Options V

1
Options V

• Stock option pricing
• Black-Scholes as a limit of the Binomial model
• How to use Black-Scholes
• Volatility and Implied VolatilityGeneral case
• n trading periods

2
Previously Binomial formula for a Call

• C Ep (S/Rn) S K - (K/Rn)
  ProbpS K
• S Fa n, p - (K/Rn) Fa n , p
  
• where p (R-d)/(u-d), p (u/R)p
• - a minimum number of ups such that the call is
  in the money, S Suadn-a ? K
• p risk neutral probability of an up
• - Fa n, q Sja,n n!/j!(n-j)! q j 1-
  q n-j prob. of at least a ups in n periods
  using the probability q

3
Black Scholes as a limit of the Binomial model

• h time between transactions
• T maturity of the call
• n number of transactions (trading periods)
• h (T/n) o (n h) T
• n ( o h0) fixing T
• Keep constant
• Stock return variance per unit of time (e.g.
  annual)
• Interest rate per unit of time (e.g. annual)

4
Using the Binomial model, when h 0 ( n 8),
well get the Black-Scholes formula

• C S N( d1 ) - Ke-rT N(d2 )
• d1 log(S/K e-rT )/s T1/2 ½s T1/2
• d2 d1 - sT1/2
• T s2 return variance until maturity.
• s return standard deviation per unit of time.
• - r interest rate (continuously compounded)
• - N(z) probability that a standard normal
  random variable takes a value less or equal to z
  (cumulative distribution).

5
Black Scholes as a limit of the Binomial model
C Ep (S/Rn) S K - (K/Rn)
ProbpS K C S Fa n, p -
KR-n Fa n , p C S N( d1 )
- Ke-rT N(d1 - s T1/2 ) When n 8
( or h 0) Fa n, p N(d1) Fa n, p
N(d1 - s t1/2) R-n e-rT ( by
the Central Limit Theorem log(S/S) is
normally distributed with mean mT, and variance
s2T )
6
Black-Scholes Formula Numerical Example
7
Numerical Example Call
8
Using Put-Call Parity P C - S K e-rT
and symmetry of N(.) 1 N(z) N(-z)
9
Symmetry of the Normal N(.) (skip)

• Remind that by definition of a CDF
• N(z) Prob x lt z
• 1 N(z) Prob x gt z
• N(-z) Prob x lt -z Prob-x gt z
• Since x is a standard Normal, by symmetry, x and
  x have the same distribution
• N(-z) Prob-xgt z 1 N(z)
• 1 N(-z) N(z)

10
Numerical Example Put
11
Rest of the Notes

• Detail of Black-Scholes as a limit of the
  Binomial.
• Estimating the parameters of the Binomial (u and
  d for each n) given data on the underlying and
  the maturity of the derivative.
• Estimating volatility s, and implied volatility.
• Calculating the deltas ?

12
Statistical Model for the price of S

• Stock Price
• S/S Sn / S0 ujdn-j total gross return (n
  periods)
• log(Si/Si-1) return (in logs), between i and
  i-1
• log(S/S) log(S1/S0) (S2/S1) (Sn-1/Sn-2)
  (Sn/Sn-1)
• Si1,..n log(Si/Si-1), total
  (log) return
• Varlog(Si/Si-1) s2 variance per trading
  period

13
Statistical Model for the price of S

• s2 Varlog (Si/Si-1) q(1-q)log(u/d)2
• q (statistical) probability of an up u.
• i.i.d returns imply
• Varlog (S/S) n s2 (total variance)
• Important The variance is adjusted with the
  length of the time period, whereas the standard
  deviation is adjusted with the square root of the
  time period.

14
Variance per period and Total variance (skip)

• i.i.d returns means that they are independent
  and identically distributed
• Var(log(S/S)) Var(log(S1/S0) (S2/S1)
  (Sn/Sn-1) )
• Var(Si1,..n log(Si/Si-1) ) lt by
  independencegt
• Si1,..n Var(log(Si/Si-1) ) ltby
  identical distributiongt
• n s2

15
Algebra for the formula of the variance (skip)

• s2 Varlog (Si/Si-1)
• Elog (Si/Si-1)2-E log (Si/Si-1)2
• q log(u) 2(1-q)log(d)2- q log(u)
  (1-q)log(d)2
• q log(u)2 (1-q)log(d)2 - q log(u)2
  -(1-q)log(d) 2
• - 2q(1-q)log(u)log(d)
• q(1-q) log(u)2 (1-q)(1-(1-q))log(d)2
• - 2q(1-q)log(u)log(d)
• q(1-q)log(u)2 q(1-q)log(d)2 -
  2q(1-q)log(u)log(d)
• q(1-q)log(u)2 log(d)2 - 2log(u)log(d)
• q(1-q)log(u) - log(d)2
• q(1-q)log(u/d)2

16
Statistical model for S Parameterization

• Parameterize returns with 3 values (m, s, q)
• u exp( m (T/n) s (t/n)½ )
• d exp( m (T/n) - s (t/n)½ )
• In logs
• log(u) m (T/n) s (T/n)½
• log(d) m (T/n) - s (T/n)½
• n (s)2 (s)2 T , s2 variance per unit of
  time
• n bigger smaller trading periods ( h T/n
  small)
• m is, momentarily, fixed at an arbitrary value.

