# Notes 4

```Derivatives Valuation
Fundamental assumption concerning the functioning of security markets:
In a well functioning market, equilibrium prices should not present pure arbitrage
opportunities.
A pure arbitrage opportunity is a set of contemporaneous prices that allow asset
combinations that


produce positive current payoff and non-negative payoff in all future states. An
arbitrage opportunity of the first type.
require no initial investment, provide positive payoff(s) in at least one state and nonnegative payoff(s) in all other states. An arbitrage opportunity of the second type.
Any meaningful set of equilibrium prices is characterized by the absence of arbitrage
opportunities.
A derivative asset is one whose end-of-period payoff is exactly determined by the payoffs
on one or more assets or the value of a measurable quantity.
Arbitrage Theorem:
 1   (1  r ) (1  r ) 
 S (t )    S (t  1) S (t  1)    1 
2

  1
  
C (t ) C1 (t  1) C 2 (t  1)  2 
1. If positive constants 1 and 2 can be found such that asset prices satisfy this
representation then there are no arbitrage opportunities.
2. If there are no arbitrage opportunities then positive constants 1 and 2 can be
found.
Intuitively if positive constants, , satisfying the representation exist, linear combinations
of assets producing identical payoffs in state (1) and identical payoffs in state (2) have the
same cost.
1 and 2 are referred to as state prices. They are the price of Arrow-Debreu securities.
Define: It = information set at t It  {S(t), S1(t+1), S2(t+1), C(t), C1(t+1), C2(t+1), r}
~
Some authors refer to risk neutral probabilities with notation, P
From first row of representation;
1
~ ~
1 = (1+r)1 + (1+r)2 or 1 = P1  P2 , i.e. risk neutral probabilities sum to one. Risk
neutral probabilities equal the product of the future value factor, (1+r), and the state
prices i .
From second row of representation;
St = S11 +S22 then multiplying by (1+r)/(1+r)
~
~
~
~
~
St = (1/(1+r))[ P1 S1 + P2 S2] = P1 (S1/(1+r)) + P2 (S2/(1+r)) = (1/(1+r))* E P [St+1]
~
~
~
~
~
Ct = (1/(1+r))[ P1 C1 + P2 C2] = P1 (C1/(1+r)) + P2 (C2/(1+r)) = (1/(1+r))* E P [Ct+1]
Expectations calculated with the risk neutral probabilities and discounted at the risk free
rate equal the current values of the assets.
A martingale is a stochastic process such that;
EtP X t  s I t   X t
~
Using the risk neutral probability measure the expectation of the normalized future price
is equal to the current price, i.e. the risk neutral probability measure converts normalized
asset prices to a martingale processes.
Define X t  s 
Sts
1  r s
From the representation if we define B(t) = 1, then Bt+s=(1+r)s , represents the
accumulation in a money market account earning r per period. Notice that it is the
normalized asset value (ratio of S/B) that behaves like a martingale.
~S
~ C


St
C
 E P  t  s  or more to the point of derivative pricing, t  E P  t  s 
1
1
 Bt  s 
 Bt  s 
No arbitrage condition; (Simple 3 security, two-state representation)
S (t  1)
S (t  1)
Define gross return on S in states (1,2); R1 (t  1)  1
R2 (t  1)  2
St
St
Then from the first and second row of the representation;
1 = (1+r)1 + (1+r)2
1 = R11 + R22
Subtracting one from the other;
0 = [(1+r)-R1] 1 + [(1+r)-R2] 2
2
if I &gt; 0 then
R1 &lt; (1+r) &lt; R2
With no payouts the arbitrage theorem states that the risk neutral probability measure
produces expected return (1+r) for all asset prices. If an asset has a known positive
payout the expected return under the risk neutral measure is reduced by the size of the
payout.
An example:
A stock with current value \$50 will have value of either \$50 or \$60 at the end of one
year. The risk free rate is 10% per annum. The arbitrage free price of a European call
option with exercise price of \$55 and time to expiration is Ct = \$2.273.
 1  1.1 1.1
 50    50 60    1 

 
  
C (t )  0 5   2 
If C(t) = \$1, positive constants  satisfying the representation do not exist and arbitrage
opportunities are possible.



From the third row of the representation, 2 = 1/5 = 5/25.
