analytical option pricing models: introduction and general concepts finance 70520, spring 2002 risk...
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Analytical Option Pricing Models:introduction and general concepts
Call & Put prices:Black-Scholes-Merton model
$-
$10
$20
$30
$40
$50
$60
Asset price
Opt
ion
Val
ue
call
put
Finance 70520, Spring 2002Risk Management & Financial EngineeringThe Neeley SchoolS. Mann
Binomial European model: one periodInputs Asset Price Dynamics
Initial Stock Price, S0 85.000 up factor, U :
Option Strike Price, K 85.00 U = exp[r -2/2)h +(h) 1/2]annual volatility, 37.0% down factor , D :T-bill ask discount rate= 5.10% D = exp[r 2/2)h -(h) 1/2]valuation date: 18-Mar-02
option expiration date 17-Jun-02 U = 1.1983D = 0.8277
generated from above: R(h) = 1.01310.25091.25 output:
17-Jun-02 B(0,T) 0.98707 8.32continuous riskless rate, r = 5.20% 7.22
101.85 16.85
Su = S0U Cu
85.00 8.32
S0 Call Value: C0
70.35 0.00
Sd = S0D Call delta= Cd
0.53507
Call ValuePut Value
maturity (days)period length h (years)
Binomial European model: two periodInputs Asset Price Dynamics
Initial Stock Price, S0 85.000 up factor, U :
Option Strike Price, K 85.00 U = exp[r -2/2)h +(h) 1/2]annual volatility, 37.0% down factor , D :
T-bill ask discount rate= 5.10% D = exp[r 2/2)h -(h) 1/2]valuation date: 18-Mar-02
option expiration date, T: 17-Jun-02 U = 1.1374D = 0.8756
generated from above: R(h) = 1.00650.12591.25 output:
17-Jun-02 B(0,T) 0.98707 6.16continuous riskless rate, r = 5.20% 5.06
109.97 24.97
Suu = S0 uu Cuu
96.68 12.40
Su = S0 u Cu; u =85.00 84.65 6.16 0.986 0.00
S0 Sud=S0ud =S0du Call Value: C0 Cud
74.42 0.00
Sd = S0 d = Cd; d =65.16 0.557 0.000 0.00
Sdd = S0 dd Cdd
maturity (days)
period length h (years)
Call ValuePut Value
Binomial European model: three periodInputs Outputs
Initial Stock Price, S0 85.00 Call Value 7.30
Option Strike Price, K 85.00 Put Value 6.23 32.11
annual volatility, 37%riskless rate 5.1% 20.60
period length t (years) 0.083312.49 1.000 9.58
117.117.30 0.785 4.71
105.240.560 2.31 0.527 0.00
94.58 94.580.288 0.00
85.00 85.000.000 0.00
76.39 76.390.00
68.650.00
valuation date 3/18/2002 61.70Maturity 6/17/2002 2.19 0.000 0.00
6.23 -0.215 4.35
-0.440 10.20 -0.473 8.61
-0.712 15.99
-1.000 23.30
Call Option Dynamics
Put Option Dynamics
Stock Price Evolution
Binomial Convergence to Black-Scholes-Merton periods call value
inputs: output: 1 $8.31
Current Stock price (S) $85.00 binomial call value $6.73 2 $6.16
Exercise Price (K) $85.00 Put Value $5.65 3 $7.30
valuation date 18-Mar-02 4 $6.47
Expiration date 17-Jun-02 Black-Scholes Call value $6.78 5 $7.09
riskless rate (continuous) 5.10% 6 $6.58
estimated volatility (sigma) (s) 37% 7 $7.00
periods (lattice dimension) 20 8 $6.64time until expiration (years) 0.250 9 $6.95
10 $6.6711 $6.9212 $6.6913 $6.8914 $6.7115 $6.8816 $6.7217 $6.8618 $6.7219 $6.8620 $6.7321 $6.8522 $6.7423 $6.8424 $6.7425 $6.8426 $6.7427 $6.8328 $6.7529 $6.8330 $6.75
Binomial model : convergence to BSM value
$0.00
$1.00
$2.00
$3.00
$4.00
$5.00
$6.00
$7.00
$8.00
$9.00
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Periods
Black-Scholes-Merton model assumptions
Asset pays no dividendsEuropean callNo taxes or transaction costsConstant interest rate over option life
Lognormal returns: ln(1+r ) ~ N ()reflect limited liability -100% is lowest possible
