1. Advanced Finance 2005-2006 Black Scholes
Professor André Farber
Solvay Business School
Université Libre de Bruxelles

2. Binomial option pricing model
Used to value derivative securities: PV=f(S)
Evolution of underlying asset: binomial model
u and d capture the volatility of the underlying asset
Replicating portfolio: Delta × S + M
Law of one price: f = Delta × S + M uS fu S dS fd Δt
M is the cash position M>0 for investment M<0 for borrowing
Consider the following example. You want to value a call option with three months to maturity. The strike price of the option is 20 and the spot price of the underlying asset is 20. The risk-free interest rate (with continuous compounding) is 5%. The volatility of the underlying asset is 30%.
You want to use the binomial option pricing model with Δt=3/12=0.25
The up and down factors are: u = and d = The two possible stock prices three months later are uS = and dS = 17.21
The values of the call option at the end of the period are: fu = Max(0, )=3.24 and fd = Max(0, )=0
The composition of the replicating portfolio is: Delta = and M = You should buy shares and borrow
As a consequence, the value of the call option is: f = ×20 – = 1.61.
As an exercise, you can verify that similar calculations for a put option would lead to the following results:
fu = 0, fd = 2.79
Delta = , M =
f = 1.36
r is the risk-free interest rate with continuous compounding
Advanced Finance 2006 Black Scholes

3. Risk neutral pricing
The value of a derivative security is equal to risk-neutral expected value discounted at the risk-free interest rate
p is the risk-neutral probability of an up movement
Risk-neutral pricing is a very powerful technique to price derivative securities. The difficulty is to be able to calculate the risk neutral probabilities. This calculation is straightforward in the binomial model. It requires more advanced mathematical techniques in more general model.
To continue the example in the previous slide, the risk neutral probability of an up movement is:
p = (e.05×0.25 – 0.861)/( ) = 0.504
The value of the call and the put options are:
Call f = (0.504 × 3.24)/ e.05×0.25 = 1.61
Put f = (0.504 × 2.79)/ e.05×0.25 = 1.36
Advanced Finance 2006 Black Scholes

4. Road map to valuation Binomial model uS S dS
Model of stock price behavior
Binomial model uS S dS
discrete time, discrete stock prices
Geometric Brownian Motion
dS = μSdt+σSdz
continuous time continuous stock prices
Create synthetic option
Based on elementary algebra
Based on Ito’s lemna to calculate df
p fu + (1-p) fd = f erΔt
PDE:
Pricing equation
Numerical methods
Black Scholes formula
Advanced Finance 2006 Black Scholes

5. Modelling stock price behaviour
Consider a small time interval t: S = St+t - St
S = E(S) + u
2 components of S:
Expected change (drift) : E(S) =  S t [ = expected return (per year)]
Unexpected change u: a normal random variable
Expected value: E(u) = 0
Variance: E(u²) = ² S² t (Variance proportional to t)
u =  S z with
z independent of past values (Markov process)
S =  S t +  S z
If t "small" (continuous model)
dS =  S dt +  S dz
Advanced Finance 2006 Black Scholes

6. Multiperiod binomial valuation
Risk neutral probability
u4S p4
Risk neutral discounting
(European option)
(1) At maturity, calculate
- firm values;
- equity and debt values
- risk neutral probabilities
(2) Calculate the expected values in a neutral world
(3) Discount at the risk free rate
u3S
u²S
u3dS
4p3(1 – p) uS
u2dS S
udS
6p²(1 – p)²
u2d²S dS
ud²S Δt
ud3S
4p (1 – p)3
d²S
Recursive method
(European and American options)
Value option at maturity
Work backward through the tree. Apply 1-period binomial formula at each node
The price change in one period in independent of the price change in previous periods.
The probability of a path with k ups and n-k downs (where n is the total number of steps) is pk (1 – p)n-k
The number of path is given by the binomial coefficient:
The Excel function COMBIN(N,n) provides the desired result.
For small binomial tree, Pascal’s triangle is useful:
Each subsequent row is obtained by adding the two entries diagonally above.
This is known in China as the Yanghui triangle as it had been studied earlier by the Chinese mathematician Yanghui about 500 years earlier.
Reference mathworld.wolfram.com
d3S
d4S
(1 – p)4
Advanced Finance 2006 Black Scholes

