Properties and Bounds of Option Prices¶
Option prices satisfy fundamental mathematical properties that follow from no-arbitrage principles. These properties—monotonicity, convexity, and bounds—provide constraints that any valid pricing model must respect and enable arbitrage detection in practice.
This section rigorously establishes these properties for European options under the Black-Scholes framework.
Fundamental Bounds¶
1. Call Option Bounds¶
For a European call option on a non-dividend-paying stock:
Lower bound: The call cannot be worth less than its discounted intrinsic value (otherwise arbitrage exists)
Upper bound: The call cannot exceed the stock price (option gives right to buy stock, not worth more than owning it)
Tighter lower bound:
This is the no-arbitrage lower bound.
2. Put Option Bounds¶
For a European put:
Lower bound: Put worth at least discounted intrinsic value
Upper bound: Put cannot exceed present value of strike (max possible payoff)
3. Verification for Black-Scholes¶
Lower bound for call:
Since \(\mathcal{N}(d_1), \mathcal{N}(d_2) \in [0,1]\):
When \(S > Ke^{-rT}\):
But more carefully, when deeply ITM, \(\mathcal{N}(d_1), \mathcal{N}(d_2) \approx 1\):
Upper bound:
Both bounds satisfied. ✓
Monotonicity in Stock Price¶
1. Call Options¶
Interpretation: Call price is strictly increasing in stock price.
Proof: Differentiating the Black-Scholes formula:
Using \(\frac{\partial d_1}{\partial S} = \frac{1}{S\sigma\sqrt{T}}\) and \(\frac{\partial d_2}{\partial S} = \frac{\partial d_1}{\partial S}\):
After substitution and using \(\mathcal{N}'(d_1) = \frac{1}{\sqrt{2\pi}}e^{-d_1^2/2}\):
The cross-terms cancel (due to Black-Scholes PDE), leaving:
Since \(\mathcal{N}(d_1) \in (0,1)\) for all finite \(d_1\), the call is strictly increasing in \(S\). ✓
2. Put Options¶
Interpretation: Put price is strictly decreasing in stock price.
Proof: From put-call parity:
Differentiate:
Since \(\mathcal{N}(d_1) < 1\), we have \(\frac{\partial P}{\partial S} < 0\). ✓
Monotonicity in Time¶
1. Time Decay (Theta)¶
For European calls on non-dividend-paying stocks:
Interpretation: Call price decreases as time passes (all else equal), a phenomenon called time decay.
Theta formula:
Since both terms are negative, \(\Theta < 0\) (time decay).
Exception: Deep ITM European calls can have slightly positive theta near expiration due to interest earned on deferred strike payment.
2. Increasing with Time to Maturity¶
Interpretation: Longer-dated options are more valuable (more time for favorable price movement).
Proof: Options with more time have:
- Greater probability of finishing ITM
- Higher optionality value
- More uncertainty (higher variance \(\sigma^2 T\))
From the formula, as \(T\) increases, \(d_1\) and \(d_2\) shift, and the net effect is positive:
Monotonicity in Volatility¶
1. Vega (Volatility Sensitivity)¶
Interpretation: Call (and put) prices are strictly increasing in volatility.
Vega formula:
Economic intuition: Higher volatility increases the probability of large moves. For options (with capped downside but unlimited upside), this increases value.
Key property: Vega is the same for calls and puts with identical strike and maturity:
This follows from put-call parity (volatility affects both equally).
Monotonicity in Strike¶
1. Call Options¶
Interpretation: Call price is strictly decreasing in strike.
Proof: Higher strike means: - Lower intrinsic value - Lower probability of exercise - Less favorable payoff structure
From the formula:
2. Put Options¶
Interpretation: Put price is strictly increasing in strike (higher strike = more valuable right to sell).
Convexity in Stock Price¶
1. Second Derivative (Gamma)¶
Interpretation: Option value is convex in stock price.
Gamma formula:
Implication:
- Options benefit from large stock price moves in either direction
- Delta increases as stock rises (accelerating gains)
- Delta decreases as stock falls (decelerating losses)
Put gamma: From put-call parity,
Both calls and puts have the same gamma.
2. Graphical Interpretation¶
The option value curve \(C(S)\) is:
- Concave up (shaped like ⌣)
- Slope (delta) increases with \(S\)
- Tangent line always lies below the curve
This convexity property is fundamental to option hedging and risk management.
