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I am struggling in interpreting results of my simulations. I use Monte Carlo algorithm to simulate stock paths and calculate option price. The notation: $r$ is a risk free interest rate, $T$ is time to maturity, $K$ - strike of an option, $S_T$ - stock price at time $T$, $\sigma$ - volatility of stock log-returns, $c$ - price of a call option, $t$ is a time step.

First I simulate stock paths assuming that stock price follows GBM process and calculate option price. Lets assume that $r = 0.05$, $K = 70$, $S_0 = 70$, $\sigma = 0.3$, $T = 0.28 $ (100 days). Number of sims $N = 1000$ and there will be 700 stock movements in each path (7 each day). I get $c = 5.008$.

BSM model with same parameters gives $c = 4.88$

Now I assume that log returns can be described by NIG disrtibution with parameters $\mu$, $\beta$, $\delta$, $\alpha$ and that log returns, $r_i$ can be calculated as (see this question): $$r_i = \hat\mu + \hat\beta\sigma_i^2 + \sigma_i\varepsilon_i$$ where $\sigma^2 \sim IG(\hat\delta/\hat\gamma, \hat\delta^2)$, $\varepsilon \sim N(0, 1)$.

I estimated parameters of NIG distribution using Google hourly returns for 2015 year ($\sigma = 0.29$). After running simulation I get $c = 8.77$ which is far bigger than BSM price and price obtained by GBM simulation.

The reason is that growth rate in the latter model bigger than in GBM. For the NIG model expected value for growth rate $R_T$ for 100 days (700 movements, 7 each day) is $$\mathbb{E}(R_T^{NIG}) = \left(\exp\left[\hat\mu + \hat\beta\frac{\hat\delta}{\hat\gamma}\right]\right)^{700} = 1.083$$

while GBM expected growth rate is $$\mathbb{E}(R_T^{GBM}) = \left[\exp(r - 0.5\sigma^2)t\right]^{700} = 1.00139$$

Finally, the question: if I want to sell an option which model should I use? Intuitively it should be GBM because it prices under $\mathbb{Q}$ measure. But why one want to construct different models that prices under real world $\mathbb{P}$ measure e.g. NIG model above? And what are applications of them?

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  • $\begingroup$ I'm interested in a detailed answer to this question as well. I'm curious though about what you mean with 'GBM prices under P-measure'. In the Black-Scholes (BS) model, options are priced under a (unique) risk-neutral measure. Pricing options under the real-world measure $\mathbb{P}$ requires simulation since you rely on time series. However, the whole idea about the BS model was the concept of risk-neutral pricing. $\endgroup$ – Cavents May 23 '16 at 21:01
  • $\begingroup$ I confused P and Q notations. Sure, BS does not price under $\mathbb{P}$ $\endgroup$ – tosik May 24 '16 at 5:30
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I believe that the confusion arises because of the wrong treatment of NIG. The answer to the question you link is misleading, as it simulates under P which is not appropriate for option pricing. None of the NIG parameters under P carries over to Q in general, but especially the drift is the problem here. First use the mom gen function of NIG to find the appropriate drift correction, and then it will work.

Edit: Also it seems to me that you are mixing time horizons here. To put vol 30% in BS looks like an annualized vol to me, but then you put time to maturity equal to 100, which means 100 years. It is quite basic to get the units sorted out. As you have presented it I cannot understand which units NIG is expressed into.

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  • $\begingroup$ Thanks for your answer. $T$ was expressed in days. I made an edit to avoid misunderstanding. I will correct my NIG drift term so that $\mathbb{E}(R_T^{NIG}) = \mathbb{E}(R_T^{GBM})$. After doing so am I able to claim that my NIG model prices option under $\mathbb{Q}$ measure? $\endgroup$ – tosik May 24 '16 at 18:51
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    $\begingroup$ No you cannot. You can claim that you price under one arbitrage-free equivalent measure, which might or (most probably) might not be the correct arbitrage-free equivalent probability measure that the options market uses (presumably this is what you mean by Q although it is not clear in your post). Levy models are incomplete, therefore changing the drift is not sufficient. Afaik there is no reason to assume that Q is NIG if P is NIG at all. $\endgroup$ – Kiwiakos May 24 '16 at 20:21
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You should see this as a comment to @Kiwiakos answer which already hit the bull's eye.

In the SE question you're referring to and to which I have answered, the idea was simply to provide you with a sound way of simulating returns out of a NIG distribution.

It so happens that, for whatever your reason was, you decided to calibrate your NIG parameters based on historical time series, hence information provided under $\mathbb{P}$.

Pricing options under $\mathbb{P}$ is possible. Yet, it is generally preferred to use an equivalent martingale measure $\mathbb{Q}$ where option prices can be expressed as discounted expectations (intuitive interpretation + nice mathematical treatment based on stochastic calculus and martingale theory).

Of course, the option price is itself a unique quantity: it does not change whether it is expressed under $\mathbb{P}$ or under $\mathbb{Q}$. Using the measure $\mathbb{Q}$ should merely be seen as a smart trick to get rid of the risk aversion issue.

In general, moving from $\mathbb{P}$ to $\mathbb{Q}$ requires the specification of a form of market risk premium. Mathematically, it is equivalent to choosing an equivalent martingale measure from the set of all possible measures rendering discounted asset prices as martingales, which relies on Girsanov theorem. It so happens that if the market is complete, there is only one unique measure per numéraire.

Anyway, there is no straightforward relationship which can tell you how to move from $\mathbb{P}$ to $\mathbb{Q}$, except in very particular modelling settings, e.g. a GBM under $\mathbb{P}$ remains GBM under $\mathbb{Q}$ but with modified drift (ie $S_T$ is lognormally distributed under both measures). This is because the market is complete in that case. But if you introduce stochastic volatility for instance, the relationship becomes much more complicated as the market is not complete anymore (this has motivated the use of empirical techniques such as entropy minimisation).

In any case, by absence of arbitrage opportunities, under $\mathbb{Q}$ the underlying asset price should always drift at the risk-free rate $r$. Your NIG model is not consistent with that as pointed by Kiwiakos. But simply making the drift $r$ is not enough, see the discussion in the comments and what I just said above. The best alternative would probably be to calibrate your NIG parameters to observed option prices (hence directly under $\mathbb{Q}$) the same way you probably derived your BS dynamics (I mean how did you get $r=5%$ and $\sigma=30%$ in the first place?).

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  • $\begingroup$ Great answer Quantuple. Clear and insightful. I would be grateful if you could also give us your views in this related question: quant.stackexchange.com/questions/8274/… $\endgroup$ – sets May 24 '16 at 10:35
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    $\begingroup$ Wow last answer dates back to 2013, surely you have gathered some ideas since then :) ? Will add my 2 cents when I find some time this afternoon. $\endgroup$ – Quantuple May 24 '16 at 10:48

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