7. Multiperiod binomial valuation: example
2-year European option S = 100 Strike price = 105 Int.Rate = 5% (annually compounded) Volatility = 40%
4-step binomial tree Δt = 0.50 u = 1.332, d = rf = 2.47% per period =(1.05)1/2-1 p = 0.471
We calculate the values using a binomial tree with 4 steps.
We first calculate the parameters of the binomial tree.
As the final maturity is 2 years, the length of one period is Δt =2/4=0.50
The up and dow factors are: and d = 1/1.332=0.751
The risk-free interest rate is 5% per annum with annual compounding. The risk-free interest per period in the binomial tree is thus rf = (1.05)1/2-1=2.47%
The risk-neutral probability of an up movement is:
p = ( – 0.751)/(1.332 – 0.751) = 0.471
The next step is to calculate the possible values of the firm at maturity and the associated probabilities.
We then calculate the values of equity and debt at maturity and their risk-neutral expected values
We get the present value by discounting the expected values at the risk-free interest rate.
Advanced Finance 2006 Black Scholes

8. Multiperiod valuation: details
Binomial models are easier to implement in Excel using diagonal matrices instead of branching trees.
V uV u²V u3V …
dV udV u²dV …
d²V ud²V … …
The upper part of the slide shows the binomial evolution of the value of the company. The lower part shows the evolution of the value, the delta and the beta of the equity and of the debt.
For the equity, the delta increases (decreases) and the beta decreases (increases) when the value of the company goes up (down). The opposite applies for the debt.
Note that, as the value of the firm goes down, the debt becomes more and more similar to equity.
Advanced Finance 2006 Black Scholes

9. From binomial to Black Scholes
Consider:
European option
on non dividend paying stock
constant volatility
constant interest rate
Limiting case of binomial model as t0
Advanced Finance 2006 Black Scholes

10. Convergence of Binomial Model
Advanced Finance 2006 Black Scholes

11. Toward Black Scholes formulas
Value
Increase the number to time steps for a fixed maturity
The probability distribution of the firm value at maturity is lognormal
Call in the money
What happens if we increase the number of steps of a given maturity?
The size of the binomial tree increases but the underlying logic remains unchanged. In a binomial tree with n steps (the length of each step is Δt=T/n), the firm can take n+1 different values at maturity depending on the number of ups (k) and downs (n – k):
Vk = ukdn – k k = 0, 1, …,n
The risk-neutral probability of Vk is:
Let fk be the value at maturity of a derivative if VT = Vk.
Risk-neutral pricing implies that the value at time 0 is:
f = e-rT (p0 f0 + p1 f1 + … + pk fk + … + pn+1 fn+1) [1]
As n→∞, the probability distribution of VT tends converges to a lognormal distribution.
ln(VT) ~ N[ln(V)+(r-0.5σ²)T, σ√T]
The Black-Scholes formulas is the continuous equivalent of [1] when the probability distribution of the underlying value at maturity is lognormal.
Put in the money
Today
Maturity
Time
Advanced Finance 2006 Black Scholes

12. Lognormal property of stock prices
If: dS=  S dt +  S dz
ln(ST) – ln(S0) = ln(ST/S0)
Continuously compounded return between 0 and T
This is a straightforward application of Ito’s lemna with G(S,t) = ln(S)
GS = 1/S, Gt = 0, GSS = -1/S²
The parameters of the Ito process are:
a = S, b = S
Replace in Ito’s lemna:
dG = (a Gx + Gt + ½ b² Gxx) dt + b Gx dz
to obtain:
d ln(S) = [(S)(1/S) ½ (²S²)(-1/S²)] dt + (S)(1/S) dz
The expression simplifies:
d ln(S) = ( - ½ ²) dt +  dz
ln(ST) is normally distributed. Hence the distribution of ST is lognormal
Advanced Finance 2006 Black Scholes

13. Understanding the PDE Assume we are in a risk neutral world
Expected change of the value of derivative security
Change of the value with respect to time
Change of the value with respect to the price of the underlying asset
Change of the value with respect to volatility
Advanced Finance 2006 Black Scholes

14. Black Scholes’ PDE and the binomial model
We have:
Binomial model: p fu + (1-p) fd = ert
Use Taylor approximation:
fu = f + (u-1) S f’S + ½ (u–1)² S² f”SS + f’t t
fd = f + (d-1) S f’S + ½ (d–1)² S² f”SS + f’t t
u = 1 + √t + ½ ²t
d = 1 – √t + ½ ²t
ert = 1 + rt
Substituting in the binomial option pricing model leads to the differential equation derived by Black and Scholes
BS PDE : f’t + rS f’S + ½ ² f”SS = r f
This result is explained in Rubinstein Derivatives: A PowerPlus Picture Book, Vol 1, Part B.
The proof involves a lot of tedious algebra.
Substituting the Taylor approximations in the binomial option pricing model:
f + [pu+(1-p)d]Sf’S – Sf’S + ½[p(u-1)² + (1-p)(d-1)²]S²f”SS + ft = f(1+rt)
But:
p u + (1-p) d = 1 + rt
and:
p (u-1)² + (1-p) (d-1)² = ² t
so that:
f + rt S f’S + ½ ² t S² f”SS + f’t t = f + r f t
Simplify and rearrange to obtain the PDE.
Advanced Finance 2006 Black Scholes

