binomial option pricing model (bopm) references: neftci, chapter 11.6 cuthbertson & nitzsche,...
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Binomial Option Pricing Model Binomial Option Pricing Model (BOPM)(BOPM)References:
Neftci, Chapter 11.6
Cuthbertson & Nitzsche, Chapter 8
1
Linear State PricingLinear State Pricing
A 3-month call option on the stock has a strike price of 21. Can we price this option?
◦ Can we find a complete set of traded securities to price the option payoffs?
◦ If we make the simplifying assumption that there are only 2 states of the world (up and down), then we only need the prices of two independently distributed traded assets, e.g. the underlying stock and the risk-free asset
Linear State PricingLinear State Pricing
Algebraically,
If S = 2, this is a system of equations in two unknowns
To get a unique solution for it, we need at least 2 independent equations
S
ss
f
sS
sq
sff
sS
s
s
R
dRR
dP
s
10
11 00
1
1
)1(1
1
The Binomial ModelThe Binomial Model
A stock price is currently S0 = $20
In three months it will be either S0u = $22 or S0d = $18
Stock Price = $22
Stock Price = $18
Stock price = $20
Stock Price = $22Option Price = $1
Stock Price = $18Option Price = $0
Stock price = $20Option Price=?
A One-Period Call OptionA One-Period Call Option
Option tree:
6
Risk-Neutral Valuation Risk-Neutral Valuation ReminderReminderUnder the RN measure the stock price
earns the risk-free rateThat is, the expected stock price at time T
is S0erT
When we are valuing an option in terms of the underlying the risk premium on the underlying is irrelevant
See handout on RNV
Risk-Neutral TreeRisk-Neutral Tree
f = [ q f u + (1 – q ) f d ]e-rT
The variables q and (1– q ) are the risk-neutral probabilities of up and down movements
The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate
S0u ƒu
S0d ƒd
S0
ƒ
q
(1– q )
Risk-Neutral ProbabilitiesRisk-Neutral ProbabilitiesS0u = 22 ƒu = 1
S0d = 18 ƒd = 0
S0
ƒ
q
(1– q )
Since q is a risk-neutral probability,
22q + 18(1 – q) = 20e0.12 (0.25)
q = 0.6523
RN Binomial Probabilities RN Binomial Probabilities FormulaFormula
RN probability of up move:
RN probability of down move:
1 – qWith this probabilities, the underling grows
at the risk-free rate (check it out)
du
deq
rT
6523.09.01.1
9.00.250.12
e
du
deq
rT
Using the RN Binomial Using the RN Binomial Probabilities FormulaProbabilities Formula
In above example, u = 22/20 = 1.1 and d = 18/20 = 0.9
So, assuming r = 12% p.a.,
Valuing the OptionValuing the OptionS0u = 22 ƒu = 1
S0d = 18 ƒd = 0
S0
ƒ
0.6523
0.3477
The value of the option is
e–0.12(0.25) [0.6523 1 + 0.3477 0]
= 0.633
A Two-Step ExampleA Two-Step Example
Each time step is 3 months
20
22
18
24.2
19.8
16.2
Valuing a Call OptionValuing a Call Option
Value at node B = e–0.120.25(0.65233.2 + 0.34770) = 2.0257
Value at node A = e–0.120.25(0.65232.0257 + 0.34770)
= 1.2823
201.2823
22
18
24.23.2
19.80.0
16.20.0
2.0257
0.0
A
B
C
D
E
F
A Put Option Example; K=52A Put Option Example; K=52
504.1923
60
40
720
484
3220
1.4147
9.4636
A
B
C
D
E
F
15
What Happens When an Option What Happens When an Option is American is American (see spreadsheet)(see spreadsheet)
505.0894
60
40
720
484
3220
1.4147
12.0
A
B
C
D
E
F
And if we did not have u and d?And if we did not have u and d?
One way of matching the volatility of log-returns is to set
where is the volatility andΔt is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein ◦ Handout on Asset Price Dynamics
t
t
ed
eu
17
The Probability of an Up MoveThe Probability of an Up Move
contract futures afor
rate free-risk foreign the is wherecurrency afor
index the on yielddividend the is whereindex stock afor
stock paying dnondividen afor
1
)(
)(
a
rea
qea
ea
du
dap
ftrr
tqr
tr
f
EXOTICS PRICING (examples)EXOTICS PRICING (examples)
a) Average price ASIAN CALL
payoff = max {0, Sav – K}
Remember: cheaper than an ‘ordinary’ option
b) Barrier Options (e.g. up and out put)pension fund holds stocks and is worried about fall in price but does not think price will rise by a very large amount◦ Ordinary put? - expensive◦ Up and out put - cheaper
04/10/23
Pricing an Asian Option (BOPM)
Average price ASIAN CALL(T = 3)
1. Calculate stock price at each node of tree2. Calculate the average stock price Sav,i at expiry, for each of
the 8 possible paths (i = 1, 2, …, 8).3. Calculate the option payoff for each path, that is
max[Sav,i – K, 0] (for i = 1, 2, …, 8)
The risk neutral probability for a particular path is
qi* = qk(1 – q)n-k
q = risk neutral probability of an ‘up’ movek = number of ‘up’ moves(n – k) = the number of ‘down’ moves
04/10/23
Pricing an Asian Option (BOPM), cont’d
4. Weight each of the 8 outcomes for the call payoff max[Sav,i – K, 0] by the qi* to give the expected payoff:
5. The call premium is then the PV of ES*, discounted at the risk free rate, hence:
8
1,
* ]0,max[)(ˆi
iavi KSqSE
3)(ˆ rAsian eSEC
04/10/23
Pricing Barrier Options (BOPM)
Down-and-out callS0 = 100. Choose K = 100 and H = 90 (barrier)
Construct lattice for SPayoff at T is max {0, ST – K }
Follow every ‘path’ (ie DUU is different from UUD) If on say path DUU we have any value of S < 90 , then the value at T is set to ZERO (even if ST – K > 0).
Use BOPM risk neutral probabilities for each path and each payoff at T
04/10/23
Example: Down-and-out call
S0 =100, K= 100, q = 0.857, (1 – q) = 0.143
H = 90
UUU ={115, 132.25, 152.09} Payoff = 52.09 (q* = 0.8573, 0.629)
DUU ={80, 92,105.8} Payoff = 0 NOT 5.08 (q* = 0.105)
C = e-rT ‘Sum of [q* payoffs at T]’
where qi* = qk(1 – q)n-k,