Derivatives Pricing & Risk Engine
Different pricing methods should converge on the same economics when their assumptions align. The project tests that agreement, then explores what changes when volatility and hedging become more realistic.
Central findingClosed-form, tree and simulation methods reconcile closely on shared benchmarks, while variance reduction and more frequent hedging materially improve numerical precision and risk control.
Evidence
The result in context
- Black–Scholes pricing error
- <3e−5
- Put–call parity error
- 0
- Control-variate standard-error reduction
- 61.75%
- Heston calibration RMSE
- 3.12e−05
On this page
Question
Can several pricing methods be implemented, cross-checked and used to explain how option value responds to market inputs?
A pricing and risk system spanning closed-form options, trees, simulation, stochastic volatility, calibration and hedging.
System
One engine, several views of option value
Black–Scholes provides an analytical benchmark; a Cox–Ross–Rubinstein tree makes early-exercise logic and convergence visible; Monte Carlo supports flexible simulation; Heston introduces stochastic volatility.
Greeks measure how an option’s value responds to changes in the market or model inputs. The explorer below calculates price, Delta, Gamma and Vega locally from the tested Black–Scholes implementation.
Local calculation
Option sensitivity explorer
Greeks measure how an option’s value responds to market and model inputs.
- Call price
- £10.45
- Put price
- £5.57
- Call Delta
- 0.6368
- Put Delta
- -0.3632
- Gamma
- 0.0188
- Vega · per 1 vol point
- 0.3752
Call value across 65%–135% of the selected spot.
Black–Scholes assumes constant volatility and frictionless trading. The calculation is an explanatory benchmark, not a market quote.
Validation
Methods are checked against each other
Black–Scholes pricing error is below 3e−5 with zero put–call parity error. A 1,000-step tree comes within approximately 0.002 of the analytical benchmark.
Cross-method reconciliation is useful because a plausible-looking option price is not, by itself, evidence that an implementation is correct.
Interactive evidence
Monte Carlo error contracts as the path count increases
Absolute pricing error versus simulation paths; logarithmic path axis.
Read: At 50,000 paths the absolute error falls to £0.0141; sampling noise means the 100,000-path point is not mechanically the minimum.
Data table · 5 verified rows
| Paths | Estimate | Absolute Error | Standard Error | Ci Lower | Ci Upper |
|---|---|---|---|---|---|
| 1,000 | 10.968869 | 0.518285 | 0.485538 | 10.017232 | 11.920505 |
| 5,000 | 10.633841 | 0.183257 | 0.213147 | 10.21608 | 11.051602 |
| 10,000 | 10.318653 | 0.131931 | 0.146225 | 10.032056 | 10.605249 |
| 50,000 | 10.464657 | 0.014073 | 0.066052 | 10.335197 | 10.594116 |
| 100,000 | 10.479967 | 0.029384 | 0.046573 | 10.388685 | 10.571249 |
Simulation
Variance reduction turns computation into precision
Antithetic sampling reduces standard error by 29.16%; a control variate reduces it by 61.75%. The objective is not a more impressive simulation count, but a tighter estimate for the same computational budget.
Risk
Hedging frequency leaves a measurable residual
Daily rebalancing produces hedge-error standard deviation of 0.4257 versus 1.9104 for monthly rebalancing in the tested setup. The comparison makes discrete hedging risk tangible without implying frictionless implementation.
Limitations
What this evidence does not establish
- Black–Scholes Greeks inherit the model’s assumptions, including constant volatility and frictionless trading.
- Heston calibration uses a deterministic synthetic option chain; calibration quality is conditional on that surface and objective function.
- The hedge comparison is an idealised Black–Scholes benchmark with zero transaction costs and a shared random seed, not live-trading evidence.
Source and reproducibility
Trace the evidence
Source code, evaluation outputs and supporting material are available in the repository.
View repository- Volatility surfacedocs/images/dashboard_volatility_surface.pngCommit / evidence ID: 60c6c85a25df2291dba13ee6de53652d6a711d19
- Numerical convergence resultsreports/numerical_convergence_results.csvCommit / evidence ID: 60c6c85a25df2291dba13ee6de53652d6a711d19
- Delta-hedging simulator artefactdocs/images/dashboard_delta_hedging_simulator.pngCommit / evidence ID: 60c6c85a25df2291dba13ee6de53652d6a711d19