On BlackScholes Equation, Black Scholes Formula and Binary Option Price


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1 On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II. The BlackScholes Formula (the price of European call option is calculated) is calculated using two methods: (1) riskneutral pricing formula (expected discounted payoff) (2) directly solving the BlackScholes equation with boundary conditions III. The two methods in II are proved to be essentially equivalent. The BlackScholes formula for European call option is tested to be the solution of BlackScholes equation. IV. The value of digital options and share digitals are calculated. The European call and put options are be replicated by digital options and share digitals, thus the prices of call and put options can be derived from the values of digitals. The putcall parity relation is given. 1. The derivation(s) of BlackScholes Equation Black Scholes model has several assumptions: 1. Constant riskfree interest rate: r 2. Constant drift and volatility of stock price: 3. The stock doesn t pay dividend 4. No arbitrage 5. No transaction fee or cost 6. Possible to borrow and lend any amount (even fractional) of cash at the riskless rate 7. Possible to buy and sell any amount (even fractional) of stock A typical way to derive the BlackScholes equation is to claim that under the measure that no arbitrage is allowed (riskneutral measure), the drift of stock price equal to the riskfree interest rate. That is (usually under riskneutral measure, we write Brownian motion as, here we remove Q subscript for convenience ) (1)
2 Then apply Ito s lemma to the discounted price of derivatives, we get (2) Still, under riskneutral measure we can argue that drift. So is martingale. It should have zero (3) This is the BlackScholes equation for the price of any derivatives on the underlying BlackScholes model., under the Now I am going to use another method ( hedge method) to derive the BlackScholes equation. This method avoids to directly use the claim (1). The price of stock follows (4) Apply Ito s lemma to derivative on : Now construct a portfolio called hedge portfolio, which shorts one derivative above and holds shares of stock at time. The value of this portfolio at time is (5) (6) Before applying Ito s lemma on (6), there is one thing that needs to be emphasized: is considered to be constant, that is and, although varies with time. This result comes from the fact that is determined by at time and thus should not be considered to be a
3 timedependent variable when we calculate the change/differential of our portfolio. The differential of our portfolio at time is (7) The Brownian motion term has vanished! This is a portfolio with riskless return rate of ( ). Now we use the assumption of no arbitrage, which requires (8) This yields (9) Here comes the conclusion of BlackScholes equation Note that (10) doesn t use the riskneutral measure. Because (10) (or (3)) is a deterministic PDE, it will hold regardless of which measure is used. However, we can see that the use of riskneutral measure does greatly simplify the derivation. (10) 2. The European vanilla call/put option price The typical way to derive the European (vanilla) call/put option in many textbook is to calculate the expected discounted payoff of option, in other words to integrate the discounted payoff the riskneutral measure. This procedure has nothing to do with the BlackScholes equation we got in (3) or (10). Below I will follow this procedure to get the price of a call option on stock at time. The call option will mature at time with striking price. The price of stock at time will be
4 The expected discounted payoff of the call option (which is also the price of the call option, from the assumption of no arbitrage) is (11) (12) Let. Here with zero mean and variance. So, where is a normal distribution (13.a) (13.b) (13.c) (13.d) (13.e) So the price of call option at time is
5 Equation (14) is also called BlackScholes formula for vanilla call option, because it can also be derived from BlackScholes equation (10) with appreciated boundary conditions: (14) By the change of variable transformation: (15.a, 15.b, 15.c) The BlackScholes equation (10) becomes the diffusion equation with initial condition (16.a, 16.b, 16.c) ( { } ) (17.a, 17.b) The solution for diffusion equation is (18) After some math, we have (19) Changing the variables of { } back to { } yields the BlackScholes formula (14). 3. Does the BlackScholes formula satisfy BlackScholes equation?
6 The first method used to derive BlackScholes formula (14) doesn t use the BlackScholes equation (10). But it so happens to give the solution of BlackScholes equation (10). This is the good property of call/put options: the expected discounted payoff of option is exactly the solution of the BlackScholes equation. This property can be extended to other derivatives with different forms of payoffs. For example, if you have a call option on the square of a lognormal asset (like stock price),. What equation does the price satisfy? The answer is still BlackScholes equation, as long as the derivative price is a function of the current time and stock price. If we derive the price using expected discounted payoff, this price will also satisfy the BlackScholes equation, i.e. the price from expected discounted payoff is also a solution of BlackScholes equation. That is, the solution of BlackScholes equation for the price any derivative is: (20) The mathematical reason behind this is, equation (10): first of all needs to satisfy the BlackScholes We can transform this equation into typical diffusion equation with change of variables The solution of this diffusion equation with initial boundary condition given in (18): is Changing the variables of { } back to { } yields
7 This is exactly the expected discounted payoff as defined in (20)! So the price of any derivative on will satisfy BlackScholes equation, and the solution (BlackScholes formula) can be calculated from expected discounted payoff (with much easy math). Now I am going to show in straightforward method that BlackScholes formula of the price of vanilla call option really satisfies BlackScholes equation. Recall the price of such call option is (21) Define Then (22.a) Also we have (22.b) (22.c, 22.d, 22.e) Calculate
8 (22.f, 22.g, 22.h) After some math, we have 4. Binary option (also called Digital option) A binary option pays a fixed amount ($1 for example) in a certain event and zero otherwise. Consider a digital that pays $1at time if. The payoff of such a option is Using riskneutral pricing formula here and are same as defined in (13.b, 13.e). It is not difficult to check that (24) satisfies BlackScholes equation (10). There are 4 kinds of digitals (if we consider dividend ) Name Definition Value digit call Option that pays $1 when digit put Option that pays $1 when share call Option that pays 1 share when share put Option that pays 1 share when { (23) (24)
9 For nonzero dividend, are slightly different from previous definition (13.c, 13.e) (25.a, 25.b) With these four digitals, we can easily recover the price of European call and put options. For European call option, use the definition of in (23), the payoff of this call can be written as This is equivalent to one share call minus K digital call. The combined price of this call option will be (26) Similarly, a European put option is equivalent to K digital put minus one share put. The price of the European put option is (27) The putcall parity is (28) This parity follows from the fact that both the left and the righthand sides are the prices of portfolios that have value at the maturity of the option. (29) 5. Greeks; hedging; hedging
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