Convexity in Strike (Butterfly Spread)¶
1. Butterfly Constraint¶
For strikes \(K_1 < K_2 < K_3\) with \(K_2 - K_1 = K_3 - K_2 = \Delta K\):
Interpretation: Call prices are convex (downward) in strike.
Proof: Consider a butterfly spread: - Buy 1 call at \(K_1\) - Sell 2 calls at \(K_2\) - Buy 1 call at \(K_3\)
Cost:
Payoff: Non-negative in all states (can verify by cases).
No-arbitrage: Since payoff \(\geq 0\), cost must be \(\geq 0\):
Rearranging:
This is the convexity constraint for call prices. ✓
2. Continuous Version¶
For infinitesimally close strikes:
From Black-Scholes:
Strict convexity holds. ✓
Calendar Spread Inequality¶
1. Time Spread Constraint¶
For maturities \(T_1 < T_2\) with identical strike \(K\):
Interpretation: Longer-dated calls are always worth at least as much as shorter-dated calls.
Proof: The longer-dated option can replicate the shorter-dated option by:
- Hold the longer option until \(T_1\)
- Exercise if optimal at \(T_1\)
- Or continue holding until \(T_2\)
This flexibility makes it at least as valuable.
Black-Scholes verification: From the formula, \(C\) is increasing in \(T\). ✓
Arbitrage Bounds Summary¶
1. Comprehensive Bounds¶
For European options on non-dividend-paying stocks:
| Property | Call | Put |
|---|---|---|
| Lower bound | \(\max(S - Ke^{-rT}, 0)\) | \(\max(Ke^{-rT} - S, 0)\) |
| Upper bound | \(S\) | \(Ke^{-rT}\) |
| Monotonicity in \(S\) | \(\frac{\partial C}{\partial S} > 0\) | \(\frac{\partial P}{\partial S} < 0\) |
| Monotonicity in \(T\) | \(\frac{\partial C}{\partial T} > 0\) | \(\frac{\partial P}{\partial T} > 0\) |
| Monotonicity in \(\sigma\) | \(\frac{\partial C}{\partial \sigma} > 0\) | \(\frac{\partial P}{\partial \sigma} > 0\) |
| Monotonicity in \(K\) | \(\frac{\partial C}{\partial K} < 0\) | \(\frac{\partial P}{\partial K} > 0\) |
| Convexity in \(S\) | \(\Gamma > 0\) | \(\Gamma > 0\) |
| Convexity in \(K\) | \(\frac{\partial^2 C}{\partial K^2} > 0\) | \(\frac{\partial^2 P}{\partial K^2} > 0\) |
Practical Implications¶
1. Arbitrage Detection¶
Market prices violating these properties suggest:
- Arbitrage opportunity: Execute butterfly, calendar, or conversion spreads
- Transaction costs: Apparent violations within bid-ask spread
- Dividend effects: Non-dividend assumptions violated
- Early exercise: American features not captured
2. Model Validation¶
Any pricing model must satisfy:
- All monotonicity conditions
- Convexity constraints
- Upper and lower bounds
Black-Scholes satisfies all of these (as verified above).
3. Greeks Consistency¶
The properties imply relationships among Greeks:
- \(\Delta \in (0,1)\) for calls (from monotonicity and bounds)
- \(\Gamma > 0\) (from convexity)
- \(\Theta < 0\) typically (from time decay)
- \(\nu > 0\) (from volatility benefit)
These must be consistent for any valid pricing model.
Comparison: American vs. European¶
For American options on non-dividend-paying stocks:
1. Call Options¶
Reason: Never optimal to exercise early (forgo time value and interest on strike).
2. Put Options¶
Reason: Early exercise can be optimal when deep ITM (capture strike immediately rather than wait).
Bound:
Note: American put lower bound is intrinsic value, not discounted intrinsic value.
