Option Portfolio Optimization When Physical and Pricing Measures Differ
Summary
The document frames a one-month portfolio problem involving cash, a stock, and a six-month European put. Returns are evaluated under the physical measure, but the option’s future market value depends on risk-neutral pricing. The author explains why valuing the option under the physical measure alone may not estimate that market value and considers avoiding a direct, potentially fragile mapping between the measures.
A proposed workaround maps selected physical-measure Heston parameters to risk-neutral parameters using historical calibration pairs, estimates the risk-neutral drift from current option prices, and simulates stock paths under the physical measure while pricing options with mapped risk-neutral parameters. The discussion highlights the difficulty of estimating rare extreme option moves and the risk of sparse data and model error. It offers no empirical validation of the mapping. A cited presentation is said to discuss Kelly optimization when both measures are known, but its details are not included.
Key ideas
- Option portfolio optimization over a short holding period requires estimating the option’s future market price, not only its physical-measure expected value.
- A proposed heuristic maps selected physical-measure Heston parameters to risk-neutral parameters using historical calibration pairs.
- The suggested workflow simulates stock paths under the physical measure and values options using mapped risk-neutral parameters.
- Rare, extreme option moves make calibration data scarce and model estimates potentially fragile.
- The proposed parameter mapping is a rough solution and is not empirically validated in the document.
Tags
Full text
# Portfolio Optimization with Options Using Only the P Measure # Portfolio Optimization with Options Using Only the P Measure Optimize a portfolio of [Cash, Stock A, European Put on A (K, T = 6 months)] over a 1-month horizon. The portfolio is constructed today and held unchanged for one month. I have a stochastic volatility model for A calibrated to historical data (physical measure P). Optimisation is straightforward a) without the option or b) option valued only as ITM ignoring the remaining time value. The problem - expected market price in 1 month (Q measure) of 6month option. I can get its expected value (P measure) but it's not the same as expected market price. The proper solution - calibrate another model mapping P -> Q and use it to price options. Are there other solutions, workarounds, heuristics? My best idea - use the option value computed under P, possibly with a penalty on large option positions. Notes I'm trying to avoid the explicit P → Q mapping, because it seems very difficult to model reliably. Another problem - I'm interested in >x10 option moves (I'm using options - as put insurance or speculative explosive far otm calls). Such situations are rare and likely depend on unusual market regimes (panic) and extreme implied volatility. so there is little data and many parameters to estimate, making calibration difficult. I feel the resulting model would be fragile, with errors that are difficult to detect. UPDATE: Rough Solution Convert P parameters to Q parameters: - Estimate $\mu_Q = r-q$ as a free parameter when calibrating Heston to today's option prices. - Map $(\Theta_Q,k_Q)=f(\Theta_P,k_P,\sigma_P)$ using a low-order polynomial fitted to historical pairs of P- and Q-measure Heston calibrations. For simplicity, the same mapping could be used across stocks. - Set the remaining Q parameters equal to their P counterparts. Simulate stock-price paths under P, but price options using the mapped Q parameters. Seems it has been known for quite a while in the Original Heston Paper, page 9 ## Answer by user93883 (score 0) https://quant.stackexchange.com/a/85793 You may find it useful to take a look at this presentation on options portfolio optimization here In particular it discusses Kelly optimal option portfolio construction when P and Q measures are known.
Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.