Mathematical Modeling in Finance with Stochastic Processes
7.1. DERIVATION OF THE BLACK-SCHOLES EQUATION 221 The derivation of the Black-Scholes equation above uses the fairly intu- itive ...
222 CHAPTER 7. THE BLACK-SCHOLES MODEL (a) V(S,t) =AS,Asome constant. (b) V(S,t) =Aexp(rt) Explain in financial terms what each ...
7.2. SOLUTION OF THE BLACK-SCHOLES EQUATION 223 7.2 Solution of the Black-Scholes Equation Rating Mathematically Mature: may con ...
224 CHAPTER 7. THE BLACK-SCHOLES MODEL equation on the real line is well-posed. That is, consider the solution to the partial di ...
7.2. SOLUTION OF THE BLACK-SCHOLES EQUATION 225 withV(0,t) = 0,V(S,t)∼SasS→∞and V(S,T) = max(S−K,0). Note that this looks a litt ...
226 CHAPTER 7. THE BLACK-SCHOLES MODEL Solution of the Black-Scholes Equation First we taket=T−(1/τ2)σ 2 andS=Kex, and we set V( ...
7.2. SOLUTION OF THE BLACK-SCHOLES EQUATION 227 The first derivatives are ∂V ∂t =K ∂v ∂τ · dτ dt =K ∂v ∂τ · −σ^2 2 and ∂V ∂S =K ...
228 CHAPTER 7. THE BLACK-SCHOLES MODEL Cancel theS^2 terms in the second derivative. Cancel theSterms in the first derivative. ...
7.2. SOLUTION OF THE BLACK-SCHOLES EQUATION 229 Gather like terms: uτ=uxx+ [2α+ (k−1)]ux+ [α^2 + (k−1)α−k−β]u. Chooseα = −(k−1)/ ...
230 CHAPTER 7. THE BLACK-SCHOLES MODEL We will evaluateI 1 ( the one with thek+ 1 term) first. This is easy, completing the squa ...
7.2. SOLUTION OF THE BLACK-SCHOLES EQUATION 231 of the exponentials neatly combine and cancel! Next putx = log (S/K), τ= (1/2)σ^ ...
232 CHAPTER 7. THE BLACK-SCHOLES MODEL Figure 7.1: Value of the call option at maturity ...
7.2. SOLUTION OF THE BLACK-SCHOLES EQUATION 233 Figure 7.2: Value of the call option at various times ...
234 CHAPTER 7. THE BLACK-SCHOLES MODEL Figure 7.3: Value surface from the Black-Scholes formula For a fixed time, as the stock ...
7.2. SOLUTION OF THE BLACK-SCHOLES EQUATION 235 ideas are also taken from Chapter 11 ofStochastic Calculus and Financial Applica ...
236 CHAPTER 7. THE BLACK-SCHOLES MODEL 7.3 Put-Call Parity Rating Mathematically Mature: may contain mathematics beyond calculus ...
7.3 Put-Call Parity With the additional terminal conditionV(S,T) given, a solution exists and is unique. We observe that the Bla ...
238 CHAPTER 7. THE BLACK-SCHOLES MODEL Buy one share of stock at priceS= 100. Sell one call option atC=V(100,0) = 11.84. Buy on ...
7.3. PUT-CALL PARITY 239 have the same price as the initial outlay. Therefore we obtain the put-call parity principle: −C+P=Kexp ...
240 CHAPTER 7. THE BLACK-SCHOLES MODEL and then present the solution as an algorithm putting the pieces together to obtain the f ...
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