pricing engine into modular components and add a more explicit credit-risk layer.

• Calibrate discounting and issuer-credit curves separately.
• Use CDS-implied hazard rates or credit-spread curves for credit adjustment.
• Compare historical and implied volatility inputs.
• Run sensitivity analysis across volatility, dividends, rates and credit spreads.
• Implement scenario analysis for issuer-credit deterioration.
• Compare Black-Scholes replication with alternative equity-volatility models.

Technologies and Methods Used

• R for pricing implementation, data preparation and visualization.
• Black-Scholes option pricing for vanilla option components.
• Replicating portfolio logic for structured certificate valuation.
• Nelson-Siegel-Svensson yield curve for maturity-matched spot rates.
• Historical volatility estimation from equity-index log returns.
• Business-day calendars for residual maturity calculation.
• Root-finding for implied-volatility diagnostics.
• Credit-risk modeling concepts including CDS-implied default intensities.

Resources

Code and raw market data are not public.

An anonymized technical note can be prepared upon request.

Technical Context

• Black and Scholes option-pricing framework for vanilla derivatives.
• Nelson-Siegel-Svensson term-structure modeling for yield-curve fitting.
• Hull and White reduced-form credit-risk ideas for defaultable instruments.
• Brigo and Mercurio interest-rate and credit-risk modeling references.