Physical principles in option pricing: analysis and valuation based on the Black-Scholes model

Gradu amaierako lana / Trabajo fin de grado Fisikako gradua / Grado en F´ısica Physical Principles in Option Pricing Analysis and valuation based on the Black-Scholes model Egilea/Autor: I˜nigo Jos´e P´erez G´amiz Zuzendaria/Director: Hegoi Manzano Moro Leioa, 2022eko ekainaren 24a / Leioa, 24 de junio de 2022 Contents 1 Introduction 2 2 Objectives 3 3 Basic financial concepts 4 3.1 Whatisanoption?............................... 4 3.2 Vanillaoptions ................................. 4 3.2.1 Calloptions............................... 5 3.2.2 Putoptions ............................... 6 3.3 Interestrate................................... 6 3.4 Arbitrage .................................... 7 3.5 Dividends .................................... 7 3.6 Portfolio..................................... 7 3.7 Long position, short position . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Brownian motion 8 4.1 Definition .................................... 8 4.2 Diffusionequation................................ 8 4.3 Wienerprocess ................................. 10 5 Asset pricing 10 5.1 Randomwalkmodel .............................. 10 5.2 Itˆo’sLemma................................... 11 5.3 Geometric Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6 The Black-Scholes model 15 7 The Black-Scholes formula for European options 17 7.1 Derivation.................................... 17 7.2 Numericalexample ............................... 21 8 American options 22 8.1 Generalconcepts ................................ 22 8.2 Americanputoptions.............................. 23 8.3 Solution to the free boundary problem . . . . . . . . . . . . . . . . . . . . 25 8.4 Finite-difference formulation: The Crank-Nicolson method . . . . . . . . . 27 8.5 Matrixformulation ............................... 30 8.6 TheLUmethod................................. 31 8.7 Pythoncode................................... 34 8.8 Numericalexample ............................... 36 9 Conclusions 38 1 1 Introduction Nowadays the world of finance is increasingly based on mathematical and physical models. That is why many financial companies hire mathematicians and physicists to build and analyse those models. The branch of finance that deals with these tasks is known as quantitative finance. It is said that quantitative finance was born in 1900 when Louis de Bachelier published his Ph.D. thesis The theory of speculation. He introduced the concept of Brownian motion to approximate asset prices random path. The asset could be, for instance, a stock. From that moment, many other theories have been developed in an attempt to predict the behaviour of financial markets and instruments. In 1973 Fischer Black and Myron Scholes introduced the Black-Scholes model to price derivatives within their publication The pricing of options and corporate liabilities. They were awarded in 1997 with a Nobel Prize for this work. Although it had certain limitations, The Black-Scholes model became a reference for many other subsequent models. The constant changes in markets require the evolution of models and the appearance of new updated ones. The main focus field has always been to manage the volatility and most of the built models are based on stochastic processes [1]. This project explores an application of physics to study financial systems. In particular, we will focus on the physical principles in option pricing. Options are financial instruments whose price depends on the price of an underlying asset. This means that we will have to model the behaviour of the underlying asset to be able to model the price of the option. Some concepts like Brownian motion and diffusion process can be extrapolated to those tasks as we will see. There is more than one model to price options, but we will only analyse the Black-Scholes model as it can be related to physical concepts. This model makes certain assumptions that are not very realistic in real markets, but it gives a good qualitative overview of how options are priced. In fact, many subsequent more sophisticated models are based on it. First of all, we will introduce in Section 3 some basic financial notions which are necessary to understand the rest of the project. It is very important to assimilate what is an option and how does it work. In Section 4 we will explain the physical concepts. This includes the definition of Brownian motion, its relation with diffusion processes, finding a solution to the diffusion equation and making a mathematical formulation of the Brownian motion (this is known as a Wiener process). These aspects will be extrapolated later to finance throughout the project. Once the financial and physical concepts are clear, we will try to model the behaviour of the underlying asset price in Section 5. We will assume a random walk model for it. This model is based on the definition of Wiener process made in Section 4.3. We will find out that the asset price follows a Geometric Brownian motion and that the probability density function of its logarithm suffers a diffusion process. However, we will previously need to introduce a mathematical concept called Itˆo’s Lemma. Knowing the behaviour of the underlying asset price, we will be able to build the Black-Scholes model that leads to the partial differential equation used to price options. This is done in Section 6. Our next step will be to find a solution to the Black-Scholes equation to price European options in Section 7. This is known as the Black-Scholes Formula. We will transform the Black-Scholes equation into a diffusion equation and use the solution found in Section 2 4.2. The problem of valuing the option will be treated as a diffusion process. We will also analyse a numerical example for the derived solution. In Section 8 we will study the case of pricing American options. We will see that we cannot explicitly solve the Black-Scholes equation and that we have to deal with a free boundary problem. We will transform the free boundary problem into a linear complementarity problem to eliminate the dependence on the free boundary. After that, we will write the problem using the finite-difference formulation and the Crank-Nicolson scheme. An algorithm which includes the LU method will be built to solve the problem. Finally, we will translate this algorithm into our own Python code and run some simulations. To finish the project, we will make some conclusions based on the analysed topics. 2 Objectives The main objective of the project is to establish a connection between physics and pinance, analysing the physical principles in Option Pricing. Moreover, the projects also has the following particular objectives: 1. Learn that the asset price behaviour can be modelled with a Brownian motion and visualize the diffusion process suffered by the probability density function of its logarithm. 