# Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem

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1 Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial Services Analytics Chandrakant Maheshwari Manager Genpact Smart Decision Services Financial Services Analytics 1

2 Table of Contents 1. Abstract 3 2. Overview 4 3. Deriving the Black-Schoels PDE 4 4. A brief note about constant volatilities and interest rates 7 5. Finite Difference Method 8 a. Boundary conditions 12 b. The method to change the volatilities and interest rates 14 2

3 1. Abstract It is a common belief that for using the famous Black-Scholes framework for Option Pricing, we need to assume that Stock Volatility and Risk Free Interest Rate have to be constant. We prove that this belief is only partially true and there are work arounds, in which the volatility and interest rates can be held as nonconstant parameters. Key Words Black Scholes, PDE, Finite Difference Methods Paper Type Mathematical Finance 3

4 2. Overview It is a common belief that for using the famous Black-Scholes framework for Option Pricing, we need to assume that Stock Volatility and Risk Free Interest Rate have to be constant. We prove that this belief is only partially true and there are work arounds, in which the volatility and interest rates can be held as nonconstant parameters. In this white paper, we derive the famous Black-Scholes PDE (partial differential equation) first, from the first principles of no arbitrage. Thereafter we stop at a point where we take a de-tour and apply a famous numerical technique for solving the PDEs, i.e. FDM (Finite Difference Method) to prove that indeed volatilities and rates do not have to be constant, to solve the Black-Scholes PDE. 3. Deriving the Black-Scholes PDE We start by considering the following Geometric Brownian motion: We also set up a special portfolio, called Π, which consists of one long option and one short stock. We short only quantities of this stock. So our special portfolio Π becomes:, Now as the time passes, we need to consider the change in our portfolio. We consider the change in our portfolio in a very small time step i.e. t t + dt. Therefore the change in our portfolio is: Now we consider the Taylor Series expansion for dv, i.e.:,,,, 1 2, We know that: 2 4

5 However, since the time step dt is a very small time step, so: So, we have: 0 0 Substituting this form of ds 2 and ds in the Taylor Series expansion for the Option Price yields:,,, 1 2,, Substituting the above expression in our expression for change in portfolio i.e. dπ, we have:,, 1 2,, Note that the above expression contains terms in dx, which is an increment in Brownian Motion. Therefore any risk present in our portfolio is because of this random Brownian increment. So we need to make the coefficients of dx equal to zero, to make our Portfolio Risk less. So, we must have the following, to achieve a risk free portfolio: This gives:, 0, Therefore the above choice of, makes our portfolio completely risk less. The above term is also called Delta. It determines the quantity of stock to be shorted. 5

6 Now we will substitute the expression for in our expression for change in portfolio, to get:, 1 2, This change in portfolio is completely riskless, as we do the Dynamic Hedging using. Now we know that if we manage to create a completely risk less portfolio, then its growth should be same as of a growth rate of an amount which is put in a risk free interest bearing account. Hence we should have:, 1 2,,, 1 2,,, On dividing by dt and rearranging, we get the famous Black-Scholes equation:, 1 2,,,0 The Black-Scholes equation is a Linear Parabolic Partial Differential Equation. In the subsequent sections, we examine if we need to keep the volatility constant, to solve the above PDE. We also solve this equation using a famous numerical method, i.e. FDM (Finite Difference Method). 6

7 4. A brief note about constant volatilities and interest rates In the derivation of the above Black-Scholes PDE we have not assumed the volatilities to be constant anywhere. However, simply having the Black-Scholes PDE is no good for practical purposes. We need a solution of the above PDE, given the particular Final Conditions and Boundary Conditions. There are two broad methods to find the solution to the above PDE, analytical methods and numerical methods. Analytical methods are more popular than the numerical methods, as they give very elegant closed form solutions to the PDE. For e.g. the following is the closed form solution to value a Call Option on a Stock:, log log However, the problem in the analytical methods to solve the Black-Scholes PDE is that, they are not very simple if we assume the volatility to change with time. At most we can use some kind of average volatility, but the above formula really needs one hard number for the volatility. Further, the above formulae are not valid for the American Options, where the user has the right to exercise the option anytime till its maturity. Worst still, if one has to value American Options, using a volatility which is changing with time, then the closed form solutions do not remain very elegant and approachable. These arguments remain true for interest rates as well, i.e. if we were to make interest rates, a function of time, then analytical methods do not remain very approachable. To overcome this limitation the trading industry has resorted to numerical methods to solve the above PDE. In the next section, we explain one of the basic numerical techniques to solve the Black-Scholes PDE and show that we can use this powerful method to value American Options as well and change the volatility and interest rates throughout the process as we wish. 7