15. And now, the Black Scholes formulas
Closed form solutions for European options on non dividend paying stocks assuming:
Constant volatility
Constant risk-free interest rate
Call option:
Put option:
N(x) = cumulative probability distribution function for a standardized normal variable
Advanced Finance 2006 Black Scholes

16. Standardized normal cumulative probability distribution
If X~N(0,1) N(x) =Probability(X ≤ x)
Advanced Finance 2006 Black Scholes

17. Black-Scholes using Excel
Comments:
Stock price: for dividend paying stocks, the stock price should be reduced by the present value of the dividends paid before the option’s maturity.
Interest rate: the easiest way is to use the interest rate with continuous compounding. In that case, the t-year discount factor is exp(-rFt).
ln(S/PV(Strike)): this number is equal to minus the total excess return (with continuous compounding) in order for the stock price to reach the exercise price at maturity. To see this, note that:
PV(Strike) = S e-x → x=ln(S/PV(Strike)) and Strike = S e(rT – x)
Adjusted sigma: σ √T is the standard deviation of the total return until the option’s maturity.
Distance to exercice:
Advanced Finance 2006 Black Scholes

18. Understanding Black Scholes
Remember the call valuation formula derived in the binomial model:
C =  S0 – B
Compare with the BS formula for a call option:
Same structure:
N(d1) is the delta of the option
# shares to buy to create a synthetic call
The rate of change of the option price with respect to the price of the underlying asset (the partial derivative CS)
K e-rT N(d2) is the amount to borrow to create a synthetic call
The BS formula for the European call option is a generalization of the formula that we studied previously for a long forward contract on a non dividend paying stock:
f = S0 – K e-rT
To see this, simply remember if you are certain to exercise the option, a call is in fact a forward contract. For a forward contract:
N(d1) = +1: to replicate you should buy one unit of the underlying asset
N(d2) = +1: the risk neutral probability of exercising the option is 1.
N(d2) = risk-neutral probability that the option will be exercised at maturity
Advanced Finance 2006 Black Scholes

19. Black-Scholes: European put option
European call option: C = S N(d1) – PV(X) N(d2)
Put-Call Parity: P = C – S + PV(X)
European put option: P = - S [N(d1)-1] + PV(X)[1-N(d2)]
P = - S N(-d1) +PV(X) N(-d2)
Delta of call option
Risk-neutral probability of exercising the option = Proba(ST>X)
Delta of put option
The next step is to use the Black-Scholes formula to value the securities. This slide is a reminder of the formulas.
Risk-neutral probability of exercising the option = Proba(ST<X)
(Remember: 1-N(x) = N(-x))
Advanced Finance 2006 Black Scholes

20. A closer look at d1 and d2 S0 / Ke-rt 2 elements determine d1 and d2
A measure of the “moneyness” of the option. The distance between the exercise price and the stock price
S0 / Ke-rt
Note that if the current stock price is equal to the present value of the exercise price, the formulas for
Time adjusted volatility. The volatility of the return on the underlying asset between now and maturity.
Advanced Finance 2006 Black Scholes

21. Example
Stock price S0 = 100
Exercise price K = 100 (at the money option)
Maturity T = 1 year
Interest rate (continuous) r = 5%
Volatility  = 0.15
ln(S0 / K e-rT) = ln(1.0513) = 0.05
√T = 0.15
d1 = (0.05)/(0.15) + (0.5)(0.15) =
In this example, the call option is at the money (the exercise price is equal to the current stock price).
The delta of the option is 0.66 (more precisely, ) which means that an increase of 1 of the stock price (from 100 to 101) would increase the value of the call option from 8.60 to 9.26.
The risk-neutral probability that the option will be exercised is 0.60.
N(d1) =
European call :
100    = 8.60
d2 = – 0.15 =
N(d2) =
Advanced Finance 2006 Black Scholes

22. Relationship between call value and spot price
For call option, time value > 0
Advanced Finance 2006 Black Scholes

23. European put option European call option: C = S0 N(d1) – PV(K) N(d2)
Put-Call Parity: P = C – S0 + PV(K)
European put option: P = S0 [N(d1)-1] + PV(K)[1-N(d2)]
P = - S0 N(-d1) +PV(K) N(-d2)
Delta of call option
Risk-neutral probability of exercising the option = Proba(ST>X)
Delta of put option
Risk-neutral probability of exercising the option = Proba(ST<X)
Remember: N(x) – 1 = N(-x)
Advanced Finance 2006 Black Scholes