2. Derive the Black-Scholes equation used to price options and solve it as a diffusion process to price European options. 3. Understand the free boundary problem for American options and solve it using the Crank-Nicolson formulation and an algorithm based on the LU method. 4. Write our own Python code with the mentioned algorithm and run a simulation with a numerical example to analyse results. 5. Deepen in mathematical resolutions, understanding the mathematical developments beyond the level of detail given by the bibliography. 3 3 Basic financial concepts Before we start developing our project, we need to introduce some basic financial concepts that will help us to understand the rest of the topics. 3.1 What is an option? An option is a contract between two parties, the holder and the writer, on an underlying asset. The holder pays a compensation (the premium) to the writer to have the right, but not the obligation, to buy or sell the underlying asset at an agreed price (the strike or exercise price) by a specific date (expiration date or maturity) [2]. In case the holder wanted to exercise his right to buy or sell the underlying asset, the writer would have the obligation to do the opposite movement, that is, sell or buy it. At the same time, the holder of the option can sell his right of execution to a third party in exchange of a premium before the expiration date. The underlying asset is usually a stock or a bond, but it could also be an index, an interest rate or even commodities. To clarify concepts, we are going to explain them one by one: •The option is the contract that depends on the underlying asset, which can be a stock, a bond,... •The holder of the option is the one who has the right to execute the option to buy or sell the underlying asset. The holder can also sell that right to another person. •The writer of the option is the one who receives a premium for writing the option but later depends on what the holder decides. •The premium is the value that has to be paid to acquire the position of holder, that is, it is the price of the option. Pricing an option means finding the premium. •The strike or exercise price is the agreed amount of money that the holder pays/receives when executing the option to buy/sell the underlying asset. •The expiration date or maturity is the date by which the holder can exercise the option. There are various types of options attending to different parameters. The first main differentiation is made between call and put options. Call options give the holder the right to buy the underlying asset, whereas put options give the chance to sell it. The simplest ones are the European call and put options, where the holder can exercise his right just at expiration date. There are also American call and put options. In this case, the right to buy or sell the underlying asset can be executed at any time before expiration date. European and American options are typically called Vanilla options if they have no other special condition to exercise the option. Apart from Vanilla options, there are other types of options like Exotic options, where the conditions of execution are different [3]. However, we are not going to work with them in this project. 3.2 Vanilla options Vanilla options are the simplest type of options and are usually traded on an exchange (marketplace where financial instruments are traded). They are divided into call and put options. 4 3.2.1 Call options As it has been explained before, call options give the holder the right to buy the underlying asset. Depending on whether it is American or European, the purchase can be done before or just at expiration date. Call options are executed when the price of the underlying asset is higher than the strike price. This means we are buying it cheaper than the actual price. We are now going to see an example to better understand how do call options work. Let us consider a holder A and a writer B. They agree on an European call option over a stock whose current price is 100 $. The expiration date is 6 months and the strike price is also 100 $. The holder pays 5 $ to the writer as the premium for the option. As it is an European option, there are two possible scenarios at expiration date. If the stock price goes up to, for instance, 110 $, the holder would execute his right to buy it for 100 $. He would get a payoff of 10 $ and a profit of 5 $ taking into consideration the 5 $ he had paid as the premium. If, on the contrary, the stock value goes down to 90 $, the holder would not buy the stock for 100 $. In that case, he would lose the 5 $ initially paid to the writer. If the option were American instead of European, it could be exercised at any time before expiration date, but we will see later that it should only be done at maturity for call options. We can deduce and plot in Figure 1.a a general expression for the payoff and profit of an European call option at maturity from the point of view of the holder [3]. Payoff : max{S(T)−K, 0},Profit : max{S(T)−K, 0}−C(ti, T, K) (1) where the 0 represents the case of no execution and •T is the expiration date or maturity •K is the exercise or strike price •S(T) is the price of the underlying asset at maturity •C(ti,T,K) is the price of the call option (the premium) when it is bought at initial time ti Regarding Figure 1.a, we can visualize that the option has to be exercised in some cases despite the profit is negative. In those situations, exercising the option means reducing the loss of money. In our example, we had that K= 100 $, ti=0, T= 0.5 year, C(ti, T, K) = 5 $. If we consider the first case in which the stock price raised to 110 $, then S(T) = 110 $ and so the payoff and profit are Payoff : max{110 −100,0}= 10 $, Profit : max{110 −100,0}−5 = 5 $ (2) and, in the other case, where the stock price fell to 90 $, Payoff : max{90 −100,0}= 0 $, Profit : max{90 −100,0}−5 = −5 $ (3) We can also define the payoff function in general at a time tfor both European and American call options from the point of view of the holder: Payoff: max{S(t)−K, 0}(4) The shape of this payoff function will be the same as the shape of the payoff function in Figure 1.a, but for another S(t). 5 3.2.2 Put options Put options grant the holder the right to sell the underlying asset. Once again, depending on if it is American or European, the sale can be done before or at maturity. In this case, the holder would only sell the underlying asset if its price went below the strike price. We can also gather an expression for the payoff and profit of an European put option at expiration date from the point of view of the holder and plot them in in Figure 1.b [3]. Payoff : max{K−S(T),0},Profit : max{K−S(T),0}−P(ti, T, K) (5) The variables K,T,ti,S(T) are the same as in the case of the call option and P(ti, T, K) is the price of the put option (the premium) when it is bought at initial time ti. For European and American put options, the general payoff function at a time tfrom the point of view of the holder would be: Payoff: max{K−S(t),0}(6) Once more, the shape of this payoff function will be the same as the shape of the payoff function in Figure 1.b, but for another S(t). (a) Call option (b) Put option Figure 1: Payoff function and profit function for European option at time Tfrom the point of view of the holder. 