stable return variance over option life
Black-Scholes-Merton Model
C = S N(d1 ) - KB(0,t) N(d2 )
d1 =ln (S/K) + (r + 2 )t
t
d2 = d1 - t
Note that B(0,T) = present value of $1 to be received at T define r = continuously compounded risk-free ratefind r by: exp(-rT) = B(0,T) so that r = -ln[B(0,T)]/T
e.g. T = 0.5B(0,.5) = 0.975 r = -ln(.975)/0.5 = 0.02532/.5 = 0.05064
Call value : Black-Scholes-Merton Model
output:Current Stock price (S) 85.00 Theoretical Call Value 6.783 Stock Call Exercise Price (X) 85.00 Theoretical Put Value 5.696 price Valuevaluation date 3/18/2002 57 0.08Expiration date 6/17/2002 60 0.16riskless rate 5.15% 62 0.31
volatility (sigma) (s) 37% 65 0.54
observed call price 6.500 68 0.89time until expiration (years) 0.250 71 1.39
74 2.0677 2.9379 4.0082 5.2985 6.7888 8.4891 10.3693 12.4196 14.6099 16.92
102 19.35105 21.87108 24.45110 27.10113 29.79
model inputs:
Call option Value
$0
$5
$10
$15
$20
$25
$30
$35
57 60 62 65 68 71 74 77 79 82 85 88 91 93 96 99 102
105
108
110
113
Mann VBA function:scm_bs_call(S,K,T,r,sigma)
Mann VBA function:scm_bs_put(S,K,T,r,sigm
Function scm_d1(S, X, t, r, sigma) scm_d1 = (Log(S / X) + r * t) / (sigma * Sqr(t)) + 0.5 * sigma * Sqr(t)End Function
Function scm_BS_call(S, X, t, r, sigma) scm_BS_call = S * Application.NormSDist(scm_d1(S, X, t, r, sigma)) - X * Exp(-r * t) * Application.NormSDist(scm_d1(S, X, t, r, sigma) - sigma * Sqr(t))End Function
Function scm_BS_put(S, X, t, r, sigma) scm_BS_put = scm_BS_call(S, X, t, r, sigma) + X * Exp(-r * t) - SEnd Function
Code for Mann’s Black-Scholes-Merton VBA functions
To enter code:tools/macro/visual basic editorat editor:insert/moduletype code, then compile by:debug/compile VBAproject
N( x) = Standard Normal (~N(0,1)) Cumulative density function:
N(x) = area under curve left of x; N(0) = .5 coding: (excel) N(x) = NormSdist(x)
Black-Scholes-Merton Model: Delta
C = S N(d1 ) - KB(0,t) N(d2 )
N(d1 ) = Call Delta (call hedge ratio
= change in call value for smallchange in asset value
= slope of call: first derivative of call with respect to asset price
output:Current Stock price (S) 85.00 Theoretical Call Value 6.783Exercise Price (X) 85.00 Theoretical Put Value 5.696 Stock Call valuation date 3/18/2002 implied volatility 32.3% price Value deltaExpiration date 6/17/2002 47.26 0.00 0.00riskless rate 5.15% 0.5644 51.03 0.01 0.00
volatility (sigma) (s) 37% 54.81 0.05 0.01
observed call price 6.000 58.58 0.13 0.03time until expiration (years) 0.250 62.36 0.31 0.07
66.13 0.64 0.1269.90 1.21 0.1973.68 2.06 0.2777.45 3.26 0.3781.23 4.84 0.4785.00 6.78 0.5688.77 9.09 0.6592.55 11.71 0.7396.32 14.60 0.80
100.10 17.72 0.85103.87 21.02 0.89107.64 24.45 0.92111.42 27.99 0.95115.19 31.60 0.96118.97 35.26 0.98122.74 38.96 0.98
model inputs:
call delta
Delta: call slope
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
47.26 54.81 62.36 69.90 77.45 85.00 92.55 100.10 107.64 115.19 122.74
Mann VBA function:delta(S,K,T,r,sigma)
1 day
20 days
40 days
60 days
0.00
5.00
10.00
15.00
20.00
25.00
asset price
Theoretical call value and time to expiration
1 d
ay
10
da
ys
20
da
ys
30
da
ys
40
da
ys
50
da
ys
60
da
ys
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
asset price
Call Delta and time to expiration
Call and Deltaover time
Gamma
