Finance-Theory · Unit 13 · Video 2 · Interactive Practice

Pricing an Option by Replication: Delta, Bonds, and Why p Disappears

IKey Formulas

Formula	Name	What it means
$V_0 = S_0\Delta + B$	Cost of the portfolio today	$\Delta$ shares plus $B$ dollars of riskless bonds; $\Delta < 0$ is a short sale, $B < 0$ is borrowing
$uS_0\Delta + rB = C_u,\quad dS_0\Delta + rB = C_d$	Matching conditions	Two linear equations in $\Delta$ and $B$ ; the bond leg pays $rB$ in both states, and $u > d$ makes the solution unique
$\Delta^* = \dfrac{C_u - C_d}{(u-d)S_0},\quad B^* = \dfrac{uC_d - dC_u}{(u-d)\,r}$	Replicating portfolio	$\Delta^$ is the call's payoff spread over the stock's price spread; for a call $\Delta^ \ge 0$
$C_0 = S_0\Delta^* + B^* = \dfrac{1}{r}\left(\dfrac{r-d}{u-d}\,C_u + \dfrac{u-r}{u-d}\,C_d\right)$	Call price by no arbitrage	Uses $S_0$ , $K$ (through $C_u$ , $C_d$ ), $u$ , $d$ and $r$ , and no $p$

Key Insight: $r$ is the gross riskless return ( $r = 1.03$ for a 3% rate): a dollar of bonds today pays $r$ dollars tomorrow, so discounting divides by $r$ , never by $1 + r$ . The portfolio matches the call in the up state and in the down state, so how likely each state is never enters its cost.

IIOne Line Through Both Call Payoffs

The portfolio pays $\Delta S_1 + rB$ tomorrow, a straight line in $S_1$ ; replication means hitting both call payoffs.

Shares Δ 0.25 Bonds repay tomorrow, rB −$10

IIIDisagree About p, Agree on the Price

Two investors who disagree about $p$ but agree on $u$ and $d$ must agree on the call's price.

A: chance of up, p 0.80 A: u 1.10 A: d 0.90

B: chance of up, p 0.20 B: u 1.10 B: d 0.90

IVQuiz Questions

Problem 1 · How Many Shares?

Given: A stock trades at $25 today and tomorrow is worth either $u S_0$ or $d S_0$ , with $u = 1.4$ and $d = 0.8$ . A call on it has strike $26 and expires tomorrow; the gross riskless return is $r = 1.05$ . Find $\Delta^*$ , the number of shares in the portfolio that replicates the call.

✅ Correct! The call's payoffs differ by $9 and the stock's prices by $15, so three fifths of a share moves exactly as much as the call.

❌ Missing a factor of $S_0$ . Dividing $9 by

u - d = 0.6

leaves out

S_0

; the stock's price spread is

(u - d)S_0

= $15.

❌ Wrong denominator. Dividing by

S_0

compares the call's spread with today's price;

\Delta^*

compares it with the stock's spread between the two states,

uS_0 - dS_0

❌ That divides by the up price only.

C_u / (uS_0)

ignores the down state; replication must match the spread between both states.

❌ Not quite. Use

\Delta^* = (C_u - C_d)/((u-d)S_0)

Show solution

Tomorrow's stock prices are $uS_0$ = $35 and $dS_0$ = $20, so the call pays $C_u = \max[35 - 26,\ 0] = 9$ dollars and $C_d = \max[20 - 26,\ 0] = 0$ .

\Delta^* = \frac{C_u - C_d}{(u-d)S_0} = \frac{9 - 0}{(1.4 - 0.8)\cdot 25} = \frac{9}{15} = 0.6

Call spread over stock spread: $9 over $15. The rate $r$ does not enter $\Delta^*$ , because the bond terms cancel when the two matching conditions are subtracted.

Problem 2 · The Bond Leg and the Gross Rate

Given: A stock trades at $48; tomorrow it is worth $uS_0$ or $dS_0$ with $u = 1.25$ and $d = 0.75$ . A call with strike $48 expires tomorrow, so $\Delta^* = 0.5$ . The gross riskless return is $r = 1.02$ (a 2% rate). Find the bond position $B^*$ and the call's price $C_0$ .