3.3 Interest rate When borrowing money or depositing it in a bank, there is a charge or benefit for the operation at the end. The parameter that measures this change of money is the interest rate. For instance, if an individual introduces an amount of money X(t0) in the bank at time t0, the money at time tconsidering a continuous and constant interest rate rwill be [4]: X(t) = X(t0)er(t−t0)(7) This relation can also be expressed in differential form: dX X=rdt (8) 6 The opposite calculation is known as the present value: X(t0) = X(t)e−r(t−t0)(9) When an investment has zero risk, the interest rate is called risk-free rate. 3.4 Arbitrage The concept of arbitrage refers to the possibility of making instantaneous risk-less profit with an investment. This is obviously not a desirable situation in financial markets. In fact, most of the financial theories are developed assuming the absence of arbitrage. It is sometimes possible to make risk-less profit in an investment, for instance, depositing money in the bank at a risk-free rate (8). However, this is not a situation of arbitrage since the profit is not instantaneous [4]. 3.5 Dividends Dividends are earnings that someone receives for owning an asset. The typical case is the company that distributes some of its earnings as dividends between its shareholders. Dividends are received at a specific time or periodically. In our topic of study, dividends will be given by the underlying asset. 3.6 Portfolio Aportfolio is a set of financial instruments like options, stocks, bonds or commodities, which aims to provide benefits. Portfolios typically tend to diversify investments in order to reduce the risk of loss [5]. 3.7 Long position, short position When talking about options, a long position refers to the situation of being the holder of an option. By contrast, a short position means that the investor sells the right of execution of the option. In the case of stocks, the long position is to buy the stock, but the short position, which is usually known as short selling, consists on selling a stock that the investor does not own. The objective of short selling is to benefit from a fall in the price of the stock. The investor borrows a stock and immediately sells it to another person and, after some time, pays the lender the price of the stock at that time. If the price of the stock has fallen, the investor obtains profit [6]. 7 4 Brownian motion Once we have the main financial concepts, we are going to explain now several physical notions that will be used throughout the project. First of all, we need to known what is a Brownian motion and how does it govern a diffusion process. We will also find a solution to the diffusion equation. Moreover, we are going to introduce the mathematical formulation of the Brownian motion, known as the Wiener process. 4.1 Definition The Brownian motion is the movement of particles in a fluid, which can be a gas or a liquid. Collisions with the molecules of the fluid make this movement random and unpredictable. It was the botanist Robert Brown the first to observe the movement in 1827 when looking at pollen grains suspended in water. That is why it is called Brownian motion. He discovered minuscule particles randomly moving in a water drop. From that moment, many qualitative hypothesis were proposed by scientists, but it was not until 1905 when Albert Einstein developed a quantitative model. He based his theory in three main principles [7]: •The existence of the particles. •The movement of the particles in a fluid is due to the enormous number of collisions with the fluid molecules. •The movement of the molecules is so complex that it can just be probabilistically described as a result of many independent hits. As a consequence of the huge number of collisions per unit time that suffers a particle, Einstein studied the problem as a whole rather than individually for each particle. He worked with a density function of the particles u(x, τ) and discovered that it satisfied the diffusion equation [8]: ∂u ∂τ =D∂2u ∂x2(10) where xis the position variable that follows a Brownian motion and τis the time. This equation suggests us that diffusion processes are governed by the Brownian motion. The density function of the particles suffers a diffusion as a result of the Brownian motion followed by the particles. The diffusion process can be seen as the macroscopic manifestation of the microscopic Brownian motion of particles in a fluid [9]. We are now going to analyse the diffusion equation in detail because it will be essential throughout the project. 4.2 Diffusion equation The diffusion equation is a partial differential equation with several applications in physics, engineering and even finance as we will see in this work. It is widely used in physics to model the flow of heat in a continuous medium [4]. That is why it is sometimes known as the heat equation. The diffusion equation gives the time evolution (τ) of the probability density function uof a variable xthat follows a Brownian motion, as we have recently seen. We are going to derive a solution for the equation. 8 6 The Black-Scholes model Once we have studied the behaviour of the asset price, we are going to apply the BlackScholes analysis that leads to the partial differential equation used to price options. The Black-Scholes model was developed in 1973 by Fischer Black and Myron Scholes and became a reference for many other subsequent models. They received a Nobel Prize in 1997 for their work. Nevertheless, as we have said before, the model makes certain assumptions that are not very realistic in real markets, but qualitatively is a great first approximation. Before introducing it, we are going to explain the assumptions [4]: •The underlying asset price follows the Geometric Brownian motion. This does not mean that the Black-Scholes analysis cannot be applied with any other model that is not the one of the random walk previously explained. •The risk-free interest rate rand the volatility σare known during the life of the option. •The underlying asset pays no dividends. •It is possible to buy or sell a fractional number of the underlying asset and short selling is allowed. •There is no possibility of arbitrage. •There are no transaction costs for buying or selling an option or underlying asset. Let us suppose that the