**In option pricing, two technical terms often create confusion. One term is “risk-neutral” and the other “real-word”. You hear these terms in the context of option pricing, backtesting, risk management and hedging. In this article I try to clarify the terminology.**

**Background**

First, we start with “risk-neutral”. The term risk-neutral refers to option pricing: The option pricing is based on the cost of a hedging strategy which ideally replicates the option without any risk. Technically, it turns out that one can price an option as the expected value denoted by E_Q() of the option’s payoff where the drift of the simulation equals the risk-free interest rate. In a risk-neutral valuation, the free simulation parameters like volatility are estimated in a way that the theoretical price and the traded prices match. This way, one obtains market consistent option prices for similar options, which are not traded.

On the other hand, the term “real-world” or “physical” simulation refers to a simulations of market values which creates a realistic behavior. This can be done using many techniques. Usually, one tries to capture as many “stylized facts” as possible. For example, one can match the historical drift, volatility and correlation of the simulated asset. Now, this real-world simulation can be used for testing trading strategies, for optimizing portfolios and for minimizing hedging errors. In the literature an expected value using a physical simulation is denoted by E_P().

One can also price options based on physical scenarios. But, this pricing is just some expected value of a speculator and not based on a risk-free hedging strategy as the risk-neutral valuation. This can still be useful in illiquid markets where hedging is not possible – in all other market the risk-neutral valuation provides the correct consistent price.

**Portfolio Market Risk Estimation**

A combination of the two simulation techniques is required for the portfolio market risk estimation. Also, a potential future exposure – the likely outstanding amount against a counter party – might require a combination of the two simulation techniques. In these cases, an option value based on a risk-neutral simulation has to be computed for each simulated physical scenario. This is usually done in a “nested simulation”: First, a physical simulation of the options underlying is done. Then, at each simulated time step and each scenario, a risk-neutral simulation with many scenarios is started.

This nested simulation is very costly, it requires many computations. In some cases, the risk-neutral option price might be available as a simple formula. In this case, one can leave out the nested simulation and use this formula instead, which is much more efficient. Another possibility avoiding the nested simulation combines the risk-neutral and the real-word simulation in a smart way and is implemented in Theta Proxy RM. This is a very quick way for estimating portfolio market risk of complex structured products.

## Conclusion

A physical or real-world simulation is used in risk-management, back-testing and portfolio optimization. The risk-neutral simulation is required in Monte Carlo simulations for market consistent option pricing. Thus, a simulation based market risk management of an options portfolio requires a physical simulation and within this simulation a nested risk-neutral valuation.

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