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Who is Afraid of Black Scholes A Gentle Introduction to Quantitative Finance Day 2 July 12 th 13 th and 15 th 2013 UNIVERSIDAD NACIONAL MAYOR DE SAN MARCOS

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Ito Calculus Suppose the stock price evolved as Problem with this model is that the price can become negative

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Ito Calculus A better model is that the ‘relative price’ NOT the price itself reacts to market fluctuations Q: What does this integral mean?

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Constructing the Ito Integral We will try and construct the Ito Stochastic Integral in analogy with the Riemann-Stieltjes integral Note the function evaluation at the left end point!!! Q: In what sense does it converge?

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Stochastic Differential Equations Consider the following Ito Integral We use the shorthand notation to write this as This is a simple example of a stochastic differential equation

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Convergence of the Integral We have noted the integral converges in the ‘mean square sense’ To see what this means consider This means

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Convergence of the Integral So we have (in the mean square sense) OR

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How to Integrate? A detour into the world of Ito differential calculus Q: What is the differential of a function of a stochastic variable? e.g. If what is Is it true that in the stochastic world as well? We will see the answer is in the negative We will construct the correct Taylor Rule for functions of stochastic variables This will help us integrating such functions as well

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Taylor Series & Ito’s Lemma Consider the Taylor expansion The change in F is given by We note that behaves like a determinist quantity that is it’s expected value as i.e. formally!!

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Taylor Series & Ito’s Lemma We consider when So the change involves a deterministic part and a stochastic part

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Ito’s Lemma We consider a function of a Weiner Process and consider a change in both W and t Ito’s Lemma

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Obtain an SDE for the process We observe that So by Ito’s Lemma

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Integration Using Ito we can derive E.g. Show that

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Example Evaluate

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Extension of Ito’s Lemma Consider a function of a process that itself depends on a Weiner process What is the jump in V if ?

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Extension of Ito So we have the result

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Example If S evolves according to GBM find the SDE for V Given

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Stochastic Differential Equation We will now ‘solve’ some SDE Most SDE do NOT have a closed form solution We will consider some popular ones that do

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Arithmetic Brownian Motion Consider dX=aXdt+bdW To ‘solve’ this we consider the process From extended Ito’s Lemma

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Ito Isometry A shorthand rule when taking averages Lets find the conditional mean and variance of ABM

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Mean and Variance of ABM We have using Ito Isometry

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Geometric Brownian Motion The process is given by To solve this SDE we consider Using extended form of Ito we have

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Black Scholes World The value of an option depends on the price of the underlying and time It also depends on the strike price and the time to expiry The option price further depends on the parameters of the asset price such as drift and volatility and the risk free rate of interest To summarize

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Assumptions The underlying follows a log normal process (GBM) The risk free rate is known (it could be time dependent) Volatility and drift are known constants There are no dividends Delta hedging is done continuously No transaction costs There are no arbitrage opportunities

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A Simple One Step Discrete Case

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The Payoff

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Short Selling

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Hedging with the Right Amount

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And the value is…….

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Drift and Volatility

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Delta Hedging How did one know the quantity of stock to short sell? Let’s re do the example: – Start with one option – And short on the stock The portfolio at the next time is worth – if the stock rises – if the stock falls

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Delta Hedging We want these to give the same value In general we should go

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The Stock Price Model Is out stock price model correct?

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Derivation of Black Scholes Equation We assumed that the asset price follows Construct a portfolio with a long position in the option and a short position in some quantity of the underlying The value of this portfolio is

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Derivation Q: How does the value of the portfolio change? Two factors: change in underlying and change in option value We hold delta fixed during this step

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Derivation We use Ito’s lemma to find the change in the value of the portfolio The change in the option price is Hence

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Derivation Plugging in Collecting like terms

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Derivation We see two type of movements, deterministic i.e. those terms with dt and random i.e. those terms with dW Q: Is there a way to do away with the risk? A: Yes, choose in the right way Reducing risk is hedging, this is an example of delta-hedging

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Derivation We pick Now the change in portfolio value is riskless and is given by

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Derivation If we have a completely risk free change in we must be able to replicate it by investing the same amount in a risk free asset Equating the two we get

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Black Scholes Equation We know what should be This gives us the Black Scholes Equation

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Black Scholes Equation This is a linear parabolic PDE Note that this does not contain the drift of the underlying This is because we have exploited the perfect correlation between movements in the underlying and those in the option price.

