Constructing an Arbitrage in a Two-Asset Black–Scholes Model

Summary

The document asks for a concrete self-financing arbitrage portfolio in a two-asset Black–Scholes setting. It specifies two correlated risky assets driven by Brownian motions, with constant expected returns, volatilities, and correlation, alongside a risk-free asset growing at a constant rate. The second asset’s noise is written as a combination of two independent Brownian motions, corresponding to a Cholesky-style representation of correlated shocks.

The sole answer points out that the requested two-asset construction is unnecessary: an arbitrage example for the classical one-dimensional Black–Scholes model can be used without trading the second asset. It refers readers to an example involving fractional processes, but gives no portfolio holdings, derivation, or payoff details in the document itself. Thus, it contributes a modeling observation and a pointer, rather than a worked arbitrage construction; the reader cannot assess the example’s assumptions or replicability from this text alone.

Key ideas

• The model contains two risky assets with correlated Brownian shocks and a constant-rate risk-free asset.
• The second asset’s stochastic term expresses correlation through independent Brownian drivers.
• The response says a one-asset Black–Scholes arbitrage example would also answer the question.
• No self-financing holdings or arbitrage payoff are derived in the supplied discussion.

Tags

# Arbitrage portfolio example


# Arbitrage portfolio example












Can you give me a concrete example of a self financing portfolio which gives arbitrage opportunity in the two-dimensional Black-Scholes model?

By the two-dimensional Black-Scholes model I mean

$$dS_{1}\left(t\right)=S_{1}\left(t\right)\left[\mu_{1}dt+\sigma_{1}dW\left(t\right)\right]$$ $$dS_{2}\left(t\right)=S_{2}\left(t\right)\left[\mu_{2}dt+\rho\sigma_{2}dW_{1}\left(t\right)+\sqrt{1-\rho^{2}}\sigma_{2}dW_{2}\left(t\right)\right]$$

where $S_{1}\left(t\right)$ and $S_{2}\left(t\right)$ are the underlyings; $W_{1}\left(t\right)$ and $W_{2}\left(t\right)$ are independent Wiener processes; $\mu_{1}$, $\mu_{2}$, $\sigma_{1}$, $\sigma_{2}$ and $\rho$ are constants; and the risk-free interest rate is also constant in the dynamic of the risk-free product: $$dB\left(t\right)=rB\left(t\right)dt.$$

In the previous underlying dynamic: $\rho\sigma_{2}dW_{1}\left(t\right)+\sqrt{1-\rho^{2}}\sigma_{2}dW_{2}\left(t\right)$ represents a kind of Cholesky decomposition.

## Answer by mortenmcfish (score 2)

https://quant.stackexchange.com/a/73628

You can choose an arbitrage for the classical 1-dimensional Black-Scholes model, and not use $S_2$ at all.

Such an arbitrage is e.g. presented in Example 3.5 in "Fractional Processes As Models In Stochastic Finance" by Bender et al (2011).

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.