Finance-Theory · Unit 13 · Video 3 · Interactive Practice

Risk-Neutral Probabilities and Arbitrage Enforcement in the Binomial Model

IKey Formulas

Formula	Name	Meaning
$p^* \equiv \dfrac{r-d}{u-d}, \quad 1-p^* = \dfrac{u-r}{u-d}$	Risk-neutral probability	The numerators add to $u-d$ , so the two weights sum to one; built from $r$ , $u$ , $d$ only ( $r$ is the gross riskless return per period)
$C_0 = \dfrac{1}{r}\left[p^\,C_u + (1-p^)\,C_d\right]$	Binomial price	The expected payoff under $p^*$ , discounted one period at the riskless rate
$d < r < u$	Equilibrium condition	Both weights positive; if $r \le d$ nobody holds bills, if $r \ge u$ nobody holds the stock
$C_0 = V_0 = S_0\,\Delta^* + B^*$	No arbitrage	$C_0 > V_0$ : sell the call, buy the portfolio. $C_0 < V_0$ : buy the call, sell the portfolio. Either way a riskless profit today

Key Insight: $p^*$ is not the true probability $p$ . The true $p$ never enters the formula; it is already inside today's stock price $S_0$ .

IIVisualization 1 — When the Weights Are Probabilities

The two weights always sum to one; both are positive exactly when the riskless rate lies between $d$ and $u$ .

d 0.50 r 1.03 u 1.50

IIIVisualization 2 — Your $p$ Versus $p^*$

Your probability $p$ moves your forecast of the stock price but leaves the call price untouched.

your p 0.55

IVVisualization 3 — Arbitrage Enforces $C_0 = V_0$

Any market price other than $V_0$ lets you sell the dear side, buy the cheap side, and keep the gap riskless.

market price of the call 6.00

Leg	Position	Today	Tomorrow, up ( $S_1 = 30$ )	Tomorrow, down ( $S_1 = 10$ )
Call
Stock, $\Delta^* = 0.5$
Bonds, $B^* \approx -4.85$
Net

💡 The money machine runs only while the model's assumptions hold: exactly two possible stock prices tomorrow, and borrowing or lending at the riskless rate $r$ .

VQuiz Questions

Problem 1 · The Risk-Neutral Probability

Given (illustrative figures): a stock trades at $75. Over one period it moves up by $u = 1.35$ or down by $d = 0.85$ , and the riskless rate is $r = 1.03$ (gross, per period). Analysts put the probability of the up move at $0.65$ . Find the risk-neutral probability $p^*$ .

✅ Correct!

p^* = (1.03 - 0.85)/(1.35 - 0.85) = 0.18/0.5 = 0.36

, and the analysts'

0.65

plays no part.

❌ Close, but…

0.64 = (u - r)/(u - d)

is the down-state weight,

(1 - p^*)

. The up-state weight has

r - d

on top.

❌ Check the rate. That uses

r = 0.03

. In this formula

r

is the gross return

1.03

; a net rate put in its place gives a meaningless negative weight.

❌ Not quite.

0.65

is the true (subjective) probability

p

. The risk-neutral probability is built from

r

u

and

d

alone.

❌ Not quite. Use

p^* = (r - d)/(u - d)

with the gross rate.

Show solution

With $r = 1.03$ , $u = 1.35$ , $d = 0.85$ :

p^* = \frac{r - d}{u - d} = \frac{1.03 - 0.85}{1.35 - 0.85} = \frac{0.18}{0.5} = 0.36, \qquad 1 - p^* = \frac{u - r}{u - d} = \frac{0.32}{0.5} = 0.64

Both weights lie between 0 and 1 because $0.85 < 1.03 < 1.35$ . The stock price of $75 and the analysts' probability of $0.65$ are not needed: $p^* = 0.36$ .

Problem 2 · Can Stocks and Bills Coexist?

Given (illustrative figures): four candidate one-period economies, each with gross stock moves $u$ , $d$ and gross riskless rate $r$ . Which one is consistent with investors holding both the stock and Treasury bills?

✅ Correct!

0.90 < 1.04 < 1.15

, so

p^* = 0.14/0.25 = 0.56

and

1 - p^* = 0.44

are both positive.

❌ Not quite. Here

r = 1.03 < d = 1.05

: even the stock's worst outcome beats bills, so nobody holds bills. The weight

p^* = (1.03 - 1.05)/0.20 = -0.1

is negative.

❌ Not quite. Here

r = 1.10 > u = 1.08

: even the stock's best outcome falls short of bills, so nobody holds the stock. The weight

1 - p^* = (1.08 - 1.10)/0.16 = -0.125

is negative.

❌ Close, but… With

r = d

the stock never does worse than bills and sometimes does better, so nobody holds bills. The condition is strict, and here

p^* = 0

❌ Not quite. Check

d < r < u

strictly.

Show solution

Stocks and bills can both be held only when $d < r < u$ , strictly. Check each economy:

(a) $0.90 < 1.04 < 1.15$ ✓: $p^* = \frac{1.04 - 0.90}{1.15 - 0.90} = 0.56$ .
(b) $r = 1.03 < d = 1.05$ : bills are dominated; $p^* = -0.1$ .
(c) $r = 1.10 > u = 1.08$ : the stock is dominated; $1 - p^* = -0.125$ .
(d) $r = d = 1.00$ : the stock weakly dominates bills; $p^* = 0$ .

Only economy (a) works.

