Why Leveraged Stock Growth Is Not a Risk-Free Arbitrage
Summary
The document considers whether borrowing at the risk-free rate to buy a stock modeled by geometric Brownian motion creates arbitrage when the stock eventually rises above the loan balance. The responses distinguish an expected return advantage from a guaranteed profit: if the stock grows faster on average than the borrowing cost, the leveraged position may have positive expected returns, but it can still lose money and fail to cover the debt.
The accepted explanation frames the proposed exit as a random first-passage time. Although the stock-to-loan threshold may eventually be crossed, the crossing time has no finite upper bound that can guarantee success by a chosen date. In the standard arbitrage argument, an equivalent martingale measure also supports the model's no-arbitrage property. These conclusions rely on the idealized diffusion and financing assumptions; they do not address practical leverage limits, default, or real-world price dynamics.
Key ideas
- A higher expected stock return than the financing rate does not guarantee a profit.
- Arbitrage requires a payoff that is nonnegative in every state and positive in at least one state.
- A threshold-crossing exit time can be unbounded, so eventual crossing does not provide a finite guaranteed horizon.
- The geometric Brownian motion model can admit an equivalent martingale measure under standard assumptions.
- Unbounded volatility accumulation and the absence of default are limitations of the idealized setup.
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Full text
# Is this arbitrage?
# Is this arbitrage?
Assume the stockprice as in the Black-Scholes model (Geometric Brownian Motion):
$$S_t=S_0e^{(\mu-\sigma^2/2)\cdot t+\sigma W_t}$$
Wouldn't there be an immediate arbitrage opportunity, to just buy the stock and wait until it reaches level above the riskfree asset (then sell stock to repay loan and gain remainder as profit)?
As we know, the Black-Scholes model is assumed to be arbitrage-free with unlimited debt and time horizon.
## Answer by ano (score 2, accepted)
https://quant.stackexchange.com/a/14346
It does not seem you feel the question is answered so I will try to elaborate over what I think seems to bother you.
Let $S_t = e^{(\mu -\sigma^2/2) t + \sigma W_t}$ be the stock price process and $B_t=e^{rt}$ be the risk free. The arbitrage you describe is then choosing a nice $\varepsilon >0$ and setting $\tilde{T}=\inf \{t>0 : (\mu -r -\sigma^2/2)t +\sigma W_t> \varepsilon\}$. Then one would have an "arbitrage" at $\tilde{T}$, as you say this will eventually happen which is true. In fact one even know the distribution of when your "arbitrage" will occur see
http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution
which is unfortunately also the problem. Since the inverse Gaussian distribution has mass over entire $\mathbb{R}^+$ you will not be able to choose $T\in \mathbb{R}$ such that $P(T \geq \tilde{T})=1$, ergo you can not in this way find an arbitrage.
It is in fact very easy to find an equivalent martingale measure in this model formally implying that the model is arbitrage free.
It is a different issue whether your "mini arbitrage strategy" is a attractive feature of a model. It is as you say simply a consequence of having a model where volatility accumulates without bound together with no possibility of default.
## Answer by meh (score 3)
https://quant.stackexchange.com/a/14329
No this is not a risk free arbitrage. What you are talking about is modeling a stock price with GBM and it has nothing to do with Black-Scholes. Black-Scholes is an option pricing formula that assumes that stocks follow GBM (which is a bad assumption to begin with but we won't get into that). What you are talking about doing is taking on leverage.
$ E[S_T]= S_0e^{ut} $ where $u$ is the growth rate of the stock. So if you take a loan out at time 0 for $S_0$ then at time T you will owe back $S_0e^{rt}$ where r is the risk free rate.
Now if $u > r$ it is true that you would EXPECT to make money. This is not arbitrage. Arbitrage is when you are guaranteed to make money with no risk. In the situation you are describing there will be times when you lose money and will not be able to pay back your loan.
## Answer by SmallChess (score -2)
https://quant.stackexchange.com/a/14323
This is not arbitrage. Your construction of the portfolio is zero but there is no guarantee that your stock will cover your loan. The GBM is a stochastic process, you have a probability that it'll not cover up your loan.
By the way, the question is invalid because you misunderstood the BS model. Under BS, GBM is simply an assumption. The BS model is used to price and hedge options not the underlying asset.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.