The classic case of a stock and bond

Well known in the literature is the Kelly-Criterion in terms of Merton's portfolio problem with log-utility. To recall the specific framework, let $S_t$ be a stock following a GBM with drift $\mu$ and volatility $\sigma$, driven by a Brownian motion $B_t$, and $M_t$ be a risk-free bond with rate $r$. One sets up a self-financing portfolio: $$dX_t^\alpha = n_t^S dS_t+n_t^M dM_t,$$ with $n_t^S := X_t \alpha_t /S_t$ and $n_t^M = X_t(1-\alpha_t)/M_t$. This implies the dynamics of the portfolio with investment-fraction $\alpha_t$ is given by the SDE $$dX_t^\alpha = (r+(\mu-r)\alpha_t)X_t^\alpha dt + \alpha_t \sigma X_t^\alpha dB_t.$$ Here the superscript $\alpha$ in $X_t^\alpha$ is just to remind ourselves of the dependency on $\alpha$ for the portfolio, i.e. it is not an exponent.

From here we can maximize $g(\alpha)=\mathbb{E}(\log X_T^\alpha | X_t = x)$ to find the optimal leverage-fraction $\alpha_t=\alpha^* = \frac{\mu-r}{\sigma^2}$. This can be done by applying Martingale optimality to $Y_t = \log X_t^\alpha - \delta t$ with a suitably chosen $\delta$, or using dynamic programming and deriving the HJB. In any case, the optimal log-wealth is $$g(\alpha)=\log x + (T-t)(r+0.5 \lambda^2)$$ where $\lambda := (\mu-r)/\sigma$.

An extension to trading an European option and bond

For some time I have wondered how this might extend to a portfolio consisting of a European option on a stock and the same bond. I think I now have a potential generalization and would like to share it for feedback.

Take a GBM, $$dS_t = \mu S_t dt + \sigma S_t dB_t,$$ and an option $C_t= v(t, S_t)$, where $v\in C^{1, 2}([0, T]\times (0, \infty))$. Then the dynamics for the option is given by the SDE $$dC_t =(v(t, S_t)-S_t v_s(t, S_t))r dt + v_s(t, S_t) dS_t.$$ This SDE is derived from rewriting the generator using the Black-Scholes PDE. Set up a portfolio holding a number of contracts of the option and the risk-free asset (money-market account). Then, we can see that the dynamics of our portfolio $X_t^\alpha$ is given by the SDE $$dX_t^\alpha = \left(r+(\mu-r) \frac{v_s(t, S_t) S_t \alpha_t}{v(t, S_t)} \right)X_t dt +\sigma \alpha_t \frac{v_s(t, S_t) S_t X_t}{v(t, S_t)} dB_t,$$ where $\alpha$ is our control process, representing, again, the percentage of our bankroll invested in the option.

We set $$u(t, x, s)=\sup_{\alpha_t} \mathbb{E}(\log X_T^\alpha | X_t = x, S_t = s),$$ and as always, tackle the HJB-PDE, for $\alpha_t=a$, $$u_t + \sup_a \mathscr{L}_{x, s}^a u =0,$$ with terminal condition $u(T, x, s)= \log x$. Here the generator $\mathscr{L}_{x,s}^a$ is given by: $$\left[r+(\mu-r) \frac{v_s(t,s)}{v(t,s)}s a\right]xu_x+\frac 12 a^2 \sigma^2 \frac{v_s(t,s)^2}{v(t,s)^2} s^2 x^2 u_{xx}+\mu s u_s + \frac 12 \sigma^2 s^2 u_{ss}+a \sigma^2 s^2 \frac{v_s(t,s)}{v(t,s)} x u_{xs}$$

We use the ansatz $u(t, x, s)=\log x +w(t)$ (the solution suprisingly does not depend on $s$, but the control does, as we shall see). Then $$w'(t)+r+ \sup_a \left[(\mu-r)\frac{v_s s }{v} a -\frac 12 a^2 \sigma^2 \frac{v_s^2 s^2}{v^2}\right]=0,$$ and $w(T)=0$. Here $v= v(t, s)$ and $v_s = v_s(t, s)$ and $S_t = s$, is given, of course. The first order condition, $$(\mu-r) \frac{v_s s}{v} -a \sigma^2 \frac{v_s^2 s^2}{v^2} = 0,$$ gives the optimal control as $$a =\frac{(\mu-r) v}{\sigma^2 v_s s}.$$ Therefore, the optimal control is a function of $s$. Entering this back in, gives $$w'(t)+r+\frac12 \lambda^2=0,$$ where $\lambda = \frac{\mu-r}{\sigma}$ is the Sharpe-ratio, and this is solved by $$w(t)=(r+\frac 12 \lambda^2)(T-t),$$ so that our original solution is simply $$u(t, x, s)=u(t, x) = \log x +(r+\frac 12 \lambda^2)(T-t),$$ which is just as it is in the GBM case on the stock proper. Curious! This is an interesting extension, for we achieve the same expected log-growth as trading in the stock with $(\mu-r)/\sigma^2$ per-cent of our wealth, as we do by trading in the option with the option-adjusted fraction above: $$\alpha_t = \frac{(\mu-r) v(t, S_t)}{\sigma^2 v_s(t, S_t) S_t}.$$ Consequently, when $\mu > r$ we short puts and go long calls since the delta $\Delta_t = v_s(t, S_t)$ is negative for puts and positive for calls, and when $\mu < r$ we do the opposite. The optimal fraction is dependent on the strike price of the option through the price of the option and its delta, and at a fixed time, it is proportional to the classical fraction, and over time it is proportional to the ratio of the option price to the dollar-delta, i.e. the delta multiplied by the stock price. Thus, it would be informative to the study the behavior of this ratio, in general, over the lifetime of an option.

Naturally, since American calls on non-dividend stocks are equivalent to European calls, we may apply this strategy in this special case with no changes. For American calls on dividend paying stocks and American puts, this analysis does not necessarily hold, although it will be approximately applicable for short maturities where the difference in prices is negligible. Also, we may have $v(t, S)$ represent the aggregate values of (European) spreads, and this analysis goes through just as well.

My questions

  1. Can someone verify my derivation and analysis? I admit this derivation cannot be fully rigorous since I have not appealed to any verification theorem yet, but as far as a formal or heuristic derivation, I cannot find any disastrous error with it on my own.
  2. Is this result already known in the literature? The most easily found paper I have seen with regards to log-utility portfolios with options is the one Wilmott article about free-lunch and implied volatility, which is not quite the same case here.
  3. How can this be extended to American options, if at all?

Please comment if you would like me to add more details in the derivation, if there are any typos or if there are any mistakes I have made. Thank you.

  • 1
    $\begingroup$ In both cases the market is complete so the utility should be the same. $\endgroup$
    – fes
    Jul 4, 2021 at 20:34
  • $\begingroup$ @fesman ah! That makes so much sense now, thank you for that clarification. $\endgroup$ Jul 19, 2021 at 12:56
  • $\begingroup$ Surprised this didn’t get more traction, perhaps I should edit it down to be less verbose? $\endgroup$ Oct 8, 2021 at 11:47


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