Given 2 = 1/5, the second row implies that 1 = 19/25
These obviously do not satisfy the first row, 1  1.1(19/25)+1.1(5/25)=1.1(24/25).
An arbitrage opportunity of the first kind exists.
A position of &frac12; share of the stock and a short position in the call option produces a
payoff of \$25 in either future state. This position costs \$24. The present value of \$25/1.1
= \$22.73. Arbitrage profit obtained by selling &frac12; share and purchasing one call option,
proceeds \$24 and lending \$22.73. Proceeds of the loan, \$25, will cover the terminal
value of the position. Current income (\$24 – 22.73) = \$1.27.
Notice, if C(t) = \$2.273



From the third row of the representation, 2 = 2.273/5.
Given 2 = 1/5, the second row implies that 1 = 22.2724/50
From the first row, 1 = 1.1(22.2724/50)+1.1(2.273/5).
How was C(t) = \$2.273 determined. Recall that under the risk neutral measure
normalized asset prices behave as a martingale or
3
50 ~  60 
~  50 
~
 Pu    (1  Pu )   Pu  1 / 2
1
 1.1 
 1. 1 
C (t )
 5 
 0 
 1 / 2   1 / 2   2.273
Hence,
1
 1.1 
 1.1 
Replication of a derivative security in the absence of arbitrage;
A derivative asset is replicable with a pre-visible process in the absence of arbitrage.
Ct (St) is a european derivative with expiration date t+. Let  =1 for this illustration.
Let w tS, w tB be the units of the underlying asset S and risk free borrowing/lending held
in a portfolio at time t. The portfolio’s value at t;
Vt  wtS S t  wtB Bt
Find wS and wB that satisfy the following equations.
wtS S t11  wtB Bt (1  r )  C ( S t11 )
wtS S t21  wtB Bt (1  r )  C ( S t21 )
wtS 
C ( S t11 )  C ( S t21 )
S t11  S t21

 C ( S t11 )  C ( S t21 ) 
1

wtB  Bt (1  r ) 1  C ( S t11 )  S t11 
S t11  S t21



To prevent arbitrage opportunities the value of the derivative at (t), Ct ,must equal the
value of the replicating portfolio. The derivative is attainable (replicable) with a previsible process (w tS, w tB).

Ct = Vt  wtS St  wtB B = (1  r ) 1 P  C ( S t11 )  (1  P)  C ( S t21 )
where P 

S t (1  r )  S t21
= risk neutral probability of up-tick in binomial model.
S t11  S t21
In the multi-period binomial model, the value of the local replicating portfolio, Vt, is
equivalent to the discounted expectation of the normalized derivative expiration value
under the risk neutral measure. But a sequence of local replication strategies is a global
replication strategy guaranteeing an arbitrage induced value for the derivative at the root
node of the binomial tree.
4
The replicating strategy is self financing. The value of the replicating portfolio
constructed at (t-1) changes to exactly the value necessary to construct the replicating
portfolio at (t) without requiring addition/subtraction of wealth.
Separation of process and measure (probability set) – The size and interrelation of
possible changes affects the values of derivatives, the actual probabilities of achieving
realized values does not.
Measure(2) equivalent to measure (1) if measure (2) associates positive probability to all
events (1) that have positive probability under measure (1).
Risk neutral probabilities  equivalent martingale measure
1. Under the risk neutral measure the expected return of all assets in the system is
the risk free rate.
2. Risk neutral probability measure may be identified from information set, It
3. A stochastic process is said to be a martingale if the expected change in the value
of the process is always zero. Under the risk neutral measure, normalized
(discounted at the risk free rate) asset prices behave as a martingale.
4. A model does not permit arbitrage opportunities if and only if the model admits at
least one risk neutral probability measure(s).
5. Price of derivative obtained by “replication” method can be recovered simply by
calculating discounted value (risk free rate) of expected payoff using risk neutral
probabilities. The replicating portfolios are specific to the claim being valued, the
risk neutral probabilities depend only on the primary assets.
6. The replicating portfolios in the binomial setting are a process of self-financing
replicating portfolios.
7. Further a market is said to be complete if a derivative security can be replicated
using only the primary securities. In a complete market any derivative security
has a unique arbitrage-free price. A market is complete if and only if the model
admits a unique risk-neutral probability measure.
