Black and Scholes solved their now-famous PDE V/t + 1/2sigma^2 S^2^2V/S^2 + rS V
ID: 2721090 • Letter: B
Question
Black and Scholes solved their now-famous PDE V/t + 1/2sigma^2 S^2^2V/S^2 + rS V/S - V = 0 with the right boundary conditions to get a solution for the value of the European Call option. this value is given by the formula C(S, t) = N(d_1)S - N(d_2)Ke^-r(T-t) d_1 = 1/sigmaSquarerootT- t[In(s/K)+(r+sigma^2/2) (T - t)d_2 = d_1 - sigmaSquarerootT-t I used the notation V instead of C, and T instead of T-t. N is the cumulative distribution function of the standard normal distribution N(x)= 1/Squareroot2pi integral_-infinity^x e^t^2/2 dt this This formula turns out to be equivalent to our expected payoff formula: V: = e^-r.T. integral_V^infinity (S - k)middot pdf(s)dSExplanation / Answer
The Black-Scholes PDE from Scratch chris bemis November 27, 2006 0-0 Goal: Derive the Black-Scholes PDE To do this, we will need to:
Come up with some dynamics for the stock returns
Discuss Brownian motion
Look at Ito’s lemma
Discuss replicating and self-financing portfolios
Cleverly put some pieces together 1 In the (additive) binomial tree model, we are led to model the returns from a stock as S t S t = µt + t.
(1) We may like to find the continuous version of (1). To do this, we need to use Brownian motion.
2 What is Brownian motion? A Brownian motion is a stochastic process; i.e. a family of random variables indexed by t: { Wt } t 0 such that The function t Wt is almost surely continuous
The process has stationary, independent increments The increment Wt + s Ws is normally distributed with variance t.
3 How does this apply to the trees we have already seen? For n 1, consider the stochastic process { W n t } t 0 given by W n t = 1 n X 1 jnt j with each 1, 2, . . . a sequence of independent standard normal random variables ( j N(0, 1)). W n t is a random walk that takes a new step every 1/n units of time. For n large, we can see the connection to trees.
4 By the Central Limit Theorem 1 p nt X 1 jnt j converges (in distribution) to a standard normal random variable, Z.
Now W n t = p nt n 1 p nt X 1 jnt j And since lim n nt n = t, in the limit we have Wt = tZ 5 One may rigorously define the infintesimal increment of a Brownian motion.
We won’t. But we will use it. Before doing so, we notice that for s, t { 0, 1/n, 2/n, . . . }, W n t + s W n t = 1 n X 1 j n(t + s) j 1 n X 1 j nt j = 1 n X nt+1 j n(t + s) j
Again we have that 1ns Pnt+1 j n(t + s) j N(0, 1) in distribution. So that W n t + s W n t N(0, s). Or, Wt + s Wt = sZ.
It therefore seems plausible that dWt is like dt
6 From the binomial tree with drift equation (1), we could guess that dS t S t = µdt + dW (2) is a reasonably similar model. In fact, this model is the continuous time analogue of the binomial tree.
7 To derive the Black-Scholes PDE, we will need the dynamics of (2) we just stated. We will also find that we need to take differentials of functions, f ( S t, t), where S t has the dynamics of (2). This is handled using Ito’s lemma. Before looking at this lemma, though, we will see why we need t o take differentials of such functions. We’ll first talk about arbitrage, and then see how arbitrage can determines prices.
8 We have already seen how to determine the price of a contingen t claim using risk-neutral probability (martingales, change of measure, etc.). Just to be clear, examples of contingent claims are call options and put options. A call option gives the holder the right (but not the obligation) to buy a specified item for an agreed upon price at an agreed upon time. A put option gives the holder the right (but not the obligation) to sell a specified item for an agreed upon price at an agreed upon time.
9 We may also use arbitrage arguments. Arbitrage is simply (risk-free) free money. And an arbitrage argument says that there should be no (risk-free) free money. How do we ’use arbitrage’ to price a claim? We try to replicate the claim with stocks and bonds. We call stocks and bonds securities.
10 A contingent claim, f, is replicable if we can construct a portfolio such that • The values of and f are the same under every circumstance. • is self financing. As time goes on, we only shift money around within the portfolio, we don’t put anymore in (or take any out). We will call the replicating portfolio (of f).
11 Why does arbitrage work? Let’s do an example with gold. Suppose the price of gold today is $200 and the risk-free interest rate is 3%. You don’t want gold today (because it’s out of fashion), but you do want gold in 6 months (when, of course, it will be all the rage). You therefore buy a forward contract. This says that you will receive gold in 6 months. You are locking in a price today for something you’ll buy in half a year. How much should you pay for this wonderful opportunity?
12 Suppose the forward contract costs $250. You should then go t o the bank, and borrow $200. Use this money to buy some gold righ t now. Then short (sell) the forward (to a sucker). In six months, what happens? • You sell your gold for $250 • You pay back your loan with your newly received funds • You are left with $250-$200 e .5(.03)=$46.97 Which is a lot of free money.
