Post on 01-Jan-2016
transcript
1
Derivatives & Risk Management: Part II
Models, valuation and risk management
2
What we are going to do
– Value a derivative in a one period binomial model:
• classical approach vs. risk neutral valuation
– The binomial and the random walk models for the stock price.
3
What we are going to do II
– The Black & Scholes model– Risk neutral vs classical valuation in
continuous time
– Applications of the Black Scholes model• Portfolio insurance
4
What we’re going to do III
– When B&S is inapplicable• Monte Carlo• binomial trees
– Risk management• The partial derivatives• Value at Risk
5
This lecture
• More abstract than earlier lectures but will quickly lead to practical applications
• Concepts of valuation for derivatives on traded assets
• Illustrative tool:– 1 period binomial– 2 period binomial model
6
• Goal: to provide the insights necessary to understand valuation theory
• Contrast the “Classic” approach to that of “Risk Neutral Valuation”
• Novelty: introduce assumptions about the statistical properties of security prices (binomial, GBM)
7
Relative pricing
• A derivatives is a financial instrument whose value can be derived from that of one or more underlying securities
• The derivative will be priced RELATIVE the underlying.– Information about market conditions expectations,
risk preferences, etc... will be incorporated into the value of the derivative insofar as they are compounded in the current prices of the underlying.
8
Arbitrage and risk free portfolios
• Derivative valuation relies on the assumption of no arbitrage– the idea is to construct a portfolio of the derivative
and the underlying which yields a known risk free payoff regardless of future states of the world.
– Given that this is the case we can value the portfolio by discounting at the risk free rate
– and then we can solve for the price of the derivative
9
More concretely
• To make matters more concrete we will think of the derivative as the a call option and the underlying as a non-dividend paying stock
• The idea is however applicable to any derivative security. Most complications will be purely technical
10
Preliminaries
• Basic assumptions– frictionless markets– borrowing and lending– investors prefer more to less– no arbitrage
• Then we add the binomial stock dynamics - unrealistic simple one-period model.
11
Preliminaries II
t
t
ed
eu
2
1
60
%10
81.0
24.130.0
70
tT
X
r
d
u
S
12
The classical approach
70
542.86 uSS uT
620.56 dSS dT
The stock price tree
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The classical approach II
542.26)0,max( XSc uT
uTOption price tree
0)0,max( XSc dT
dT
14
The classical approach III
• We will now construct a portfolio of stocks and one short call option.
• The value of this portfolio in 6 months will be the (random) value
TTT cS
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The classical approach IV
• The trick is to design this portfolio so that its value turns out to be non-random.
• In other words we want to select so that
dT
uT
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The classical approach V
• More precisely
8870.0
062.56542.26542.86
dT
dT
uT
uT
dT
uT
cScS
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The argument
• Now since
22.5062.5689.054.2654.8689.0 d
TuT
with certainty we must have thatwith certainty we must have that
77.4722.505.010.0 ee TtTr
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The result
• And since
32.14
77.47708870.0
c
ccS
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The hedge ratio
T
TdT
uT
dT
uT
S
c
SS
cc
Binomial equivalent to the “delta” which we will talk about in more detail later on when we’ve introduced the B&S model.
20
A summary of the argument
• Set up a portfolio in the call and the stock (long in one and short in the other) which has a certain payoff in both states of the world
• Argue that since its payoff is certain the appropriate discounting rate is the risk free rate
• Solve for the option price using today’s stock price, the hedge ratio and r
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A slightly different way of seeing it
• Instead of setting up risk free portfolios in the asset and the option we could replicate the derivative by borrowing and investing in the stock
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A different way
uT
uT cKS
dT
dT cKS
cKeS rT
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A different way II
542.26542.86 uTcK
0620.56 dTcK
224.50
8870.0
K
32.14224.50708870.0 2
110.0
ec
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IMPORTANT
• This approach is by no means limited to options.
• It is the basic approach (which from case to case may require some alteration) to the valuation of any derivative security on a traded financial security
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An alternative approach:“Risk-neutral” valuation
• In principle we could value an option by discounting the expected future cash flow at the appropriate rate
dT
uT
tTT
tT cqqcecEec 1
• Problem: the discount rate is extremely hard to estimate
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“Risk-neutral” valuation
• Suppose that we turn the question around:– suppose we insist on discounting the
expected payoff at the risk free rate, then what probabilities should we use to obtain the arbitrage free (correct) option price
• Answer:
du
dep
tr
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A sketch of the proof
• Start with the composition of the initial “classical portfolio”
dT
dT
tTr cSeS
Sc
1u d u dT T T Tu dT T
c c c c
S S u d S
• Substitute
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Sketching on
• Then after some reshuffling you will find that:
dT
uT
tTr cppcec 1
where
du
dep
tr
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Interpretation
• One way of interpreting this result is that it is the present value of the expected payoff in a risk neutral world
• p are the risk free probabilities
• and since there is no compensation for risk the appropriate discounting rate is r
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Interpretation II
• This interpretation is dangerous since it may lead to the misconception that in order to use the result one implicitly assumes that investors are risk neutral
• But we have seen that this is not so, all we require are frictionless markets and agents that prefer more to less
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Interpretation III
• Furthermore this interpretation is not fully accurate since it neglects the fact that the volatility that we will use is the one estimated on actual markets and this quantity may reflect market preferences
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Bottom line
• The risk neutral valuation result should be considered a purely mathematical tool that will prove extremely useful for computing derivative prices in more complex situations
• It does not require any assumptions about investor risk preferences.
