Seven
The Week Binomial Option Pricing
Model
!
Copyright Bruce N. Lehmann 2000-2011
!
Goals
See how stock prices evolve if returns
follow the binomial distribution
Learn how to hedge options with the
binomial model
Understand how to price options with that
model
Find out how to hedge complex payoffs
using that model
Learn the basics of the Black-Scholes model
!
Copyright Bruce N. Lehmann 2000-2011
!
The Binomial Model for Stock
Price Evolution
Notation
+ r = riskless rate
+ S = underlying asset price
+ C = call option value
- Put value given by put/call parity
Binomial model
+ S can take on only two values next period
- S u = S x u
- S d = S x d
!
+ option values next period are either Cu or Cd
Copyright Bruce N. Lehmann 2000-2011
!
The Magic Formula for the
Binomial Option Pricing Model
Asset
Buy 1 period call
Sell h shares
Borrow $B
Total
!
Initial
Cash Flow
C
HS
B
B + HS
C
Payoff on Expiry in:
Up state Down state
Cu
Cd
HSu
HSd
B(1+r) B(1+r)
Cu HSu Cd HSd
B(1+r) B (1+r)
Magic formula holds if payoff is zero in
both states find right values for H and B
Copyright Bruce N. Lehmann 2000-2011
!
The Magic Formula for the
Binomial Model, Continued
Set
+ Cu HSu B(1+r) = 0
+ Cd HSd B(1+r) = 0
Find right value of hedge ratio h
C u ! Cd
+ Cu HSu = B(1+r) = Cd HSd " H =
S u ! Sd
Find right value of borrowing $B
!
+ CuHSuB(1+r) = 0 (or CdHSdB(1+r) = 0
C u ! Cd
C d S u ! C u Sd
+ B(1 + r ) = C u !
Su " B =
S u ! Sd
(1 + r )(Su ! Sd )
Copyright Bruce N. Lehmann 2000-2011
!
The Magic Formula for the
Binomial Model, Continued
Magic formula
+ Ct+1 = [B (1+r)] + HSt+1 = Bt + HtSt+1
+ Note the absence of basis risk
Perfect hedge implies call option price is:
+ Ct = Bt + HtSt
Is the model complete?
!
+ No, we do not know the up and down market
call values Cu and Cd that determine H
+ There is one date on which we know Cu and Cd,
the expiration date
Copyright Bruce N. Lehmann 2000-2011!
The Magic Formula for the
Binomial Model, Continued
Valuation on last date before expiration
+ CT-1 = BT-1 + HT-1ST-1
+ Valueless if call cannot expire in the money
+ Riskless bond if it can t expire out of the
money
Consider intermediate strike prices
!
+ i.e., Cu = ST X = ST-1 u X and Cd = 0
(ST !1u ) ! X
+ " H T !1 =
ST !1 ( u ! d )
 !d[(ST !1u ) ! X ]
" BT !1 =
=
Copyright r )( u Lehmann 2000-2011!
(1 + r )S ( u ! d )
(1 + Bruce N. ! d )
The Structure of Valuation in the
Binomial Option Pricing Model
!
Pick the option pricing problem for an
arbitrary date
Find correct hedge ratio and borrowing for
given values of the option next period
Use hedge ratio and borrowing on the last
date before expiration to determine option
values on that date
Work your way back from expiry to find all
of the possible stock and option prices along
the binomial tree
Copyright Bruce N. Lehmann 2000-2011
!
Risk Neutral Valuation
Consider again the value of a call option on
an arbitrary date
+ C = B + HS
B and HS can be simplified using the
following manipulations
C d S u ! C u Sd
S(Cd u ! C u d ) (Cd u ! C u d )
=
=
+ B =
(1 + r )(Su ! Sd ) S(1 + r )( u ! d ) (1 + r )( u ! d )
C u ! Cd
C u ! Cd (1 + r )(C u ! Cd )
+ H " S =
S=
=
S u ! Sd
(u ! d)
(1 + r )( u ! d )
!
Copyright Bruce N. Lehmann 2000-2011
!
Risk Neutral Valuation,
Continued
So the call option value is given by:
(Cd u ! C u d ) (1 + r )(C u ! Cd )
+
+ C =
(1 + r )( u ! d )
(1 + r )( u ! d )
(1 + r ! d )C u + [ u ! (1 + r )]Cd )
=
(1 + r )( u ! d )
Define the risk neutral probability
1 + r " d er " d
#
+ ! =
u"d
u"d
Which makes the call option value:
!
