In continuous time, the discount factor is e-iT.
Also note the integration boundaries, K to ∞,
since the payoff is zero between 0 and K.
Fin 440 Lecture 5: Option Pricing Models
1) The value of a European call option with a strike K and maturity T:

1
Call K ,T 
( ST  K ) f ( ST )dST
1  iT K
-
This is just the present value 1  iT  of the range of payoffs ST  K  , for all stock
price realizations above the strike K ,  , multiplied by the corresponding
probability of a given realization f (ST )dST
1
-
This definition embeds an assumption of risk-neutrality—that is the value of an
uncertain payoff to you is linearly related to the payoff. If you were risk averse
the value would not increase proportionately with the size of the payoff, e.g. the
1
value of a payoff ST  K might actually be ST  K  2 .
-
The definition makes no distributional assumptions, e.g. f (ST ) could be the
normal, lognormal, uniform, bi-modal, jump-diffusion, double exponential, or
anything you choose. However, the distribution should represent the actual riskneutral probability for each realization (e.g. stock price) above the strike.
Payoff
S-K
K
Only the portion of the
distribution exceeding the strike
price contributes to the value of
the call option.
∞
2) Assumptions of the Black and Scholes Model
-
Either Risk-neutral valuation of payoffs, or must establish a risk-free portfolio
-
Returns distributed log-normally: the log of asset returns is distributed normally—
f(ST) is log-normal, in the equation above (no negative asset prices)
-
Complete markets (which basically says there will always be a buyer or seller at
any given price)
-
Ignores transaction costs and frictions
-
Model uses continuous-time discounting
3) Black and Scholes Model for the Price of a European Call Option
C K S 0 , T   S 0  N d1   Ke  rT N d 2 
d1 
2
S 0    

ln 
T
 r 
 K  
2 
 T
and d 2 
2
S 0    

ln 
T
 r 
 K  
2 
 T
 d1   T
-
The key parameters determining the value of an option (an option’s premium) are:
o spot price (S0)
o strike (K)
o time to maturity (T)
o continuous-time risk-free rate (r)
o expected volatility (standard deviation) of returns over a year (σ)
-
Model depends on probabilities from the normal distribution that correspond to
two Z-statistics, d1 and d2, computed from the parameters. Note: N d1   N d 2  .
4) Background mathematics:
-
Continuous time discounting:
o If we discount using the discrete one year rate, the PV factor is:
o If we discount using the daily rate, the PV factor is:
1
1  rAnnual
1
1  r 
365
Daily
o If we allow the periods to become infinitely small, the PV factor is:
1
lim
 e rT
T  1  r T
T
o Thus, given a discrete risk-free rate (rAnnual), we must first convert it to the
corresponding continuous time rate. Mechanically: 1  rAnnual  e r1  e r .
-
Natural Logs:
o If e x equals a number, the natural log of that number is x.
o Any number has a unique corresponding natural log. E.g. if e x  1.05 ,
then x  .04879 . This is also written as: x  ln 1.05 .
o Therefore, x  ln e x and e lnnumber  number .
o Since e x  e y  e x  y , then ln e x  e y  ln e x y  x  y  ln e x  ln e y
Thus:
S
o ln  0   ln e lnS0 ln K   ln S 0   ln K 
 K
o ln S 0   rT  ln S 0  e rT 
 






   
Note: S0∙erT/K is the ratio of the
Forward/Strike. Thus, ln(S0∙erT/K) is the
percentage that the forward is less than
or exceeds the strike.
5) Re-writing Black and Scholes equation
CK S 0 ,T   S 0  N d1   PV K N d 2 
 S  e rT 
ln  0
K   T

and d 2  d1   T ;
d1 

2
 T
-
-
So d1 is this percentage plus ½ std. dev.
of at time T, and d2 is that percentage
less ½ std. dev. at time T.
Because N d1   N d 2  , CK S 0 , T   max 0, S 0  PV ( K )  or intrinsic value.
o If the stock price is deep out-of-the-money: N d1  and N d 2  are zero.
Thus, CK S 0 ,T   0 .
o If the stock price is deep in-the-money: N d1  and N d 2  approach one.
Thus, CK S 0 , T   S 0  PV K  , the PV of the payoff.
o The intuition is that the Black and Scholes price exceeds intrinsic value
because it captures the additional time value of the option
The term S 0  e rT is the forward price of the stock at maturity (T), thus, d1 and d2
are relative to the forward price of the stock—not the current spot price
Payoff
BS price is
near Zero
BS price is
S – PV(K)
PV(K)
S0 – PV(K)
S0
Since the BS price approaches
the intrinsic value in the
limits, the BS price always
exceeds intrinsic value.
6) Z statistics:
-
-
-
Recall that the Z statistic is a point on the normal distribution, and it is found by
standardizing a (random) variable that is normally distributed. E.g. if x is normal:
x
Z

