1. Option Valuation The Black-Scholes-Merton Option Pricing Model

2. 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)

3. 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

4. Estimating Volatility
(Check P 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

5. 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

6. 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”)

7. 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 )

8. 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

9. 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

10. 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

11. 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.)

12. 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.

13. 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)

14. 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 * * e -0.5*.5 = 0.98
Revalue call using S* = $  c = $6.32 (6.89 previously)

15. 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]

16. 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