8. Multiple risk-neutral probabilities can exist if and only if there are multiple stateprice vectors. This means that at least one Arrow-Debreu security has many
possible prices consistent with no-arbitrage, and this is possible only if the ArrowDebreu security is not replicable. Thus, the market must be incomplete.
5
Derivative valuation methods:
Development of Discrete time – Discrete value methods (binomial):



Given dynamics of underlying, generate tree with desired time step from the
pricing date till the derivative expiration date,
uncover the risk neutral probability measure from the underlying asset’s dynamic
process and the risk free rate.
Recursively apply the martingale property working backward (expiration to
pricing date) through the tree to recover the no-arbitrage value of the derivative.
Underlying asset price dynamics (no payout):
Multiplicative binomial recombining process:
u  e t , d  e  t , where  is the annualized volatility of logarithmic returns of the
underlying asset and t = T/n.
T = time in years till derivative expiration date and
r = annualized continuously compounded risk free rate.
Risk neutral probability measure {P, (1-P)} (per time step): P 
S 0 e rt  S 0 e 
S 0 e
t
 S 0 e 
t
t
Recursive valuation through tree as necessary given T, n
~
~
~
Ct = e  rt [ P1 C1 + P2 C2] = e  rt * E P [Ct+1]
Expectations calculated with the risk neutral probabilities and discounted at the risk free
rate equal the current values of the assets.
Numerical derivatives are calculated naturally from perturbations of the input factor.

c  c
2dS
 c   c0 c0  c  


 dS
dS 
 dS
6
Development of the partial differential equation approach to derivative valuation.
Ito’s Lemma:
Let C(St,t) be a twice-differentiable function of t and of the random process St:
dS t  at dt   t dWt then
2
dCt  SCt dSt  Ct dt  12 SC2  t2 dt after substituting for dSt
t
and applying multiplication rules: dt2 = 0, dt*dW = 0, dW2 = dt;
2
dCt  SCt at  Ct  12 SC2  t2 dt  SCt  t dWt

t

Forming Risk Free portfolios:
Let dSt be the SDE (equation of motion) for the underlying asset. Because Ito’s lemma
indicates the SDE of the derivative is a function of the same stochastic process, dWt, it
will in general be possible to form a portfolio of the derivative and underlying asset such
that the stochastic component’s cancel out. To prevent arbitrage opportunities the hedge
portfolio must earn the risk free rate of return.
Given portfolio weights, w1 , w2 the value of the portfolio is then; Pt = w1C + w2S
Assuming the portfolio weights are previsible, i.e knowable at the start of the time step;
dPt = w1dC + w2dS. Choosing weights such that w1 = 1, w2 = -Cs makes the portfolio
stochastic components, dWt, cancel out.

dP   C

dPt  C s at  12 C ss t2  Ct dt  C s t dWt  C s at dt   t dWt 
t
1
2
  Ct dt
2
t
ss
To prevent arbitrage opportunities this portfolio must earn the risk free rate; dPt = rPtdt .
rPt dt 
C
1
2
 t2  Ct dt
ss
r C  C s S dt 
C
1
2
 t2  Ct dt
ss
Fundamental PDE for a European derivative asset on non-payout asset;
 rC  rC s S  Ct  1 C ss t2  0
2
s.t.C ( ST , T )  G ( ST , T )
C s  , C ss  , Ct  
Where G(ST,T) Is the payout rule for the derivative contract being valued.
7
The fundamental PDE for European derivative asset on a non-payout asset is a linear
second-order parabolic PDE. Each derivative is unique in the specification of the
dependence of the terminal payoff on the ST, G(ST , T) (boundary constraints).
The Black-Scholes option pricing model is the most prominent continuous time European
option pricing model.
The Black-Scholes model, Black Fischer and Scholes Myron, (1973), price derivative
security through “replication.”