price of an option V(S, t) (does not matter if it is a call or put option) depends only on the underlying asset price Sand time t. Applying Itˆo’s Lemma (30), taking into consideration that X≡S,λ≡µS and β≡σS, we can write [4]: dV =∂V ∂t dt +∂V ∂S dS +1 2σ2S2∂2V ∂S2dt (44) If we introduce relation (18) for dS into (44), we reach to the expression that gives the random walk followed by the option price V: dV =∂V ∂t +µS ∂V ∂S +1 2σ2S2∂2V ∂S2dt +σS ∂V ∂S dW (45) Now we are going to build a portfolio made up of one option (long position) and the short sell of ∆ shares of the underlying asset. The value of the portfolio, Π, in this case is: Π = V−∆S(46) and the differential of this value will be: dΠ = dV −∆dS (47) Substituting expressions (18) and (45) into (47), we get that the value of the portfolio Π also follows a random walk: dΠ = ∂V ∂t +µS ∂V ∂S +1 2σ2S2∂2V ∂S2−µ∆Sdt +σS ∂V ∂S −∆dW (48) 15 Regarding this expression, we can choose ∆ so that we eliminate the random component containing dW: ∆ = ∂V ∂S (49) As we can see, in this case ∆ is the rate of change of the option price Vwith respect to the underlying asset price S. The result is that the change in the value of the portfolio is deterministic: dΠ = ∂V ∂t +1 2σ2S2∂2V ∂S2dt (50) We are now going to consider some arguments related to arbitrage. We suppose that there are no transaction costs for buying or selling an option or underlying asset, as said before. According to relation (8), when investing an amount Π at a risk-free rate r, the return of the investment after a time twill be: dΠ Π=rdt (51) As a result, the change in Π is: dΠ = Πrdt (52) The right-hand side of (50) has to be equal to the right-hand side of (52) so that there is no arbitrage. If it was greater, anyone could borrow an amount Π at a risk-free rate r and invest it in the portfolio. The result would be a riskless profit because the increment in the value of the portfolio (right-hand side of (50)) is greater than the amount that has to be returned for borrowing (right-hand side of (52)). This means that there is arbitrage. If, in contrast, the right-hand side of (50) was smaller than the right-hand side of (52), it would be possible to short sell the portfolio (sell the option Vand buy ∆ shares of S) and invest that Π amount in a bank at a risk-free rate r. Once again, there is riskless profit and, as a result, arbitrage. As we have stated that there is no place for arbitrage, we must impose that: Πrdt =∂V ∂t +1 2σ2S2∂2V ∂S2dt (53) Dividing by dt at both sides and introducing expressions (46) and (49) into (53): V−∂V ∂S Sr=∂V ∂t +1 2σ2S2∂2V ∂S2(54) we reach to: ∂V ∂t +1 2σ2S2∂2V ∂S2+rS ∂V ∂S −rV = 0 (55) This is known as the Black-Scholes partial differential equation (PDE). It is a linear, parabolic and backward in time equation. Any option whose price only depends on time tand the underlying asset price S, must verify this equation, as long as the assumptions made so far are verified. Consequently, solving the Black-Scholes equation gives way to pricing options. 16 It is interesting to remark that µ, the average rate of growth of the underlying asset price S, does not appear in the equation. The only parameter from the stochastic differential equation (18) that affects the price of the option is the volatility σ. This means that, although there might be a discrepancy between people in the estimation of µ, the price of the option would be the same anyway [4]. 7 The Black-Scholes formula for European options 7.1 Derivation We have already found a PDE to price options, so now it is time to look for solutions that satisfy it. The Black-Scholes formula is a well-known expression that gives price to European options. It can be obtained solving the Black-Scholes equation (55) with boundary and final conditions. We are going to derive the formula for an European put option P(S, t). Remember that an European put option granted its holder the right to sell the underlying asset only at expiration date T. We start from the Black-Scholes equation ∂P ∂t +1 2σ2S2∂2P ∂S2+rS ∂P ∂S −rP = 0 (56) The equation is backward in time due to the opposite sign of term ∂P/∂t with respect to the other partial derivatives ∂2P/∂S2and ∂P/∂S. For that reason, we need a final condition to solve the equation, this is, a condition at expiration date T. Regarding expression (5) for the payoff of an European put option, we determine that the option price at Thas to be equal to the payoff to avoid arbitrage [4]: P(S, T) = max{K−S, 0}(57) We also require boundary conditions. According to the final condition (57), when the asset price is 0 at expiration date T, the price of the option is K:P(0, T) = K. If we want to determine P(0, t), we just have to calculate the present value (9) of P(0, T). Assuming a constant interest rate r, we get that the first boundary condition is: P(0, t) = Ke−r(T−t)(58) On the other hand, when the asset price tends to ∞, the put option is very unlikely to be exercised, so it loses its value and the second boundary condition is: P(S, t)−→ 0,as S−→ ∞ (59) Once we have the final and boundary conditions, we are going to solve the equation. We are going to make some transformations to try to convert the Black-Scholes equation for the put option (56) into a diffusion equation(10). The solutions for the forward diffusion equation are already known for us. However, we have seen that the BlackScholes is backward. Since we want to convert it into a forward diffusion equation, we introduce a new time variable τ=T−t. Let us now assume the following transformations [10]: x= ln S K+r−1 2σ2τ(60) 17 P(S, t) = p(x, τ) (61) p(x, τ) = e−rτ g(x, τ) (62) With this modifications, we have that ∂P ∂t =−∂p ∂τ +∂p ∂x ∂x ∂τ =−∂(e−rτ g) ∂τ −∂(e−rτ g) ∂x ∂x ∂τ =−e−rτ −rg +∂g ∂τ +r−1 2σ2∂g ∂x(63) ∂P ∂S =∂p ∂x ∂x ∂S =e−rτ ∂g ∂x 1 S(64) ∂2P ∂S2=∂ ∂S e−rτ ∂g ∂x 1 S=∂ ∂S e−rτ ∂g ∂x1 S+e−rτ ∂g ∂x −1 S2= =∂ ∂x e−rτ ∂g ∂x∂x ∂S 1 S−e−rτ ∂g ∂x 1 S2=e−rτ S2∂2g ∂x2−∂g ∂x(65) and so equation (56) becomes e−rτ −rg +∂g ∂τ +r−1 2σ2∂g ∂x=1 2σ2S2e−rτ S2∂2g ∂x2−∂g ∂x+rS e−rτ S ∂g ∂x −re−rτ g (66) Simplifying we get: ∂g ∂τ =σ2 2 ∂2g ∂x2(67) We have reached a diffusion equation as we wanted. We had previously found that Green’s function (15) is a fundamental solution to this equation, so, in this case, we can write (we take ˆ G0= 1): g(x, τ) = 1 √2πσ2τe−x2 2σ2τ(68) Adding the factor e−rτ given by (62) to this result, we get Green’s function for the Black-Scholes equation [10]: G(x, τ) = 1 √2πσ2τe−x2 2σ2τ−rτ (69) If we reverse the transformations made, it can be verified that (69) satisfies equation (56). However, it does not satisfy all the conditions. For instance, the final condition (57), which turns into an initial condition with the change τ=T−t, is not verified as G(x, 0) −→ δ(x). In Section 4.2 we saw that thanks to the linearity of the diffusion equation, there was another way to obtain a solution using an initial distribution and Green’s function. 