The concept of "arbitrage-free" pricing refers to the process of determining the value of a financial derivative in a way that there are no risk-free opportunities to make a profit by trading the underlying assets and the derivative. In other words, an arbitrage opportunity occurs when an investor can take advantage of mispricing or inefficiencies in the market by buying low and selling high, thereby making a riskless profit. In an arbitrage-free market, any such opportunities would be rapidly exploited and eliminated as other investors react to the mispricing.
The principle of "no arbitrage" is fundamental to the pricing of financial derivatives using stochastic calculus, as it helps us establish the appropriate pricing model to value these complex financial instruments derived from other underlying assets such as stocks or bonds. By ensuring that the derivative’s price does not create arbitrage possibilities, we ensure that the model is consistent with observed market phenomena.
One of the most prominent methods of derivative pricing using stochastic calculus that utilizes the concept of arbitrage-free pricing is the Black-Scholes-Merton (BSM) model for option pricing. This model is based on the idea that the price of an underlying asset follows a geometric Brownian motion—a continuous-time stochastic process in which an asset’s price evolves based on the combination of a deterministic trend and a random component.
To demonstrate the connection between arbitrage-free pricing and stochastic calculus, let’s consider the BSM model for European call options. Let St denote the price of the underlying asset at time t, with the initial price S0. Furthermore, let C(St, t) denote the price of a corresponding European call option with a strike price K and maturity time T.
In the BSM model, the price of the underlying asset is assumed to follow the stochastic differential equation (SDE):
dSt = μStdt + σStdWt,
where μ represents the expected return, σ represents the volatility of the asset, and dWt is the increment of a Wiener process, which captures the random component of the price evolution.
The BSM model aims to find a fair price for the call option that precludes any arbitrage opportunities. To achieve this, the model relies on constructing an investment portfolio that replicates the option’s payoff by dynamically trading the underlying asset and investing in a risk-free asset with a continuously compounded interest rate r.
Suppose at time t, we hold Δ(St, t) units of the stock and Bt units of the risk-free asset in an investment portfolio. Then, the value of the portfolio, denoted by Πt, is given by:
Πt = Δ(St, t)St + Bt
The key idea of the BSM model is to find a hedging strategy, i.e., a choice for Δ(St, t), such that the portfolio’s value Πt perfectly replicates the option’s payoff. This is expressed as having the portfolio’s value satisfy the following SDE:
dΠt = (Δ(St,t)μSt+rBt)dt + Δ(St, t)σStdWt
However, we know that the option price C(St, t) also depends on the stochastic process St, and we can apply Ito’s Lemma to find the SDE governing its evolution:
$$dC(S_t,t) = \left( \frac{\partial C}{\partial t} + \mu S_t \frac{\partial C}{\partial S_t} + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 C}{\partial S_t^2} \right) dt + \sigma S_t \frac{\partial C}{\partial S_t} dW_t$$
By comparing the SDE of the call option C(St, t) with the SDE of the replicating portfolio Πt, we can apply the no-arbitrage principle by imposing the self-financing condition:
$$\Delta(S_t,t) = \frac{\partial C}{\partial S_t} \quad \text{and} \quad B_t = \frac{C(S_t,t) - \Delta(S_t,t) S_t}{e^{rt}}$$
When we substitute these conditions back into the SDEs for Πt and C(St, t) and equate the two, the random terms (dWt) cancel out, and we obtain a deterministic partial differential equation (PDE) known as the Black-Scholes equation:
$$\frac{\partial C}{\partial t} + r S_t \frac{\partial C}{\partial S_t} + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 C}{\partial S_t^2} - rC = 0$$
This equation, along with appropriate boundary conditions, can be used to find the unique arbitrage-free price of the European call option C(St, t).
In summary, the concept of "arbitrage-free" pricing ensures that financial derivatives are priced such that no risk-free profits can be made in the market. Stochastic calculus is an indispensable tool for modeling the random behavior of financial asset prices and helps us derive the correct pricing model based on the assumption of no arbitrage opportunities. In the case of the Black-Scholes-Merton model, this involves constructing a replicating portfolio and using self-financing conditions to obtain a deterministic PDE to find the arbitrage-free price of a European call option.