output:Current Stock price (S) 85.00 Theoretical Call Value 6.778Exercise Price (X) 85.00 Theoretical Put Value 5.701 Stock Call valuation date 3/18/2002 implied volatility 32.4% price Value gammaExpiration date 6/17/2002 47.26 0.00 0.000riskless rate 5.10% 0.0250 51.03 0.01 0.001
volatility (sigma) (s) 37% 54.81 0.05 0.003
observed call price 6.000 58.58 0.13 0.007time until expiration (years) 0.250 62.36 0.31 0.011
66.13 0.64 0.01669.90 1.20 0.02173.68 2.06 0.02477.45 3.26 0.02681.23 4.83 0.02685.00 6.78 0.02588.77 9.08 0.02292.55 11.70 0.01996.32 14.59 0.016
100.10 17.71 0.012103.87 21.01 0.010107.64 24.44 0.007111.42 27.98 0.005115.19 31.59 0.004118.97 35.25 0.003122.74 38.95 0.002
model inputs:
call gamma
Call Gamma: change in delta
0.000
0.005
0.010
0.015
0.020
0.025
0.030
47.26 54.81 62.36 69.90 77.45 85.00 92.55 100.10 107.64 115.19 122.74
Mann VBA function:gamma(S,K,T,r,sigma)
10%
15%20%
25%30%
35%40%
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
asset price
Call gamma as a function of volatility: 4 months left
54 60 66 72 79 85 91 98 104 110 116
5 days
10 days
20 days
30 days
40 days
50 days
60 days
0.00
0.02
0.04
0.06
0.08
0.10
0.12
asset price
Call gamma as a function of time to expiration
Call gamma(curvature)
Vega
output:Current Stock price (S) 85.00 Theoretical Call Value 6.778Exercise Price (X) 85.00 Theoretical Put Value 5.701 Stock Call valuation date 3/18/2002 implied volatility 32.4% price Value vegaExpiration date 6/17/2002 47.26 0.00 0.1riskless rate 5.10% 16.7356 51.03 0.01 0.4
volatility (sigma) (s) 37% 54.81 0.05 0.9
observed call price 6.000 58.58 0.13 2.1time until expiration (years) 0.250 62.36 0.31 4.0
66.13 0.64 6.569.90 1.20 9.373.68 2.06 12.277.45 3.26 14.681.23 4.83 16.185.00 6.78 16.788.77 9.08 16.492.55 11.70 15.296.32 14.59 13.5
100.10 17.71 11.6103.87 21.01 9.5107.64 24.44 7.6111.42 27.98 5.9115.19 31.59 4.5118.97 35.25 3.4122.74 38.95 2.4
model inputs:
call vega
Call vega: sensitivity to volatility
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
47.26 54.81 62.36 69.90 77.45 85.00 92.55 100.10 107.64 115.19 122.74
Mann VBA function:vega (S,K,T,r,sigma)
Call Values:S(t) 10% 15% 20% 25% 30% 35% 40%
53.55 (0.00) 0.00 0.00 0.00 0.01 0.05 0.1256.70 0.00 0.00 0.00 0.01 0.04 0.11 0.2559.84 0.00 0.00 0.00 0.03 0.10 0.24 0.4562.99 0.00 0.00 0.01 0.08 0.22 0.45 0.7766.13 0.00 0.00 0.05 0.19 0.44 0.80 1.2369.28 0.00 0.03 0.16 0.43 0.82 1.31 1.8672.42 0.01 0.11 0.40 0.84 1.39 2.01 2.6975.57 0.06 0.36 0.86 1.49 2.20 2.95 3.73
78.71 0.31 0.92 1.65 2.45 3.28 4.12 4.9881.86 1.07 1.94 2.84 3.74 4.65 5.55 6.4685.00 2.61 3.52 4.45 5.37 6.30 7.23 8.1688.15 4.92 5.64 6.46 7.33 8.23 9.14 10.0691.29 7.74 8.18 8.82 9.58 10.40 11.26 12.1594.44 10.79 11.01 11.46 12.08 12.79 13.58 14.4197.58 13.92 14.01 14.29 14.76 15.36 16.06 16.82
100.73 17.06 17.09 17.26 17.60 18.08 18.68 19.36103.87 20.20 20.21 20.30 20.54 20.91 21.42 22.02107.02 23.35 23.35 23.40 23.55 23.84 24.25 24.77
110.16 26.49 26.49 26.51 26.61 26.82 27.16 27.60
113.31 29.64 29.64 29.65 29.71 29.86 30.12 30.50
116.45 32.78 32.78 32.79 32.82 32.93 33.13 33.44
years (T) 0.00 0.027 0.055 0.082 0.110 0.137 0.164
Call vega values:S(t) expiration10 days 20 days 30 days 40 days 50 days 60 days
