Why Market-Calibrated Derivative Parameters Are Risk-Neutral

# Are quantities implied by the market always risk neutral?


# Are quantities implied by the market always risk neutral?












Let's imagine I'm working with the Black-Scholes model which under the statistical measure postulates that the stock obeys $$\frac{dS_t}{S_t} = \alpha dt + \sigma dW_t.$$ Under the risk-neutral measure the real drift $\alpha$ can be shown to be replaced by the risk-free-rate $r$, so $$\frac{dS_t}{S_t} = rdt + \sigma dW_t^{\mathbb{Q}}$$ in the risk-neutral world.

If I knew all the Black-Scholes parameters and tried to imply the drift from market prices by inverting the Black-Scholes formula for $r$, would I get the risk-neutral drift $r$, not $\alpha$, since the market is providing risk-neutral option prices?

Now let us suppose I'm working with the Heston model. If I tried calibrating the Heston model to market prices, would I be calibrating risk-neutral Heston parameters or statistical parameters?

To summarise, the way that I currently view derivatives modelling is:

- Create some kind of dynamics for the underlying (Black-Scholes, Heston, so on). Once these dynamics have been specified, calculate an associated risk-neutral measure to identify the new risk-neutral parameters that describe the underlying dynamics.

- Price derivatives using those risk-neutral dynamics (Monte-Carlo, PDEs, so on) derived in step (1).

- Using the above pricing function, create a calibration procedure that returns the model parameters from step (1) that best fit the market.

What I'm asking is whether I can only ever work in the risk-neutral setting and whether the parameters implied by the calibration procedure in (3) are always the risk-neutral ones (as in the Black-Scholes example at the start of my question).

Thanks!

## Answer by Daneel Olivaw (score 3, accepted)

https://quant.stackexchange.com/a/82115

> Are quantities implied by the market always risk neutral?

Generally speaking if you are calibrating a pricing model to the market then by definition your model quantities are risk-neutral.

Indeed if you work with a given pricing model $\mathcal{P}$ and you assume there is no arbitrage in the market, then your model’s calibrated parameters will preclude any theoretical arbitrage; by the fundamental theorem of asset pricing this implies there exists (at least) one risk-neutral pricing measure.

It is also worth noting in practice, for vanilla derivatives, you tend to calibrate volatility parameters. This implicitly assumes that any drift parameters are captured by the underlying’s forward (for which there tends to be a liquid market and therefore observable prices) yet as has been understood more acurately since e.g. Piterbarg (2010) this entails the existence of a risk-neutral (or more abstractly, pricing) measure under which the underlying’s drift equals its funding cost - typically the repo rate for a stock, or the collateral rate when transactions are fully collateralized.

This circles back to the completeness assumption underpinning pricing models, which in real life translates more or less into the existence of a liquid funding market (ie tradeable) for the derivative’s underlying.

## Answer by Con Fluentsy (score 1)

https://quant.stackexchange.com/a/82143

Ed Thorp in the seventies concluded that risk neutral was a fallacy and that you should use an appropriate rate r according to whether you are short or long, however he concluded that r in most instances is not as significant as time and volatility and can be substituted with the risk free rate. There is the alternative quants Taleb, Thorp, Ziemba,Haag,Wilmott, and a lot more, who make money. The risk neutral principal is central to those who subscribe to the CAPM. However the alternative of Roll; Arbitrage Pricing Theory is not based on risk neutral, using no arbitrage you can derive the Black Scholes from much simpler assumptions and is real world usable, no arbitrage is not always synonymous, with risk neutral, most people on stack overflow finance assume as given the CAPM baggage, when the smart money knows it is bereft of any actual real world analogue, and is mostly a recipe for mediocre returns at best. However if you have to pass an MBA as many people on here are posting questions to solve papers and assignments, then follow the orthodoxy, but when you get out and start trading don't take that stuff too seriously.