Physical and Risk-Neutral Measures in FX Pricing and Forecasting

# $\mathbb{P}$ vs. $\mathbb{Q}$ measure in FX options


# $\mathbb{P}$ vs. $\mathbb{Q}$ measure in FX options












### Background

I want to look at this FX rate with the risk-neutral measure. I read that we strip away the real-world historical trend ($\mu=−2.88\%$). In the risk-neutral world, my assumption is that investors do not demand a risk premium for holding volatile assets, which means the expected return on any asset must equal the risk-free rate.

### My attempt

For FX, the solution with $\mathbb{Q}$ is governed by Garman-Kohlhagen framework (the FX extension of Black-Scholes).

$$dX_t = (r_d - r_f) X_t dt + \sigma X_t dW_t^{\mathbb{Q}}$$ where:

- $X_t$ is the exchange rate (how many USD per 1 EUR).

- $r_d$ is the domestic interest rate (USD rate, since USD is the pricing currency).

- $r_f$ is the foreign interest rate (EUR rate, acting exactly like a continuous dividend yield $q$ in the classic Black-Scholes formula.

Applying Itô's Lemma under the $\mathbb{Q}$ measure gives the exact solution for the exchange rate at time $T$:

$$X_T = X_0 \exp\left( \left((r_d - r_f) - \frac{1}{2}\sigma^2\right)T + \sigma W_T^{\mathbb{Q}} \right)$$

To see how this differs from the real-world calculation, let's look at the risk-neutral expected value, which is used directly to calculate forward rates and price options:

$$E^{\mathbb{Q}}[X_T] = X_0 e^{(r_d - r_f)T}$$

- Current Rate ($X_0$): $1.14376$.

- US Interest Rate ($r_d$): let's assume a standard risk-free rate of $4.5\%$ ($0.045$).

- Eurozone Interest Rate ($r_f$): let's assume the European Central Bank rate is at $3.25\%$ ($0.0325$).

- Time ($T$): $1$ year.

\begin{align*} E^{\mathbb{Q}}[X_T] &= 1.14376 \cdot e^{(0.045 - 0.0325) \cdot 1}\\ &= 1.14376 \cdot e^{0.0125}\\ &= 1.14376 \cdot 1.01258\\ &= \boxed{1.1581} \end{align*}

### My questions

Would this be correct?

Also, do quants/analysts typically separate the entire financial universe into two distinct domains:

- the $\mathbb{Q}$-World (Risk-Neutral)?

- and, the $\mathbb{P}$-World (Physical/Real-World)?

So, if we are market-making options traders or a derivatives pricing quants, we'd work exclusively with $\mathbb{Q}$ (Risk-Neutral Measure)? If we are risk managers or a directional algorithmic traders, we would use without $\mathbb{Q}$ ($\mathbb{P}$-Measure)?

Also, to find the explicit solution to this SDE without using $\mathbb{Q}$, we apply Itô's Lemma directly to the real-world process. The exact solution for the exchange rate at a future time $T$ is:$$X_T = X_0 \exp\left( \left(\mu - \frac{1}{2}\sigma^2\right)T + \sigma W_T^{\mathbb{P}} \right)$$

- Current Rate ($X_0$): $1.14376$

- Real-World Annual Drift ($\mu$): $-0.0288$ (from the $-2.88\%$ 1-year return on the screen)

- Annual FX Volatility ($\sigma$): let's assume a standard $\approx 7\%$ ($0.07$) for G10 currencies.

- Time ($T$): $1$ year

If we want to find the expected mathematical baseline for the exchange rate next year without pricing options, we take the expected value

\begin{align*} E^{\mathbb{P}}[X_T] &= X_0 e^{\mu T}\\ &= 1.14376 \cdot e^{-0.0288 \cdot 1}\\ &= 1.14376 \cdot 0.9716\\ & = \boxed{1.1113} \end{align*}

As such?

## Answer by almost_surely_ (score 2, accepted)

https://quant.stackexchange.com/a/85811

Both calculations are arithmetically right, and the accepted answer's replication framing is the correct way to think about $\mathbb{Q}$. Two things it doesn't cover, though, and both bite specifically on what you're doing here.

### 1. Your $\mathbb{P}$ number isn't wrong — it carries no information

You take $\mu = -2.88\%$ from one year of screen data. Here's what that estimate is worth.

