Transcript No Slide Title

Version 1/9/2001
FINANCIAL ENGINEERING:
DERIVATIVES AND RISK MANAGEMENT
(J. Wiley, 2001)
K. Cuthbertson and D. Nitzsche
Lecture
Asset Price Dynamics
© K.Cuthbertson, D. Nitzsche
TOPICS
Stochastic Processes:
Weiner, Ito, GBM, Black-Scholes PDE
RNV and Monte Carlo Simulation
Finite Difference Methods
© K.Cuthbertson, D. Nitzsche
Stochastic Processes:
Weiner, Ito, GBM
Black-Scholes, PDE
© K.Cuthbertson, D. Nitzsche
Weiner Process
Weiner Process
[17.3]
z =  t
[17.4a] Expected Value
[17.4b] Variance
[17.4c] Standard Deviation
E(z) = 0
var(z) = E(z)2 = t
std(z) = t
Generalised Weiner Process
 x = a t + b  z = a t + b  t
E( x) = a t
var( x) = b2 t
Ito Process
dx = a(x,t) dt + b(x,t) dz
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GBM and Ito’s Lemma
Geometric Brownian Motion
[17.16b]
dS/S =  dt +  dz
GBM is Ito process with a = S and b = S
dS
~ N (  dt , 
S
2
dt )
Ito’s Lemma
If S follows an Ito process then the stochastic differential
equation (SDE) for any function (S,t)~ option premium)
 f

f
1 2 2 f
df 
dt  
dS  b
dt

2
t

S
2

S


Substitute for dS from [17.20]
[17.22]
 f
f 1 2  2 f 
 f 
df    a
 b
dt   b dz
2 
S 2 S 
 S 
 t
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Table 17.1 : SDE for Future’s Prices using Ito’s Lemma
Ito’s eqn 17.22 for f(S,t)
 f
f 1 2  2 f 
 f 
df    a
 b
dt

 b dz
2 

t

S
2

S
 S 


a = S, b= S
Futures price:
F = Ser(T-t)
The stock price follows a GBM:
dS = (S)dt + (S) dz
Substituting in Ito’s eqn above:
dF = [ er(T-t) (S) – r S er(T-t) ] dt + er(T-t) (S) dz
Substituting F = Ser(T-t)
dF = ( - r)F dt +  F dz
which is a GBM for dF/F with drift rate ( - r) and variance rate .
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Black-Scholes PDE
Replication portfolio of stocks and bonds
mimics the payoff of derivative security thus offsetting any
uncertainty dz inherent in the derivative security, taken in
isolation.
The resulting equation (see appendix 17.2) for the price of
the derivative security is deterministic (ie. non-stochastic)
and is known as the Black-Scholes PDE
[17.48a]  f
f
12f
2
 S    r f (S , t )
  (rS ) 
2
2 S
 t S

This PDE can be solved by standard methods, to give the
B-S closed form solution for the derivatives price.
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Does a Forward Contract Obey B-S, PDE ?
Value of the forward contract:see chapter 2
[17.55]
f
  r Ke  r (T t )
t
f = S - Ke-r(T-t)
f
1
S
 2 f
 0
2
S
Substituting Black-Scholes PDE and using [17.55]
The LHS of the Black-Scholes equation becomes :
[17.57]
r[-Ke-r(T-t) + S] = r f
Hence f satisfies the PDE
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Table 17.2 : From GBM for dS/S to the properties of S
Assume dS/S follows a GBM with drift rate  and variance rate, 2
dS
 dt  dz
S
Use Ito’s lemma to obtain the stochastic process for d( ln S )
d(ln S) =  dt +  dz
where  =  - 2/2.
Because dz is N(0,1) then the distribution of ln(S) is normal (and S is
lognormal) with
ln(ST/So) ~ N(T, 2T
Statistical distribution theory then indicates that the level of ST has
mean and variance
EST = So eT