What is $B^*$ ?

What is $C_0$ ?

✅ Correct! Borrow about $17.65 today (repaying $18 tomorrow) and buy half a share for $24: the portfolio costs about $6.35, and that is the call's price.

❌ Check the sign. The call pays nothing in the down state, where half a share is still worth $18; the bonds must cancel that $18, so they are a debt:

B^* < 0

❌ That is the repayment tomorrow, $rB^*$ . Divide by the gross return:

B^* = -18/1.02

❌ $r$ is already gross. Dividing by

1 + r = 2.02

counts the one twice; the formula divides by

r = 1.02

❌ $B^*$ is negative. The cost is

S_0\Delta^* + B^*

= $24 + (−$17.65), not $24 + $17.65.

❌ That uses $B = -18$ . $18 is what the loan repays tomorrow; today's loan is $18 divided by

r = 1.02

❌ That is the shares alone. Half a share costs $24, but the portfolio also borrows, which lowers its cost.

❌ Not quite. Use

B^* = (uC_d - dC_u)/((u-d)\,r)

with

r = 1.02

❌ Not quite. The call's price is the portfolio's cost,

S_0\Delta^* + B^*

Show solution

Tomorrow's stock prices are $60 and $36, so $C_u$ = 12 dollars and $C_d$ = 0. Then

B^* = \frac{uC_d - dC_u}{(u-d)\,r} = \frac{1.25\cdot 0 - 0.75\cdot 12}{0.5 \cdot 1.02} = \frac{-9}{0.51} \approx -17.65

so the portfolio borrows about $17.65 today and repays $rB^*$ = $18 tomorrow. Check the down state: half a share is worth $18, minus the $18 repayment, equals $0 = $C_d$ . Up state: $30 − $18 = $12 = $C_u$ .

C_0 = S_0\Delta^* + B^* = 48 \cdot 0.5 - \frac{9}{0.51} \approx 24 - 17.65 = 6.35

The closed form agrees: $\frac{1}{1.02}\left(\frac{1.02 - 0.75}{0.5}\cdot 12 + \frac{1.25 - 1.02}{0.5}\cdot 0\right) = \frac{6.48}{1.02} \approx 6.35$ .

Problem 3 · Price It, and Ignore the Forecast

Given: A stock trades at $60; tomorrow it is worth $uS_0$ or $dS_0$ with $u = 1.2$ and $d = 0.9$ . Analysts put the chance of the up move at $p = 0.7$ . A call with strike $50 expires tomorrow, and the gross riskless return is $r = 1.04$ . Find $\Delta^*$ and the call's price $C_0$ .

What is $\Delta^*$ ?

What is $C_0$ ?

✅ Correct! The call finishes in the money in both states, so it is replicated by one full share and a loan that repays $50:

C_0 = 60 - 50/1.04 \approx 11.92

dollars, whatever

p

is.

❌ Wrong denominator. The stock's spread is

uS_0 - dS_0

= $18, not

S_0

= $60.

❌ That is $C_u / S_0$ . Replication matches the spread between the states:

(C_u - C_d)/(uS_0 - dS_0)

❌ Missing a factor of $S_0$ .

(C_u - C_d)/(u - d) = 18/0.3

still has to be divided by

S_0

= 60.

❌ That weights the payoffs by $p = 0.7$ . The replicating portfolio pays

C_u

and

C_d

whichever state occurs, so its cost does not involve

p

; the forecast is a distraction.

❌ Not discounted. 12.40 is the bracket in the closed form; it still has to be divided by

r = 1.04

❌ That is $S_0 - K$ . The loan repays

K

tomorrow, so today it is worth only

K/r

; the call is worth

S_0 - K/r

here.

❌ Not quite. Use

\Delta^* = (C_u - C_d)/((u-d)S_0)

❌ Not quite. Use

C_0 = S_0\Delta^* + B^*

or the closed form; neither contains

p

Show solution

Tomorrow's stock prices are $72 and $54, so $C_u$ = 22 dollars and $C_d$ = 4 dollars: the call pays off in both states.

