A novel portfolio optimization method and its application to the hedging problem

1 Introduction

Assume that an investor has decided to invest the amount Y at a d different assets. The portfolio optimization problem is: how much money should the investor spend on each asset?

One way (see [16]) is to buy w i ⁢ Y S 0 shares of the asset S i in a way that the variance of the return of the portfolio will be as minimum as it can be, subject that the mean value of the return equal to some given m > 0 . This method of portfolio optimization is a static one since the investor buy some shares at time 0 and hold them until time T. There are of course other, more complicated, methods for the portfolio management problem as one can see for example in [19]. In the next section we will describe a dynamic way to manage a portfolio which is based on the well known principle “sell high – buy low”. This trading strategy is self financing and path dependent, that is, the investor construct a portfolio investing the amount Y and then she manage that portfolio in a way that she does not need to put money until the time T. Therefore, the possible loss is limited.

We will also discuss the problem of option pricing. The existing pricing methods attempt to define the concept of the fair price of an option based on an assumption about the underlying asset. It is well known that the price of an option is formed according to the law of supply and demand, but the calculation of the fair price can be useful for investors. For example, if an investor wants to buy or sell an option with a complicated payoff function written on several underlying assets may want to calculate the fair price in order to have the order of magnitude for the value of the contract. This fair price can be used then at the trading stage.

Suppose that an investor make a guess about the future movement of the price of the asset, i.e., for example that follows the geometric Brownian motion with some parameters m , σ . Then we can calculate the 𝔼 ⁢ f ⁢ ( S T ) and supposing that the payoff of an option equal to f ⁢ ( S T ) then this is the mean value of the payoff. Can we define this mean value as the fair price of the option? Of course, we cannot because this guess is not accepted by all the other investors. Suppose now that we change the probability measure to another equivalent one. Can we define the mean value under the new measure as the fair price of the contract? Again, of course we cannot because this equivalent probability measure depends on the first one. As one can see, this is precisely what the Black–Scholes theory do. Even worst, the hedging portfolio that this theory suggests cannot be constructed in real life. Is there any possibility to assume that the price of the underlying asset is the solution of a more complicated stochastic differential equation? Of course, we can assume anything we want but in order to be accepted by all the other investors this equation should not include any free parameter. But this is impossible since the price of the underlying change behavior and this change cannot be modelled by an equation. One has to take into account the behavior of the investors and future and unknown events that will change this behavior. On the other hand the binomial, trinomial, etc., option pricing model do not need any assumption about the probability measure but need an assumption about the number of possible values of the asset at the next period. This of course is not true in real life.

Since existing option pricing theories are based on an assumption about the future movement of the stock, we do not expect all investors to agree on these prices. In this article we will define another notion of the fair price of an option without making an assumption about the behavior of the underlying asset. Moreover, the proposed hedging portfolio is a static one and therefore can be constructed easily by the investors.

2 A dynamic portfolio optimization method

In this section we will describe a dynamic trading strategy based on the “sell high – buy low” (and “borrow high – return low”) principle. Assume that an investor has decided to spend the amount Y buying a shares of a stock borrowing the amount b from his own bank account with risk free rate equal to r. Let also that she has decide to reconstruct her portfolio at the times 0 < t 1 < t 2 < ⋯ < t N = T and that the today price of the asset is S 0 . We will construct a sequence a k which will be the number of shares at each time t k , with a 0 = a . If a > 0 , the writer can buy or sell at time t k a number of shares so that

while if a < 0 ,

We can choose for example δ ⁢ ( x ) as follows:

for some z 1 , z 2 , z 3 > 0 . There are, of course, infinitely many choices of the function δ and the choice has to be made by the investor. For example one can choose as δ the following function:

where p ⁢ ( x ) = a n ⁢ x n + ⋯ + a 0 such that p ⁢ ( M ) = 1 and p ⁢ ( 1 ) = 0 .

If we take into account the transaction costs then we should find a suitable ε > 0 and define the function δ ⁢ ( x ) as follows:

Suppose that at the time t the price S t is below than S 0 , i.e., S t < S 0 . If the investor at this time buy some shares that she already sold (if any), then she can redefine S 0 accordingly to a lower price. We can construct also a different path dependent trading strategy by setting, when a > 0 ,

and when a < 0 ,

Finally, another similar to the above trading strategy is by setting, when a > 0 ,

and when a < 0 ,

At the above hedging strategy one can choose different functions δ in each case.