For $dX = \mu X\,dt + \sigma X\,dW$, the drift estimator over a calendar span $T$ has

$$\operatorname{SE}(\hat\mu) = \frac{\sigma}{\sqrt{T}}.$$

With $\sigma = 7\%$ and $T = 1$ year, $\operatorname{SE} = 7\%$ — the same size as the estimate itself.

|  |  |
| $\hat\mu$ | $-2.88\%$ |
| standard error | $7.00\%$ |
| t-statistic | $-0.41$ |
| 95% CI | $[-16.6\%,\ +10.8\%]$ |

The estimate is not distinguishable from zero, or from $+10\%$. And the part that surprises people: sampling more finely does not help. Tick data, hourly, daily — irrelevant. Only calendar span enters. To pin the drift to $\pm 1\%$ you need $(\sigma/0.01)^2 = 49$ years of EURUSD, over which the underlying monetary regime has changed several times. This is Merton (1980), and it is the single most useful fact to know about $\mathbb{P}$ drifts.

Now contrast with $\sigma$, which converges beautifully as you sample more finely, and which the market quotes directly. That asymmetry — vol estimable, drift not — is the real engine behind the $\mathbb{P}/\mathbb{Q}$ split. It's why there is a liquid implied volatility market and no "implied drift" market, and why $\mathbb{Q}$ parameters are pinned by traded prices to the basis point while the $\mathbb{P}$ drift is a matter of opinion.

Propagating that uncertainty into your forecast, your "mathematical baseline" of 1.1113 comes with:

|  | 90% interval for $X_T$ | width |
| $\mu$ treated as known | $[0.9880,\ 1.2439]$ | 26 big figures |
| $\mu$ estimated from 1Y | $[0.9420,\ 1.3046]$ | 36 big figures |

The point estimate sits inside a band a third of a big figure wide per trading day for a year. (Also worth noting: 1.1113 is the mean; the median is 1.1086. For a lognormal those differ, and "the expected rate" is ambiguous between them.)

### 2. In FX there is no single $\mathbb{Q}$ — and your question doesn't say which one

This is the part that's genuinely FX-specific and I think the most useful thing to take away.

$F = X_0 e^{(r_d - r_f)T} = 1.1581$ is a traded price, fixed by covered interest parity through a pure replication argument: borrow EUR, spot into USD, lend USD. No measure appears anywhere in that construction. It is measure-free.

The statement $\mathbb{E}^{\mathbb{Q}}[X_T] = F$ is only true under $\mathbb{Q}^{\mathrm{USD}}$ — the measure with the USD money-market account as numeraire. A EUR-based desk uses $\mathbb{Q}^{\mathrm{EUR}}$, and gets a different answer for the same economic quantity:

|  | value |
| $1/F$ | 0.863448 |
| $\mathbb{E}^{\mathbb{Q}^{\mathrm{EUR}}}[1/X_T]$ | 0.863448 |
| $\mathbb{E}^{\mathbb{Q}^{\mathrm{USD}}}[1/X_T]$ | 0.867690 |

$$\frac{\mathbb{E}^{\mathbb{Q}^{\mathrm{USD}}}[1/X_T]}{1/F} = e^{\sigma^2 T} = 1.00491 \quad (49\text{ bp at }\sigma=7\%)$$

So the USD desk and the EUR desk agree exactly on the forward — it's a price, they'd arbitrage each other otherwise — and disagree on "the expected exchange rate." This is Siegel's paradox, and it is the same $e^{\sigma^2 T}$ that shows up as the quanto adjustment when you price a payoff in the wrong currency.

At $\sigma = 7\%$ it's 49bp, small enough to ignore in a back-of-envelope. At $\sigma = 15\%$ — an EM cross, or G10 in stress — it's 228bp, which is not a rounding error on a forward.

The takeaway for your framing: "what does the risk-neutral measure say EURUSD will be?" is not a well-posed question in FX until you name the numeraire currency. Ask instead "what is the forward?", which is unambiguous, tradeable, and the same number for everyone.

### 3. On the $\mathbb{Q}$ -world / $\mathbb{P}$ -world split

The accepted answer is right that it splits jobs rather than the universe. I'd sharpen it one notch: it splits by question, not by job title.

- What is this worth, and how do I hedge it? → $\mathbb{Q}$

- What might actually happen, how much could I lose, how much capital do I hold? → $\mathbb{P}$

A market maker uses $\mathbb{P}$ every day — for their own VaR, position limits and inventory risk. A risk manager uses $\mathbb{Q}$ every day — to value collateral and compute haircuts.

And in the area I work in, the two appear inside a single calculation. An XVA engine simulates counterparty exposure paths; calibrated to market-implied vols and forwards, those paths feed the CVA that goes into the P&L, which is a $\mathbb{Q}$ number. The same engine, recalibrated to historical dynamics, produces the PFE used for credit limits and the paths used to backtest the exposure model — $\mathbb{P}$ numbers, and regulators require the backtest be done that way precisely because a $\mathbb{Q}$-calibrated model has no claim to describe what actually happens. Same code, same trade, two measures, both mandatory.

