3.4 Continuous-time Kelly and fractional Kelly

Motivation

Stocks and futures are not binary bets. The continuous-time version of Kelly gives the optimal leverage on an asset with drift and volatility, and its closed forms (growth, volatility and drawdown probabilities as functions of the Kelly multiple) are the cleanest guide to sizing a portfolio.

Optimal leverage under GBM#

Setup

A risky asset follows GBM, $dS_t/S_t = \mu\,dt + \sigma\,dB_t$ ; cash earns $r$ . The investor keeps a constant fraction $f$ of wealth in the risky asset (so $f > 1$ is leverage, $f < 0$ a short), rebalancing continuously. Then

\frac{dW_t}{W_t} = r\,dt + f\Big(\frac{dS_t}{S_t} - r\,dt\Big) = \big(r + f(\mu - r)\big)dt + f\sigma\,dB_t. \tag{3.8}

Let $S = (\mu - r)/\sigma$ be the asset's Sharpe ratio.

Proposition 3.6continuous-time Kelly

Wealth is GBM with drift $r + f(\mu - r)$ and volatility $f\sigma$ . Its log grows at

g(f) = \lim_{t\to\infty}\frac1t\ln\frac{W_t}{W_0} = r + f(\mu - r) - \tfrac12 f^2\sigma^2 \quad \text{a.s.}, \tag{3.9}

maximised at

f^* = \frac{\mu - r}{\sigma^2}, \qquad g(f^*) = r + \tfrac12 S^2. \tag{3.10}

Moreover $g(2f^*) = r$ , and $g(f) < r$ for every $f > 2f^*$ ; with $r = 0$ , every $f > 2f^*$ has negative growth and $W_t \to 0$ a.s.

Proof

By M02 Proposition 2.3 applied to (3.8), $\ln(W_t/W_0) = (r + f(\mu - r) - \frac12 f^2\sigma^2)t + f\sigma B_t$ . Divide by $t$ and use $B_t/t \to 0$ a.s. (strong law for Brownian motion). $g$ is a concave quadratic in $f$ with derivative $(\mu - r) - f\sigma^2$ , zero at $f^*$ , and $g(f^*) = r + (\mu-r)^2/(2\sigma^2)$ . Finally $g(f) - r = f\sigma^2(f^* - f/2)$ , which is zero at $f = 2f^*$ and negative beyond.∎

Equation (3.9) is the discrete second-order picture of 3.3 made exact: $m = \mu - r$ per unit time and $v = \sigma^2$ .

Worked example 3.4.1Kelly leverage on the Nifty

Take $\mu = 11\%$ (an arithmetic mean close to the Nifty sample's 10.8%, M02 · 2.2), $\sigma = 20\%$ , and $r = 6.5\%$ (an assumed risk-free rate in the range of Indian T-bill yields; check the current 91-day yield on RBI's site). Then $f^* = 0.045/0.04 = 1.125$ : Kelly says hold 112.5% of capital in Nifty, financed by borrowing. Growth $g^* = 0.065 + 0.5(0.225)^2 = 9.031\%$ against $g(1) = 0.065 + 0.045 - 0.02 = 9.000\%$ for an unlevered holding. The extra 0.03% a year comes with 12.5% more volatility: $g$ is flat near its maximum, so the last part of the way to $f^*$ buys almost nothing. 3.5 shows the inputs are in any case too uncertain to justify it.