Problem 3 · Price a Call With $p^*$

Given (illustrative figures): a stock trades at $150 and over one period moves up by $u = 1.24$ or down by $d = 0.92$ ; the riskless rate is $r = 1.04$ (gross, per period). Investors believe the up move has probability $0.6$ . Find the price of a one-period European call with strike $156.

✅ Correct! The expected payoff under

p^* = 0.375

0.375 \cdot 30 = 11.25

; one period's discount at

1.04

gives about $10.82.

❌ Close, but…

11.25

is the expected payoff under

p^*

, which arrives tomorrow. Divide by

r = 1.04

to bring it to today.

❌ Not quite. That weights the payoffs with the believed probability

0.6

. The price uses

p^* = (r - d)/(u - d)

, never the true

p

❌ Check the weights. That puts

1 - p^* = 0.625

on the up state. The up-state weight is

p^* = (r - d)/(u - d) = 0.375

❌ Not quite. Find

C_u

C_d

and

p^*

, then

C_0 = [p^* C_u + (1 - p^*) C_d]/r

Show solution

Payoffs: $uS_0 = 1.24 \cdot 150 = 186$ and $dS_0 = 0.92 \cdot 150 = 138$ , so

C_u = \text{Max}[186 - 156,\ 0] = 30, \qquad C_d = \text{Max}[138 - 156,\ 0] = 0

Risk-neutral probability ( $0.92 < 1.04 < 1.24$ , so it is valid):

p^* = \frac{1.04 - 0.92}{1.24 - 0.92} = \frac{0.12}{0.32} = 0.375, \qquad 1 - p^* = 0.625

Price:

C_0 = \frac{1}{1.04}\left[0.375 \cdot 30 + 0.625 \cdot 0\right] = \frac{11.25}{1.04} \approx 10.82

The call is worth about $10.82. The believed probability of $0.6$ never enters.

Problem 4 · Exploit a Mispriced Call

Given: the stock and call of Problem 3 ( $S_0 = 150$ , $u = 1.24$ , $d = 0.92$ , gross $r = 1.04$ , strike $156, model price about $10.82). The call trades in the market at $10.20. Find the arbitrage trade and the riskless profit it locks in today, per call.

Which trade?

Riskless profit today, per call?

✅ Correct! The call is cheap: buy it for $10.20, sell its replica for about $10.82, keep about $0.62 today, and owe nothing tomorrow.

❌ Check the direction. That trade profits when the call is dear. At $10.20 the call is below

V_0 \approx 10.82

, so it is the side to buy.

❌ Check the bond sign. The replica borrows about $82.93 (

B^* \approx -82.93

), so selling the replica means lending that amount. Borrowing as well would leave you owing $86.25 on the loan tomorrow on top of buying back the shares, far more than the call pays.

❌ Not quite. One share is not the replica: it pays 186 or 138 tomorrow against the call's 30 or 0. The replica holds

\Delta^* = 0.625

share and a loan.

❌ Check the trade. Buy the cheap side, sell the dear side, using the replicating portfolio

\Delta^*

B^*

❌ Close, but…

11.25 - 10.20

compares with the undiscounted expected payoff. The replica is worth

V_0 = 11.25/1.04 \approx 10.82

today.

❌ Not quite.

17.31

is the call valued with the believed probability

0.6

, which is not a price the replica delivers. Compare with

V_0 \approx 10.82

❌ Close, but…

0.62 \times 1.04 \approx 0.64

is that profit carried a period forward. The profit is collected today, and nothing more arrives tomorrow.

❌ Check the profit. It is the replica's value today minus the market price,

V_0 - 10.20

Show solution

Replicating portfolio ( $C_u = 30$ , $C_d = 0$ ):

\Delta^* = \frac{C_u - C_d}{(u - d)S_0} = \frac{30}{0.32 \cdot 150} = 0.625, \qquad B^* = \frac{uC_d - dC_u}{(u - d)r} = \frac{0 - 0.92 \cdot 30}{0.32 \cdot 1.04} \approx -82.93

V_0 = 150 \cdot 0.625 - 82.93 \approx 93.75 - 82.93 \approx 10.82

Trade: the market price 10.20 is below $V_0$ , so buy the call and sell the replica: short $0.625$ share (receiving 93.75) and lend about $82.93. Cash today: $93.75 - 82.93 - 10.20 \approx 0.62$ .

Tomorrow: the loan repays $|B^*|\,r = \frac{27.6}{0.3328} \cdot 1.04 = 86.25$ .

Up ( $S_1 = 186$ ): buying back $0.625$ share costs $116.25$ ; the loan repays $86.25$ and the call pays 30, so $30 + 86.25 - 116.25 = 0$ .
Down ( $S_1 = 138$ ): buying back $0.625$ share costs $86.25$ ; the loan repays $86.25$ and the call pays 0, so the net is 0.

The positions cancel in both states, so the $0.62 per call is riskless profit.

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Risk-Neutral Probabilities and Arbitrage Enforcement in the Binomial Model

IKey Formulas

IIVisualization 1 — When the Weights Are Probabilities

IIIVisualization 2 — Your ppp Versus p∗p^*p∗

IVVisualization 3 — Arbitrage Enforces C0=V0C_0 = V_0C0​=V0​

VQuiz Questions

IIIVisualization 2 — Your $p$ Versus $p^*$

IVVisualization 3 — Arbitrage Enforces $C_0 = V_0$