Wednesday 10 Jun 2026. The Nifty 22,500 put of the 30 Jun expiry settled at 94.55 and the 22,250 put at 60.55. Sell one and buy the other and the credit is 34.00 a unit, with a loss capped at 216.00. One lot of 65 units brings in Rs 2,210 and can lose Rs 14,040. The question this lesson asks is not whether that trade is good. It is how much of an account one such loss may be, and what the Kelly formula can and cannot say about it. Kelly is a maximum on size under assumptions that are never quite true, and option strategies break those assumptions in a particular way. The numbers are made up or dated, and nothing here is a view on where Nifty goes.
3By the end you can
- Compute the Kelly fraction and say what it assumes.
- Find p and b for a defined risk put spread and read the break even win rate.
- Explain why users stay below Kelly and what a lot does to the answer.
The idea, plainly
What does the Kelly formula say?
For a bet that wins with chance p and pays b times the amount risked, and otherwise loses the amount risked, the Kelly fraction is the share of the account that grows it fastest in the long run if p and b are known exactly. It is p less the loss chance divided by b. When the edge is zero or negative it gives zero or less, which means do not bet. A fair coin paying even money has p of one half and b of one, and Kelly of nothing. A coin that wins 55% of the time at even money gives 10% of the account on each toss. Position sizing and risk of ruin introduced the formula. This lesson asks what happens when it meets an option position.
Maths: where the formula comes from
g(f) = p × ln(1 + f × b)
+ (1 − p) × ln(1 − f)
g'(f) = 0 gives
f = p − (1 − p) ÷ b
Here f is the share of the account risked on a bet and g is the average growth of the logarithm of the account per bet. The slope of g is zero at the Kelly fraction. Around twice that fraction g falls to zero, so a size well above twice Kelly shrinks the account on average, even with a positive edge.
What are p and b for a put spread?
For a defined risk spread the bet is easy to read. The win is the credit and the loss is the width less the credit, so b is credit over maximum loss. Here b is 0.157. If every losing trade lost the full maximum, the win rate needed just to break even is the maximum loss over the width, 86.4%. Real losses are often partial, so the true figure is lower, but the gap is the point: a credit spread needs a high win rate before it has any edge. The delta of the short put, 0.19, is a rough read of the chance that it finishes in the money, and so a rough read of the market's own odds that the spread keeps its credit, near 80.9%. A trader who believes the market's odds has no edge by that reading, and Kelly returns zero or less. An edge is a belief that the odds differ from the chain's.
The credit spread has a floor on its loss, which makes Kelly's loss term computable
How fast does Kelly move when the win rate moves?
Fast. At even money Kelly is twice the win rate less one, so each point of win rate is two points of size. At a win rate of 50 percent Kelly is 0 percent of the account, at 52 it is 4, at 55 it is 10, at 58 it is 16 and at 60 it is 20. A win rate estimated from a few dozen trades carries an error of several points, so the formula's answer carries an error of many. That is the main reason it is a ceiling and not a setting.
Ten points of win rate move Kelly from nothing to a fifth of the account
What happens when the odds are wrong?
Take a trader who believes the win rate is 55% at even money and risks the full Kelly size, 10%. If the belief is right, the chance of touching a loss of half the account within 250 trades is 41.0 percent. If the true win rate is 50%, so there is no edge at all, the same size touches that level in 84.8 percent of paths. If it is 45%, in 98.8 percent. Betting half the size, five percent, softens it, to 8.6 percent when the odds are right and 50.8 percent with no edge. The error in p is the whole problem, and no size removes it. It only decides how much of the account the error costs.
A wrong estimate at full Kelly puts most paths through a loss of half
Why do people use a fraction of Kelly?
Because the growth curve is flat near its top and falls away fast on the right. Half of Kelly keeps about three quarters of the growth and gives up much of the swing, and the chart shows what that does to ruin. At the right odds, a risk of two and a half percent touches a loss of half the account in 0.3% of paths, five percent in 8.6%, ten percent, the full Kelly size, in 41.0%, and fifteen and twenty percent, beyond it, in 66.3% and 81.2%. The jump after the Kelly size is the practical reason to stay under it. Whether to use half, a quarter or less is a matter of how wrong the inputs may be, not of arithmetic.