Drawdown risk of fractional Kelly#

Proposition 3.7Brownian hitting probability

Let $X_t = mt + sB_t$ with $m > 0$ , $X_0 = 0$ , and $d > 0$ . Then

P\big(\inf_{t \ge 0} X_t \le -d\big) = e^{-2md/s^2}. \tag{3.11}

Proof

For any $\lambda$ , $Z_t = \exp(\lambda X_t - (\lambda m + \frac12\lambda^2s^2)t)$ is a martingale (the exponential martingale of Brownian motion). Choose $\lambda = -2m/s^2$ , for which $\lambda m + \frac12\lambda^2 s^2 = -2m^2/s^2 + 2m^2/s^2 = 0$ , so $Z_t = e^{-2mX_t/s^2}$ . Fix $u > 0$ and let $\tau$ be the first exit time of $X$ from $(-d, u)$ ; $\tau < \infty$ a.s. On $[0, \tau]$ , $0 < Z_t \le e^{2md/s^2}$ , so the stopped martingale is bounded and optional stopping gives $1 = E[Z_\tau] = P_u e^{2md/s^2} + (1 - P_u)e^{-2mu/s^2}$ , where $P_u = P(X \text{ hits } -d \text{ before } u)$ . Solve: $P_u = (1 - e^{-2mu/s^2})/(e^{2md/s^2} - e^{-2mu/s^2})$ . Letting $u \to \infty$ , $P_u \uparrow P(\inf X \le -d) = e^{-2md/s^2}$ .∎

Proposition 3.8fractional Kelly

Let $r = 0$ (or measure wealth in units of the cash account) and bet $f = cf^*$ , $c > 0$ . Then (a) log-wealth is Brownian motion with drift $m_c = S^2(c - \frac12c^2)$ and volatility $s_c = cS$ ; (b) the growth rate is $(2c - c^2)$ times the full-Kelly rate $S^2/2$ , and the volatility of wealth is $c$ times the full-Kelly volatility; (c) for $0 < c < 2$ and $x \in (0, 1)$ ,

P\big(W_t \le xW_0 \text{ for some } t\big) = x^{2/c - 1}. \tag{3.12}

Proof

(a) Substitute $f = c(\mu - r)/\sigma^2$ into (3.9) with $r = 0$ : drift $cS^2 - \frac12c^2S^2$ ; volatility $f\sigma = cS$ . (b) Divide $m_c$ by $S^2/2$ . (c) Apply Proposition 3.7 to $X_t = \ln(W_t/W_0)$ with $d = -\ln x$ : $2m_cd/s_c^2 = 2S^2(c - c^2/2)(-\ln x)/(c^2S^2) = (2/c - 1)(-\ln x)$ , so the probability is $e^{(2/c - 1)\ln x} = x^{2/c-1}$ .∎

$c$	Growth share	Vol share	$P$ (ever lose 20%)	$P$ (ever halve)	$P$ (ever lose 90%)
0.25	0.44	0.25	0.21	0.0078	$10^{-7}$
0.5	0.75	0.5	0.51	0.125	0.001
0.75	0.94	0.75	0.69	0.31	0.022
1	1	1	0.80	0.50	0.10
1.5	0.75	1.5	0.93	0.79	0.46
2	0	2	1	1	1

A full-Kelly bettor halves their wealth at some point with probability one half, and loses 90% with probability one in ten. Half Kelly cuts these to one in eight and one in a thousand, at the cost of a quarter of the growth. These are the numbers Thorp (2006, 7) gives for fractional Kelly, and the tests of Exercise m03_kelly check them by simulating log-wealth with a Brownian-bridge correction for barrier crossings between grid points.

Practitioner notes

The table assumes the model is right. In practice $\mu$ and $\sigma$ are estimated, returns have fat tails and jumps, and rebalancing is discrete. Each of these moves the true $c$ above the intended $c$ , which is the argument for betting well below full Kelly.
Leveraged index products are fixed- $f$ strategies. A 2× daily-rebalanced Nifty fund has $f = 2$ against $f^* \approx 1.1$ in the example above: $c \approx 1.8$ , near the zero-growth boundary before fees.
Kelly is equivalent to maximising log utility. Fractional Kelly with $c$ is approximately optimal for power utility $W^{1-\gamma}/(1-\gamma)$ with relative risk aversion $\gamma = 1/c$ (MacLean, Thorp & Ziemba 2010). Half Kelly is the choice of an investor with $\gamma = 2$ .