Option Valuation The Black-Scholes-Merton Option Pricing Model

How Do Stock Prices Evolve Over Time? Random walk: proportional changes in prices over a short period of time are normally distributed Let S = current stock price m = expected return on stock = volatility of stock price Over a short time interval Dt, the mean proportional change in stock price is m*Dt and the standard deviation of the proportional change is s *  Dt Formally, DS/S ~ N (m*Dt , s *  Dt ) and DS /S = m*Dt + e*s  T (used for simulation: DS = S[m*Dt + e*s  T ]) Where e is a random draw from the standard Normal dist. (mean=0, s=1)

The Lognormal Distribution Random Walk implies that stock price at any future date, ST, has a lognormal distribution. This means that the logarithm of the stock price at any future time is normally distributed: Ln(ST) is normal with Mean = Ln(S0) + (m - 0.5s2)T St. Dev. = s  T Thus, E(ST) = S0emT Ln(ST) - Ln(S0) = Ln(ST/S0), which is the continuously compounded return realized over the period from now till time T, is normal with Mean = (m - 0.5s2)T

Estimating Volatility (Check P. 238-239 of textbook) Let: Si = Stock price at end of ith time interval t = length of time interval in years Compute return over interval i: ui = ln (Si / S i-1) An estimate, s, of the standard deviation of the ui’s is: s = S (ui - u)2 where u is the mean of the ui ‘s The annual estimate of volatility, s, is s /  t n 1 n - 1 i = 1

Assumptions Underlying Black-Scholes-Merton Stock prices follow a random walk with constant m and s No transactions costs or taxes No dividends over the life of the option Trading is continuous Investors can borrow and lend at risk free rate, r, which is constant over the life of the option There are no riskless arbitrage opportunities

The Black-Scholes-Merton Analysis Similar logic to that for binomial option pricing Create riskless portfolio consisting of position in stock and option The return on the portfolio must be the risk free rate Complication: must adjust hedge portfolio every instant (“portfolio rebalancing”)

The Black-Scholes-Merton Formula where N(d) = The standard cumulative normal density function (i.e. the probability of standardized normal variable being < d) d1 = [ ln(S0/X)+(r + 1/2s2) t ] / [s Ö t ] d2 = d1 - s Ö t c = S * N(d1 ) - X* e -r * t * N(d2 )

The Black-Scholes-Merton Formula: Puts p = X* e-r*T * N(-d2) – S0* N(-d1) Comparing the two models: p+S = c + X*e-r * T  Put-call parity

Measurement of Inputs Current Stock Price S - observable Strike Price X - observable Time to Expiration T - observable Risk-free rate r Annual yield on risk-free, short-term Bill (using continuous compounding) Volatility s Standard deviation of historical returns on the stock Implied volatility

Implied Volatility Set market price today = model price Know S, X, t and r Solve for s S=100, X=100, t = 0.5, r = 5%, c = 6.89  s = 20% Use 20% to value OTC options with similar maturity

Option Valuation: General Intuition Call payoff at T = CT = ST – X if ST > X 0 if ST < X E(CT) = Expected payoff = pr(ST>X)*E(ST – X | ST>X) + 0 = Pr(ST>X)*E(ST | ST>X) – Pr(ST > X)*X = N(d1)*E(ST) – N(d2)*X Notice: N(d1) > Pr(ST>X) but E(ST) < E(ST | ST>X) E(ST) = S0emT Hence, E(CT) = N(d1)* S0emT – N(d2)*X And C0 = PV(CT) = [N(d1)* S0emT] e-mT – N(d2)*X* e-rT (Notice we discounted the stock position at the rate of return of the stock m while the riskless cash flow X at the risk free rate on interest.)

General Intuition, Continued Hence, C0 = N(d1)* S0 – N(d2)*X* e-rT Notice: we get this formula regardless of what m is. We might as well assume m = r, i.e., we are valuing the option in a risk neutral world. This is one more way to see the appropriateness of the risk neutrality assumption: It does not hurt, but it adds a lot of convenience.

Effect of Dividends (dividend yield) If the underlying asset pays a continuous dividend yield, q: E(ST) = S0e(r-q)T PV of E(ST) = e-rT S0e(r-q)T = S0e-qT, then c = S0e-qT * N(d1) – X* e -r*T * N(d2) p = X*e-r*T * N(-d2) – S0* N(-d1) Where d1 = [ ln(S0/X)+(r –q + 1/2s2) t ] / [s Ö t ] d2 = d1 - s Ö t N(-d) = 1 – N(d)

Known Dividend Amount Break stock price into two components: S* + PV(div) riskless component, PV(div), used to pay dividends during option life risky component S* Black-Scholes-Merton formula correct if S* = S – PV(div) In previous example, assume firm pays one dividend of $0.5 in 3 months and $0.5 in 6 months S=100, X=100, T = 0.5, r = 5%, s = 20% PV(div) = 0.5 * e-.05 *.25 + 0.5 * e -0.5*.5 = 0.98 Revalue call using S* = $99.02  c = $6.32 (6.89 previously)

American Calls with Dividends It may be optimal to exercise an American call immediately before an ex-dividend date C = Max[c, c1, c2,…..,cn] c is value assuming exercise at maturity T ci is value assuming exercise at ith ex-dividend date In practice, calls are most likely to be exercised just before last dividend Black approximation: C = Max [c , cn]

The Black Approximation: An Example t= 3 months, ex-dividend in 1 month, dividend = $1, Stock price = $20, r = 10%, Volatility = 30%, X = 20 S=20, X=20, T=0.0833, r=0.1,  = 0.3 c1 = 0.77 S*= 20 – PV($1) = 19.01, T=0.25 c = 0.92 Call value = Max[0.77, 0.92] = 0.92 now Div = $1 T 1 month 2 months