Checking Short-Horizon Volatility in Lognormal Monte Carlo Paths

# Volatility in simulated paths different to monte carlo parameters


# Volatility in simulated paths different to monte carlo parameters












I am trying to convince myself that I have set up my monte carlo simulation correctly by looking at the results and trying to get them to agree with the model parameters. Please help me understand what am I doing wrong.

I use the following code to generate each simulated price path.

```
def create_paths_changes_normal(mu, sigma, path_len, sample_size):
    dt = 1 / BUSINESS_DAYS_IN_YEAR

    r_daily = (1 + mu) ** dt - 1
    iv_daily = sigma * np.sqrt(dt)

    drift = (r_daily - 0.5 * iv_daily ** 2)
    diffusion = iv_daily

    # Generate random daily returns
    return np.exp(np.random.normal(loc=drift, scale=diffusion, size=[sample_size, path_len]))
```

Afterwards, I run the following code

```
def check_statistics(mu, sigma, days, sample_size=100000):
    path_changes = create_paths_changes_normal(mu, sigma, days, sample_size)

    returns = np.prod(path_changes, axis=1)
    annal_returns = returns.mean() ** (BUSINESS_DAYS_IN_YEAR / days) - 1
    avg_return = annal_returns.mean()

    daily_returns = path_changes - 1.0
    vol = daily_returns.std(axis=1).mean() * np.sqrt(BUSINESS_DAYS_IN_YEAR)

    # assert mu == avg_return  # This is ok
    # assert sigma == vol  # This fails for short paths e.g. days=5.
```

For paths of length 5 (days) the average returns agree, but the vol does not. For longer paths both agree.

Please help me understand if I have made a wrong assumption somewhere and whether I am doing this correctly.

## Answer by THATS MY QUANT MY QUANTITATIVE (score 1)

https://quant.stackexchange.com/a/79943

The issue is with how you are calculating your mean and volatility. According to Wikipedia, $$ \begin{align} \mathbb{E}(S_t) &= S_0 e^{\mu t} \\ \text{Var}(S_t) &= S_0^2 e^{2\mu t} \left( e^{\sigma^2 t} - 1 \right) \end{align} $$

Then adapting your code:

```
import numpy as np

def create_paths_changes_normal(mu, sigma, BUSINESS_DAYS_IN_YEAR, path_len, sample_size):
    dt = 1 / BUSINESS_DAYS_IN_YEAR

    r_daily = (1 + mu) ** dt - 1
    iv_daily = sigma * np.sqrt(dt)

    drift = (r_daily - 0.5 * iv_daily ** 2)
    diffusion = iv_daily

    # Generate random daily returns
    return np.exp(np.random.normal(loc=drift, scale=diffusion, size=[sample_size, path_len]))

mu = 0.05
sigma = 0.2
days = 5
sample_size = 10000000
BUSINESS_DAYS_IN_YEAR = 252.

path_changes = create_paths_changes_normal(mu, sigma, BUSINESS_DAYS_IN_YEAR, days, sample_size)
returns = np.prod(path_changes, axis=1)

S_0 = 100. 
t = days / BUSINESS_DAYS_IN_YEAR

S_mc = S_0 * returns 

EV_S_theo = S_0 *np.exp(mu * t)   # 100.09925557498198
EV_S_MC = S_mc.mean() # ~100.09840608411352

VAR_S_theo = (S_0**2) * (np.exp(2 * mu * t)) * (np.exp(t * sigma**2) - 1) # 7.955427106400656
VAR_S_MC = S_mc.var() # ~7.955427106400656
```

## Answer by Mahavir Bhattacharya (score 0)

https://quant.stackexchange.com/a/79840

In the second code snippet, in this line: path_changes = create_paths_changes_normal(mu, sigma, days, sample_size)

you are changing the arguments from what you had declared in the previous code:

create_paths_changes_normal(risk_free_rate_difference, implied_vol, path_len, sample_size)

Essentially, the first argument is getting changed.