1
1
!
+ C =
[! C u + (1 " ! )Cd ] =
E! [ C ]
1+ r
1 + r Copyright Bruce N. Lehmann 2000-2011!
Risk Neutral Valuation,
Continued
Probabilities figured nowhere in the pricing
formula but and 1 just like probabilities
+ they are positive
+ they sum to one
is called a risk neutral probability because
stock and option sell for expected present
value at the riskless rate r
+ S=[(Su)+(1)(Sd)]/(1+r) = E[ST]/(1+r)
+ C = [(Cu) + (1)Cd]/(1+r) =E[CT]/(1+r)
!
Copyright Bruce N. Lehmann 2000-2011
!
Risk Neutral Valuation,
Continued
Only a risk neutral investor would believe
that the up state occurs with probability
+ That is, r is not a risk-adjusted discount rate
+ is all probabilistic information needed to
value the option that is already embedded in S
Arrow-Debreu securities
!
+ /(1+r) is the price of security that pays one
dollar if up state occurs and zero otherwise
+ This is the price of insurance against the
occurrence of this state
Copyright Bruce N. Lehmann 2000-2011
!
Multiperiod European Option
Pricing via the Binomial Model
Setting
+ Binomial model assumptions hold for N periods
with constant parameters
+ Seek value of N period call option strike at X
Hard way
+ There are 2N possible states in N periods
- Not distinct: N possible terminal values
!
+ Solve all one period problems at time N-1
+ Using implied call prices at time N-1, solve all
one period problems at time N-2, etc.
Copyright Bruce N. Lehmann 2000-2011
!
Multiperiod Binomial European
Option Pricing, Continued
Easy way: compute risk neutral expected PV
Interim mathematics for them what wants it
Option value
C(S, X, N ) = e " rN E! RN {Max[SN " X, 0]}
+
Possible distinct stock prices at expiration
+ SN(j) = S0 uj dN-j, j=0,,N
Call pays off for all j such that SN(j) > X
+ Suj*dNj* X
- j* = smallest j s. t. jlog + u (N-j)log d + log S log X
!
Copyright Bruce N. Lehmann 2000-2011
!
Multiperiod Binomial European
Option Pricing, Continued
Option value = E{PV of RN stock price if in
the money less that of the strike price}
N
C(S, X, N ) = e ! rN " Prob[ j up moves][Su jd N ! j ! X ]
+
j= j*
Risk neutral probability of j up moves
N!
j
Prob! RN [SN = SN ( j)] =
! RN (1 " ! RN ) N " j
+
j!( N " j)!
- n! = n x (n-1) x x 2 x 1
Key risk neutral fact
!
j
e " rN! RN (1 " ! RN ) N " j u jd N " j = (e " r! RN u ) j [e " r (1 " ! RN )d ]N " j
j
# ! RN (1 " ! RN ) N " j
Copyright Bruce N. Lehmann 2000-2011
!
Multiperiod Binomial European
Option Pricing, Continued
Resulting call option pricing formula
N
+ C(S, X, N ) = e " rN & Prob RN [SN = SN ( j)][Su jd N " j " X ]
N
j= j*
N
j
j
= S&! RN (1 " ! RN ) N " j " Xe " rN &! RN (1 " ! RN ) N " j
j= j*
j= j*
# S $ Bi[ j*; N ,! RN ] " X $ e " rN $ Bi[ j*; N ,! RN ]
#S $ %+B
+ Bi[j;N,] = 1 Binomdist(j,N,p) in Excel
!
Note that, for all of the mathematical
complications, the essential structure and
risk neutral reasoning is the same
Copyright Bruce N. Lehmann 2000-2011
!
Objections to the Binomial
Model and a Simple Response
What is the length of a period?
Trading takes place almost continuously
Stock prices take on more than two values
each period?
Response: Can take period to be very short
and value options over many periods
!
+ T = 30 230 1.07 MMM sample paths
+ Black-Scholes is limit if u = et and d = 1/u
+ Real question is whether riskless hedge
Copyright Bruce N. Lehmann 2000-2011!
available over short time steps
The Value of General Derivatives
in the Binomial Model
Cash flows known function of underlying
asset price and time
+ CFt = CF(St,t)
St evolves on binomial tree
Derivative value is discounted risk neutral
expected presentNvalue of cash flows
E! [CF(St ,t)] = $ Pr ob[St = St ( k )]CF(Su k d T " k , t )
+
k#t
+ V0 = tT E[CF(St,t)]/(1+r)t
!