Given S0 and dSt  u * St * dt   * St * dWt , then;
dW ~ N(0,dt)
ST  S 0 exp{( u  0.5 * 2 )T  WT }
W ~ N(0, T)
Given this assumption concerning the dynamics of the underlying assets’ price, the price
of a European call/put option with strike K, time to expiration, T, risk free rate, r, and 
that precludes arbitrage opportunities is
c  S 0 * N (d1 )  e  r *T * K * N (d 2 )
p  K * e  rT N (d 2 )  S 0 * N (d1 )
d1 
ln(
2
S0
)  (r   ) * T
K
2
 T
d 2  d1   T
Derivatives of the Black-Scholes model
c
S
p
S
   N (d1 )  0
 2c    (2 )  12 * exp(  1 d 2 ) * ( S T ) 1  0
2 1
S 2
c     1
t
2
same as call option
p
1
t
2
1
S exp(  12 d12 )  rKe  rT N (d 2 )  0
2T
1
c
 Vega  S T * (2 ) 2 * exp(  1 d12 )

2
c  rho  KTe  rT N (d )  0
2
r
   N (d1 )  1  0
1
S exp(  12 d12 )  rKe  rT N (d 2 )  0
2T
same as call option
0
p
r
8
 rho   KTe rT N (d 2 )  0
Development Monte Carlo valuation methods:



Recognizing that under the risk neutral measure the expected rate of return of all
assets in the system is the risk free rate, substitute the risk free rate for the
expected rate of return (drift) of the risk free asset. Simulate many paths of length
T or realized values at T for the underlying asset.
Allowing the relative frequency of each realized value to approximate the risk
neutral probability of that state, calculate the payoff on the derivative for each
realized value of the underlying asset and determine the expected (average) value
of the derivative’s payoff.
Discount the derivative’s average payoff to the present at the risk free rate.
Risk neutral valuation with multiplicative binomial process (no payouts):
Given GBM with drift m and volatility, : dS = mSdt + Sdz
u  e t , d=1/u
d &lt; e r  t &lt; u .
e r  t  d
P
(u  d )
European option on an underlying asset with a continuous payout (=dividend
yield)
c  S * e T * N (d1 )  e  rT * K * N (d 2 )
ln( S / K )  (r     2 )T
d1 
 T
2
d 2  d1  T
Risk neutral valuation with multiplicative binomial process (continuous payout ):
Given GBM with drift m and volatility, : dS = mSdt + Sdz
u  e t
d=1/u
d &lt; e ( r  )t &lt; u .
9
e ( r  ) t  d
P
(u  d )
For options on a foreign currency, S = FX = (\$/unit), replace  with foreign country risk
free rate, i.e. yield on zero coupon sovereign debt of foreign country with maturity less
than or equal to option expiration date.
European option on currency S = (\$/&yen; , \$/&pound;) rd and rf are the domestic (\$) and
foreign (&yen;)risk free rates respectively.
c  S *e
r f T
* N (d1 )  e  rd T * K * N (d 2 )
ln( S / K )  (rd  r f   2 )T
d1 
 T
2
d 2  d1   T
Black’s Model: European option on futures contract (F=futures price)
c  F * e  rT * N (d1 )  e rT * K * N (d 2 )
ln( F / K )  ( 2 )T
d1 
 T
2
d 2  d1   T
Given GBM with drift m and volatility, : dF = mSdt + Sdz
u  e t
d=1/u
d &lt; e r  t &lt; u .
P
1 d
(u  d )
10
1. Find the value for each of the five European digital contracts defined below. For
this problem =0.25 and the digital contracts have time to maturity, T = 1 year. A
stock has current price, S0 = \$100. The stock dynamics are modeled with a
multiplicative binomial process such that in the up state St+ = u*St and down
state St+ = u-1*St. For this stock u = 1.10. The risk free rate is r = 0.05.
Digital Contracts;
C4U – has payoff \$1 if the stock moves up 4 times and zero otherwise.
C3U – has payoff \$1 if the stock moves up 3 times and zero otherwise.
C2U – has payoff \$1 if the stock moves up 2 times and zero otherwise.
C1U – has payoff \$1 if the stock moves up 1 time and zero otherwise.
C0U – has payoff \$1 if the stock moves up 0 times and zero otherwise.
Find the value of a European call options with strike price, K = \$100, and time to
maturity T = 1 year.
