Finance-Theory · Unit 12 · Video 8 · Interactive Practice

Why an Option's Price Ignores Risk Preferences

IKey Formulas

Formula	Name	What it says
$C_T = \max[S_T - K,\ 0]$	Call payoff at maturity	What the call pays at date $T$ for a stock price $S_T$ and strike $K$
$E_t\big[\max[S_T - K,\ 0]\big]$	Expected payoff	Needs a probability law for $S_T$ ; it is an expected payoff, not yet a price
$\tfrac{1}{2}\sigma^2 S^2 \dfrac{\partial^2 C}{\partial S^2} + rS\dfrac{\partial C}{\partial S} + \dfrac{\partial C}{\partial t} = rC$	Black–Scholes/Merton equation (1973)	Holds the stock price, time, the interest rate $r$ and the volatility $\sigma$ ; the stock's expected return and risk preferences do not appear
$C(S,T) = \max[S - K,\ 0], \quad C(0,t) = 0$	Boundary conditions	At maturity the call is worth its payoff; if the stock price is zero the call is worth zero at every date

Key Insight: A hedged position that carries no risk must earn the riskless rate, or cost zero if it pays zero, or it opens an arbitrage, so two investors who agree on the volatility agree on the price whatever their outlook or attitude to risk.

IIAn Expected Payoff Is Not Yet a Price

Turning an expected payoff into a value today takes a discount rate, and that rate carries the investor's attitude to risk.

Investor A: required return 20% Investor B: required return 8%

IIISeventy Years of Investor-Specific Values

From 1961 to 1969 every option value contained ingredients that differ from investor to investor; 1973 removed them.

Year 1900

IVRiskless Must Earn the Riskless Rate

A riskless position priced to earn anything other than the riskless rate is a free profit for every investor.

Position Price today $1.50

💡 The 1973 equation comes from imposing this condition on a portfolio of the option and the stock, rebalanced so that it carries no risk; the next unit works through the simplest version of that argument, the binomial model.

VQuiz Questions

Problem 1 · Expected Payoff of a Call

Given: an illustrative one-year call with strike $50. Investors agree that the stock price in one year, $S_T$ , will be $40, $50, $60 or $70, each with probability $\tfrac{1}{4}$ . Find the expected payoff $E_t\big[\max[S_T - K,\ 0]\big]$ .

✅ Correct! The call pays $0, $0, $10 and $20 in the four outcomes, and a quarter of their sum is $7.50. That is an expected payoff; it is not yet a price.

❌ Not quite. That is the payoff at the average stock price, $55 − $50. The call pays nothing in the low outcomes rather than a negative amount, so average the payoffs, not the prices.

❌ Not quite. That averages only the two outcomes where the call pays. The outcomes where it pays $0 still carry half the probability.

❌ Not quite. That adds the payoffs without weighting them; each outcome has probability

\tfrac{1}{4}

❌ Not quite. Find the payoff in each outcome, then weight each by its probability.

Show solution

With $K$ equal to $50, the payoff $\max[S_T - K,\ 0]$ in the four outcomes is $0 (stock at $40), $0 (stock at $50), $10 (stock at $60) and $20 (stock at $70).

Weighting each by $\tfrac{1}{4}$ : expected payoff = ¼ × ($0 + $0 + $10 + $20) = $7.50.

The payoff at the average stock price, $55 − $50 = $5, is smaller, because the call's payoff is floored at zero: the low outcomes cost the holder nothing.

Problem 2 · From Expected Payoff to Value

Given: two investors agree on every probability and on an option's expected payoff of $12 in one year. The cautious investor demands a 25% return for bearing the option's risk; the bold investor demands 10%. Find the value the cautious investor puts on the option today.

✅ Correct! $12 ÷ 1.25 = $9.60, while the bold investor gets $12 ÷ 1.10 ≈ $10.91: one set of probabilities, two values, and no single market price.

❌ Not quite. $12 × 1.25 carries the payoff forward a year. A value today discounts it: divide by

1 + k

❌ Not quite. That treats the expected payoff as the price. It still has to be discounted, and the rate depends on how much return the investor demands for the risk.

❌ Not quite. That discounts at the bold investor's 10%. The cautious investor demands 25%, so values the option less.

❌ Not quite. Discount the expected payoff by one year at the cautious investor's required return.