- Can value American options with valuable early
Copyright Bruce N. Lehmann 2000-2011!
exercise features too
A Simple Model of Firm Cash
Flows with Quantity Risk
Consider a firm the following costs and
revenues
+ Fixed costs per period: $FC
+ Constant marginal costs: $MC
+ Random prices: $P
+ Random production quantities: Q
Interpretations of random production
!
+ True randomness as in agriculture
+ Uncertain demand as in electricity generation
Copyright Bruce N. Lehmann
+ Uncertain dollar revenues due to currency risk 2000-2011!
Price/Quantity Relations and
Quantity Risk
Cash flows per period
+ CFt +1 = ( Pt +1 ! MC)Q t +1 ! FC
Quantity can always be written in the form
of a mechanical regression model
+ Q t +1 = ! 0 + !1Pt +1 + " t +1
+ Just a definition of parameters and residuals
Cash flows are quadratic in prices
!
+ CFt +1 = ( Pt +1 # MC)( ! 0 + !1Pt +1 + " t +1 ) # FC
+ Price risk is assumed to be hedgeable
+ Residual quantity risk is basis risk Copyright Bruce N. Lehmann 2000-2011!
in this case
Quantity Risk and the Magic
Formula for Cash Flows
The magic formula for cash flows in the
presence of quantity risk
CFt +1 = Bt + H t $ Pt +1 + % t $ Pt2+1 + ! t +1
+
Bt = &( " 0 $ MC + FC)
H t = ( " 0 & "1 $ MC)
% t = "1
! t +1 = ( Pt +1 & MC) $ # t +1 ; Cov[ Pt +1 , # t +1 ] = 0
Note that:
!
+ Riskless borrowing covers expected costs
ignoring quantity risk
+ New hedging term with quadratic derivative
Copyright Bruce N. Lehmann 2000-2011
!
Quantity Risk and the Valuation
of Cash Flows
Expected cash flows
+ E[CFt +1 | I t ] = "( ! 0 # MC + FC)
+ ( ! 0 " !1 # MC) # E[ Pt +1 | I t ]
+ !1 # {E[ Pt +1 | I t ]2 + Var[ Pt +1 | I t ]}
A trio of valuation intuitions
+ CFs require a risk premium if price risk is
systematic risk
+ Likely that E[ Pt +1 | I t ] ! Pt
+ Volatility component of expected cash flows is
given by: SD[ Pt +1 | I t ] = Pt ! SD[ Pt +1 Pt | I t ]
!
- i.e., proportional to returns on goods Copyright Bruce N. Lehmann 2000-2011!
Risk Management in the
Presence of Quantity Risk
Price risk Pt can be hedged with futures
+ i.e., short ! 0 " !1 # MC futures contracts
Volatility risk can be hedged with power
options
+ Exotic option with payoff given by the square
of the difference between the underlying asset
and the strike price
+ This power option pays off the square of the
price since the implicit strike price is zero
+ What should the power option sell for?
!
Copyright Bruce N. Lehmann 2000-2011
!
Hedged Cash Flows
Asset
Initial Cash Flow
Payoff on Expiry
Bt + H t Pt +1 + " t Pt2+1 + ! t +1
Firm cash flow
PV(CFt)
! H t ( Pt +1 ! Ft )
Sell forward Ht
0
! t " C( Pt2+1 , 0,1)
! t " Pt2+1
Sell t options
Borrow PV of
Bt + H t Ft
!( Bt + H t Ft )
Bt + H t Ft
1+ r
Cash flow PV discounted at riskless rate if
basis risk does not command a risk premium
!
Bt + H t ! Ft
PV(CFt ) =
+ " t ! C( Pt2+1 , 0,1)
+
1+ r
Copyright Bruce N. Lehmann 2000-2011!
A Binomial Model for the Pricing
of Quantity Risk
How should power option be priced?
Risk neutral pricing works for power
options
1
C( P , 0,1) =
E! [ Pt2+1 ]
+
1N+ r
2
2 2 k 2(T "k )
E! [ Pt +1 ] = $ Pr ob[St = St ( k )]S u d
2
t +1
k#t
+ Risk neutral probabilities unchanged from
before
!
Note that power option might have been
undervalued in the absence of hedging
Copyright Bruce N. Lehmann 2000-2011
!