8 5. Finite Difference Method Recall that we have to solve the following Black-Scholes PDE: The above PDE is a Linear Parabolic PDE and given the particular final and boundary conditions, we need to find the value of V in this PDE. In this section we explain how to solve this PDE using a robust numerical technique called Finite Difference Method. First we introduce some notation. We let the Stock price vary from 0 to N and we divide the Stock price into small increments of δs. Similarly we vary the time from 0 to M and we divide the time into small increments of δt. So, we have, 0, 0 Now we consider the different terms of our PDE. We first consider the first term i.e. the term dv/ dt. It is called the Theta. This is the derivative of the Option price w.r.t time. It follows that:,, lim To see how accurate we are in the above approximation, we consider the Taylor Series of Option value at the asset value S and time t, as follows:,,, 8

9 We translate the above differential equation into the following difference equation: So now that we have the difference equation for the first term i.e. Theta of our PDE, let us consider the next term i.e. Delta dv/ ds. Consider the following Taylor Series first, Similarly,,,,,,, Subtracting from one another, dividing by 2δS and rearranging gives: 1 2, 1 2, 2 The above is the second term of our BS PDE, i.e. Delta. Finally let us consider the third term of our PDE i.e. Gamma d 2 V/ d 2 S. Consider the Forward difference and Backward difference forms of the Delta, i.e. Take the difference between the forward difference and backward difference and divide it by δs, to see how this difference is changing w.r.t change in underlying stock price. So, we have: 2 9

10 So now we finally write all the above results at one place: We write 1 as: ,, Now substitute 2, 3 and 4 into 5 to get 2 Multiply by and take on LHS 2, 2 2, 10

11 Collect the terms in, and , Substitute, so , Where Eqn 6 is of the form This equation has an error of,,, 11

12 5.a Boundary conditions Boundary Condition 1: For a call option when S = 0 then V = 0, so 0. Boundary Condition 2: We know that the price of the call option is, log Now when,, 1, 1 So, we have, So boundary condition is Boundary Condition 3:, As S = 0, the BSE becomes 0 12

13 Numerically this becomes 0 1 Boundary Condition 4: From BC 2, we know that when So, we have, This is linear payoff. So, This means, Or, This is a boundary condition for vanishing gamma.,, 1, 1,

14 5.b The method to change the volatilities and interest rates The expression for, can be further simplified. Recall that, Substitute the values for V 1,V 2, a, b and c,, 1 2,, Also we use So we have

15 Now that we have the expressions for,, examine our earlier difference equation, The above difference is of the form, 1,, Note that the above difference equation implies that, we can calculate the option value one step back in time, from three option values which are one step ahead in time, but at three different points of the underlying Stock price. This is really the crux of the Finite Difference Methods. We can travel back in time, using option values at different points, which are one step ahead in time. Now examine the expressions for, Note that only and, need to be held constant (because we have to define these from the start of our algorithm). Rest of the parameters in the above equations are volatility(σ) and interest rate(r). So we can vary these as we like while we are moving back in time, calculating option values. In this framework, we can even make very realistic assumptions that volatilities are some functions of the underlying stock price and time, and we can implement some external econometric processes like GARCH/ ARIMA etc., which can give us volatilities and interest rates at different points of underlying stock price and time. Our objective in this paper was to show that Black-Scholes framework does not really imply that volatilities and interest rates have to be held constant to compute the prices of the derivatives. We hope that we are able to demystify this common misbelief. 15

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