risk neutral valuation the black-scholes model and monte carlo
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Risk Neutral ValuationTRANSCRIPT
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Risk Neutral Valuation, the Black-
Scholes Model and Monte Carlo
Stephen M SchaeferLondon Business School
Credit Risk Elective
Summer 2012
Objectives: to understand
1
2 12
2
1
1 (.) : cumulative standard normal distribution
ln( / ( ))= T and = T
T
( )-PV( ) ( ) N
S
C SN d X N
Pd d
d
V X d σ σ
σ + −
=
The Black-Scholes formula
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 2
• The Black-Scholes formula in terms of risk-neutral valuation
• How to use the risk-neutral approach to value assets using
Monte Carlo (next week: the binomial method)
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Valuing Options using Risk Neutral Probabilities
• In a complete market we can calculate:
unique risk neutral probabilities (and A-D prices)
the no-arbitrage price of an option as the risk-neutralexpected pay-off, discounted at the riskless rate
1 ˆPayoff on call
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 3
1
1
1( ,0), for a callˆ
1
: probability of stateˆ
f
S
s T
s f
s
r
Max S X r
RN s
π
π
=
+
= −+
∑
• Black-Scholes formula can be obtained in exactly this way
Definition: Lognormal Distribution
• If a random variable x has a normal distribution with mean µ
and standard deviation σ then:21
2 has a lognormal distribution with mean xe e µ σ +
0.003
0.004
0.004
0.005
n s i t y
Returns
mean (mu) 10%SD (sig) 40%
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 4
0.000
0.001
0.001
0.002
0.002
0.003
0 200 400 600
stock price
p r o b a b i l i t y d e _
Time Period (years) 4
Mean S _0*Exp(mu + .5 *sig^2) 205.44
Current value * Exp(mu) 149.18
• A lognormal distribution (like
the normal) has two parameters –
can take these as mean µ andstandard deviation σ ( of “ x”)
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Assumptions of the Black-Scholes Model
• The continuously compounded rate of return (CCR) on the
underlying stock over a length of time T has a normal
distribution (if the expected return is constant over time)
2
0
1ln ( )
2
C T T
S R T T u
S µ σ σ
= = − +
–
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 5
–
stock price (the expected return over a very short period – “dt ”)
– σ is the volatility of the CCR per year, so σ ⌦ T is the volatility of
CCR over the period of length T
• “u” is a normally distributed random variable with a mean of
zero and standard deviation equal to one.
Assumptions of the Black-Scholes Model
• Since the continuously compounded rate of return (CCR) has
a normal distribution the stock price itself (S T ) is lognormal
( ) ( )
21( )
20 0
2 2
and the mean and variance of are:
1( ) and var2
C T
T T u R C
T T
C C T T
S S e S e R
E R T R T
µ σ σ
µ σ σ
− +
= =
= − =
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 6
• rom t e propert es o t e ognorma str ut on t s means
that the expected future value of the stock price is:
( ) ( ) ( ) ( ) 2 21 1 1
var ( )2 2 2
0 0 0 0 C C
C T T T
E R R T T R T
T E S S E e S e S e S e µ σ σ
µ + − +
= = = =
• So the expected stock price just grows at the rate µ –CORRECT!
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Why does setting the expected value of the continuously compounded return equal to the
true (continuously compounded) expected return give the wrong answer?
• The reason is that the future stock price isa non-linear (and convex) function ofthe continuously compounded return.
• This means that the expected stock priceincreases with the variance of thereturn).