var(ST) = S 2e 2 T e 2T  1
o
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Risk Neutral Valuation, RNV
and
Monte Carlo Simulation, MCS
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Risk Neutral Valuation, RNV
When pricing an option
it is valid to use q as the probability of an ‘up’ move
~this is equivalent to the stock price growing at the risk
free rate r
However, the resulting value for the option premium, is
valid in the real world.
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Risk Neutral Valuation, RNV
Step 1:
Assume the expected return of the underlying asset (eg.
stock) equals the risk free rate.
(For example, for the BOPM this involves using q as the
probability of an ‘up’ move, which is consistent with S
growing at the risk free rate r)
Step 2:
Calculate the expected payoff from the derivative at
maturity
Step 3:
Discount the expected payoff at the risk free rate to
obtain the price of the derivative
© K.Cuthbertson, D. Nitzsche
MCS for Option Premia (Excel T17.4+Gauss)
Under RNV the call premium is :
[17.74]
C = e-rT E* [max(ST - K, 0)]
Generate S
[17.77a]
St = [1 +  t +  t t ]St-1
[17.77b]
St = St-1 exp[( - 2/2)t +  t t]
[17.80]
Payoff-C(1) = max {0, - 100} =
10.12
m
After m-runs: [17.81]
Cˆ  e 0.05( 0.3)  Payoff[C (i ) ] / m
i 1
Hedge Parameters
f ( S  S )  f ( S  S )

2(S )
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Stochastic Volatility and MCS
dS = ( r- ) S dt
+ S (  V ) dzs
dV = b (Vm - V) dt + ( V ) dzv
Vm = long run value of volatility of stock return
dzs dzv =  dt
b ,  and  are parameters to be estimated
© K.Cuthbertson, D. Nitzsche
Variance Reduction Methods
Antithetic Variables
Each time we draw a value for  we also use -
Both are used to generate new values for the stock price.
We therefore get “two stock prices for the ‘price’ of one
random draw”
This technique can be applied to any symmetric
distribution
So in any run we have two option payoffs using + and
using -
We then take the average of the two payoffs as the payoff
for that simulation
© K.Cuthbertson, D. Nitzsche
Variance Reduction Methods
Control Variate Method
To calculate value of a ‘complex’ option-A, AMCS
‘Simple’ option-B whose value by B-S = BBS
Now value option-B using MCS giving BMCS
(its value would be close to but not equal to BBS (because
of MCS sampling error)
Control variate technique adjusts AMCS depending on how
big the error is in the MCS valuation of option-B
The ‘new and improved’ estimate for option-A is
f Anew  f AMCS  ( f BBS  f BMCS )
© K.Cuthbertson, D. Nitzsche
Finite Difference Methods
© K.Cuthbertson, D. Nitzsche
Finite Difference Methods
Approximate the continuous time B-S, PDE using
numerical derivatives on a ‘grid’
Impose boundary conditions
Put Option
At S=0 then f = K e-rt and at S>>K then then f = 0
Solve the PDE using numerical methods
© K.Cuthbertson, D. Nitzsche
Figure 17.3 :Approximations for f / S
f
fi+1
Central
difference
Forward
difference
Backward
difference
fi
Derivative required
for this point
fi-1
S
S
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S
Figure 17.2 :Use of grid points
Differential with respect to S
(index for S is i )
fki+1 
Central
difference
fik

fki-1 
Value of option, f(S,t)
(index for t is k)
forward
difference
backward
difference
Differential with respect to time
(Note: as k increases t decreases)
fik+1 
 f ik
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 fki+1
fik+1 
 fik
 fki-1
Figure 17.1 : Finite difference grid
Stock Price (index i)
Current Stock Price: S = 4(DS).
Hence value of f40 will be solution for the option premium.
f36 is determined by the values of f
at points A, B and C
5
A
4
B
3
C
2
S
1
0
1
t
2
3
4
5
6
© K.Cuthbertson, D. Nitzsche
T
time, t
(index k = T - k t )
Finite Difference Methods
One of the nodes on the left vertical axis will coincide with
the current stock price and the solved value for f at this
same node will be the option premia
fi k  f i(S ), T  k t ]
S = i(S), t = T-k(t)
at k = 0, then t = T (ie. expiry) and as k increases, real time
t decreases.
 2 f  f i k1  f i k f i k  f i k1 


 / S
2
S
S 
 S
fi  fi
 f

t
t
k
[17.87]
k 1
fi k 1  Aik fi k1  (1  Bik ) fi k  Cik fi k1
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Finite Difference Methods
From [17.87] note that we can calculate the value of fik+1
once we know the values of f at time k for the three nodes
i, i-1 and i+1 (figure 17.1).
Solve for fik+1 by working backwards through the grid (once
we have the terminal conditions) ~ explicit finite
difference method.
American
Compare fik+1 with the payoff to early exercise K-S = K - i
S at each node, and take the max. value.
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LECTURE ENDS HERE
© K.Cuthbertson, D. Nitzsche