17
Statistical model for q 1/2

• 1) T s 2 n q(1-q)log(u/d)2 ( T s 2
  n s2 )
• 2) log(u) m (T/n) s (T/n)½
• 3) log(d) m(T/n) - s (T/n)½
• Given n and m, plugging 2) and 3) in 1)
• q ½

18
Statistical model for S q1/2 (skip)
Plug log(d) and log(u) in equation 1) T s 2 n
q(1-q)2s (T/n)½2 T s 2 n q(1-q)4s2(T/n) T
s 2 q(1-q)4s2T T s 2 1 q(1-q)4
Which implies q ½
19
Choice of m

• Well consider 2 alternative ways to choose m
• m r ½ s2 which implies that the risk
  neutral probability p ½ , equals the
  statistical probability q.
• m 0, which implies that u x d 1
• We can prove that for big n (h T/n small) the
  value of m does not alter the results
• p( m, T/n ) ? ½ when T/n ? 0
• where p(m, T/n) risk neutral probability for m,
  T/n.

20
Numerical example of p( m, T/n ) ? ½ when T/n ?
0
21
Statistical prob. q vs. Risk Neutral prob. p
(skip)
(Gross) expected stock return q exp(
m(T/n)s(T/n)½ ) (1-q) exp( m(T/n)-s(t/n)½
) Recall using p implies that the expected
stock return should be equal to the return of the
bond. exp( r(T/n) ) p exp(
m(T/n)s(T/n)½ ) (1-p) exp( m(T/n)-s(T/n)½ )
22
Statistical prob. q vs. Risk Neutral prob. p
(skip)
p exp( m(T/n)s(T/n)½ ) (1-p) exp(
m(T/n)-s(T/n)½ ) exp( r(T/n) ) Ignoring the
terms with power in (T/n) greater than 1 r
(T/n) (2p-1) s (T/n)½ ½ s (T/n)
m(T/n) Dividing by (T/n)½ r (T/n)½ (2p-1) s
½ s (T/n)½ m (T/n)½ lim p( m, T/n ) ? ½
when T/n ? 0
23
Choosing m such that the risk neutral probability
is p ½ (skip)
Using the expansion exp(a) 1 a ½ a2
1/3! a3 exp( r (T/n) ) 1 r (T/n) ½
(r (T/n))2 1/3! (r (T/n))3 exp(m(T/n)s(T/n)
½) 1 (m(T/n)s(T/n)½) ½
(m(T/n)s(T/n)½)2 1/3!( ... exp(m(T/n)
-s(T/n)½) 1 (m(T/n) -s(T/n)½) ½ (m(T/n)
-s(T/n)½)2 1/3!( ... Ignoring the terms with
power in (T/n) greater than 1 we get r (T/n)
m(T/n) ½ s2(T/n) .
24
Summary Statistical Model for S
Inputs s per period standard deviation of
(log) return of S T maturity N number of
trading periods 2 sets of parameters for the
Binomial model ia) m r ½ s2 implies
risk neutral prob. p ½ ib) m 0
implies u x d 1 ii) u exp( m(T/n) s
(T/n)½ ) iii) d exp( m(T/n) - s (T/n)½ )
25
Black Scholes as a limit of the Binomial model
log(S/S) Si1,..n log(Si/Si-1) sum of
binomials When n goes to the (standardized)
distribution goes to a normal. (CLT) Central
Limit Theorem log(S/S) m T / s T1/2
Is normally distributed, N(0,1)
26
Black Scholes as a limit of the Binomial model
C Ep (S/Rn) S K - (K/Rn)
ProbpS K C S Fa n, p -
KR-n Fa n , p C S N( d1 )
- Ke-rT N(d1 - s T1/2 ) When n 8
( o h 0) Fa n, p N(d1) Fa n, p
N(d1 - s t1/2) R-n
e-rT log(S/S) Si1,..n log(Si/Si-1) is
normally distributed, (mT, s2T )
27
Estimating Volatility ?

• 1. Consider observation S0, S1, . . . , Sn in
  intervals of h years (typically h1/52)
• 2. Define the continuously compounded return as
• 3. Calculate the standard deviation std of the
  zi
• 4. The estimator for the volatility is

28
Estimating the volatility ? Example

• Suppose we have a year of weekly data
• S1/S0, S2/S1,,S52/S51
• Where Si is, say, the price in the Wednesday of
  week i.
• Net (log) returns
• z1 log(S1/S0) ,, z52 log(S52/S51)
• h T / n 1 / 52
• s std(z) ( 52 )½

29
Volatility (cont)

• Usually, the volatility is bigger when the market
  is open than when the market is closed (e.g.
  weekends, holidays, etc)
• For this reason, in order to price an option,
  the volatility is measured in trading days
  instead of calendar days.

30
Implied Volatility

• The implied volatility of an option is the
  volatility that results from the Black-Scholes
  formula that gives the market price of the
  stock.
• There is a one-to-one correspondence between
  prices and implied volatility (why?)
• Traders/brokers frequently use implied volatility
  instead of the prices in

31
Implied Volatility

• s satisfies the following equation
• C S N( d1(s) ) - Ke-rT N(d1(s) - s T1/2 )
• d1(s) log(S/K e-rT )/s T1/2 ½s T1/2
• Where we take into account the following data
• Price of the underlying S
• Interest rate r
• Price of the Call C, with maturity T and strike K

32
Put Delta vs. Call Delta

• Put Call Parity
• P C - S K e-rT
• Then

33
Calculating the Delta of the equivalent
portfolio Binomial Model

• Delta is calculated at each node