53.55 0.00 0.00 0.00 0.00 0.01 0.06 0.1756.70 0.00 0.00 0.00 0.01 0.07 0.24 0.5359.84 0.00 0.00 0.00 0.07 0.29 0.71 1.3262.99 0.00 0.00 0.04 0.30 0.89 1.75 2.7866.13 0.00 0.00 0.22 0.99 2.19 3.60 5.0869.28 0.00 0.05 0.91 2.56 4.46 6.36 8.1872.42 0.00 0.42 2.68 5.29 7.70 9.85 11.7875.57 0.00 2.04 5.92 8.99 11.47 13.57 15.4078.71 0.00 5.91 10.10 12.86 15.02 16.85 18.4681.86 0.00 10.80 13.69 15.77 17.53 19.09 20.5085.00 10.72 13.11 15.12 16.88 18.47 19.92 21.2688.15 0.00 11.05 13.91 15.99 17.75 19.32 20.7491.29 0.00 6.71 10.86 13.58 15.73 17.55 19.1694.44 0.00 3.03 7.32 10.46 12.94 15.02 16.8597.58 0.00 1.05 4.32 7.37 9.96 12.20 14.17
100.73 0.00 0.29 2.26 4.80 7.22 9.44 11.46103.87 0.00 0.06 1.06 2.90 4.96 7.00 8.93107.02 0.00 0.01 0.45 1.65 3.25 4.99 6.74110.16 0.00 0.00 0.18 0.88 2.03 3.43 4.94113.31 0.00 0.00 0.06 0.45 1.22 2.28 3.52116.45 0.00 0.00 0.02 0.21 0.71 1.48 2.45
54 60 66 72 79 85 91 98 104
110
116
10%
15%
20%25%
30%35%
40%
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
posi
tion
val
ue
asset price
Call: value as function of asset price and volatility
54 60 66 72 79 85 91 98 104 110 116
expiration
10 days
20 days
30 days
40 days
50 days
60 days
0.0
5.0
10.0
15.0
20.0
25.0
Call vega as function of asset price and time to expiration
Implied volatility (implied standard deviation)
annualized standard deviation of asset rate of return, or volatility.
=
Use observed option prices to “back out” the volatility implied by the price.
Trial and error method:1) choose initial volatility, e.g. 25%.2) use initial volatility to generate model (theoretical) value 3) compare theoretical value with observed (market) price. 4) if:
model value > market price, choose lower volatility, go to 2)
model value < market price, choose higher volatility, go to 2)eventually,
if model value market price, volatility is the implied volatility
Call implied volatility
output:Current Stock price (S) 85.00 Theoretical Call Value 6.783Exercise Price (X) 85.00 Theoretical Put Value 5.696 Call
valuation date 3/18/2002 volatility Value
Expiration date 6/17/2002 implied volatility 38.3% 10.0% 2.28riskless rate 5.15% 12.0% 2.61
volatility (sigma) (s) 37.0% 14.0% 2.94observed call price 7.000 16.0% 3.27time until expiration (years) 0.250 18.0% 3.61
20.0% 3.9422.0% 4.2724.0% 4.6126.0% 4.9428.0% 5.2830.0% 5.6132.0% 5.9534.0% 6.2836.0% 6.6238.0% 6.9540.0% 7.2842.0% 7.6244.0% 7.9546.0% 8.2948.0% 8.6250.0% 8.96
model inputs:
Call Value as a function of volatility
$0
$1
$2
$3
$4
$5
$6
$7
$8
$9
$10
10%
14%
18%
22%
26%
30%
34%
38%
42%
46%
50%
this is the Mann VBA function:scm_bs_call_ isd(S,K,T,r,sigma)
Historical annualized Volatility Computation
1) compute daily returns2) calculate variance of daily returns3) multiply daily variance by 252 to get annualized variance: 2
4) take square root to get
or:1) compute weekly returns2) calculate variance3) multiply weekly variance by 524) take square root
annualized standard deviationof asset rate of return =
1 day
10 days
20 days30 days
40 days50 days
60 days
(2.00)
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
16.00
po
siti
on
va
lue
asset price
Call: value as function of asset price and time to expiration
72 75 77 80 82 85 88 90 93 95 98
60 days50 days
40 days30 days
20 days
10 days
1 day
(140.0)
(120.0)
(100.0)
(80.0)
(60.0)
(40.0)
(20.0)
0.0
asset price
Call theta and time to expiration:note reversal of vertical axis
Call Theta:Time decay