Options, exercise and learned hedging
A practical primer: contracts, probability, financing, Greeks, exercise decisions, learning and the application tools. Worked examples and a guided reading list.
DERIVATIVES RESEARCH
Derivative portfolio risk
Volt calculates option prices, Greeks and portfolio exposures. Jormungandr trains and serves hedging policies. These papers document their models, application interfaces and completed experiments.
APPLICATION DEMONSTRATION
Option analytics replay
HISTORICAL CAPTURE · AMERICAN OPTIONS
Observed option quotes and paired stock prices: 64 American contracts across eight stocks and ETFs, captured over about 24 seconds. Volt supplies exercise-aware implied volatilities and Greeks. The example book holds one long contract per row; the final browser scene edits one quantity. Carry uses a flat 4% rate, zero yield and an empty cash-dividend schedule. Playback uses presentation pacing. Raw quote files and the replay producer remain outside the public source checkout.
01
Market observations
Bring option quotes, stock observations and contract terms into one dated calculation.
02
Portfolio exposures
Calculate contract Greeks, signed book exposures and the effect of specified market moves.
03
Hedge holdings
Train a policy on financed trading paths, including the liability payment and transaction costs.
METHODS AND APPLICATIONS
Papers and reading guide
The primer covers the financial and mathematical background. The application papers document pricing methods, hedge-learning experiments and software interfaces.
Pricing and hedging a multi-asset option
A five-stock Bermudan contract: continuation learning, independent price estimates, the vector hedge account and measured CPU/GPU execution.
Pricing and risk analytics
Prices, Greeks, portfolio units and CPU/GPU calculations for a call-and-put book.
Learning to hedge in Volt
The Heston market, financed trading account, loss preferences and completed hedging experiments.
Learning to hedge with stocks and options
The control problem and the training, evaluation and holdings-inference API calls.
Live option analytics
Quote handling, implied volatilities, contract Greeks and portfolio scenarios in an updating risk view.
European straddles
Gamma, theta, hedge frequency and transaction costs in a European straddle.
USE THE APPLICATION INTERFACE
Training and inference API
The application creates a learner and uploads separate training and validation batches. Each update advances a model version. Inference takes current observations and holdings, then returns target quantities.
Volt retains the financial calculation, tensors and optimizer in its worker. Jormungandr supplies the HTTP interface, dataset partitions, model lifecycle and checkpoints. Other compatible engines can use the same interface.
Python client · configured model and uploaded batches
from jormungandr.client import JormungandrClient
client = JormungandrClient("http://127.0.0.1:8080")
version = client.get_native_model("hedge")["version"]
client.native_command("hedge", "update", {
"batch_id": "train", "expected_version": version})
risk = client.native_command("hedge", "evaluate", {
"batch_id": "validation"})
holdings = client.native_command("hedge", "infer", {
"step": 0, "paths": 1, "features": [0., 1.],
"previous_holdings": [0., 0.]})EXERCISE AND SETTLEMENT
Early exercise and settlement
A European contract settles at expiry. A Bermudan holder chooses among stated exercise dates; an American holder can exercise throughout the permitted interval. The writer’s hedge must respond to the resulting cash payment or stock delivery. That timing enters the contract model, pricing method and trading account.
Becker, Cheridito and Jentzen develop learned stopping rules and dynamic hedges for these problems. Volt supplies the dated pricing reference, learned exercise policies, price-bound estimates and a writer account that settles and closes its hedge at exercise. A one-year Bermudan put study with monthly hedging reduced mean squared loss from 61.44 unhedged to 4.01 without fees and 4.08 with 10-basis-point stock fees; numerical delta reached 3.94 and 3.98. The application guide includes a runnable writer example, and the learned-stopping guide explains the two holder algorithms and interval hedging. A five-stock Bermudan max-call comparison now recovers prices 26.1576–26.1609 across three independently initialized models, against the published 26.154. Each model was assessed on 4,096,000 fresh lower-estimate paths and 2,048 nested outer paths; the reported confidence intervals are wider than the paper’s. The executed configurations and comparison record accompany the multi-asset application paper. The original studies cover direct optimal stopping and American-style pricing and hedging.