Therefore, at time T the value of the portfolio will be equal to, concerning for example the trading strategy (2.3) and (2.4) and taking into account the transaction costs,

The investor can buy suitable call and put options in order to maximize her profit. Let K = { K 1 , … , K q } be the available strike prices of the tradable call options, L = { L 1 , … , L d } be the available strike prices of the tradable put options and T = { t 1 < t 2 < ⋯ < t M = T } the available expiration dates. Of course, if there are available call and put options of American type, then the investor can use them accordingly. The investor can solve the following minimization problem in order to construct her dynamic portfolio, given that she has decided to invest the amount Y:

where

for some D ∈ ℝ . Let us note here that we require Π ≥ - D for any possible S T but not almost surely! The almost surely assumption depends on the investor’s guess and thus is not an efficient assumption in practice. The investor should compute the transaction costs for the asset, for the call and put options, the dividends and other costs for the above construction. We denote by r the risk-free rate. In order the borrowing rate to be equal to the deposit rate, one should borrow from her own bank account with interest rate r. Here Δ is a quantity chosen by the investor. For example can be as follows:

The parameters w 1 , w 2 , w 3 has to be defined by the writer using historical data and of course her intuition. More general, we can define

for a suitable chosen function f, A ⊆ ℝ + . The investor may want to maximize a quantity like the above making a further guess, i.e., that the price of the underlying will be in A.

The investor may want to solve a more complicated optimization problem which is as follows:

where Δ can be of the form

for example Δ = ℙ ( S T ∈ B ) - D + 𝔼 ( Π | S T ∈ A ) and for some M > 0 specified by the investor. Solving the above optimization problem, the investor will construct a portfolio which will be such that Π ≥ 0 when S T ∈ B ⊆ A , where A ⊆ ℝ + is her guess and Π ≥ - D for some D > 0 for all S T > 0 , that is,

By solving the above optimization problem, one can construct well known hedging strategies (see for example [8, Section 9.3]) but also more complex ones by appropriately selecting the set A ⊆ ℝ + .

A purely deterministic optimization problem that can be solved by the investor is the following:

for some Y , D specified by the investor.

We denote by Π ⁢ ( x → , t ) : ℝ + N × ℝ + → ℝ the profit function which is the following:

Here A t i ⊆ ℝ + are some sets which are a guess of the investor in the case where the investor want to use this guess. Solving this problem, the investor will construct a static portfolio that will be profitable when S t i ∈ A t i . This kind of construction will be useful to hedge path dependent options. Note that the portfolios that minimizes the above problem can be more than one. In this case we can choose that with the smallest possible loss D. One can add also the dividend function and forward and futures.

Other similar deterministic optimization problems are, for example, finding the smallest D, i.e., constructing a portfolio with the smallest possible loss, or maximizing the quantities

for a specified G ⊆ ℝ + N . One can see [7, Section 7.4] for some similar optimization problems.

One can construct an even more complicated optimization problem by assuming that the “buy low – sell high” strategy can be applied more often than the times 0 < t 1 ⁢ ⋯ < t N = T .

In the above optimization problems one can add more requirements such as

The requirement Π ≥ - D holds for all S T > 0 and therefore is a much stronger result than the well-known value at risk results because the latter depends on the choice of the distribution of the random variables.

All the above optimization problems should be converted to more suitable mathematical forms. Let us see some examples on this direction.

The simplest problem from a mathematical point of view seems to be the following:

The solution will be a portfolio with the least possible loss which is the value of the maxmin problem above. If the maximum possible loss of this portfolio is nonnegative, then we have an arbitrage opportunity.

If we want to add the deterministic property that Π → + ∞ as S T → ∞ , then one can solve the following optimization problem:

The solution will be a portfolio with the deterministic property that Π → + ∞ as S T → ∞ and of course with limited possible loss.

Solving the following optimization problem:

we can construct all the possible portfolios which have the property Π ≥ - D for all S T > 0 for some given D > 0 . If we want to choose one of them, we can make a guess about the future behavior of the asset price, for example that follows the geometric Brownian motion, and then choose the portfolio that maximizes a certain quantity Δ.

For a given D > 0 consider the following optimization problem:

where

This problem will give us a portfolio (if any) with the deterministic property that Π > 0 when S T ∈ B ⊆ ℝ + , where B is a guess of the investor and that Π ≥ - D for all S T > 0 .

In this spirit we can construct various optimization problems searching portfolios with deterministic properties. If there exists several portfolios with the desired properties then we can choose that one which maximizes some Δ as above.

In all the above optimization problems we set

where Y is the value of the portfolio at time zero.