Model, not market data
Past the Kelly size, ruin climbs steeply even with a real edge
Maths: growth at a fraction of Kelly
Growth at c × Kelly
≈ Kelly growth
× (2c − c × c)
Here c is the fraction of Kelly used. At c = 0.5 the factor is 0.75, so half Kelly keeps three quarters of the growth. At c = 2 it is zero, the point past which the average growth is negative. The result is an approximation that holds for small bets, but the shape is general: a plateau on the left and a cliff on the right.
Why are lots a problem for sizing?
A lot is the smallest bet, and the Kelly figure rarely lands on a whole number of lots. Take an assumed account of Rs 5,00,000. At 10% of it, the budget is Rs 50,000, and one lot of the spread can lose Rs 14,040. That is 3.56 lots, so three. Half Kelly gives Rs 25,000 and 1.78 lots, so one, and a quarter gives 0.89, so none. An account has to be large before the arithmetic can be followed at all: one lot at the full Kelly share needs an account of Rs 1.40 lakh. Rounding up risks more than the formula allows and rounding down risks less, so either way the size is no longer the one it computed.
Can Kelly size a short option position?
Only with care. A naked short option has no cap on the loss, so there is no b to put in the formula. A defined risk spread has one, and the payoff above is how to find it. And priced fairly, a seller's expected result is zero, so Kelly says zero. In the model of the weekly seller of one far put, over 100 weeks the wins add up to Rs 77,177 and the losses to Rs −77,246, a net of Rs −70 on a lot whose premium is Rs 822 a week. There is no edge in the model by design. Any edge a seller believes in comes from a view that the market overprices the risk, and the loss weeks that prove it wrong are the ones a short sample often misses. The question on option selling as income shows that on a model.
The term this section turns on: Defined risk
Worked example: how many lots does a budget allow?
The account is an assumption, chosen to show the arithmetic and not as a view on how much to risk. The odds are borrowed from the even money coin with a win rate of 55%, not the spread's own: at the chain's odds the spread's Kelly is zero or less, as shown above. The spread, from 10 Jun 2026, supplies only the loss of a lot, because the point here is the lot arithmetic.
| Step | What we work out | Value |
|---|---|---|
| 1 | Account, an assumption | Rs 5,00,000 |
| 2 | Kelly at 55% and even money | 10% |
| 3 | Budget at full Kelly: step 1 times step 2 | Rs 50,000 |
| 4 | Maximum loss of one lot: 216.00 times 65 | Rs 14,040 |
| 5 | Lots at full Kelly: step 3 over step 4 | 3.56 |
| 6 | Lots at half Kelly | 1.78 |
| 7 | Account that makes one lot the full Kelly share: step 4 over step 2 | 1.40 lakh |
Read one more paragraph, 91 words
Step 5 is not a whole number, so the account holds three lots and leaves part of the budget unused. Step 6 holds one. The belief behind step 2 is that the win rate is 55% and not 50%, and the section above showed what the second possibility does to the same size. SEBI's August 2026 study of individual traders in equity derivatives found 87.7% of them made a net loss in FY26. A formula that sizes a trade when the edge is real does not make the edge real.
The common mistake
The belief: Kelly is the size that makes the most money, so it is the size to use.
Kelly is the size that grows an account fastest if the odds are known exactly. They are estimates, often from a short sample, and the result is steep: at the right odds, full Kelly gives ruin in 41.0 percent of paths, and with no edge, 84.8 percent. A strategy that cannot say what its win rate is, within a few points, cannot be sized by this formula. Counting the trades behind p, using a fraction of the formula and sizing for the loss that only the lot can cut are the usual habits of those who use it.
Key takeaways
- Kelly is the size that grows an account fastest if the odds are known.
- Kelly moves two points for each point of win rate at even money.
- A wrong win rate at full Kelly puts most paths through a deep fall.
- Half Kelly keeps three quarters of the growth and much less swing.
- A lot is a lump, so rounding changes the size the formula meant.
- A short option has no b until the loss is capped.
Two terms to hold on to from 17.22
Before you trade. Derivatives trading carries a high risk of loss. SEBI's study "Profitability of Individual Traders in the Equity Derivatives Segment (FY25 to FY26)" (August 2026) found that 87.7% of individual traders in equity F&O made net losses in FY26, with aggregate net losses of Rs 91,685 crore. Read the SEBI study