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## Chapter 5

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**Chapter 5**European Option Pricing ------ Black-Scholes Formula**Introduction**• In this chapter, • we will describe the price movement of an underlying asset by a continuous model --- geometrical Brownian motion. • we will set up a mathematical model for the option pricing (Black-Scholes PDE) and find the pricing formula (Black-Scholes formula). • We will discuss how to manage risky assets using the Black-Scholes formula and hedging technique.**History**• In 1900, Louis Bachelier published his doctoral thesis ``Thèorie de la Spèculation", - milestone of the modern financial theory. In his thesis, Bachelier made the first attempt to model the stock price movement as a random walk. Option pricing problem was also addressed in his thesis.**History-**• In 1964, Paul Samuelson, a Nobel Economics Prize winner, modified Bachelier's model, using return instead of stock price in the original model. • Let be the stock price, then is its return. The SDE proposed by P. Samuelson is: • This correction eliminates the unrealistic negative value of stock price in the original model.**History--**• P. Samuelson studied the call option pricing problem (Ć. Sprenkle (1965) and J. Baness (1964) also studied it at the same time). The result is given in the following. (V,etc as before)**History---**• In 1973, Fischer Black and Myron Scholes gave the following call • option pricing formula • Comparing to Samuelson’s one, are no longer present. Instead, the risk-free interest r enters the formula.**History----**• The novelty of this formula is that it is independent of the risk preference of individual investors. It puts all investors in a risk-neutral world where the expected return equals the risk-free interest rate. The 1997 Nobel economics prize was awarded to M. Scholes and R. Merton (F. Black had died) for this brilliant formula and a series of contributions to the option pricing theory based on this formula.**Basic Assumptions**• (a) The underlying asset price follows the geometrical Brownian motion: • μ – expected return rate (constant) • σ- volatility (constant) • - standard Brownian motion**Basic Assumptions -**• (b) Risk-free interest rate r is a constant • (c) Underlying asset pays no dividend • (d) No transaction cost and no tax • (e) The market is arbitrage-free**A Problem**• Let V=V(S,t) denote the option price. At maturity (t=T), where K is the strike price. • What is the option's value during its lifetime (0< t<T)?**Δ-Hedging Technique**• Construct a portfolio • (Δ denotes shares of the underlying asset), choose Δ such that Π is risk-free in (t,t+dt).**Δ-Hedging Technique -**• If portfolio Π starts at time t, and Δ remains unchanged in (t,t+dt), then the requirement Π be risk-free means the return of the portfolio at t+dt should be**Δ-Hedging Technique --**• Since where the stochastic process satisfies SDE*, hence by Ito formula So**Δ-Hedging Technique ---**• Since the right hand side of the equation is risk-free, the coefficient of the random term on the left hand side must be zero. Therefore, we choose**Δ-Hedging Technique ----Black-Scholes Equation**• Substituting it, we get the following PDE: • This is the Black-Scholes Equation that describes the option price movement.**Remark**• The line segment {S=0, 0< t< T} is also a boundary of the domain . However, since the equation is degenerated at S=0, according to the PDE theory, there is no need to specify the boundary value at S=0.**Well-posed Problem**• By the PDE theory, above Cauchy problem is well-posed. Thus the original problem is also well-posed.**Remark**• The expected return μ of the asset, a parameter in the underlying asset model , does not appear in the Black-Scholes equation . Instead, the risk-free interest rate r appears it. As we have seen in the discrete model, by the Δ---hedging technique, the Black-Scholes equation puts the investors in a risk-neutral world where pricing is independent of the risk preference of individual investors. Thus the option price arrived at by solving the Black-Scholes equation is a risk-neutral price.**An Interesting Question**• 1) Starting from the discrete option price obtained by the BTM, by interpolation, we can define a function on the domain Σ={0< S<∞,0<t<T}; • 2) If there exists a function V(S,t), such that • 3) if V(S,t) 2nd derivatives are continuous in Σ, What differential equation does V(S,t) satisfy?**Answer of the Question**• V(S,t) satisfies the Black-Scholes equation in Σ. i.e., if the option price from the BTM converges to a sufficiently smooth limit function as Δ 0, then the limit function is a solution to the Black-Scholes equation.