\Delta^* = \frac{22 - 4}{(1.2 - 0.9)\cdot 60} = \frac{18}{18} = 1, \qquad B^* = \frac{1.2\cdot 4 - 0.9\cdot 22}{0.3\cdot 1.04} = \frac{-15}{0.312} \approx -48.08

One share plus a loan that repays $rB^* = -15/0.3 = -50$ dollars tomorrow: up, $72 − $50 = $22; down, $54 − $50 = $4. Both match.

C_0 = 60 \cdot 1 - \frac{15}{0.312} \approx 60 - 48.08 = 11.92, \qquad \frac{1}{1.04}\left(\frac{1.04 - 0.9}{0.3}\cdot 22 + \frac{1.2 - 1.04}{0.3}\cdot 4\right) = \frac{12.4}{1.04} \approx 11.92

The analysts' $p = 0.7$ is used nowhere.

Problem 4 · Replicating a Put

Given: A stock trades at $30; tomorrow it is worth $uS_0$ or $dS_0$ with $u = 1.3$ and $d = 0.9$ , and the gross riskless return is $r = 1.02$ . A put with strike $30 expires tomorrow and pays $\max[K - S_1,\ 0]$ . The same two matching conditions apply with the put's payoffs $P_u$ , $P_d$ in place of $C_u$ , $C_d$ . Find the replicating portfolio and the put's price $P_0$ .

Which portfolio replicates the put?

What is $P_0$ ?

✅ Correct! The put pays when the stock falls, so its replica is short the stock and long bonds:

\Delta^* = -0.25

and

B^* \approx 9.56

dollars, costing about $2.06.

❌ Those are a call's signs. Here

P_u - P_d = 0 - 3 < 0

, so

\Delta^*

is negative: the replica gains when the stock falls.

❌ Check $B^*$ .

B^* = (uP_d - dP_u)/((u-d)\,r) = (1.3\cdot 3 - 0)/(0.4\cdot 1.02) > 0

: the replica lends.

❌ Check $\Delta^*$ . A long position gains when the stock rises, but the put pays only in the down state;

\Delta^* = (P_u - P_d)/((u-d)S_0) < 0

❌ Not discounted. The bracket in the closed form is 2.10; divide it by

r = 1.02

❌ $\Delta^*$ is negative. The short sale brings in $7.50, which offsets the $9.56 of bonds:

30\cdot(-0.25) + 9.56

, not

7.50 + 9.56

❌ That weights the states equally. Averaging the payoffs with probability one half and discounting is not replication; the weights in the closed form are

(r-d)/(u-d)

and

(u-r)/(u-d)

❌ Not quite. Work out the signs of

\Delta^*

and

B^*

from the put's payoffs.

❌ Not quite. The price is the replica's cost,

S_0\Delta^* + B^*

Show solution

Tomorrow's stock prices are $39 and $27, so the put pays $P_u = \max[30 - 39,\ 0] = 0$ and $P_d = \max[30 - 27,\ 0] = 3$ dollars.

\Delta^* = \frac{P_u - P_d}{(u-d)S_0} = \frac{0 - 3}{0.4\cdot 30} = -\frac{3}{12} = -0.25

B^* = \frac{uP_d - dP_u}{(u-d)\,r} = \frac{1.3\cdot 3 - 0.9\cdot 0}{0.4\cdot 1.02} = \frac{3.9}{0.408} \approx 9.56

Short a quarter of a share and lend about $9.56, which repays $rB^* = 3.9/0.4 = 9.75$ dollars tomorrow. Up: $-0.25\cdot 39 + 9.75 = 0$ ; down: $-0.25\cdot 27 + 9.75 = 3$ . Both match.

P_0 = S_0\Delta^* + B^* = 30\cdot(-0.25) + \frac{3.9}{0.408} \approx -7.50 + 9.56 = 2.06

Closed form: $\frac{1}{1.02}\left(\frac{1.02 - 0.9}{0.4}\cdot 0 + \frac{1.3 - 1.02}{0.4}\cdot 3\right) = \frac{2.1}{1.02} \approx 2.06$ .

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