Learning Module 8
Pricing and Valuation of Options
Key Outcomes Summary & Practice Problems
What you must be able to do
Curriculum Year: 2026
Explain the exercise value, moneyness, and time value of an option.
Contrast the use of arbitrage and replication concepts in pricing forward commitments and contingent claims.
Identify the factors that determine the value of an option and describe how each factor affects the value of an option.
1 · Exercise Value
Exercise (intrinsic) value — what an option would be worth if exercised right now, incorporating the time value of money on the exercise price.
This focuses only on the value of the right to transact — not the option's full market price, which also includes time value.
Exercise Value = Max[0, St − X(1+r)−(T−t)]
Exercise Value = Max[0, X(1+r)−(T−t) − St]
2 · Moneyness
Moneyness | Call Option | Put Option |
|---|---|---|
In the money (ITM) | St > X | St < X |
At the money (ATM) | St = X | St = X |
Out of the money (OTM) | St < X | St > X |
Deep ITM options have option-value sensitivity to the underlying that approaches 1-for-1 (high exercise likelihood).
Deep OTM options have low sensitivity (low exercise likelihood).
ATM options are most sensitive to whether a small underlying move pushes the option in- or out-of-the-money.
3 · Time Value
Time value = option's current market price minus its exercise value.
Always ≥ 0; declines to zero at maturity — known as time value decay.
Driven by the chance that the underlying's future price moves will increase the profitability of exercise — increases with both more time to expiry and higher volatility.
ct = Exercise Value + Time Value | pt = Exercise Value + Time Value
4 · No-Arbitrage Price Bounds
Max[0, St − X(1+r)−(T−t)] < ct ≤ St
Max[0, X(1+r)−(T−t) − St] < pt ≤ X
Call upper bound = St — a buyer would never pay more for the right to buy than the underlying itself costs outright.
Put upper bound = X — the maximum a put can ever be worth is the exercise price itself (when ST → 0).
An option trading below its lower bound (exercise value) creates a riskless arbitrage opportunity.
5 · Replication — Forwards vs. Options
Instrument | Replication at t=0 | Adjustment Needed Over Time? |
|---|---|---|
Long forward | Borrow S0, buy underlying | No — fixed strategy throughout |
Long call | Borrow a proportion of X(1+r)−T, buy a proportion of the underlying | Yes — proportion shifts with moneyness/likelihood of exercise |
Long put | Short sell a proportion of the underlying, lend proceeds | Yes — same reason |
Because option exercise is uncertain (unlike a forward's certain settlement), replication uses only a fraction of the full borrowing/lending and underlying position — that fraction is tied to the likelihood of exercise (moneyness).
This is the core conceptual difference in applying arbitrage/replication logic to contingent claims vs. forward commitments.
6 · Factors Affecting Option Value
Factor (↑ increase) | Call Value | Put Value |
|---|---|---|
Underlying price (St) | + | − |
Exercise price (X) | − | + |
Time to expiration | + | +/− (usually +, but can be − for deep ITM puts with high rates) |
Risk-free rate | + | − |
Volatility of underlying | + | + |
Income/cost of owning underlying | −/+ (income lowers, cost raises) | +/− (income raises, cost lowers) |
Volatility and time to expiration usually move call and put values in the same direction — more uncertainty/time benefits the asymmetric (capped-loss) payoff of both option types.
Exercise price, risk-free rate, and income/cost have opposite effects on calls vs. puts.
A higher risk-free rate lowers PV(X), raising call exercise value and lowering put exercise value — but doesn't directly affect time value.
Income/non-cash benefits (dividends, convenience yield) accrue to the underlying's owner, not the option holder — lowering call value, raising put value; carry costs have the reverse effect.