Analytical Option Pricing: Black-Scholes –Merton model; Option Sensitivities (Delta, Gamma, Vega , etc) Implied Volatility Call& Putprices: Black-Scholes-M erton m odel $- $10 $20 $30 $40 $50 $60 A ssetprice O ption Value call put Finance 30233, Fall 2010 The Neeley School S. Mann
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Binomial European model: one periodInputs Asset Price Dynamics
Initial Stock Price, S0 100.00 up factor, U :
Option Strike Price, K 100.00 U = exp[(r - s2/2)h + s (h) 1/2]annual volatility, s 30.0% down factor , D :T-bill ask discount rate= 0.25% D = exp[(r - s2/2)h - s (h) 1/2]valuation date: 31-Oct-10
option expiration date 31-Oct-11 U = 1.2937D = 0.7100
generated from above: R(h) = 1.00251.000
365 output:31-Oct-11 B(0,T) 0.99747 14.68
continuous riskless rate, r = 0.25% 14.43
129.37 29.37
Su = S0U Cu
100.00 14.68
S0 Call Value: C0
71.00 0.00
Sd = S0D Call delta= Cd
0.50322
Put Value
maturity (days)period length h (years)
Call Value
Binomial European model: two periodInputs Asset Price Dynamics
Initial Stock Price, S0 100.00 up factor, U :
Option Strike Price, K 100.00 U = exp[(r - s2/2)h + s (h) 1/2]annual volatility, s 30.0% down factor , D :
T-bill ask discount rate= 0.25% D = exp[(r - s2/2)h - s (h) 1/2]valuation date: 31-Oct-10
option expiration date, T: 31-Oct-11 U = 1.210D = 0.792
generated from above: R(h) = 1.0010.500
365 output:31-Oct-11 B(0,T) 0.99747 11.61
continuous riskless rate, r = 0.25% 11.36
146.49 46.49
Suu = S0 uu Cuu
121.03 23.24
Su = S0 u Cu; Du =100.00 95.84 11.61 0.918 0.00
S0 Sud=S0ud =S0du Call Value: C0 Cud
79.19 0.00
Sd = S0 d D0= Cd; Dd =62.71 0.555 0.000 0.00
Sdd = S0 dd Cdd
maturity (days)
period length h (years)
Call ValuePut Value
Binomial Convergence to Black-Scholes-Merton periods call value
inputs: output: 1 $14.68
Current Stock price (S) $100.00 binomial call value $12.03 2 $11.61
Exercise Price (K) $100.00 Put Value $11.78 3 $12.86
valuation date 31-Oct-10 4 $11.99
Expiration date 31-Oct-11 Black-Scholes Call value $12.03 5 $12.48
riskless rate (continuous) 0.25% 6 $12.08
estimated volatility (sigma) (s ) 30% 7 $12.32
periods (lattice dimension) 25 8 $12.12time until expiration (years) 1.000 9 $12.23
Or:1) Compute monthly returns2) calculate variance3) Multiply monthly variance by 124) take square root
annualized standard deviation of asset rate of returns =
Implied volatility (implied standard deviation)
annualized standard deviation of asset rate of return, or volatility.s =
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) 100.00 Theoretical Call Value 12.034Exercise Price (X) 100.00 Theoretical Put Value 11.784 Call
this is the Mann VBA function:scm_bs_call_ isd(S,K,T,r,sigma)
Implied volatility (implied standard deviation):VBA code:
Function scm_BS_call_ISD(S, X, t, r, C) high = 1 low = 0 Do While (high - low) > 0.0001 If scm_BS_call(S, X, t, r, (high + low) / 2) > C Then high = (high + low) / 2 Else: low = (high + low) / 2 End If Loop scm_BS_call_ISD = (high + low) / 2End Function
1 day
10 days
20 days30 days
40 days50 days
60 days
(2.00)
0.00
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po
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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)
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(20.0)
0.0
asset price
Call theta and time to expiration:note reversal of vertical axis