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Using it
• Back to our example
5671.0
du
dep
tr
32.14043.0542.265671.0
1
ˆ
5.010.0
e
cppce
cEecdT
uT
tTr
ttTr
34
Constraints on probabilities
• For p to qualify as a probability we require that
ued
dude
du
de
p
tr
tr
tr
0
10
10
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A general statement of the result
• For an arbitrary derivative f we can write its price as
TtTr feEf ˆ
And in the special case of non random interest rates
TtTr fEef ˆ
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Comments
• In practice the question is how to compute the expectation under risk neutral probabilities.
• This task will differ in degrees of difficulty that depend on assumptions about– the dynamics of underlying security, interest rates– features of the derivative security (path
dependence...)
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Example
• Consider a forward on a security paying no dividends
• What is the “risk neutral” expectation of the underlying security at expiration?
1T
r tr t
E S pSu p Sd S p u d d
e dS u d d Se
u d
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Example II
• What is the value of a long forward?
trT
tr KeSKSEef ˆ
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A 2 period example
t
t
ed
eu
60
%10
86.0
16.130.0
704
12
2
1
X
r
d
u
S
tntT
40
Novelties
• On the path to more realistic models
• The hedge ratio will have to be adjusted dynamically as time goes by and the stock price moves.
• This in contrast to the portfolio we have to set up to value e.g. a forward contract.
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The binomial tree
10
70
udT
duT
udT
c
SS70
81.33
60.25
49.34
49.94
uuT
uuT
c
S
0
86.51
ddT
ddT
c
S
uuu cS 25.025.025.0 ,,
ddd cS 25.025.025.0 ,,
ddc 25.025.0 ,
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Computations
We begin at t=0.25, after an up move
0.1
107049.3449.94
25.0
25.025.0
5.05.025.05.05.025.0
5.05.0
u
uu
ududuuuuuu
uduu
cScS
81.22
52.58
25.0
5.04
10.0
25.0
u
uuu
c
e
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Computations II
Then at t=0.25, after a down move
55.0
086.511070
25.0
25.025.0
5.05.025.05.05.025.0
5.05.0
d
dd
dddddududd
ddud
cScS
33.5
88.27
25.0
5.04
10.0
25.0
u
ddd
c
e
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The binomial tree
7081.22
33.81
25.0
25.0
u
u
c
S
?, 25.025.0ddc
33.5
25.60
25.0
25.0
u
u
c
S
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The final step
83.0
33.525.6081.2233.81
0
00
25.025.0025.025.00
25.025.0
dduu
du
cScS
52.1452.4370
52.43
025.0
25.04
10.0
u
d
c
e
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Risk Neutral valuation 2 periods
• Begin by computing the risk-neutral probability
55.0
du
dep
tr
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The binomial tree
70
81.33
49.34uuTc
0ddTc
10udTc
81.22
10)55.01(49.3455.0
ˆ
4
10.0
5.025.0
e
cEec utru
p
p
p
(1-p)(1-p)
(1-p)
33.5
ˆ5.025.0
cEec u
tru
52.14
ˆ25.000
cEec tr
48
An even quicker alternative
20.01
50.012
30.055.0
2
22
pp
ppp
pp
dd
ud
uu
52.14
020.01050.049.3430.0
ˆ
2
10.0
5.02
e
cEec t
49
Replicating portfolios and dynamic arbitrage valuation
• The idea that the “hedge ratio” must be changed as the stock price evolves brings us to the concept of dynamic replication which is the underlying concept for the development of the Black & Scholes model
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Replication
• This should be contrasted to “static” replication– put-call-parity– coupon bonds as portfolios of discount bonds
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Models for the behavior of security prices
• Discrete time - discrete variable– binomial model
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Another model
• Discrete time - continuous variable
tS tttS ,1
1tS
112 , tttS
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Other models
• Continuous time - discrete variable
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The geometric Brownian motion
• Continuous time; continuous variable
0
20
40
60
80
100
120
140
160
180
200
1 30 59 88 117 146 175 204 233 262 291 320 349 378 407 436 465 494
Markov Processes
• In a Markov process future movements in a variable depend only on where we are, not the history of how we got where we are
• In the developent of the B&S model, we will assume that stock prices follow Markov processes
Weak-Form Market Efficiency
• The assertion is that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices. In other words technical analysis does not work.
• A Markov process for stock prices is clearly consistent with weak-form market efficiency
Variances & Standard Deviations
• In Markov processes changes in successive periods of time are independent
• This means that variances (not standard deviations) are additive