18 We refer to expression (17). This means that we can turn our problem of valuing a put option into a diffusion problem where the final condition (57) (initial condition in time framework τ) is the initial distribution. Green’s function acts as a propagator, backwards in time tand forward in time τ, of the initial distribution. This initial distribution has the following expression: erτ max{K−S, 0}|τ=0 = max{K−S, 0}(70) The first factor erτ comes from transformation (62) and the second one, which is max{K−S, 0}, from the final condition. If we introduce a variable x′= ln(S/K), which corresponds to xwhen τ= 0, the initial distribution can be written as max{K−S, 0}=Kmax{1−S K,0}=Kmax{1−ex′,0}at τ= 0 (71) Now that we have the initial distribution, we can calculate its time τevolution integrating the contribution of all points x′with their Green’s function as in (17). Each point x′represents a possible value of Sat τ= 0 (at t=T). Green’s function G(x−x′, τ) acts as the propagator from x′to x[10]. p(x, τ) = Z∞ −∞ Kmax{1−ex′,0}G(x−x′, τ)dx′(72) Regarding the initial distribution (71), we can see that it can be divided into two parts: K(1 −ex′) when x′<0 and 0 when x′>0. This means that we can just integrate (72) from −∞ to 0. p(x, τ) = Z0 −∞ K1−ex′G(x−x′, τ)dx′=Z0 −∞ K1−ex′1 √2πσ2τe−(x−x′)2 2σ2τ−rτ dx′ =Ke−rτ √2πσ2τZ0 −∞ e−(x′−x)2 2σ2τdx′−Z0 −∞ e−(x′−x)2 2σ2τ+x′dx′ =Ke−rτ √2πσ2τ Z0 −∞ e−(x′−x)2 2σ2τdx′−Z0 −∞ e−[(x′−(x+σ2τ)]2+x2−(x+σ2τ)2 2σ2τdx′! =Ke−rτ √2πσ2τZ0 −∞ e−(x′−x)2 2σ2τdx′−ex+σ2τ 2Z0 −∞ e−(x′−x−σ2τ)2 2σ2τdx′=I1+I2 (73) In the second term we have completed the square in the exponent. Before we continue integrating, we need to introduce the standard normal cumulative distribution function: N(y) = 1 √2πZy −∞ e−1 2z2dz (74) This function represents the probability of a random variable Y, which has a normal distribution, to be less or equal to y. Once we have this expression, we are going to use it in (73). I1=Ke−rτ √2πσ2τZ0 −∞ e−(x′−x)2 2σ2τdx′=Ke−rτ √2πZ−x/√σ2τ −∞ e−z2 2dz =Ke−rτ N−x √σ2τ(75) 19 To solve this term, we have made a change of variable z=x′−x √σ2τ. For the second term, the transformation is z=x′−x−σ2τ √σ2τ. I2=−Ke−rτ √2πσ2τex+σ2τ 2Z0 −∞ e−(x′−x−σ2τ)2 2σ2τdx′=−Ke−rτ ex+σ2τ 21 √2πZ(−x−σ2τ)/√σ2τ −∞ e−z2 2dz =−Ke−rτ ex+σ2τ 2N−x+σ2τ √σ2τ(76) Joining both terms, we reach: p(x, τ) = I1+I2=Ke−rτ N−x √σ2τ−Ke−rτ ex+σ2τ 2N−x+σ2τ √σ2τ(77) If we now replace xand τby their original expressions, we get that [10]: P(S, t) = Ke−r(T−t)N −ln S K+r−1 2σ2(T−t) pσ2(T−t)!−SN −ln S K+r+1 2σ2(T−t) pσ2(T−t)! (78) This is the well-known Black-Scholes formula for a put option. It gives price to European put options as a function of the underlying asset price and time. Remember that the formula was derived considering constant interest rate rand volatility σduring the life of the option. For American put options this formula is not valid because they can be exercised at any time before maturity Tand so, as we will see later, the boundary conditions are not the same. The Black-Scholes formula for a call option can be obtained following the same process, but with different boundary and final conditions. The formula in that case is [10]: C(S, t) = SN ln S K+r+1 2σ2(T−t) pσ2(T−t)!−Ke−r(T−t)N ln S K+r−1 2σ2(T−t) pσ2(T−t)! (79) and the final and boundary conditions are [4] C(S, T) = max{S−K, 0}(80) C(0, t)=0 C(S, t)∼Sas S−→ ∞ (81) 20 7.2 Numerical example We are now going to use both Black-Scholes formulas (78)(79) for a numerical example. We consider the following parameters: •Risk-free interest rate r=0.08 (8%) •Volatility σ=0.3 √year−1(30%) •Expiration date T= 0.75 year (9 months) •Strike price K= 15$ We first plot in Figure 3.a the value of the European call option Cas a function of the price of the underlying asset Sfor a given time t= 0. The plot also contains the value of the option at expiration t=T, which according to (80), is equal to the payoff function. Regarding the curve for t= 0, we can see that when Stends to 0, the option price Calso approaches 0, and as Sincreases considerably, the value of the option linearly grows with Swith unitary slope. This means that the boundary conditions (81) are verified. On the other hand, we represent in Figure 3.b the value of the European put option Pas a function of the price of the underlying asset Sfor a given time t= 0. Once again, the plot also contains the value of the option at expiration t=T, which is equal to the payoff function as the final condition (57) enforces. In the case of t= 0, when the underlying asset price Sis 0, the value of the option is slightly smaller that the strike price K= 15$, particularly Ke−r(T−t). This value is the same as the boundary condition (58). When Sincreases significantly, the option price Ptends to 0, the same way as in the other boundary condition (59). As we have seen, in both cases the final and boundary conditions are satisfied, so the behaviour of the options value was the expected. (a) C(S) for t= 0 and t=T(b) P(S) for t= 0 and t=T. Figure 3: C(S) and P(S) for t= 0 and t=T. 21 8 American options We have solved the Black-Scholes equation for European call and put options, but we still do not have the way to price American options. We have seen that we can not explicitly solve the equation for American options due to the possibility of early exercise. It is time now to analyse the problem of pricing American options and find a solution for it. 8.1 General concepts American options, unlike European options, can be exercised at any time before expiration date. We are soon going to see that the possibility of early exercise changes the boundary conditions, so the Black-Scholes formula does not apply to those cases. In fact, the possibility of early exercise gives the holder more flexibility, so we could expect the value of an American option to be higher than the value of an European option. This can be shown using arbitrage arguments. We are going to analyse the case of a put option. There are values of the underlying asset price Sfor which the value of an European put option is less than the payoff function max{K−S, 0}. As a result, if we consider that the price of an American put option is the same as the European put option, we can buy an American option for Pand immediately exercise it for K, obtaining a profit of K−S−Pwithout risk. There is arbitrage. As an example, we consider the case of S= 0. According to the payoff function max{K−S, 0}, the payoff is K. If we remember that S= 0 was a boundary condition (58) with option value Ke−r(T−t), we can see that the payoff function is greater than the value of the option, K > Ke−r(T−t), so we are in the situation described before. This can be visualised in Figure 3.b. To avoid arbitrage, we must impose the following condition for American put options [4]: P(S, t)≥max{K−S, 0}(82) Similarly, for