We standardize a variable so that we can use the normal distribution (tables) to
compute the probability for a certain value of x.
Thus, we standardize a variable to convert its distribution to a scaling that we
know. This assumes that the variable has a normal distribution.
Probability Density
1 Sd
N(1) is the total amount of the distribution
to the left of Z = 1 (1 standard deviation).
Computing a Z-statistic for a variable—
e.g. a stock price—centers its distribution
about the mean (µ) and scales it by its
standard deviation (σ).
N(1)
Z value
Note: N(Z) is the probability that a value less then Z will occur; thus, N d1   N d 2  .
d1 = ln(forward rate/K)+ ½ Std. Dev.
d2 = ln(forward rate/K- ½ Std. Dev.
7) Understanding d1 and d2:




ln S 0  e rT  ln K   T
ln S 0  e rT  ln K   T
and d 2 
d1 


2
2
 T
 T
-
-
The first term in d1 is the deviation of the expected price from strike (K) divided
by the standard deviation over T years. This looks like an ordinary Z statistic, e.g.
x
.

The second term in d1 is the one half the expected volatility.
-
The intuition is that d1 and d2 are the Z statistics for being half a standard
deviation in and out of-the-money at maturity.
-
Note that time is linear in variance σ2T, so the standard dev. is the square root.
8) What are log returns and why do we use them?
-
Log returns are the natural log of the stock ratio, e.g. ln  ST  for a period T.
 S0 
If we assume that this ratio is normally distributed, then the stock price in period
T is normally distributed:
For example: if S0 is the reference stock price, and ST is the period T price, then:
2
ST
 1  rT ~ N 1  r , 2 or, multiplying by S0: ST ~ N S0 1  r , S0 
S0

  


The problem with the stock price being normally distributed is that ST can be
negative:
Payoff
Probability of the stock price
being negative—not a plausible
assumption.
0
 
S0 1  r
ST
-
So, instead, we assume that the log returns are normally distributed—or that
returns are log normal. This means the Log of the stock price is normally
distributed. Thus, you can never have a negative stock price.
-
Taking the natural log also converts a ratio to a percentage.
-
For example: suppose that ln ST   1.5 , then ST  e lnST   .22 or 22 cents.
-
The Big point is that if stock returns are distributed log-normally, stock prices
can never be negative. That is why BS makes that assumption.
9) So what does all this mean for the Black-Scholes equation?
-
It makes a crucial assumption about the distribution of stock returns/prices,
namely that they are log-normally distributed. If this is an incorrect assumption,
it will not properly value options.
-
It also makes other crucial assumptions about market completeness, and
transactions costs, which if these assumptions break down mean that the model
will be incorrect.
Binomial Model:
1) The idea behind the Binomial model is to approximate the movement of a stock
price using a Binomial tree
-
Permits a wide range of assumptions about the volatility of a stock price
-
Consistent with the assumptions of the Black-Scholes distribution, because when
allowed to grow to infinite, the probability distribution for the stock price
converges on the log-normal.
-
Assume that we have a stock with an original price S, and after one period the
return on that stock will be u or d, depending whether the price goes up or down
Time 0
Time 1
u∙S
S
d∙S
-
For example, S = $100, u = 1.10, and d = .90, then our one-period stock price
model would imply that after one period, our price goes up to $110, or down to
$90:
Time 0
Time 1
1.10∙100 = 110
100
.90∙100 = 90
-
Thus, this approach seeks to approximate the movement of the price of a stock by
creating a tree where the volatility is determined by the size of the up or down
move.
-
The reason we use this approach is because it allows us to price American
Options—something that the Black-Scholes model doesn’t do very well
-
The chief drawback is that without special modifications, the Binomial
distribution approaches the log-normal.
2) Two Approaches to Pricing Options using the Binomial Model
-
The first requires us to create a Risk-free Portfolio, then determine the price of
the option so that no arbitrage opportunities (risk-free profits) exist
-
The second approach uses risk-neutral probabilities to price the option (and can
be derived from the first method)
-
Either method can be extended to a multi-period framework
3) No-Arbitrage or Hedge Portfolio Approach
-
We establish a risk-free Portfolio by writing an option in combination with ∆
shares of stock. The initial cost of our portfolio is ∆S – C. Assume S = $100, u =
1.10, and d = .90. Assume that the strike on the call is 105, and that the risk-free
rate is 5%.
-
Since the strike on the call is 105, when the stock price moves up to 110, the
payoff is 5, when it moves down to 90, the payoff is zero
Time 0
Time 1
∆110 – 5
∆S – C105
∆90
-
-
Because the portfolio is risk-free, the payoff must be equal in both states, thus:
110  5  90
  5  .25
20
This is the delta of the call option—and it says that call price moves .25 with
every dollar change in the share price. It is also known as the Hedge Ratio.
-
Likewise, the payoff in each state is .25  90  22.5
-
Because the portfolio S  C105 is risk-free, its value must grow at the risk-free
rate. Therefore:
.25  100  C105  PV 22.5
25  C105 
22.5
 21.43
1.05
C105  3.57
-
Notice, we made no assumptions about the probability of an up or down move
4) An example using Puts
-
Assume the same parameters as above, but now we are working with Puts.
-
We create a portfolio using ∆ shares of stock and by writing a put with a strike at
105. Because a Put pays off when the stock price is below the strike price, it is in
the money when the stock price goes down.
-
When the stock price is 90, the value of the put is 15; when the stock price is 110,
the put is out of the money.
Time 0
Time 1
∆110
∆S – P105
∆90 – 15
-
-
Again, because the portfolio is risk-free, its payoff must be the same in both
states:
110  90  15
   15  .75
20
Notice that the delta of the put is negative, just as we observed when studying the
BS model. Also notice that:  P   C  1.
To solve for the price of the put, substitute delta to determine the payoff in Time 1
(remember, both states have the same payoff), and discount the payoff back to
Time 0:
 .75 100  P105  PV  82.5  PV  .75  90  15
 75  P105 
 82.5
 78.57
1.05
P105  3.57
-
Again, we made no assumption about the probability that our stock price would
move up or down. We didn’t need it because we created a portfolio that would
have the same value regardless.
-
Don’t worry that the call and put premia are the same; that is just coincidence.
-
Note that d < 1+r < u or the model breaks down and you get arbitrage
5) Extension to a multi-period model:
-
Consider a two period model with the same parameters:
Time 0
Time 1
Time 2
u = 1.10
u = 1.10
∆121 – Cuu
∆110 – Cu
d = .90
∆100 – C105
u = 1.10
∆99 – Cud
d = .90
∆90 – Cd
d = .90
∆81 – Cdd
-
The basic approach is to multi-period model is to start in time 2 and work
backwards. (Backwards Induction)
-
For example you will have to solve for ∆ the upper triangle:
121  16  99
  16  .7272
22
Then solve for Cu:
-
.7272 110  Cu  PV 72  PV .7272  99
72
 68.57
1.05
Cu  11.43
80  Cu 
-
Then you have to do the same for the lower triangle, which will have a different
delta, and likewise find Cd. Note that the call option is out of the money for
movements ud and dd. Thus, Cd = 0. This simplifies our problem.
-
We re-write the Binomial problem :
Time 0
Time 1
∆110 – 11.43
∆S – C105
∆90
-
We again equate the payoffs at Time 1 to compute delta, then solve for the option
price. ∆ = .5715 and C105 = 8.16. This is a painful, and there is a better way.
6) Generalized Model:
-
Now we assume the general parameters u, d, S, and r to see if we can generate a
more simple formula for the price of a call
-
Cu is the payoff of the call if the stock price moves up, Cd if it moves down
Time 0
Time 1
∆uS – Cu
∆S – CK
∆dS – Cd
-
Step one, solving for ∆:
uS  C u  dS  C d
Cu  C d
uS  dS
Notice that ∆ is a function of S. This means that delta is dependent on the price of
the stock and changes with the price of the stock.

