Estimating a Stock’s Expected Future Value Under the Real Measure

Summary

The document asks how to estimate a share’s expected value at a future date under the real-world probability measure, distinguishing that goal from risk-neutral option pricing. It notes that a zero-strike call should have the same payoff as one share, yet a calculation using a lognormal price distribution and historical daily volatility produces a value above the current share price when the assumed log-price mean is zero.

The example computes the expected call payoff by numerically integrating a lognormal density, using current price, daily volatility, and time to expiry as inputs. It contrasts the zero-mean assumption with a risk-neutral log-mean adjustment, but does not provide a model for forecasting real-world returns or resolve which expected future price is appropriate. Historical volatility alone specifies dispersion, not the expected return, so the example raises a modeling question rather than establishing a prediction method. Its truncated integration range and simplified assumptions also limit the calculation.

Key ideas

• The document distinguishes real-world price distributions from risk-neutral distributions used for derivative valuation.
• A zero-strike call has the same terminal payoff as one share, so its expected payoff depends on the expected future share price.
• The example integrates a lognormal distribution using historical daily volatility scaled over the horizon.
• A zero log-price mean can produce an expected future price above the current price because of lognormal convexity.
• Volatility alone does not determine the real-world expected return or expected future share value.

Tags

# What's the expected value of 1 share at time T the E[S_T]?


# What's the expected value of 1 share at time T the E[S_T]?












The premium for call option with strike 0 and expiration 365 days will be `1.13`, if calculated as $E[max(0,S−K)]$, assuming risk free rate as 0 and mean daily volatility `0.0263`.

That's strange, as $C(K=0)$ is same as 1 share and (intuitively) should have the same price (option premium) as 1 share and be equal to `1` not `1.13`.

So that lead me to the next question - what's the expected value of 1 share $E[S_{T=365}]$?

I'm interested in the stock price distribution at time T=365, under the real measure - the real prices that will be 1y ahead. Not synthetic distribution under risk neutral measure.

So, how the real distribution would look like at time T and what is the value of $E[S_{T=365}]$?

Avoiding the arbitrages is not the goal (I'll handle it separately), the primary goal is to model real portfolio scenarios.

Example:

Python code to calculate $E[max(0,S−K)]$ as numerical integral

```
import numpy as np
from scipy.stats import norm, lognorm

S           = 1      # Current stock price
K           = 0.001  # Strike price = 0
sigma_daily = 0.0263 # Average stock daily volatility of past 365 days
t_days      = 365    # Time to expiration

sigma = sigma_daily * np.sqrt(t_days)
mean = 0                            # Without risk neutrality
# mean = np.log(S) - 0.5 * sigma**2 # With Risk Neutrality

# E[max(0,S−K)] via numerical integration
x, dx = K, 0.01
premium = 0
while x <= 5:
  d = lognorm.pdf(x, s=sigma, scale=np.exp(mean))
  premium += max(x - K, 0) * d * dx
  x += dx

print("Premium:", round(premium, 2))
```
```

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.