13 What if the forward contract, F0, is selling for less than $200 e .5(.03)? Well, you have to be able to sell an ounce of gold today. Assuming you have gold lying around, you’ll (because you know the trick) sell your gold today and get $200. Next, you put this $200 in the bank. Finally, you go long (buy) the forward contract. So what happens at the end of 6 months? • Take your money, $200 e .5(.03) out of the bank. • Use it to buy your gold back for $ F0. You have your gold back, and $(200 e .5(.03) - F0). Since this number is positive, you are very happy.
14 Arbitrage therefore sets the price of the forward contract to be $200 e .5(.03). If the price is anything else, there is risk-free free money to be made. This is true of any forward contract on an asset with no storag e costs and which does not pay dividends and if we assume interest rates are constant. Even more generally, we have that any replicable claim will have the same price as its self-financing replicating portfolio.
15 Forward contracts are simple(!) to price. This is due in large part to the linearity of the payoffs at maturity. Options are not so easy. The payoff at maturity has a kink. However, we may construct a self-financing portfolio. Now we will need Ito’s Lemma.
16 If dSt = Stµdt + StdW, and f : (St, t) R, we would like to determine df. In Newtonian calculus, if dx = (dSt, dt), we would simply have df = (f, dx) = fS dS + ft dt = fS Stµ + ft dt + fS StdW But we observed that dW is like dt. So our first order expansion should include one second order term.
17 If we believe that (dW)2 = dt, we need to look at 1 2(dx, 2f dx) If we do, we see that: 1 2 2f S2 (dS)2 = 12 2f S2 S2t 2dt up to first order.
18 We therefore have Ito’s Lemma df = fS S t µ + ft + 1 2 2 f S 2 S 2 t 2 dt + fS S tdW (3) with the same dW from (2).
19 How will we use this? The only randomness in df is the dW term. So if we can construct a portfolio that eliminates the random part, we know exactly how the portfolio should behave. For the first showing of this derivation, we will rely on the discrete versions of (2) and (3). We can prove this with much more rigor, but it is not much more enlightening.
20 Our goal is to price a contingent claim, or derivative. We set to have 1 : derivative : shares where = f S . We get that for a small change in time, t, the corresponding change in is given = f + S
21 From the discrete versions of (2) and (3), we get = f t 1 2 2 f S 2 2 S 2 t t. (4) But this implies the change in the portfolio is riskless (no uncertainty), and so arbitrage arguments, we must have = r t f t 1 2 2 f S 2 2 S 2 t t = r ( f + S )t ft + 1 2 2 f S 2 2 S 2 t + r S t = rft f t + 1 2 2 f S 2 2 S 2 t + r S = rf (5)
22 The pde in (5) is the Black-Scholes-Merton differential equation: f t + 1 2 2 f S 2 2 S 2 t + r f S S rf = 0 with Cauchy data f ( S T , T) known.
23 By using only (2) and arbitrage, we must have that • Any function f that satisfies (5) is the price of some theoretical contingent claim. • Every contingent claim must satisfy (5).
24 When f t + 1 2 2f S2 2 + r fS S = rf is solved with boundary conditions depicting a European call option with strike K, f(S, T) = max(S K, 0), we get the Black-Scholes price of the option.
25 The BS price of a European call, c, (on a stock with no dividend) is c = c (K, r, S t, t, T, ) = S t( d 1 ) Ke r(T t)( d 2) (6) d 1 = ln( S t/K) + ( r + 2 /2)( T t ) T t d 2 = d 1 T t (7) is the cumulative distribution function of standard norma l random variable ( N(0, 1))
26 Here are a few properties of the BS price of c (a benchmark test, really) • We would expect that if S t is very large, c should be priced like a forward contract (why?). We see that if S t is large, c S t Ke r(T t) which is, in fact, the price of a forward contract (why?). • When is extremely small, we would expect that the payoff would be c max( S t e r(T t) K, 0) (8) (why?).
27 We also have • c is an increasing function of . • c S = N(d1). From the last point, we can estimate the to use in the replicating portfolio of c.
28 So we see that the price determined by risk-neutral expectation is the same as the price determined by solving the Black-Scholes pde. Everything seems to be going swimmingly.
29 Next up.... Implied Volatility, and Where Black-Scholes is Going Wrong
30 Prices are not set by the BS options price. Rather, markets se t prices (and if you believe some economists, they set prices near perfectly). We may therefore go to the market to see what a call option on a certain underlying is selling for right now at t = 0. We observe K, r, S t, T. We can’t observe , though. We solve for using (6). This is relatively easy since the BS call option price is monotonic in . The number we get is called the implied volatility.
31 If we check market data for different strike prices, K, with all else being equal, we get different implied volatilities. In fact we get what is called a volatility smile, or a volatility skew depending on the shape. Why is this a problem? We have assumed that is some intrinsic property of the underlying. It shouldn’t vary with K.
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