Since Δ contain quantities like the probability of profit, the investor has to make a guess about the future movement of the price of the asset, like the geometric Brownian motion or more complicated. The above optimization problem may be is difficult to solve, however, since this is strongly dependent on the guess of the investor, then one can sacrifice the accuracy of the calculations for the sake of the computation time. We can easily see that the quantities like the probability of profit, the mean value of profit and so on, are functions of W t 1 , … , W t N . In order to compute for example the mean value of the profit one should know the density function of the vector ( W t 1 , … , W t N ) . By [6, Proposition 2.194] we know that the density function of this vector is

where

Of course, we can compute the above quantities by doing some simulations and this will be the case if one assumes a more complicated assumption about the future movement of the price of the underlying.

Concerning the “sell high – buy low” strategy it is easy to see that the possible loss and the transaction costs are limited while the possible profit is unlimited. Comparing with the Black–Scholes hedging strategy for a call or a put option, we see that it is not satisfy the “sell high – buy low” principle, on the contrary. Moreover, the possible loss is unlimited and the transaction costs are unlimited as well!

A similar optimization problem will arise if the investor want to construct a portfolio with d different assets, adding also call and put options for each asset. Of course, the investor should use a small time interval [ 0 , T ] in order to change her guess about the future movement of the asset prices as often as possible taking into account new information from the market.

The above portfolio management method is by far more general than the existing ones. For example, one can choose δ ⁢ ( x ) = 0 for all x ∈ ℝ and for some m > 0 can define the function

Setting Δ = H m ⁢ ( 𝔼 ⁢ ( Π ) ) Var ⁡ ( Π ) , we arrive at the well-known mean-variance optimization problem.

The above optimization problem can be converted to a stochastic optimal control problem in continuous time. Closely related works can be found in the literature, see for example [17, 18]. The advantage of our trading strategy is that it depends on the price of the asset at each time step, i.e., is path dependent, it satisfies the “buy low – sell high” principle and it also assumes that the investor will rebuild the portfolio at a discrete time.

3 Option pricing and the Black–Scholes model

In this section we will discuss the option pricing problem. Consider for example the well-known Black–Scholes option pricing model (see [5]) for a call option which pays the holder the amount P T = max ⁡ { S T - K , 0 } , where S T is the price of the underlying asset at time T and K is the strike price. The basic idea of this option pricing model is first to assume that the price of the underlying asset follows the geometric Brownian motion, i.e.,

and then to compute the initial value of a replicating portfolio, i.e., a portfolio which contain a t number of shares and the amount b t at a bank account and at the time T the value of this portfolio will be equal to the payoff. The Black–Scholes theory then define the fair or the arbitrage free price of this option to be this initial value, assuming of course that all the investors have the same opinion about the behavior of the future price of the asset, i.e., that all the investors believe at the same assumption above! The problem however, for the writer of this option, is not to compute the fair price (if any!) but to design a hedging strategy and how to construct it in the real world. The portfolio proposed by the Black–Scholes theory cannot be constructed in real life and this is the main issue of this theory. Indeed, let us suppose that at time t the price of the underlying asset is S t the number of shares is a ⁢ ( t , S t ) and the amount at the bank account is b ⁢ ( t , S t ) . Then at time t + h the number of shares should be instantaneously equal to a ⁢ ( t + h , S t + h ) and the amount at the bank account should be equal to b ⁢ ( t + h , S t + h ) . This is not possible in the real world and therefore the fair price is not fair for the writer! Even if the price of the asset remains unchanged the number of shares and the amount at the bank account should change continuously in time! Assuming that one can indeed construct such a portfolio, the notion of the fair or arbitrage free price is not well posed because the volatility σ is not a universal constant (see [4, 12] for example) but it depends on the beliefs of each investor. Any attempt to define a fair price or a unique arbitrage free price will not be well defined in practice if it is based on an assumption about the future price movement of the underlying asset! This seems to be an insurmountable obstacle for the construction of an analogous theory. Since the price Y of an option follows the law of supply and demand, one has to decide the hedging strategy that she will follow using this amount selling the option.

So, it is not possible to construct the Black–Scholes hedging strategy in continuous time but only approximately in discrete time. Concerning the expiration time T (for example three months) we should transform it to actively hours in which we can make any trade we want. How often we have to reconstruct this hedging portfolio? Can we estimate the probability of profit, the mean value and the variance of profit, assuming that the underlying asset follows the geometric Brownian motion? The possible loss is unlimited or not? If the possible loss is unlimited, can we design a similar hedging strategy by using call options with different strike prices and/or maybe with different expiration dates? Note that the parameter σ at the assumption (3.1) is not a universal constant and the investors do not agree on a specific price. If someone uses historical data of six months, he/she will get a different value using historical data of one year and so on. Using the notion of the implied volatility has no meaning for the pricing problem but only for the hedging problem as we will explain below!