• Therefore, if we set the expected value ofthe continuousl com ounded return 0.95
1.15
1.35
1.55
1.75
S t o c k P r i c e = E x p ( x )
A
B
C
D
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 7
equal to the correct expected return (e.g.,given by the CAPM – 20% say) theexpected value of the stock price gives acontinuously compounded return that is
higher than 20% (point A versus point Bis the figure)
0.55
0.75
-60% -50% -40% -30% -20% -10% 0% 10% 20% 30% 40% 50% 60%
Continuously Compounded Stock Return (X)
• To make the expected price of the stock equal the point B (and therefore give anexpected return equal to the correct value) we must reduce the expected value ofthe continuously compounded return by an amount that depends on the variance.This is what the term (-½ * σ 2 * T ) does in the formulae given on the previous slide
The Black-Scholes Theory
• A lognormal distribution is a continuous distribution
the “number” of possible states is infinite (in the
binomial case it was just two per period)
• In the Black-Scholes model the number of assets is just two
(the underlying stock and borrowing / lending) exactly as in
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 8
the binomial example
• Black and Scholes’ big, surprising, deep, Nobel-
prize-winning result is that, under their assumptions,
the market is complete
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Risk Neutral Probabilities and A-D
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 9
r ces n ac - c o es
Risk Neutral Distribution in the Black-Scholes Model• In Black-Scholes, risk neutral
distribution:
is also lognormal
has same volatility
parameter (σ )
BUT mean parameter ( µ )is changed so that the
0.005
0.010
0.015
0.020
0.025
0.030
p r o b a b i l i t y d e n s i t y
Lognormal: E(returnon stock) = risklessrate (10%)
Lognormal: E(returnon stock) = 40%
risk neutral
distribution
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 10
stock is the riskless interest
rate (exactly as in the
binomial case)
v (like u) is also a normally
distributed random variable
0.000
60 80 100 120 140 160
stock price
natural distribution
• If underlying asset has positive risk premium this means that therisk-neutral distribution is shifted to the left
( ) 21( )
2
C RN
T R r T T vσ σ = − +
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A-D prices in Black-Scholes
• In the binomial case the A-D price is just the
corresponding risk-neutral probability discounted at the
riskless rate:
1ˆ
(1 )
: probability of state : price for stateˆ
s s
f
s s
qr
RN s q A D s
π
π
=+
−
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 11
• In B-S, because the distribution of the asset price is
continuous, we have a “distribution” of A-D prices
• To calculate the distribution of A-D prices in the B-S case
we just “discount” the risk-neutral distribution at theriskless interest rate (as in the binomial case).
A-D Prices in Black-Scholes
0.015
0.020
0.025
0.030
s i t y / s t a
t e p r i c e s
risk neutral
distribution
(RND)
A-D Price
“density” means:
price of A-D
security that pays
£1 if stock price is
between 122 and
124 (say) is equal
to area under curve
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 12
0.000
0.005
0.010
60 80 100 120 140 160
stock price
p r o b a b i l i t y d e
A-D prices
= RND / (1+r_f)
between 122 and
124
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The Black-Scholes “theory” vs. the Black-
Scholes “formula”
• The Black-Scholes theory – their key result – is
that (under their assumptions) the market iscomplete and that we can calculate the risk-neutral
distribution of the underlying asset.
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 13
• e - s e resu we ge
when we apply the theory to the particular problem
of valuing European puts and calls. This is much
narrower.
• There are many, many cases when we can apply the
theory without being able to use the formula.
Calculating Black-Scholes price as risk
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 14
riskless rate
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Calculating Option Values in the Black-Scholes
Model using Risk Neutral Probabilities
• To value an option (or any
asset) in EITHER the
binomial model OR the
Black-Scholes model we:
calculate the expected
cash flow usin the0.015
0.020
0.025
0.030
0.035
p r o b a b i l i t y d e n s i t
y
15
20
25
30
35
40
45
t i o n p a y o f f a t m a t u r i t y
Risk-Neutral Distribution of Stock Price
Call Option Payoff X = 120
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 15
risk-neutral
probabilities
discount at the riskless
rate of interest
• E.g., for a European call option with exercise price X = 120 we
calculate the expected value of the cash flow at maturity using the
R-N distribution and discount at the riskless rate
0.000
0.005
.
