A fully nonlinear problem arising in financial modelling

In 1973, Fisher Black and Myron Scholes suggested a model that became fundamental for the valuation of financial derivatives in a complete frictionless market. Along with the no-arbitrage possibilities, the classical Black-Scholes model assumes that in order to replicate exactly the returns of a certain derivative, the hedging portfolio is continuously adjusted by transactioning the underlying asset of the derivative. This fact can only happen if no transaction costs exist when buying or selling financial assets. Otherwise, a continuous adjustment would imply that those costs, such as taxes or fees, would become infinitely large.

Hence the introduction of transaction costs in the model is a problem that has been motivating the work of several authors and has led to the study of new models that generalise the classical Black-Scholes model.

where $0<c<d$ and p, q are positive constants (p and q can be assumed nonnegative, but the assumption is neither interesting from the mathematical viewpoint nor reasonable for applications to finance). We also consider that $V_c\le V_d$. This assumption turns out to be quite natural in some financial settings, for instance, if we are dealing with call options.

This problem is related to the study of stationary solutions of a nonlinear parabolic equation that models the valuation of a call option in presence of transaction costs. These stationary solutions give the option value V as a function of the stock price, which can be interesting when dealing with a model where the time does not play a relevant role such as, for instance, in perpetual options.

However, we will consider it in the form presented in (1) in order to keep some similarity to the financial setting problem, as we will see later.

The arguments used rely on the upper and lower solutions method (see [1, 2]).

The results we present were motivated by the work of Amster et al. (see [3]). There, the authors studied an analogous problem but without obtaining a localisation result. Then they proceeded their study using a limit of a nonincreasing (nondecreasing) sequence of upper (respectively lower) solutions. Those results were proved under some Lipschitz condition, which has some financial implications. We do not assume such a condition and, moreover, we obtain localisation information on the solution.

In this paper, we state some existence results for the problem (1), following a previous study contained in [4], and improving the localisation result. Relating to the financial model, this implies more accuracy in the information of the option regarding the behaviour of the stock value, in a model where transaction costs are considered. In Section 2, we consider an auxiliary problem. Using the results proved there, we state in Section 3 an existence and localisation theorem for (1). In Section 4, we make some comments on the financial problem that suggested our analytical study. With this section we aim to illustrate the interplay between finance and mathematics which, in fact, is very challenging and often stimulates the use of innovative mathematical and computational techniques.

The main argument to prove Theorem 2 relies on the method of upper and lower solutions.

is an upper solution of the problem (2).

where $k=\sqrt{\frac{q}{c^3}}$ .

So, the three assertions of the lemma are proved. □

which is used in the last step of the proof of assertion 2.

Now we state an existence and localisation result for the auxiliary problem (2).

So, if the function H satisfies the Nagumo condition, the thesis will follow by [1] or [2].

So, by what was said above, the first two assertions of the thesis hold.

The third assertion follows directly from obvious computations. □

Proposition 1 Consider the problem (2) and the solution V given by Theorem 1. Then V is convex and satisfies $V(x)\ge x{V}^{\mathrm{\prime }}(x)$ in $[c,d]$.

But, using Theorem 1, assertion 1, we have that $V^{\prime }(d)\le \frac{V_d}{d}$. In fact, since $\frac{V_d}{d}x\le V(x)$, it follows that $\frac{V_d}{d}\ge \frac{V(x)-V_d}{x-d}$, for $c\le x<d$. Letting $x\to d$, we obtain that $\frac{V_d}{d}\ge V^{\mathrm{\prime }}(d)$.

 □

We return to the original problem (1) and state an existence and uniqueness result. We also provide information on the localisation of the solution.

Remark 2 We remark that Theorem 2.1 of [3] guarantees the existence and uniqueness of a solution but does not give information on localisation, due to the method used. However, using that uniqueness result, from the above theorem, assertions 2 and 3, we can derive the following corollary.

Figure 1 illustrates the localisation for $x\in [c,d]$ of the convex solution for the problem (1) whose existence is given by assertion 2 of Theorem 2.

Assertion 3 of Theorem 2 improves the localisation result contained in assertion 2 as long as k is small enough. In fact, more accuracy is obtained near c.

Figure 2 is an illustration of the new localisation of the convex solution in this case.

where V is the option price, S is the stock price at time t, T is the maturity time, r is the interest short rate and σ is the volatility of the stock returns.

where the constant $k>0$ depends on the parts involved in the transaction.

Leland [5] suggested a new strategy using transaction costs proportional to the monetary value of any buy or sale of the asset. Leland’s replication strategy consisted in using the common Black-Scholes formulae in periodical revisions of the portfolio but with an appropriately enlarged volatility. This is a model widely accepted in the financial industry. However, there are some mathematical problems with this approach, as referred by Kabanov and Safarian [6]: the terminal value of the replicating portfolio does not converge to the terminal payoff of the derivative if the transaction costs do not depend on the number of revisions (tending to infinity), limiting discrepancy that can be calculated explicitly.

where ν is the number of shares traded ($\nu >0$ in a buy or $\nu <0$ in a sale) and $a,b>0$ are constants depending on the individual investor.

We consider the hedging portfolio consisting in an option of value V (short position) and Δ shares of the underlying asset of price S (long position), and consider a strategy where the portfolio is reviewed every δt, with δt a finite, fixed time step.

where μ is the drift coefficient and σ is the volatility of the underlying asset.

The equation obtained (10) is clearly an extension of the classical Black-Scholes equation. The introduction of our particular model for the transaction costs in the option pricing market led us to a partial differential equation that contains the Black-Scholes terms with an additional nonlinear term modelling the presence of transaction costs. We also use an adjusted volatility in the model $\tilde{\sigma }$, not the real volatility σ.

This equation is precisely the equation of the problem (1) studied in the previous section. Contributions to better understanding of the analytical features of the model can be useful for future developments of the financial model itself. In fact, research is full of many examples that illustrate the fruitful interplay between finance and mathematics with reciprocal advantages.