American call options the condition would be: C(S, t)≥max{S−K, 0}(83) At expiration date, the price of American options has to be equal to the payoff function, that is, the price of the American options is equal to the price of the European options given by the Black-Scholes formula. P(S, T) = max{K−S, 0}C(S, T) = max{S−K, 0}(84) American call options are special since they should never be exercised before maturity unless they pay dividends. For now, we have always considered options with no dividends and we are going to continue doing the same. This is just a remark to clarify that American call options are just exercised at expiration date. The reason for this is that early exercise requires the immediate payment of the strike price Kand it is more profitable to keep that money, for instance in the bank, with its risk-free interest rate until maturity. This can shown building two portfolios as it is explained in Ref[13]: •Portfolio A: American call option C+ money in the bank Ke−r(T−t) •Portfolio B: one share S 22 If we exercise the call option in portfolio A at early time tex < T because S > K, the payoff will be S−K+Ke−r(T−tex), which is smaller than Sin portfolio B. If otherwise we exercise the option at T, the payoff of A will be max(S−K, 0) + K= max(S, K), which is always greater than or equal to Sin B. Consequently, non-dividend paying American call options should only be exercised at maturity and its price is the same as European call options. We are now going to focus on the study of American put options since for them early exercise is optimal in some cases. 8.2 American put options As early exercise is possible for American put options, there must also be some values of Sfor which the exercise of the option is optimal before maturity. At each time t, there are two regions for S: one with the values of Sfor which the option should be exercised and the other with the values of Sfor which the option should be hold. The point that marks the boundary between those two regions at time tis referred as the optimal exercise price and it is denoted by Sf(t). This optimal exercise price is unknown to us. That is why the problem of pricing an American put option is called a free boundary problem. We have to solve a problem divided in two regions where we do not know where is the boundary. We will have have to find out this boundary as part of the solution. Free boundary problems are really common in physics. For instance, the Stefan problem that describes the joint evolution of a liquid and a solid phase is a free boundary problem [14]. Another typical example is the obstacle problem, which consists in finding the equilibrium configuration of an elastic membrane whose boundary is held fixed, and which is constrained to lie above a given obstacle [15]. We are now going to formulate the free boundary problem for American put options. Let us consider a put option with value P(S, t). As we have seen before, the value of the option verifies that [4] P(S, t)≥max{K−S, 0}(85) and the final condition is P(S, T) = max{K−S, 0}(86) An American put option should be early exercised at a time t < T when the value of S is lower than or equal to the optimal exercise price Sf(t). In those cases, the price of the option must be P(S, t) = max{K−S, 0}. Conversely, if S > Sf, the option should be hold since its price is P(S, t)>max{K−S, 0}and it is more profitable to sell the option than to execute it. When S > Sf, the option price follows the Black-Scholes equation. The combination of this two facts makes the Black-Scholes equation (56) become an inequality. We can now analyse how the Black-Scholes equation becomes an inequality. We build the same portfolio as in (46), with the same value of delta (49). For the put option, it would be: Π = P−∂P ∂S S(87) 23 As early exercise is possible with American options, the arbitrage argument used for European options, where the return of the portfolio had to be equal to the return of money deposited in the bank, is not valid. This means that expression (53) is not an equality anymore. In this case, all we can say is that the return of the portfolio cannot be greater than the money invested in the bank: Πrdt ≥∂P ∂t +1 2σ2S2∂2P ∂S2dt (88) If we introduce the value of the portfolio (87) into (88), we reach to: ∂P ∂t +1 2σ2S2∂2P ∂S2+rS ∂P ∂S −rP ≤0 (89) We can see that the Black-Scholes equation has become an inequality for American put options. The expression is an equality when the option should be hold and an inequality when the exercise is optimal, so we can write that for the region 0 ≤S < Sf(t): P(S, t) = max{K−S, 0}=K−S(90) ∂P ∂t +1 2σ2S2∂2P ∂S2+rS ∂P ∂S −rP < 0 (91) and for the other region, Sf(t)< S < ∞: P(S, t)> K −S(92) ∂P ∂t +1 2σ2S2∂2P ∂S2+rS ∂P ∂S −rP = 0 (93) We now need to impose two conditions at the free boundary Sf(t). We suppose that Sf(t) is smaller than the strike price Kso as to have a positive payoff. The first condition comes from the continuity of P(S, t) at S=Sf(t): P(Sf(t), t) = max{K−Sf(t),0}=K−Sf(t) (94) The second one, from the continuity of the rate of change or delta (49) of P(S, t) at S=Sf(t): ∂P ∂S S=Sf =−1 (95) To understand the reason for this second condition, we are going to consider the two other possible scenarios: ∂P/∂S < −1 and ∂P/∂S > −1. In the first case, an increase in Sfrom Sf(t) implies a decrease of P(S, t) below max{K−S, 0}. According to (85), P(S, t) cannot be smaller than max{K−S, 0}, so this situation is not possible. In the second case, a decrease in Sfrom Sf(t) induces a misvalued increment of the option value, giving rise to possibilities of arbitrage. This is not a desired situation. As a result, the only possible scenario is that ∂P/∂S =−1 [4]. We have already formulated the free boundary problem to price American put options. We have the problem divided into two regions and we have two conditions at the boundary of the regions. We also have a final condition. 24 8.6 The LU method The method we are going to use to solve the problem is known as the LU method. Ref[4] solves the problem with the Projected SOR method, not the LU. It uses the LU method with the implicit scheme to price European options. We are going to apply the same procedure but for American options with the Crank-Nicolson scheme. This method is used to solve systems of linear equations as (Aum+1 −bm) = 0. It is based on the LU decomposition, which consists on factorizing a matrix, in this case A, into a product of a lower triangular matrix Land an upper triangular matrix U, so that A=LU. A=               1 + α−α 20. . . 0 −α 21 + α−α 2 . . . 0−α 2 ......0 . . .......−α 2 0. . . 0−α 21 + α               =               1 0 0 . . . 0 lN−+1 1.... . . 0.........0 . . .......0 0. . . 0lN+−21               ·               yN−+1 zN−+1 0. . . 0 0yN−+2 .... . . 0.........0 . . .......zN+−2 0. . . 