-
-
Step two, compute the payoff at Time 1:
uS  Cu 
-
Cu  Cd
dC  uCd
uS  Cu  u
uS  dS
ud
Step three, equate the present value of the payoff to value of the portfolio at time
zero and solve for the price of the option:
Cu  C d
1 dC u  uC d
S  CK 
ud
1 r u  d
1  1 r  d
1 r  u
CK 
C

C
u
d
1  r 
ud
u  d 
or ,
CK 
1
1 r
 1 r  d
 1  r  d 
C u u  d  C d 1  u  d 



-
Now we have a formula for the price of a call that was created by establishing a
risk-free portfolio, and we no-longer have to solve for the parameters
individually.
-
We’re not done though…
7) Risk-Neutral Probabilities:
-
We can define:
1 r  d
ud
then,
p
CK 
1
 p  C u  1  p   C d 
1 r
-
What this means is that the price of a call option (or put option) is the present
value of its expected payoff given using risk-neutral probabilities
-
How do we know they are risk-neutral? It turns out that solving for the
probability of u and d, (the return from an up-move and down-move) so that one
gets the risk-free rate on average, yields p. E.g. solve:
p  u  1  p   d  1  r
then,
1 r  d
p
ud
-
Note: this is not the true probability that the stock price will move up or down by
u or d, it is the probability assuming risk-neutral pricing.
-
So, all that you have to do to price an option is compute the risk neutral
probabilities and know what the payoffs are.
-
However, remember that this was derived by establishing a risk-free portfolio.
-
Big Point: The reason we do this is to facilitate computation of the multi-period
model.
8) The Multi-period Model using Backwards Induction and Risk-Neutral
Probabilities
-
Same approach: to compute C105 we first need to compute Cu and Cd. But this
time we first compute the risk-neutral probability. Then compute the option’s
value at time zero.
1  r  d 1.05  .90

 .75
ud
1.10  .90
1
 p 16  1  p   0  11.43
Cu 
1.05
Cd  0
p
C105 
1
 p 11.43  1  p   0  8.16
1.05