1. Lec 12B: BOPM for Puts (Hull, Ch.12) Single-period BOPM
▸ Replicating Portfolio
▸ Risk-Neutral Probabilities
How to Exploit Arbitrage Opportunities
Multi-period BOPM
▸ Two-period Risk-Neutral Probabilities
▸ Two-period Replicating Portfolios
▸ Four Period European Puts, with R-N probabilities
▸ Four Period American Puts, with R-N probabilities
Value of early exercise
Lec 12B BOPM for Puts
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2. Single Period BOPM for Puts (Replication) (p.2)
Example: Consider a P0E(K = $50, T=1yr).
Our job is to find a “fair” price for this Put.
Stock, Put, and bond prices evolve as follows (r = 25%/yr) :
t= T= t= T= t= T=1
100 (SU ) ( PU )
S0 = P0 = ? B0 = 1.0
25 (SD) (PD )
Lec 12B BOPM for Puts
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3. Construct a Stock/Bond portfolio that replicates the Put exactly.
Replication Solution.
Construct a Stock/Bond portfolio that replicates the Put exactly.
Portfolio T= CF0
SD = SU =100
Short 1 Share
Buy a Bond (FV=100)
Sell 3 Puts P0
P0-60 = ?
At time 1, CFs from this portfolio (strategy) = 0.
Therefore, to avoid arbitrage:
3P = 0, ⇒ P0 = $10
Lec 12B BOPM for Puts
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4. Arbitrage Opportunities (p. 2)
Suppose (physical) puts sell for P0 = $15, can we make some free money? YES.
3 Actual Puts = $ ⇒ Sell
3 Synthetic Puts = {-S, +B(FV=100)} , CF0 = /1.25 = -30 ⇒Buy
Arbitrage Portfolio: {-1 share, +B(FV = $100), -3P}
Portfolio T= CF0
SD = SU =100
Short 1 Share
Buy a Bond (FV=100)
Sell 3 Puts (15)
Arb Profit
At time 1, CFs from this portfolio (strategy) = 0.
Therefore, to avoid arbitrage: ⇒ P0 = $10
Lec 12B BOPM for Puts
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5. How to create a synthetic Put?
Theory of BOPM (p. 3)
Given: t= T= t= T=1
100 (SU ) (PU )
S0 = P0 = ?
25 (SD) (PD )
How to create a synthetic Put?
Let, Δ = Shares of Stock, B = $ amount of Bonds.
If the stock ↑ Δ100+(1.25)B = 0
If the stock ↓ Δ25 +(1.25)B = Solve for two unknowns.
Δ*= (0-25)/(100-25)= -1/3 i.e., short 1/3 shares
and B* = -[100(-1/3)]/1.25 = +$ (i.e. +B(PV=$26.67)
1 Physical Put = { -(1/3) sh of Stock, +B(PV=26.67, T=1yr) }
At t=0 P0 = -1/3(50) = $10 (same as before)
Lec 12B BOPM for Puts
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6. “Risk-Neutral Probabilities” (p. 4)
Assume a risk-neutral economy. Then,
all assets are priced to yield the risk-free rate of return. Why?
For the stock: S0 = [100p + 25(1 - p)]/(1.25) = $50 ➟ solve for p: p =½
p= prob. of SU = $100, and 1 - p = ½ = prob. of SD = $25
Put price: P0 = [0(1/2) + 25(1/2)]/(1.25) = $10 (same as before).
Does Put-Call Parity hold? +S +P = +C +B
=? / Yes.
In general, Put price: P0 = [p PU + (1-p) PD ]/(1+R) , p =[(1+r)-d]/(u-d)
Lec 12B BOPM for Puts
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7. Two-periods BOPM. Use R-N probabilities (p. 3)
Let S0 = $50, need to find the Put price for P(K=$50, T=1 year).
1 pd = 6 months. r = 22.22%/year, or 11.11%/6 months; 1/(1+r)=0.9
Stock Price tree Put Price tree
D 100 A D A
S0 = B P0 =$ B
E 25 C E C
t = t =
at D, Put is worthless (Why?)
at E, R-N probability: S0 =[50p + 25(1 - p)]/(1.1111) = 35 ➟ p=5/9
➟ Put price: P0 = [0(5/9) + 25(4/9)]/(1.1111) = $10
at F (right now) u = 70/50 = 7/5; d = 35/50 = 7/10
p =[(1+r)-d] / (u-d) = [( (7/10)] / (7/5-7/10) =0.5873
➟ P0 ={0.5873(0) (10)} /1.1111=$3.71
N.B. in this example u & d change from one period to the next. In general, it is much easier to keep u, d constant.
Lec 12B BOPM for Puts
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8. Stock Price tree Put Price tree D 100 A D 0 A 70 0
Two-periods BOPM. (p. 4)
Replication: Trade a portfolio of stocks and bonds to replicate the cash flows from the Put
Stock Price tree Put Price tree
D 100 A D A
S0 = B P0 =$ B
E 25 C E C
t = t =
at D, Put is worthless. Thus, (Δ=0, B=0} no stock, no bond
at E: if S ↑ (E → B): Δ(50) + B(1.1111) = 0
if S ↓ (E → C): Δ( 25 ) + B(1.1111) = 25 ➟ Δ = -1, and B = 45
What is the math telling us?
Short 1 share at $35, buy a bond, PV=$45, (FV=45(1.1111)=$50),
CFE = = ➟ Portfolio ValueE = $10 = Value of PutE
Lec 12B BOPM for Puts
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9. Stock Price tree Put Price tree D 100 A D 0 A 70 0
Two-periods BOPM.
Replication: Trade a portfolio of stocks and bonds to replicate the cash flows from the Put
Stock Price tree Put Price tree
D 100 A D A
S0 = B P0 =$ B F
E 25 C E C
t = t =
at F (now), (F → D): Δ(70) + B(1.1111) = 0
(F → E): Δ( 35 ) + B(1.1111) =10 ➟ Δ = -10/35, B = +18
Short shares at $50, buy a bond, PV=$18,
CFF = = ➟ Portfolio ValueF = $3.71 = Put ValueE
Lec 12B BOPM for Puts
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10. European vs. American Puts (p. 5)
First, consider a European Put(K= $35); r = 25%/period
Stock Price tree Put Price tree G G
B D D
H H
S0 = E P0 =$ E
I I
C F E F
14.82J J
At G, H put is out of the money ➟ P = 0
at I put is in the money, therefore exercise. Put Value = $1.667,
same at J: Put =20.18
At A thru F, use R-N approach: u = 1.5, d = 2/3; p = 0.7
For example, at F: PF ={1.67(0.7)+20.18(0.3)} /1.25=$5.78
Lec 12B BOPM for Puts
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11. European vs. American Puts (p. 5)
Now, consider the American Put(K= $35)
Stock Price tree Put Price tree G G
B D D
H H
S0 = E P0 =$ E
I C I
C F F
14.82 J J
Notes: G thru J are the same as for the European Put.
At F the European price ($5.78) may be too low. Check for Arb. Opp. {+P, +S, and exercise immediately} ➟ CF = =+$7
To preclude this arbitrage at F we must have PF = 12.78
At all other points (A thru F) use R-N approach:
u = 1.5, d = 2/3; p = 0.7
For example, at C: PC ={0.40(0.7)+12.78(0.3)} /1.25=$3.29
Lec 12B BOPM for Puts
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12. Thank You
(a Favara)
Lec 12B BOPM for Puts
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