Worst-of Phoenix Autocallable Pricer

A practitioner-grade Monte Carlo pricer for worst-of Phoenix autocallables on Euro Stoxx 50 / S&P 500 / Nikkei 225, with Heston stochastic-volatility calibration, live market data, and a Streamlit dashboard. Built around the products that French bank equity-derivatives desks (SG, BNP Paribas, Natixis) sell to retail in volume.

What it demonstrates

Pricing a worst-of autocallable correctly requires three things that a desk does daily and a textbook rarely connects: multi-asset Monte Carlo with correlated drivers (the worst-of feature dominates the payoff), a stochastic-volatility model because the embedded knock-in put is short skew and a flat-vol price under-reserves it, and a fair-coupon solver because in practice the coupon is the output — the structurer quotes the coupon that prices the note at par. The headline result is the skew premium: pricing off a calibrated Heston surface rather than a flat vol moves the fair coupon by tens of basis points, which is what the desk earns and hedges over the life of the trade.

Left: the fair coupon under flat-vol GBM vs a calibrated Heston surface — the gap is the skew premium. Right: worst-of PV as a function of average correlation, the dominant risk axis of the structure (higher correlation → less dispersion across underlyings → fewer knock-in breaches → higher PV). Regenerate with python make_figures.py.

How the product works

The note pays an above-market coupon in exchange for the investor taking equity downside on the worst-performing of several underlyings (here SX5E, SPX, NKY). At each annual observation, every underlying's performance is measured against its strike (the level fixed at trade date), and the structure keys off the worst of them:

performance = spot / strike, and worst = min(performance) across all underlyings.

• Autocall. If the worst performer is at or above the autocall barrier (e.g. 100% of strike) on an observation date, the note redeems early at par plus any coupons due. Most notes end this way — early, and at par.
• Coupon, with memory. If the worst performer is at or above the coupon barrier (e.g. 70%), a coupon is paid; coupons missed below the barrier are remembered and paid in full on the next date the barrier is met.
• Knock-in at maturity. If the note never autocalled and the worst performer finishes below the knock-in barrier (e.g. 60%), the investor takes the equity loss of that worst performer. Otherwise par is returned.

Because the payoff always tracks the minimum across underlyings, the note is short correlation (less correlation means more dispersion, a lower worst performer, and more knock-in risk) and short volatility (it embeds a sold down-and-in put). The coupon is the compensation for bearing those two risks — and pricing it off a calibrated volatility surface rather than a flat vol is what produces the skew premium shown above.

Methodology references

• Heston, S. L. (1993). "A Closed-Form Solution for Options with Stochastic Volatility." Review of Financial Studies.
• Lewis, A. (2001). "A Simple Option Formula for General Jump-Diffusion and Other Exponential Lévy Processes." Working paper.
• Bouzoubaa, M. & Osseiran, A. (2010). Exotic Options and Hybrids. Wiley.

Project structure

autocallable-pricer/
├── autocallable_pricer.py   # Phase 1 — single-asset reference pricer
├── market_data.py           # Phase 2 — live spot/vol/rate/div + correlation
├── worstof_pricer.py        # Phase 3-5 — worst-of MC, Greeks, fair coupon
├── heston.py                # Phase 4 — char-fn pricing, calibration, MC
├── calibrate_heston.py      # Phase 7 — multi-index calibration framework
├── app.py                   # Phase 6 — Streamlit dashboard
├── run_snapshot.py          # daily snapshot writer (CI)
├── make_figures.py          # regenerate the README figure (offline)
├── tests/                   # pytest suite (offline, identity/property tests)
├── sources.md               # data provenance
├── requirements.txt / requirements-dev.txt
└── .github/workflows/       # tests.yml + daily-snapshot.yml

Status

Phase	Scope	Module	Status
1	Single-asset Phoenix autocallable: memory coupon, autocall barrier, European KI	autocallable_pricer.py	Complete
2	Live market data — spot, realized vol, div yields, €STR, cross-asset correlation	market_data.py	Complete
3	Worst-of multi-asset MC with Cholesky-correlated GBM, antithetic variates	worstof_pricer.py	Complete
4	Heston stochastic vol — Lewis char-fn pricer, calibration, full-truncation Euler	heston.py	Complete
5	Fair-coupon solver (Brent) — quantifies the GBM-vs-Heston skew premium	worstof_pricer.py	Complete
6	Streamlit dashboard — GBM / calibrated-Heston / manual-Heston, fair-coupon, ladders	app.py	Complete
7	Multi-index calibration framework — SPX live, SX5E via Eurex, vol-scaled proxies	calibrate_heston.py	Complete

Quick start

pip install -r requirements.txt

# 1. Calibrate Heston per index (S&P 500 live; others by proxy)
python calibrate_heston.py --all                       # run during US market hours
python calibrate_heston.py --all --eurex oesx.csv      # real SX5E from a Eurex CSV

# 2. Price (auto-loads calibration/heston_params.json)
python worstof_pricer.py --heston                       # static demo inputs
python worstof_pricer.py --live --heston                # live SX5E/SPX/NKY

# 3. Dashboard
streamlit run app.py

Calibration data

Heston is calibrated per index: S&P 500 live to the SPY option chain (the most liquid options market in the world, freely accessible, but requires US market hours); Euro Stoxx 50 from a Eurex OESX settlement CSV if supplied, otherwise inheriting S&P 500's skew shape with its variance level rescaled to SX5E's own realized vol. Full provenance is in sources.md.

Limitations

Honest about what the model can and cannot do:

• Inputs drive everything. The PV is only as good as the calibrated surface, the correlation estimate, and the dividend assumptions. The SX5E/NKY proxies are explicitly second-best to a directly-calibrated surface.
• One discount curve. A single EUR rate discounts all legs; a production multi-currency book would discount per currency and add quanto adjustments for the non-EUR underlyings.
• Flat correlation in scenarios. The correlation ladder shifts all pairs uniformly; real correlation risk is richer (correlation tends to rise in stress).
• Single daily-close vendor. No intraday data and no bid-ask beyond the two-sided-quote filter used in calibration.

Testing

The tests/ directory holds a pytest suite asserting the mathematical identities at the core of the pricer. It is driven by fixed-seed synthetic inputs, so it runs offline with no market-data calls.

• Black-Scholes / implied vol — implied_vol inverts bs_call to machine precision.
• Heston → Black-Scholes — the characteristic-function call collapses onto Black-Scholes as the vol-of-vol xi → 0.
• Monte Carlo convergence — the Heston MC vanilla price matches the characteristic-function price within Monte Carlo error.
• Worst-of structure — PV rises monotonically with correlation and reproduces the single-asset PV in the perfectly-correlated, identical-asset limit.
• Coupon & solver — PV is monotone in the coupon; the fair-coupon solver round-trips to par.
• Greeks — delta is positive and vega negative on every leg under GBM (the note is short vol).

pip install -r requirements-dev.txt
pytest tests/ -q          # 10 tests

Tests run automatically on every push via GitHub Actions (.github/workflows/tests.yml).