0 0 yN+−1               (147) To determine ln,ynand znwe have to multiply the matrices LU =                 yN−+1 zN−+1 0. . . 0 lN−+1yN−+1 lN−+1zN−+1 +yN−+2 .... . . 0.........0 . . .......zN+−2 0. . . 0lN+−2yN+−2lN+−2zN+−2+yN+−1                 (148) and equal the result to matrix A. Here we find that zn=−α 2ln=−α 2yn for n=N−+ 1, . . . , N+−2 (149) yN−+1 = 1 + α(150) 31 yn= (1 + α)−α2 4yn−1 for n=N−+ 2, . . . , N+−1 (151) As we can see, we are just interested in the values of ynfor n=N−+ 1, . . . , N+−1. We can now divide the problem (Aum+1 −bm) = 0 into two sub-problems: Lwm=bmUum+1 =wm(152) This is the same as doing LUum+1=bm, but with an intermediate vector wm. Substituting expressions (149), (150) and (151) into the matrices Land U, we have that the sub-problems are               1 0 0 . . . 0 −α 2yN−+1 1.... . . 0.........0 . . .......0 0. . . 0−α 2yN+−21               ·               wm N−+1 . . . wm 0 . . . wm N+−1               =               bm N−+1 . . . bm 0 . . . bm N+−1               (153) and               yN−+1 −α 20. . . 0 0yN−+2 −α 2 . . . 0 0 ......0 . . .......−α 2 0. . . . . . 0yN+−1               ·               um+1 N−+1 . . . um+1 0 . . . um+1 N+−1               =               wm N−+1 . . . wm 0 . . . wm N+−1               (154) We begin with the first system (153). We can directly derive the value of wm N−+1 and then increasing nfrom N−+ 1, obtain the terms wm nknowing the previous one wm n−1. wm N−+1 =bm N−+1 (155) wm n=bm n+αwm n−1 2yn−1 for n=N−+ 2, . . . , N+−1 (156) In the second system (154), we directly get um+1 N+−1and then decreasing n from N+−1, we can obtain the terms um+1 nknowing um+1 n+1 . um+1 N+−1=wm N+−1 yN+−1 (157) um+1 n=wm n+α 2um+1 n+1 yn for n=N+−2, . . . , N−+ 1 (158) 32 With all these expressions, we can build the algorithm for the LU method to solve our linear system (Aum+1 =bm) in (145). We suppose that Aand bmare already known. The algorithm has the following steps: 1. Find all the values ynstarting from yN−+1 (150) and using (151). 2. Calculate the vector wmstarting with the component wm N−+1 (155) and then using (156) to obtain the rest. 3. Obtain the vector um+1 starting from um+1 N+−1(157) and calculating the other components with (158). This is the algorithm that solves the linear system (Aum+1 =bm), but this is just one part of the resolution of the linear complementarity problem. The algorithm for the whole problem (145)(146) is the following: 1. In the beginning, we have the matrix Aand the initial (138) and boundary conditions (139). With the initial condition, we can obtain u0(first time step m= 0). 2. We calculate b0and solve the linear system Au1=b0using the LU method to obtain u1. The terms u1 N−and u1 N+do not appear in the solution of the linear system but are known from the boundary conditions. According to the expression um+1 ≥fm+1, every component of the vector u1has to be greater than or equal to the correspondent component of the vector f1. This means that when we are calculating u1 nusing (157) and (158), we have to check whether it is greater than or equal to f1 n. In case it was smaller, we would have to force u1 n=f1 n. 3. Once we have u1, we calculate b1to be able to solve the system Au2=b1and obtain u2. We have to check that u2≥f2the same way as before and do the necessary readjustments. 4. The process is repeated until we calculate uM. We have seen that in some cases we have to force the value of um nto be equal to fm n. This suggests us that we have to make some changes in expressions (157) and (158): um+1 N+−1= max wm N+−1 yN+−1 , fm+1 N+−1(159) um+1 n= max wm n+α 2um+1 n−1 yn , fm+1 nfor n=N+−2, . . . , N−+ 1 (160) We are now going to build our own Python code for the whole algorithm of resolution. The algorithm gives us the values of uwith respect to xand τ, but we are also going to calculate the values of P(S, t) reversing the transformations. Moreover, we are going to try to find the optimal exercise price xf(τ) at every time step and then convert it into Sf(t). To obtain xfat a time step mδτ, we have to find the value of nfor which um n=fm n and which verifies that for n′> n,um n′> fm n′, excluding n=N−and n=N+. 33 8.7 Python code We are now going to write our Python code for the algorithm. import math import matplotlib.pyplot as plt class AmericanPut: def __init__(self, r, sigma, T, K): self.__r =r self.__sigma =sigma self.__T =T self.__K =K self.__q =2*r/(sigma *sigma) self.__S =[] self.__t =[] self.__P =[[]] self.__Sf =[] def get_t(self): return self.__t def get_S(self): return self.__S def get_P(self): return self.__P def get_Sf(self): return self.__Sf def payoff(self, x, tau): #Payoff function f(x,tau) return math.exp(0.25 * (self.__q + 1)*(self.__q + 1)*tau) *\ max(math.exp(0.5 * (self.__q - 1)*x) -math.exp(0.5 * (self.__q + 1)*x), 0.0) def from_u_to_P(self, u, x, tau): #Transformation from u to P P=[] for iin range(len(u)): P.append(self.__K *math.exp(-0.5 * (self.__q - 1)*x[i] - 0.25 * (self.__q + 1)*(self.__q + 1)*tau) *u[i]) return P @staticmethod def find_y(alpha, N): #Method that finds all the y_n y=[1+alpha] for iin range(1, N): y.append(1+alpha -alpha *alpha /(4*y[i - 1])) return y def find_b(self, u, x, tau, alpha): #Method that calculates b^m b=[] for iin range(1,len(u) - 1): C=(1-alpha) *u[i] +0.5*alpha *(u[i + 1]+u[i - 1]) b.append(C) b[0]+= 0.5 * alpha *self.payoff(x[0], tau) b[-1]+= 0.5 * alpha *self.payoff(x[-1], tau) return b 34 def LU(self, b, y, x, tau, alpha, length): #LU solver w=[b[0]] check_sf =True for iin range(1, length): w.append(b[i] +0.5*alpha *w[i - 1]/y[i - 1]) u=[0for kin range(length)] u[length - 1]=max(w[length - 1]/y[length - 1], self.payoff(x[length], tau)) for jin range(length - 2,-1,-1): u[j] =max((w[j] +0.5*alpha *u[j + 1]) /y[j], self.payoff(x[j + 1], tau)) if check_sf and u[j] == self.payoff(x[j + 1], tau): xf =x[j + 1] check_sf =False self.__Sf.append(self.__K *math.exp(xf)) return u def values(self, dtau, dx, Nmin, Nplus): #Method that gives the values of P(S,t) alpha =dtau /(dx *dx) N=Nplus -Nmin + 1 M=int(0.5 * self.__sigma *self.__sigma *self.__T /dtau) self.__P =[[0for iin range(N)] for jin range(M)] self.__t =[self.__T] x=[] u=[] for lin range(N): x.append((Nmin +l) *dx) self.__S.append(self.__K *math.exp(x[l])) u.append(self.payoff(x[l], 0.0)) # Finding u^0 self.__P[0][:] =self.from_u_to_P(u, x, 0.0) y=self.find_y(alpha, len(u) - 2) for iin range(1, M): #Finding u^m from m=1 up to m=M tau =i*dtau self.__t.append(self.__T -tau /(0.5 * self.__sigma *self.__sigma)) b=self.find_b(u, x, tau, alpha) u[1:N - 1]=self.LU(b, y, x, tau, alpha, len(u) - 2) u[0]=self.payoff(x[0], tau) u[N - 1]=self.payoff(x[N - 1], tau) self.