**Generalized Black-Scholes Model (I) -----Dividend-Paying**Options • Modify the basic assumptions as follows • (a^)The underlying asset price movement satisfies the stochastic differential equation • (b^) Risk-free interest rate r=r(t) • (c^) The underlying asset pays continuous dividends at rate q(t) • (d) and (e) remain unchanged**Δ-hedging**• Use the Δ-hedging technique to set up a continuous model of the option pricing, and find valuation formulas. • Construct a portfolio • Choose Δ, so that Π is risk-neutral in [t,t+dt]. • the expected return is**Δ-hedging -**• Taking into account the dividends, the portfolio's value at is • Therefore, we have**B-S Equation with Dividend**• Apply Ito formula, and choose we have • Thus, B-S Equation with dividend is**Solve B-S Equation with dividend**• Set • choose α&βto eliminate 0 & 1st terms:**Solve B-S Equation with dividend -**• Let α&β be the solutions to the following initial value problems of ODE: • The solutions of the ODE are**Solve B-S Equation with dividend --**• Thus under the transformation, and take the original problem is reduced to**Apply B-S Formula**• Let σ=1, r=0, T=Tˆ, t=τ,**European Option Pricing (call, with dividend)**• Back to the original variables, we have**Theorem 5.1**• c(S,t) - price of a European call option • p(S,t) - price of a European put option, with the same strike price K and expiration date T. • Then the call---put parity is given by where r=r(t) is the risk-free interest rate, q=q(t) is the dividend rate, and σ= σ(t) is the volatility.**Proof of Theorem 5.1**• Consider the difference between a call and a put: W(S,t)=c(S,t)-p(S,t). • At t=T, • W is the solution of the following problem**Proof of Theorem 5.1-**• Let W be of the form W=a(t)S-b(t)K, then • Choose a(t),b(t) such that**Proof of Theorem 5.1--**• The solution is • Then we get the call---put parity, the theorem is proved.**Predetermined Date Dividend**• If in place of the continuous dividend paying assumption (c^), we assume • (c~) the underlying asset pays dividend Q on a predetermined date t=t_1 (0<t_1<T) • (if the asset is a stock, then Q is the dividend per share). • After the dividend payday t=t_1, there will be a change in stock price: S(t_1-0)=S(t_1+0)+Q.**Predetermined Date Dividend -**• However, the option price must be continuous at t=t_1: V(S(t_1-0),t_1-0)=V(S(t_1+0),t_1+0). • Therefore, S and V must satisfy the boundary condition at t=t_1: V(S,t_1-0)=V(S-Q,t_1+0) • In order to set up the option pricing model (take call option as example), consider two periods [0,t_1], [t_1,T] separately.**Predetermined Date Dividend --**• In 0≤ S<∞, t_1 ≤ t ≤ T, V=V(S,t) satisfies the boundary-terminal value problem • Obtain V=V(S,t) on t_1**Predetermined Date Dividend ---**• in 0 ≤ S<∞,0 ≤ t ≤ t_1,V=V(S,t) satisfies • By solving above problems, we can determine the premium V(S_0,0) to be paid at the initial date t=0 (S_0 is the stock price at that time).**Remark1**• Note that there is a subtle difference between the dividend-paying assumptions (c^) and (c~) when we model the option price of dividend-paying assets.**Remark1-**• In the case of assumption (c^), we used the dividend rate q=q(t), which is related to the return of the stock. Thus in [t_1,t_2], the dividend payment alone will cause the stock price By this model, if the dividend is paid at t=t_1 with the intensity d_Q, • then at t=t_1 the stock price Thus we can derive from the corresponding option pricing formula.**Remark1--**• In the case of assumption (c~), we used the dividend Q, which is related to the stock price itself. So at the payday t=t_1, the stock price • Thus we have at t=t_1 the boundary condition for the option price. • We should be aware of this difference when solving real problems.**Remark 2**• For commodity options, the storage fee, which depends on the amount of the commodity, should also be taken into account. • Therefore, when applying the Δ-hedging technique, for the portfolio: • where Δq^dt denotes the storage fee for Δ amount of commodity and period dt.**Remark 2-**• Similar to the derivation we did before, choose such that Π is risk-free in (t,t+dt). Then we get the terminal-boundary problem for the option price V=V(S,t) • This equation does not have a closed form solution in general. Numerical approach is required.**Remark 2--**• If the storage fee for Δ items of commodity and period dt is in the form of Δq^Sdt, proportional to the current price of the commodity, then • And the option price is given by the Black-Scholes formula.**Generalized B-S Model (II) ------Binary Options**• There are two basic forms of binary option (take stock option as example):**Cash-or-nothing call**• Cash-or-nothing call (CONC): • In Case: t=T: stock price < strike price, the option =0; • In Case: t=T: stock price > strike price, the holder gets $1 in cash.**Asset-or nothing call**• Asset-or nothing call (AONC): • In Case: t=T: stock price < strike price, the option =0; • In Case: t=T: stock price < strike price, the option pays the stock price.