Here, we should distinguish the asset pricing model that the Black–Scholes theory use from the asset pricing model that the investor use to guess the future movement of the price of the underlying asset. That is, in order to compute the Black–Scholes hedging strategy we should make an assumption about the asset price, like the geometric Brownian motion, Heston model, etc. Note that by using a different assumption for the asset price we arrive at a different Black–Scholes-type hedging strategy, that is, we have a whole family of Black–Scholes strategies choosing a different model. We call this asset pricing model as the embedded asset pricing model. In order to compute the parameters of the embedded asset pricing model we should first decide how much money we want to spend on this hedging strategy. Suppose that the investor decides to use the amount Y, then she should find the implied parameters which are such that the initial value of the Black–Scholes replicating portfolio equals Y. This is a calibration of the imbedded asset pricing model which is a totally different notion than the calibration suggested by [10], for example.

Obviously, in order to find the implied parameters of the imbedded asset pricing model one should assume a fairly simple model, like the geometric Brownian motion. The investor can make a guess about the future movement of the price of the asset which may be different from the imbedded asset pricing model. We call this asset pricing model as the external asset pricing model. In order to compute the parameters of the external asset pricing model, the investor should use historical data and her intuition. That is, the notion of the implied parameters has a meaning only for the imbedded asset pricing model while the historical (or other) data and information has a meaning only for the computation of the parameters of the external asset pricing model.

We arrive at the following corollary.

Using probabilistic techniques is equivalent to forecasting, therefore the notion of the fair price is not well defined under this point of view. In this spirit the authors of [3] have try to overcome this difficulty describing option pricing models without probability. However, again the hedging portfolio should be reconstructed continuously in time.

Concerning the well-known put-call parity formula one can easily observe that it is not satisfied in reality, mainly, because of the following two reasons. The first reason is the risk free rate which is not a universal constant and each investor can invest/borrow at a different rate while the second reason are the transaction costs. Therefore the put-call parity cannot be used in order to compute the price of the put knowing the price of the call. Note also that each of them are known at time zero while each of them are unknown in the future. However, the put-call parity formula can be used by the arbitrageurs (applying their own risk free rate) in order to make profit without risk.

4 Hedging strategies

Suppose that the writer wants to use the amount Y (not necessarily the price of the option) in order to construct the hedging portfolio suggested by the Black–Scholes theory with the geometric Brownian motion as the embedded pricing model. Then she has to compute the implied volatility σ Y which corresponds to the amount Y. If the writer make a guess about the future movement of the option, such as the geometric Brownian motion with parameters m g , σ g , then she can estimate the probability, the mean and the variance of profit constructing this replicating portfolio. Note that the implied volatility σ Y and the volatility σ g are in general different. Note also that the probability, the mean and the variance of profit are functions of m g , σ g , σ Y .

A different hedging strategy is to choose two possible prices S u T and S d T (see [15, Remark 2.5])) and construct a portfolio at time zero in a way that the value of this portfolio equal to the payoffs at the cases where the price of the underlying asset will be equal to the above two numbers. If the writer want to use the amount Y for this strategy, then she has to compute the implied parameters d , u so as the initial value of this portfolio equal to the amount Y. This is the well-known binomial option pricing model (see [11]) but here we do not assume that the possible values of the asset are only two. We call it the realistic binomial hedging strategy. We can prove then (see [15, Lemma 2.1 and Theorem 2.2]) that the writer will have a profit if the value of the asset will be at the interval ( S d T ⁢ T , S u T ) and loss otherwise.

This conclusion holds without assuming anything about the behavior of the price of the underlying asset. If the writer wants to compute the probability of profit, then she must assume something like the geometric Brownian motion for the underlying, i.e., to make an external asset pricing assumption. This hedging strategy can be applied in the real world and in fact the writer bets on the event S T ∈ ( S d T , S u T ) ! However, the use of the multi-period binomial model does not seem to be suitable for the writer (see [15, Theorem 4.9]).

Another hedging strategy is to choose some γ ≥ 0 and buy γ + 1 shares of the underlying asset borrowing a suitable amount of money (see [15, Theorem 4.2 and Remark 4.3]). In this case also the writer can use the amount Y and then she should compute the implied parameters which in this case is the parameter γ and the amount w that she should borrow from her account. Assuming something about the movement of the underlying asset she can compute the probability, the mean and the variance of the profit.

We can design also other hedging strategies that can be applied in the real world. The writer must decide which is the most appropriate for her case.

In all the above proposed optimization problems we can add call and put options with different expiration dates and of course we can add any other type of options that the investor can buy or sell.

5 Arbitrage and fair price of an option