__P[i][:] =self.from_u_to_P(u, x, tau) pass if __name__ == '__main__': american_put =AmericanPut(r=, sigma=, T=, K=) american_put.values(dtau=, dx=, Nmin=, Nplus=) t=american_put.get_t() S=american_put.get_S() P=american_put.get_P() Sf =american_put.get_Sf() To create an object that represents an American put option in this code, we need to set an interest rate r, a volatility σ, an expiration date Tand a strike price K. Once the object is created, we have to call the method values if we want to price the option. The inputs for this method are the time interval dtau, the xinterval dx and the minimum an maximum numbers for n, that is, Nmin and Nplus. The method applies the algorithm of resolution and fills the lists tand Sand the matrix Pwith the correspondent values. A row in Pmatrix represents a time step and contains the values of the option for all values of xat that time step. The method also fills the list Sfcontaining the optimal exercise price at each time step. The transformations from u(x, τ) to P(S, t) are internally made. 35 8.8 Numerical example We are now going to run the code with a numerical example. We consider an American put option and the following parameters: •Risk-free interest rate r=0.03 (3%) •Volatility σ=0.35 √year−1(35%) •Expiration date T= 0.5 year (6 months) •Strike price K= 10$ Firstly, we calculate the values of the option with dtau = 0.0005, dx = 0.00125, Nmin =−4000 and Nplus = 600. The program fills the lists t,S,Sfand the matrix P. If we were interested in the value of the option for some specific Sand t, we could take it from P. The same for Sfas a function of t. We are now going to plot some results. On the one hand, we represent in 3D in Figure 4 the value of the option Pas a function of the underlying asset price Sand time t. When the asset price Stends to 0, the option value Pgoes to the strike price K= 10$ the same way as the payoff function (max{K−S, 0}), and as Sincreases significantly, Papproaches 0. This means that the boundary conditions (123) are verified. The plot also contains the optimal exercise boundary, which is the red line, for the given the values of the option. The values of Pabove the line are cases for which the option would be executed and the values under the line, for which it would be hold. Figure 4: 3D plot for P(S, t). The red line represents the optimal exercise boundary. We are also going to plot the 2D function P(S) for two different times t= 0 and t=T, as well as the optimal exercise price function Sf(t). The resulting plots are the ones in Figure 5. In the first one, Figure 5.a, we observe that for t=T, the value of P(S) is equal to the payoff function (Figure 1.b) as it has to be according to the final condition (86). When t= 0, the function P(S) is greater than or equal to the payoff function, so condition (85) is satisfied. From Figures 4 and 5.a and the analysis of the boundary and final conditions we have made, we can infer that the behaviour of P(S, t) is correct. 36 Regarding Figure 5.b, we deduce that when time approaches expiration date, the optimal exercise price increases and tends to the strike price K= 10$. This is logical since at time t=T,P(S, T) = max{K−S, 0}and exercise should be done only if S < K. The region behind the line is the exercise region and the region above the line is the one to hold. As time evolves, we are more predisposed to execute the option and sell the asset despite obtaining a smaller payoff. The reason for this is that, at early times, the value of the option is greater than the payoff function for more values of S(it is visualized in Figure 5.a) and, in those cases, it is better to hold the option than to execute it. (a) P(S) for t= 0 and t=T(b) Optimal exercise price function Sf(t) Figure 5: Function P(S) for t= 0 and t=Tand optimal exercise price function Sf(t). 37 9 Conclusions In this work we have established a connection between physics and finance. Concretely, we have analysed some physical concepts in the valuation of options. First of all, we have learnt some basic financial concepts. It was crucial to understand what is an option for the rest of the project. We have also introduced some physical notions related to Brownian motion. We have seen that diffusion processes are governed by the Brownian motion and we have found different solutions for the diffusion equation based on Green’s function. Furthermore, we have explained the mathematical formulation of the Brownian motion, that is, the Wiener process. With the main financial and physical concepts already assimilated, we have analysed the behavior of the underlying asset price. We have seen that it can modelled with a random walk based on a Brownian motion. We have found that in the random walk model, the asset price follows a Geometric Brownian motion and that the probability density function of its logarithm suffered a diffusion process. To visualize this diffusion process, we have plotted some functions. Our next step has been to pose the Black-Scholes model based on the random walk model for the underlying asset. We have derived the Black-Scholes partial differential equation used to price options. Once we had the equation, we have looked for an explicit solution for European put options. This was the Black-Scholes formula. We have solved the problem as a diffusion problem with the payoff function playing the role of the initial distribution. However, first we have had to transform the Black-Scholes equation into a diffusion equation. Once we had the Black-Scholes formulas for both European call and put options, we have used them with a numerical example. We have found that the behaviour of the options was the expected. In the last section, we have studied the problem of pricing American options. We have seen that we cannot find an explicit solution and that for American call options, early exercised is never recommended. We have analysed the free boundary problem for American put options and transformed it into a linear complementarity problem to obtain a solution. Once more. we have worked with the Black-Scholes equation transformed into a diffusion equation. Writing the problem as a linear complementarity one, eliminated the dependence on the free boundary and made resolution easier. We have used the finite difference formulation and the Crank-Nicolson scheme with matrices to solve the problem. We have also built an algorithm that uses the LU method to solve linear systems of equations and translated it into our own Python code. The last step has been to run this code with a numerical example and analyse the results. We have seen that the function of the option value had the right behaviour and we have also obtained the optimal exercise price function. The value of the optimal exercise price increased and converged to the strike price as time approached expiration date. Overall, we have found that there is a relation between physics and finance. Particularly, we have found that the Brownian motion and diffusion equation play an important role in the valuation of options. 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