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Try: market making · Kelly · Black–Scholes · daily five
dS = μS dt + σS dW
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This strategy generates buy signals when a fast moving average (Fast MA) crosses above a slow moving average (Slow MA). It also takes profit when the returns reach 1% to lock in small but consistent profits. The…
Pine Script · Trend following · 3d · ChaoZhang
/agenkit build a production-grade trading bot with a backtest engine, risk controls and paper trading by default from strategies/1-Profit-Moving-Average-Cross-Strategy.mdOriginal lessons with the formulas written out, worked examples, and the mistakes people actually make.
A fresh warm-up every day. Solve it to start a streak; the daily five and the 60-second sprint keep it going.
Two fair dice are rolled. What is the probability that they sum to 7?
Six of the 36 equally likely outcomes sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). So .
On average, how many rolls of a fair die does it take to roll the first 6?
The number of trials until the first success is geometric with , so its mean is .
What is the expected number of fair coin flips until you see two heads in a row?
Let be the expected flips from scratch and after one head. and . Solving gives .
Three cards are drawn from a standard deck without replacement. What is the probability all three are hearts? (decimal)
.
What is the smallest number of people for which the chance that two share a birthday exceeds 50%? (365 equally likely days)
The probability all birthdays differ is . It first drops below 0.5 at (about 0.493).
X and Y are independent Uniform(0, 1). What is the expected value of max(X, Y)?
, so the density is and . In general the max of uniforms has mean .
You roll a die and are paid its face value in dollars. You may reroll once, but must then accept the second roll. With optimal play, what is the game worth?
A reroll is worth 3.5, so keep 4, 5 or 6 and reroll 1–3. Value .
On average, how many rolls of a fair die does it take to see every face at least once?
Coupon collector: after faces are seen, the wait for a new one is geometric with mean . Total .
What is the probability of at least one six in four rolls of a fair die? (decimal)
Use the complement: . (The Chevalier de Méré problem.)
Monty Hall: you pick one of three doors, the host (who knows where the car is) opens another door showing a goat and offers a switch. What is your win probability if you switch?
Your first pick is right with probability 1/3, and the host’s action never changes that. Switching wins whenever your first pick was wrong: 2/3.
A family has two children and at least one is a boy. What is the probability both are boys?
Of the equally likely BB, BG, GB, GG, the condition removes GG. One of the remaining three is BB: 1/3.
A drawer holds 10 black and 10 white socks. In the dark, how many must you take to guarantee a matching pair?
Pigeonhole principle: with two colours, three socks must include two of the same colour.
You start with 2 dollars and bet 1 on fair coin flips until you reach 5 or go broke. What is the probability you reach 5?
For a fair game your wealth is a martingale, so .
Ten people throw their hats in a pile and each takes one at random. What is the expected number who get their own hat?
By linearity of expectation, each person gets their own hat with probability 1/10, and ten such indicators sum to an expected 1 — for any number of people.
A crash signal fires on 95% of real crashes and 10% of normal weeks. Crashes occur in 2% of weeks. Given that it fired, what is the probability of a crash? (decimal)
. Rare events make even accurate signals mostly false alarms.
You have 100 independent daily returns with a standard deviation of 2%. What is the standard error of the mean daily return, in percent?
Corr(X, Y) = 0.6 and Y’s volatility is twice X’s. What is the OLS slope of Y regressed on X?
A strategy has an annualised Sharpe ratio of 1.0 over 4 years of data. Approximately what is its t-statistic?
An equity strategy has 1.2% daily volatility. What is its annualised volatility in percent (252 trading days)?
You combine 9 uncorrelated strategies, each with Sharpe 0.5, at equal risk. What is the Sharpe ratio of the combination?
Means add linearly while volatilities add in quadrature, so the Sharpe scales by : .
You test 20 strategies with no real edge at the 5% significance level. How many do you expect to look significant?
. Selecting it would be a pure false positive.
Same 20 worthless strategies at 5% significance, tested independently. What is the probability at least one looks significant? (decimal)
. Search widely enough and a "discovery" is almost guaranteed.
Two assets each have 10% volatility and correlation 0.5. What is the volatility of a 50/50 portfolio, in percent?
(in units of ), so .
You regress the price of one coin on the price of an unrelated coin and get R² = 0.85. What is the most likely explanation?
Regressing one random walk on another routinely produces high with no real relationship. Regress returns, or test residuals for stationarity (cointegration).
A setup wins 40% of the time for +3R and loses 60% of the time for −1R. What is its expectancy per trade, in R?
. A low win rate can still be very profitable.
A strategy risks 1 to make 2 on every trade. What win rate, in percent, is break-even (ignoring costs)?
After a 25% loss, what percentage gain is needed to get back to even?
. Losses and recoveries are asymmetric.
An even-money bet wins 60% of the time. What fraction of wealth, in percent, does Kelly say to bet?
A strategy has expected excess return 8% and volatility 20% a year. What is the full-Kelly leverage?
. Most practitioners would run half of that or less.
What long-run growth rate do you get betting exactly twice the Kelly fraction?
Growth is a parabola that crosses zero at , exactly twice Kelly.
An asset has an arithmetic mean return of 10% and volatility of 30% a year. Approximately what is its compound growth rate, in percent?
A perp pays 0.01% funding every 8 hours. What does that annualise to, in percent (365 days, no compounding)?
Equity is 50,000. You risk 1% per trade, entering at 100 with a stop at 95. How many units do you buy?
Risk is 500 and the stop is 5 away, so units.
A 1,000,000 position has normally distributed daily returns with 2% volatility and zero mean. What is its one-day 99% VaR, in thousands?
A one-day VaR is 10,000. Using the square-root-of-time rule, what is the 10-day VaR?
. The rule assumes independent, identically distributed returns.
Using the rule of 72, roughly how many years does money take to double at 9% a year?
years. (Exactly, .)
Risking 1% of current equity per trade, what is the total drawdown, in percent, after 10 straight losses?
. Risking a fixed fraction of current equity shrinks bets after losses.
Estimate the price of a one-year at-the-money call on a 100 stock with 20% volatility and zero rates.
. The exact Black–Scholes value is about 7.97.
A call struck at 100 trades at 10. The stock is at 100 and rates are zero. What should the put with the same strike and expiry cost?
Put–call parity: .
A 95-strike call costs 8 with the stock at 100 and zero rates. What is the 95-strike put worth?
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