60 80 100 120 140 160
stock price
0
5
10 o p
• The payoff at maturity is zero for S < X and (S-X) for S X
• Using the RN distribution the discounted expected payoff is*:
Calculating the Black-Scholes value of a call
( ){
0 RN densityPayoff at of
ˆ,0 ( )
ˆ
f
T
f
r T
T T T
T S
r T
C e Max S X f S dS
∞
−
∞
−
= −
= −
∫ 1442443
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 16
{
Payoff whenis zero
ˆ ˆ( ) ( )
ˆ= ( ) ( )prob (
T
f f
f
X S X
r T r T
T T T T T
X X
r T
T T T RN X
e S f S dS e X f S dS
e S f S dS PV X
≤
∞ ∞
− −
∞
−
= −
−
∫ ∫
∫)S X ≥
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• the risk-neutral expected payoff on the call discounted at the
riskless rate, i.e., the call price is therefore:
Calculating the Black-Scholes value of a call
• Evaluating this expression using the lognormal risk-neutral
ˆ ( ) ( )prob ( ) f r T
T T T RN
X
C e S f S dS PV X S X
∞
−
= − ≥
∫
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 17
distribution for the Black-Scholes model we obtain the Black-
Scholes formula
1 2
1
1 2 12
( ) ( ) ( )
(.) : cumulative standard normal distribution
ln( / ( ))= T and = T
T
C SN d PV X N d
N
S PV X d d d σ σ
σ
= −
+ −
S = 49 X = 50 r f = 5% p.a.* T = 0.25 years Volatility=20% p.a.
1 2( / ) ( )
0 2n S S r T
j f u
j T
σ
σ
− −
=
lCalculating Black-Scholes Value by
Adding up payoff x RN probability
• Working out RN probs. and average payoff on
call for stock price intervals of 0.1 and then
calculating RN expected payoff as sum of av.
Payoff x prob. we find call value of 1.77799 vs.
1.77792 from Black-Scholes formula:
u j is N(0,1) shock to continuously
compounded return corresponding to
stock price S j.
*r f is continuously compounded
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 18
j S_j u_j
RN_prob
S<=S_j
RN Prob
S_j-1 <= S = <=S_j Payoff Av_Pa yoff
RNP * Av
Payoff
1 50.00 0.127027 0.550541 0.00000000 0.000 0.000 0.000000
2 50.10 0.147007 0.558437 0.00789628 0.100 0.050 0.000395
3 50.20 0.166947 0.566294 0.00785744 0.200 0.150 0.001179
4 50.30 0.186848 0.574110 0.00781578 0.300 0.250 0.001954
5 50.40 0.206709 0.581881 0.00777133 0.400 0.350 0.002720
6 50.50 0.226530 0.589606 0.00772419 0.500 0.450 0.003476
7 50.60 0.246313 0.597280 0.00767441 0.600 0.550 0.004221
8 50.70 0.266056 0.604902 0.00762207 0.700 0.650 0.004954
9 50.80 0.285761 0.612469 0.00756724 0.800 0.750 0.005675
10 50.90 0.305426 0.619979 0.00750999 0.900 0.850 0.006383
11 51.00 0.325053 0.627430 0.00745042 1.000 0.950 0.007078
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Interpreting the Black-Scholes Formula
PV ofexpected
PV of expec
proceedsfromexercise
RN
RN 64748
cost of exercisted e
644444474444448
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 19
1 2price of bonddiscounted probabilitypaying exerciseexpected of exercisepricevalue of stock
when
-
j
RN RN
S X >
142431442443 14243
Interpreting the Merton Formula for the Value of
Credit Risky Debt
• In the same way, the
Merton formula for the
value of credit risky debt
can be interpreted as the
sum of:
the value of the a ment3
4
5
6
7
8
o n d P a y o f f a t M a t u r i y
`
default
no default
(V ) in default
The value of the payment
of the face value ( B) in no-
default
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 20
0
1
0 1 2 3 4 5 6 7 8 9 10 11
value of assets of firm at maturity (€ million)
( ) ( ) ( )1 2
Riskless PVvalue in default Risk-neutral prob
of no-default
D VN d PV B N d = − + ×12314243 123
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Numerical Methods:
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 21
on e ar o
We need numerical methods when we
cannot find a formula for the option value
• In some cases we can find a formula for the value of
an option (e.g., the Black-Scholes formula)
• BUT often, thou h we continue to use the Black-
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 22
Scholes theory (and the Black-Scholes risk-neutral
distribution) there is no formula for the option price
Example: true for almost all American options (except in
cases – such as call on non-dividend paying stock – where
American and European options are worth the same)
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Monte-Carlo
• Standard technique to calculate the expected value
of some function f(x) of a random variable x:• How it works:
1. Generate random numbers drawn from the distribution“ ”
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 23
2. For each drawing ( xi) calculate f(xi)
3. Then simply take the average value of the f(xi)’s
• Wide variety of important problems (pricing, riskassessment etc.) can be solved using Monte Carlo.
