What Can Tonight Cost?

Question

The close caught you long 2,000 shares of a 50 stock you couldn't shed. Daily volatility is 2%. Using a normal approximation, what is the 95% one-day VaR of the position in dollars (one-sided z = 1.645)?

Accepted Answer

Dollar volatility of the position: 2,000 times 50 times 2% = 2,000 text per day. 95% VaR: 1.645 times 2,000 = mathbf3,290. Read it precisely: on roughly one night in twenty, this leftover inventory loses more than 3,290; on the other nineteen it loses less (or makes money). VaR is a quantile, not a worst case, and saying so out loud is part of the answer. Why a market maker cares: quoting all day is fine, the position you fail to flatten is what risk limits are written about. Desks run the identity in reverse: given an overnight loss budget L, the inventory cap is Q_max = fracL1.645 ,sigma, S, so a 10,000 budget here caps you at about 6,000 shares. That's where position limits actually come from, they're a risk number divided by a vol number, not folklore. Caveats worth volunteering: overnight gaps are fatter-tailed than normal (earnings, news in the after-hours vacuum), so 1.645 sigmas understates the true 95% tail; correlated inventory across names compounds; and you can't stop-loss your way out while the market is shut, which is exactly why overnight inventory gets a harsher budget than intraday. The trap: quoting the VaR as "the most I can lose." The 5% tail is unbounded; VaR l

What Can Tonight Cost?

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