OPTION PRICING MODEL • Black-Scholes (BS) OPM. • Cox, Ross, and Rubinstien Binomial Option Pricing Model: BOPM
BOPM • Model is based on constructing a replicating portfolio (RP). • The RP is a portfolio whose cash flows match the cash flows of a call option. • By the law of one price, two assets with the same cash flows will in equilibrium be equally priced; if not, arbitrage opportunities would exist.
SINGLE-PERIOD BOPM • The single-period BOPM assumes there is one period to expiration (T) and two possible states at T -- up state and a down state. • The model is based on the following assumptions:
ASSUMPTION 1 • In one period the underlying stock will either increase to equal a proportion u times its initial value (So) or decrease to equal a proportion d times its initial value. • Example: assume current stock price is $100 and u = 1.1 and d = .95.
Assumption 2 • Assume there is a European call option on the stock that expires at the end of the period. • Example: X = $100.
Assumption 3 • Assume there is a risk-free security in which investors can go short or long. • Rf = period riskfree rate. • rf = (1 + Rf). • Example: Rf = .05 and rf = 1.05.
Replicating portfolio • The replicating portfolio consist of buying Ho shares of the stock at So and borrowing Bo dollars. • Replicating Portfolio is therefore a leveraged stock purchase. • Given the binomial stock price movements and the rate on the risk-free security, the RP’s two possible values at T are known.
Constructing the RP • The RP is formed by solving for the Ho and Bo values (Ho* and Bo*) which make the two possible values of the replicating portfolio equal to the two possible values of the call (Cu and Cd). • Mathematically, this requires solving simultaneously for the Ho and Bo which satisfy the following two equations.
Solve for Ho and Bo where: • Equations:
Solution • Equations:
Example • Equations
Example • A portfolio consisting of Ho* = .6667 shares of stock and debt of Bo* = $60.32, would yield a cash flow next period of $10 if the stock price is $10 and a cash flow of 0 if the stock price is $95. • These cash flows match the possible cash flow of the call option.
RP’s Cashflows • End-of-Period CF:
Law of One Price • By the law of one price, two assets which yield the same CFs, in equilibrium are equally priced. • Thus, the equilibrium price of the call is equal to the value of the RP.
Equilibrium Call Price • BOPM Call Price:
Arbitrage • The equilibrium price of the call is governed by arbitrage. • If the market price of the call is above (below) the equilibrium price, then an arbitrage can be exploited by going short (long) in the call and long (short) in the replicating portfolio.
Arbitrage: Overpriced Call • Example: Market call price = $7.35 • Strategy: • Short the Call • Long RP • Buy Ho*= .6667 shares at $100 per share. • Borrow Bo* = $60.32. • Strategy will yield initial CF of $1 and no liabilities at T if stock is at $110 or $95.
Short call Long Ho shares at So Borrow Bo CFo $7.35 -$66.67=(.6667)$100 $60.32 $1 Initial Cashflow
Arbitrage: underpriced Call • Example: Market call price = $5.35 • Strategy: • Long In call • Short RP • Sell Ho*= .6667 shares short at $100 per share. • Invest Bo* = $60.32 in riskfree security. • Strategy will yield initial CF of $1 and no liabilities at T if stock is at $110 or $95.
Long call Short Ho shares at So Invest Bo in RF CFo -$5.35 $66.67=(.6667)$100 -$60.32 $1 Initial Cashflow
Conclusion • When the market price of the call is equal to $6.35, the arbitrage is zero. • Hence, arbitrage ensures that the price of the call will be equal to the value of the replicating portfolio.
Alternative Equation • By substituting the expressions for Ho* and Bo* into the equation for Co*, the equation for the equilibrium call price can be alternatively expressed as:
BOPM Equation Alternative Equation:
BOPM Equation Example:
Note • In the alternative expression, p is defined as the risk-neutral probability of the stock increasing in one period. • The bracket expression can be thought of as the expected value of the call price at T. • Thus, the call price can be thought of as the present value of the expected value of the call price.
Realism • To make the BOPM more realistic, we need to • extend the model from a single period to multiple periods, and • estimate u and d.
Multiple-Period BOPM • In the multiple-period BOPM, we subdivide the period to expiration into a number of subperiods, n. • As we increase n (the number of subperiods), • we increase the number of possible stock prices at T, which is more realistic, and • we make the length of each period smaller, making the assumption of a binomial process more realistic.
Two-Period Example • Using our previous example, suppose we subdivide the period to expiration into two periods. • Assume: • u = 1.0488, d = .9747, • So = $100, n = 2, • Rf = .025, X = $100
Method for Pricing Call • Start at expiration where you have three possible stocks prices and determine the corresponding three intrinsic values of the call. • Move to period 1 and use single-period model to price the call at each node. • Move to period one and use single-period model to price the call in current period.
Step 1: Find IV at Expiration • Start at expiration where you have three possible stocks prices and determine the corresponding three intrinsic values of the call. • Cuu = Max[110-100,0] = 10 • Cud = Max[102.23-100,0] = 2.23 • Cdd = Max[95-100,0] = 0
Step 2: Find Cu and Cd • Move to preceding period (period 1) and determine the price of the call at each stock price using the single-period model. • For Su = $104.88, determine Cu using single-period model for that period. • For Sd = $97.47, determine Cd using single-period model for that period.
At Su = $104.88, Cu = $7.32 • Using Single-Period Model
At Sd = $97.47, Cd = $1.48 • Using Single-Period Model
Step 3: Find Co • Substitute the Cu and Cd values (determined in step 2) into the equations for Ho* and Bo*, then determine the current value of the call using the single-period model.
At So = $100, Co = $5.31 • Using Single-Period Model
Point: Multiple-Period Model • Whether it is two periods or 1000, the multiple-period model determines the price of a call by determining all of the IVs at T, then moving to each of the preceding periods and using the single-period model to determine the call prices at each node. • Such a model is referred to as a recursive model -- Mechanical.
Point: Arbitrage Strategy • Like the single-period model, arbitrage ensures the equilibrium price. The arbitrage strategies underlying the multiperiod model are more complex than the single-period model, requiring possible readjustments in subsequent periods. • For a discussion of multiple-period arbitrage strategies, see JG, pp. 158-163.
Point: Impact of Dividends • The model does not factor in dividends. If a dividend is paid and the ex-dividend date occurs at the end of any of the periods, then the price of the stock will fall. The price decrease will cause the call price to fall and may make early exercise worthwhile if the call is America.
Point: Adjustments for Dividends and American Options • The BOPM can be adjusted for dividends by using a dividend-adjusted stock price (stock price just before ex-dividend date minus dividend) on the ex-dividend dates. See JG, pp.192-196. • The BOPM can be adapted to price an American call by constraining the price at each node to be the maximum of the binomial value or the IV. See JG, pp. 196-199.
Estimating u and d The estimating equations for determining u and d are obtained by mathematically solving for the u and d values which make the statistical characteristics of a binomial distribution of the stock’s logarithmic returns equal to the characteristic's estimated value.
Binomial Process • The binomial process that we have described for stock prices yields after n periods a distribution of n+1possible stock prices. • This distribution is not normally distributed because the left-side of the distribution has a limit at zero (I.e. we cannot have negative stock prices) • The distribution of stock prices can be converted into a distribution of logarithmic returns, gn:
Binomial Process • The distribution of logarithmic returns can take on negative values and will be normally distributed if the probability of the stock increasing in one period (q) is .5. • The next figure shows a distribution of stock prices and their corresponding logarithmic returns for the case in which u = 1.1, d = .95, and So = 100.