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Black Scholes Equation The different kinds of options valued by BS are specified by the Initial (Final) and Boundary Conditions For example for a European Call we have We will discuss BC’s later

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Variations: Dividend Paying Stock If the underlying pays dividends the BS can be modified easily We assume that the dividend is paid continuously i.e. we receive in time Going back to the change in the value of the portfolio

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Variations: Dividend Paying Stock The last terms represents the amount of dividend Using the same delta hedging and replication argument as before we have

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Variations: Currency Options These can be handled as in the previous case Let be the rate of interest received on the foreign currency, then

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Variations: Options on Commodities Here the cost of carry must be adjusted To simplify matters we calculate the cost of carrying a commodity in terms of the value of the commodity itself Let q be the fraction that goes toward the cost of carry, then

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Solving the Black Scholes Equation We need to solve a BS PDE with Final Conditions We will convert it to a ‘Diffusion Equation IVP’ by suitable change of variables Method of solution depends upon the PDE and BC Considering the BC in this case we will use the Fourier Transform Methods to find a function that satisfies the PDE and the BC Using different IC/FC will give the value for different options

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Transforming the BS Equation Consider the Black Scholes Equation given by As a first step towards solving this we will transform it into a IVP for a Diffusion Equation on the real line

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Transforming the BS Equation We make the change of variables This transforms the equation into Where

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Transforming the BS Equation Choosing Letting Choose to simplify the expression

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Transforming the BS Equation i.e. we take We get the following IVP

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Introducing the Fourier Transform Our introduction will be very formal We will study only those properties that are needed to solve the IBVP We will derive a solution called the fundamental solution (Green’s function) This will allow us to find option pricing using BS

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Fourier Transform We define the Fourier transform of ‘nice’ functions as The inverse transform is defined as

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Fourier Transform: Properties We state some basic properties – Linearity : If and then for – Translation: for – Modulation: If

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Fourier Transform: Properties A very useful property for solving linear constant coefficients differential equations is Pf: Integrate by parts in the definition

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Fourier Transforms Table of some common transforms F(x)F(k)

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Solving an IBVP We will use the Fourier Transform Method to solve the heat equation on an infinite domain PDE IC BC as

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Solving the Heat Equation Take the Fourier Transform of both sides and assuming We have using the properties of FT

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The Heat Kernel

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Solving the Heat Equation For a general initial condition we note that the solution is given by The idea is that you ‘break’ your IC into tiny bits and add them (integrate)

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Back to the Black Scholes Equation We had transformed the B-S FVP to the IVP Where

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Solving the Black Scholes for a European Call Here the final condition is replaced by the IC Recalling the IVP and the fundamental solution, we have

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Solving the Black Scholes for a European Call Going back to the original variables The call option has the payoff

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Solving the Black Scholes for a European Call Substituting into the solution we have

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Solving the Black Scholes for a European Call From which we get

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Solving the Black Scholes for a European Call These are integrals of the form By doing a little algebra (HW 4) we have

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European Call Option

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The Black Scholes PDE Consider the Black Scholes Equation given by European Call European Put Binary Options American Style Options

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Solution of Black Scholes for a European Call Solution for a European Call

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Solution for a European Put We use the put call parity Using the solution for a European Call Noting We have

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Calculating the Greeks To calculate for a call we note The delta for a put is given by

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Binary Options The payoff is of a binary call option given by The price of an European type option is given by So we have

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American Options These can be exercised anytime prior to and at expiry Issue with analyzing these is ascertaining when to exercise This will lead to a free boundary problem

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American Options Note that if the previous was the value of an American Put then there would be an arbitrage opportunity. i.e. If an arbitrage opportunity exists Pf: Buy the assent in the market for S and the option for P, immediately selling the asset for K by exercising the option. We make a risk free profit of Hence for the American Option we have

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Perpetual American Put This can be exercised at any point and there is not expiry The payoff is We have already note that The option must satisfy the BS ODE

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Perpetual American Put Solving the BS equation we have We also know that as Suppose we exercise when so that the payoff is Q: What is this ‘optimal’ exercise price?

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Perpetual American Put This gives us the value of B Q: What maximizes V? Setting

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Perpetual American Put Notice the slope of the payoff function and the option value are the same at This is called the smooth pasting condition

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Perpetual American Options with Dividends Let’s consider the perpetual American Call with dividends This has solutions The perpetual put now has value It is optimal to exercise when

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Perpetual American Call with Dividends The solution for a perpetual call is Optimal to exercise when If there are no dividends then and it is NEVER optimal to exercise

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American Options with Finite Time Horizon Consider the case where the strike time is finite We construct a portfolio with one option and short Δ of the underlying Change in the value of the portfolio in access of the risk free rate is

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American Options with Finite Time Horizon We delta hedge to obtain We have three possibilities

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American Options with Finite Time Horizon We consider them each – There is an arbitrage opportunity as one can buy the option and short sell the asset and loaning out the cash – There is an arbitrage opportunity as one can sell the option and buy the asset borrowing cash In the American Option case the second strategy will not always lead to arbitrage as exercising the option in no longer in the hands of the seller

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American Options with Finite Time Horizon So we have (Payoff for early exercise) is continuous

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American Options with Finite Time Horizon Q: How to go about ‘solving’ this? A: Closed form solutions not available Will sue numerics (finite difference) Recast as a Linear Complementarity problem

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