So no — not two disjoint worlds with different inhabitants. One system that has to be honest about which question it's answering at each step.

### Two smaller things

The convexity slip. $-2.88\%$ off the screen is a simple return. Converting properly to the instantaneous drift in your exponent:

$$\mu = \frac{\ln(1-0.0288)}{T} + \frac{\sigma^2}{2} = -2.677\%,$$

not $-2.88\%$. Immaterial next to the $\pm 7\%$ standard error, but it's the same Itô correction you already applied inside the exponent, so worth being consistent about.

Covered interest parity doesn't hold exactly. Since 2008 the cross-currency basis has been persistently non-zero, typically tens of basis points depending on tenor and stress. The traded EURUSD forward embeds that basis, so your textbook $X_0 e^{(r_d-r_f)T}$ is close to but not equal to the screen forward. On a pricing desk you'd build the forward from the basis-adjusted curve rather than from two deposit rates.

### Code

```
import numpy as np
X0, rd, rf, sig, T, mu = 1.14376, 0.045, 0.0325, 0.07, 1.0, -0.0288

print(f"SE(mu) = sigma/sqrt(T) = {sig/np.sqrt(T):.4f}   t = {mu/(sig/np.sqrt(T)):+.2f}")
print(f"years for SE = 1%      : {(sig/0.01)**2:.0f}")

F = X0*np.exp((rd-rf)*T)
print(f"forward                : {F:.6f}   (measure-free)")
print(f"E^Q_EUR[1/X] = 1/F     : {1/F:.6f}")
print(f"E^Q_USD[1/X]           : {(1/X0)*np.exp(-(rd-rf)*T + sig**2*T):.6f}")
print(f"Siegel gap = exp(s^2T) : {1e4*(np.exp(sig**2*T)-1):.1f} bp"
      f"   -> {1e4*(np.exp(0.15**2*T)-1):.0f} bp at 15% vol")
```

### References

- R. Merton, On Estimating the Expected Return on the Market, Journal of Financial Economics 8(4), 1980 — the $\sigma/\sqrt{T}$ result and why sampling frequency doesn't rescue the drift.

- J. Siegel, Risk, Interest Rates and the Forward Exchange, Quarterly Journal of Economics 86(2), 1972 — the paradox, in the original.

- U. Wystup, FX Options and Structured Products, 2nd ed., Wiley, 2017 — quanto adjustments and the two-numeraire structure of FX, worked through properly.

- W. Du, A. Tepper, A. Verdelhan, Deviations from Covered Interest Rate Parity, Journal of Finance 73(3), 2018 — the persistence of the basis.

- A. Meucci, 'P' Versus 'Q': Differences and Commonalities between the Two Areas of Quantitative Finance, GARP Risk Professional, 2011 — the canonical short treatment of exactly your second question.

## Answer by Rylan (score 1)

https://quant.stackexchange.com/a/85720

Your calculation looks right to me for the price of the FX forward. It is also correct for the "mathematical baseline" with the assumption that $\mu = -2.88\%$ is the drift in the physical measure (a big assumption but not a completely unreasonable one).

The way I would personally suggest thinking of the risk neutral measure is less about making assumption about investor behaviour/risk preferences, and more a mathematical convenience allowing you to find the value of a replicating portfolio.

If we can replicate a derivative with assets we can trade (so in your example an FX forward by trading in the currency and the various money market accounts in each currency), then any price that differs from the price we find will admit an arbitrage.

We can find the replicating portfolio by hand without bothering with expectations for something like an FX forward. For something like an option, even a vanilla put or call, it's quite a bit more complicated; we have to solve a PDE, which Black and Scholes proved is equivalent to taking an expectation in something we call the "risk-neutral measure".

As you note, this price says nothing about the "physical" expectation of the payoff. (Well, nothing beyond what the prices, rates, etc we use as inputs say.)

I am a derivatives pricing quant and I do split jobs in the way you suggested, but not the "financial universe" per se. In my job, the $\mathbb{Q}$ measure is more important as we price derivatives and hedge them, not really intending to take directional risk. As you note, someone like an analyst in signal research at a hedge fund would probably be looking at the $\mathbb{P}$ measure to determine whether a trade is worth making, but if they need to price a derivative, they still need to use the $\mathbb{Q}$ measure, as any other price will result in an arbitrage which a competitor can take advantage of.