24. Example
Stock price S0 = 100
Exercise price K = 100 (at the money option)
Maturity T = 1 year
Interest rate (continuous) r = 5%
Volatility  = 0.15
N(-d1) = 1 – N(d1) = 1 – =
N(-d2) = 1 – N(d2) = 1 – =
You can check that put-call parity is verified:
C + PV(K) = S + P =
The delta of the put option is call – 1
This easily derived from put call parity by taking the partial derivatives with respect to the price of the underlying asset:
CS + 0 = 1 + PS
If the stock price increases by 1, the value of the call option increases by call In order for the put call to hold (remember, this is a no arbitrage condition), the value of the put option should change by call – 1
In our example, the delta of the put option is
The risk neutral probability of exercising the put option is This is equal to 1 minus the risk neutral probability of exercising the call option.
European put option
- 100 x x = 3.72
Advanced Finance 2006 Black Scholes

25. Relationship between Put Value and Spot Price
For put option, time value >0 or <0
One important lesson from this figure is that the time value for a European put option can be negative. This is the case if the stock price is low relative to the strike price (the option is “deep in the money”).
To understand this, remember that if you own a put option, you sell the underlying asset if you exercise. You receive the strike price. Now imagine that the option is very, very deep in the money: you are almost certain that you will exercise the option at maturity. But this like selling forward: you are short on a forward contract. The value of your put option is equal to minus the value of the forward contract. Remember that the value of the forward contract is the difference between the spot price and the present value of the exercise price. So, you get:
P = PV(K) – S = (K – S) – [K – PV(K)]
Intrinsic value – Time value
But the time value also incorporate an “insurance value”: the value of the option that you have to exercise or not. This insurance value of always positive which explains why the time value of the put option is positive if the option is at the money or out of the money.
Advanced Finance 2006 Black Scholes

26. Dividend paying stock
If the underlying asset pays a dividend, substract the present value of future dividends from the stock price before using Black Scholes.
If stock pays a continuous dividend yield q, replace stock price S0 by S0e-qT.
Three important applications:
Options on stock indices (q is the continuous dividend yield)
Currency options (q is the foreign risk-free interest rate)
Options on futures contracts (q is the risk-free interest rate)
Advanced Finance 2006 Black Scholes

27. Dividend paying stock: binomial model
t = u = 1.25, d = 0.80 r = 5% q = 3% Derivative: Call K = 100
uS0 eqt with dividends reinvested
uS ex dividend 125
fu 25
S0 100
dS0 eqt with dividends reinvested 82.44
fd 0
dS ex dividend 80
f =  S0 + M
f = [ p fu + (1-p) fd] e-rt =
Replicating portfolio:
A 1-period gives us a clearer view of the impact of dividends for the valuation of a derivative.
We assume that the dividend yield q (with continuous compounding) is known.
Remember that the trick to value a derivative is create a replicating portfolio (a synthetic derivative). In our example, the derivative is a call option. The point is that if you buy shares, you receive a dividend whereas if you buy the call option, you miss the dividend paid before maturity. The dividend shows up in the future value of the replicating portfolio. As a consequence, the numbers of shares to buy to replicate the call option is smaller. This means a lower delta. The difference appears in the formula for the delta of the option. The amount borrowed is not affected by the dividend yield.
The risk neutral probability is higher with dividend than without dividend. This is because the stock price grows at rate r-q rather than r when there is a dividend yield at rate q.
The probability of an up movement should therefore satisfy:
[p uS0 + (1 – p) dS0] eqt = S0 ert
As a consequence, the value of the call option is lower if the dividend yield is positive.
(You can check that setting q = 0 would lead to a value of for the call option instead of with q = 3%)
 uS0 eqt + M ert = fu  M = 25
p = (e(r-q)t – d) / (u – d) =
 dS0 eqt + M ert = fd  M = 0
 = (fu – fd) / (u – d )S0eqt =
Advanced Finance 2006 Black Scholes

28. Black Scholes Merton with constant dividend yield
The partial differential equation: (See Hull 5th ed. Appendix 13A)
Expected growth rate of stock
Call option
Put option
These results were first derived by Merton in 1971 who was later to share the Nobel prize in 1997 with Myron Scholes.
Those of you willing to have a option valuation spreadsheet in their computer should use these formulas. By setting q = 0, you are back to the case of an option on a non dividend paying stock.
Advanced Finance 2006 Black Scholes