Option Valuation with Monte-Carlo
• implementing Monte Carlo option pricing
1. make random drawings from the risk-neutral distribution of the stock price at the maturity of the option
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 24
2. or eac stoc pr ce ca cu ate t e payo on t e option
3. average these payoffs (this gives the risk neutral expected
payoff on the option)
4. discount the risk neutral expected payoff at the riskless
rate.
Key step: will explain how we do this
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How to calculate a random drawing of the
stock price under the risk-neutral distribution
: For trial generate drawing from normally
distribution with mean of zero and standard deviation of one ( ).
: Calculate drawing from risk-neutral distribution
of stock
S
p
te
ri
p 1
c
St
e
ep
as:
2
th
j
j
v
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 25
• Excel or @Risk will give you the random variables (the v’s)and then, for each S j we simply work out the payoff on the
option, average the payoffs and discount at the riskless rate
21( )
20
jr T T v
T S S e
σ σ − +
=
Monte-Carlo Example – first 4 samples: 3-month call
Ex. Price = 100, S 0 = 100, σ = 30%, r f = 10%
Norm Dist
(0,1) random
variable v j
Stock Price at
Maturity S j,T *
Option Payoff
Max(S j,T – 100,0)
Cumulative
average Option
Payoff
Cumulative
average
discounted at r f
1 0.7729 113.8468 13.8468 13.8468 13.5049
2 -0.2069 98.2863 0.0000 6.9234 6.7524
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 26
3 -1.6298 79.3967 0.0000 4.6156 4.5016
4 2.1393 139.7447 39.7447 13.3979 13.0671
Spreadsheet available on Portal
, 0
1 2
2* Note: j T
r T T v f jS S e
σ σ
− + =%
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Approximating the Risk-Neutral Distribution
with Monte-Carlo
0.06
0.08
0.10
b i l i t y d
e n s i t y
0.015
0.020
0.025
0.030
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 27
0.00
0.02
0.04
60 80 100 120 140 160
stock price
p r o b a
0.000
0.005
0.010
Monte-Carlo (5000 samples) Risk-Neutral Dis tr ibution of S tock Pr ice
Calculating the Option Price with Monte-Carlo
11
13
15
17
t i o n p r i c e
Monte-Carlo Price (cumulative average)
Black-Scholes Price
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 28
5
7
9
0 200 400 600 800 1000
number of Monte-Carlo samples
o
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Implementing Monte Carlo with @Risk
• The @Risk software package allows you to carryout Monte-Carlo in Excel even more simply.
• You will find @Risk essential in the exercises on
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 29
basket credit derivatives later in the course and so it
is worthwhile finding out now how to use it.
Summary
• In Black-Scholes theory market for options (and
effectively all claims on underlying asset) is complete
• This means we can calculate unique A-D prices and risk
neutral probabilities
• We can “read” the Black-Scholes formula as: RN expected payoff on option discounted at riskless rate
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 30
Cost of replicating portfolio
• In many cases (e.g., most American options) there is no
formula for the option price and we need to use a
numerical approach
idea: calculate the RN expected payoff and discount at the riskless
rate
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Key Concepts
• Market completeness in B-S
• Lognormal distribution
• RN distribution in B-S
• A-D prices in B-S
Risk Neutral Valuation, the Black-Scholes Model and Monte Carlo 31
• B-S formula as discounted RN expected payoff
• Delta (hedge ratio)
• Delta hedging strategy
• Monte Carlo valuation options