European Digital Contracts - Binomial Model
S(0)
up
down
0
\$100

0.25
1.1
T
1
0.9091
r
0.05
1
2
3
C4U
C3U
C2U
C1U
C0U
\$146.41
4
1
0
0
0
0
\$121.00
0
1
0
0
0
\$100.00
0
0
1
0
0
\$82.64
0
0
0
1
0
\$68.30
0
0
0
0
1
\$133.10
\$121.00
\$110.00
\$100
\$110.00
\$100.00
\$90.91
\$90.91
\$82.64
\$75.13
State Prices ()
P(up)
0.5417
C(4U)
0.0819
P(down)
0.4583
C(3U)
0.2772
C(2U)
0.3519
C(1U)
0.1985
C(0U)
0.0420
sum(C)
0.95152
(1+r)-4
0.95152
11
Value of European Call Option with Strike Price, K =
0
1
2
100
3
4
\$46.41
\$34.33
\$23.45
\$15.27
\$9.62
\$21.00
\$11.23
\$6.01
\$3.22
\$0.00
\$0.00
\$0.00
\$0.00
\$0.00
\$0.00
\$9.62
Notice the connections:
1. Sum of state prices = present value at risk free rate
2. Arbitrage free european call option value = \$9.62=
[0.0819
0.2772
0.3519
0.1985
0.0420]*
[\$46.41
\$21.00
\$0.00
\$0.00
\$0.00]
12
The studentized range test is a two-sided test, the null hypothesis is rejected if the test
statistic is greater in absolute value than the critical value for the chosen confidence level.
Fractiles of the distribution of the Studentized Range from samples of size, T
Source: H.A. David, H.O. Hartley and E.S. Pearson, &quot;The distribution of the ratio, in a single normal sample, of range to standard deviation,&quot;
Biometrika 61 (1954) pg. 491
0.9
0.95
0.975
0.99
0.995
T
3
1.997
1.999
2.000
2.000
2.000
3
4
2.409
2.429
2.439
2.445
2.447
4
5
2.712
2.753
2.782
2.803
2.813
5
6
2.949
3.012
3.056
3.095
3.115
6
7
3.143
3.222
3.282
3.338
3.369
7
8
3.308
3.399
3.471
3.543
3.585
8
9
3.449
3.552
3.634
3.720
3.772
9
T
0.005
0.01
0.025
0.05
0.1
10
2.470
2.510
2.590
2.670
2.770
3.570
3.685
3.777
3.875
3.935
10
11
2.530
2.580
2.660
2.740
2.840
3.680
3.800
3.903
4.012
4.079
11
12
2.590
2.650
2.730
2.800
2.910
3.780
3.910
4.010
4.134
4.208
12
13
2.650
2.700
2.780
2.860
2.970
3.870
4.000
4.110
4.244
4.325
13
14
2.700
2.750
2.830
2.910
3.020
3.950
4.090
4.210
4.340
4.431
14
15
2.750
2.800
2.880
2.960
3.070
4.020
4.170
4.290
4.430
4.530
15
16
2.800
2.850
2.930
3.010
3.130
4.090
4.240
4.370
4.510
4.620
16
17
2.840
2.900
2.980
3.060
3.170
4.150
4.310
4.440
4.590
4.690
17
18
2.880
2.940
3.020
3.100
3.210
4.210
4.380
4.510
4.660
4.770
18
19
2.920
2.980
3.060
3.140
3.250
4.270
4.430
4.570
4.730
4.840
19
20
2.950
3.010
3.100
3.180
3.290
4.320
4.490
4.630
4.790
4.910
20
30
3.220
3.270
3.370
3.460
3.580
4.700
4.890
5.060
5.250
5.390
30
40
3.410
3.460
3.570
3.660
3.790
4.960
5.150
5.340
5.540
5.690
40
50
3.570
3.610
3.720
3.820
3.940
5.150
5.350
5.540
5.770
5.910
50
60
3.690
3.740
3.850
3.950
4.070
5.290
5.500
5.700
5.930
6.090
60
80
3.880
3.930
4.050
4.150
4.270
5.510
5.730
5.930
6.180
6.350
80
100
4.020
4.000
4.200
4.310
4.440
5.680
5.900
6.110
6.360
6.540
100
150
4.300
4.360
4.470
4.590
4.720
5.960
6.180
6.390
6.640
6.840
150
200
4.500
4.560
4.670
4.780
4.900
6.150
6.380
6.590
6.850
7.030
200
500
5.060
5.130
5.250
5.370
5.490
6.720
6.940
7.150
7.420
7.600
500
1000
5.500
5.570
5.680
5.790
5.920
7.110
7.330
7.540
7.800
7.990
1000
13
```