Show solution

A value today is the expected payoff discounted at the return the investor requires for bearing the option's risk:

V = \frac{E_t[\text{payoff}]}{1 + k}

Cautious investor, $k = 25\%$ : $12 ÷ 1.25 = $9.60.

Bold investor, $k = 10\%$ : $12 ÷ 1.10 ≈ $10.91.

They agree on every probability yet reach values about $1.31 apart. This is the obstacle Sprenkle, Samuelson and Samuelson–Merton could not get around: the value depended on the investor's preferences.

Problem 3 · What the 1973 Equation Leaves Out

Given: investor A expects the stock to return 15% a year and is cautious; investor B expects 3% a year and is bold. Both agree that the volatility is $\sigma = 30\%$ and the interest rate is $r = 4\%$ , and they look at the same European call.

Which does NOT appear in the Black–Scholes/Merton equation or its boundary conditions?

What follows for the two investors' prices of the call?

✅ Correct! The equation and its boundary conditions hold only

S

K

t

r

and

\sigma

. A and B agree on all of them, so they agree on the price, whatever each thinks about where the stock is headed or about risk.

❌ Not quite.

r

and

\sigma

sit in the equation itself, and

K

enters through the boundary condition at maturity. Which input from the pre-1973 attempts is missing?

❌ Not quite. That is the pre-1973 view. The stock's expected return does not appear in the equation, so a more optimistic outlook does not raise the price.

❌ Not quite. That would be true of Sprenkle's or Samuelson's values, which carried risk preferences. The 1973 equation carries none.

❌ Not quite. Needing risk preferences is exactly what the 1973 result did away with: no measure of risk preference appears in the equation.

❌ Not quite. List what the equation needs and check which of those the investors disagree about.

Show solution

\tfrac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} + rS\frac{\partial C}{\partial S} + \frac{\partial C}{\partial t} = rC, \qquad C(S,T) = \max[S - K,\ 0], \quad C(0,t) = 0

The inputs are the stock price $S$ , time $t$ (and maturity $T$ ), the interest rate $r$ and the volatility $\sigma$ in the equation, and the strike $K$ in the boundary condition at maturity. The stock's expected return does not appear, and neither does any measure of risk preference.

A and B differ only in expected return (15% against 3%) and attitude to risk, and they agree on $\sigma$ and $r$ . So they must agree on the option's price, as long as they agree on the volatility.

Problem 4 · A Riskless Position Priced Wrong

Given: a position carries no risk and pays $52 for certain in one year. The riskless rate is 4% a year, you can borrow or lend at that rate, and the position trades today at $53. Interest is counted in this problem.

What is the arbitrage?

What riskless profit does it lock in, one year from now?

✅ Correct! Riskless, the position should cost $52 ÷ 1.04 = $50. At $53 it earns less than 4%, so write it, lend the $53 to collect $55.12, and pay the $52 owed: $3.12 for every investor, cautious or bold.

❌ Not quite. Buying it with $53 borrowed at 4% means repaying $55.12 against the $52 it pays: a certain loss. The position is too dear, not too cheap.

❌ Not quite. The position has no risk, so there is nothing for risk preferences to price. Any investor who prefers more money to less takes the same side.

❌ Not quite. $53 − $52 ignores the 4% interest earned on the $53 you lend.

❌ Close. $53 − $50 is the profit measured today. One year from now it has grown at 4%.

❌ Not quite. $2.12 is the interest earned on the $53 you lend, only part of the profit: the premium over the $52 owed counts too.

❌ Not quite. Compare $53 with the price at which the position earns exactly 4%.

❌ Not quite. Grow the $53 you receive at 4%, then subtract the $52 you owe at maturity.

Show solution

Arbitrage-free price. A position with no risk must earn the riskless rate: $52 ÷ 1.04 = $50. At $53 its return is $52 ÷ $53 − 1 ≈ −1.89%, less than 4%, so it is too dear.

Trade. Write (short) the position, receiving $53, and lend the $53 at 4%. Nothing of your own is invested today.

In one year. The loan returns $53 × 1.04 = $55.12; pay the $52 the position owes. Profit: $55.12 − $52 = $3.12, which is the $3 gap today grown at 4% ($3 × 1.04 = $3.12).

The profit needs no view on any stock's expected return and no attitude to risk. This is Merton's argument in miniature: a riskless position must earn the riskless rate, and one that pays zero must cost zero, otherwise one side of it is an arbitrage.

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