Summary¶
European option prices under Black-Scholes satisfy rigorous mathematical properties:
Bounds:
- Calls: \((S - Ke^{-rT})^+ \leq C \leq S\)
- Puts: \((Ke^{-rT} - S)^+ \leq P \leq Ke^{-rT}\)
Monotonicity:
- Increasing in \(S\) (calls), \(T\), \(\sigma\)
- Decreasing in \(K\) (calls), increasing in \(K\) (puts)
Convexity:
- Convex in \(S\) (positive gamma)
- Convex in \(K\) (butterfly constraint)
Significance:
- Arbitrage detection: Violations indicate mispricing
- Model validation: Any valid model must satisfy these
- Hedging implications: Convexity drives dynamic hedging needs
- Greeks relationships: Properties constrain sensitivities
These properties are more fundamental than the specific Black-Scholes formula—they follow from no-arbitrage alone and apply to any option pricing model.
Exercises¶
Exercise 1. A European call is quoted at \(C = 12\) with \(S = 100\), \(K = 95\), \(r = 4\%\), and \(T = 0.5\) years. Verify that the call price satisfies both the upper bound \(C \leq S\) and the lower bound \(C \geq \max(S - Ke^{-rT}, 0)\). What is the time value of the option?
Solution to Exercise 1
Given: \(C = 12\), \(S = 100\), \(K = 95\), \(r = 0.04\), \(T = 0.5\).
Upper bound check: \(C = 12 \leq 100 = S\) ✓
Lower bound: \(\max(S - Ke^{-rT}, 0) = \max(100 - 95e^{-0.02}, 0) = \max(100 - 93.12, 0) = 6.88\).
\(C = 12 \geq 6.88\) ✓
Both bounds are satisfied.
Time value: The intrinsic value is \(\max(S - K, 0) = \max(100 - 95, 0) = 5\).
The option's time value is \(\$7\), representing the value of optionality (the chance of further favorable moves and the interest savings from deferring the strike payment).
Exercise 2. Prove the butterfly spread inequality: for strikes \(K_1 < K_2 < K_3\) with \(K_2 = \frac{K_1 + K_3}{2}\), show that
by constructing a butterfly spread portfolio and arguing that its payoff is non-negative in all states.
Solution to Exercise 2
Consider the butterfly spread portfolio: buy 1 call at \(K_1\), sell 2 calls at \(K_2 = \frac{K_1+K_3}{2}\), buy 1 call at \(K_3\).
The payoff at maturity is \(\Pi(S_T) = (S_T - K_1)^+ - 2(S_T - K_2)^+ + (S_T - K_3)^+\).
Case 1: \(S_T \leq K_1\)
All calls expire worthless: \(\Pi = 0 - 0 + 0 = 0 \geq 0\).
Case 2: \(K_1 < S_T \leq K_2\)
Case 3: \(K_2 < S_T \leq K_3\)
since \(2K_2 - K_1 = K_3\). Since \(S_T \leq K_3\), we have \(\Pi = K_3 - S_T \geq 0\).
Case 4: \(S_T > K_3\)
In all cases \(\Pi \geq 0\). By no-arbitrage, a portfolio with non-negative payoff must have non-negative cost:
Rearranging: \(C(K_2) \leq \frac{C(K_1) + C(K_3)}{2}\).
Exercise 3. Compute the delta \(\Delta = \mathcal{N}(d_1)\), gamma \(\Gamma = \frac{\mathcal{N}'(d_1)}{S\sigma\sqrt{T}}\), and vega \(\nu = S\sqrt{T}\,\mathcal{N}'(d_1)\) for a call with \(S = 50\), \(K = 50\), \(r = 3\%\), \(\sigma = 20\%\), \(T = 1\). Verify that \(\Delta \in (0, 1)\), \(\Gamma > 0\), and \(\nu > 0\).
Solution to Exercise 3
Parameters: \(S = 50\), \(K = 50\), \(r = 0.03\), \(\sigma = 0.20\), \(T = 1\).
Standard normal values: \(\mathcal{N}(0.25) = 0.5987\), \(\mathcal{N}'(0.25) = \phi(0.25) = \frac{1}{\sqrt{2\pi}}e^{-0.03125} = 0.3867\).
Delta:
Check: \(\Delta \in (0, 1)\) ✓
Gamma:
Check: \(\Gamma > 0\) ✓
Vega:
Check: \(\nu > 0\) ✓
All three conditions are verified, consistent with the theoretical properties of European options.
Exercise 4. Show that gamma is the same for a European call and put with the same strike and maturity. Start from put-call parity \(P = C - S + Ke^{-rT}\) and differentiate twice with respect to \(S\).
Solution to Exercise 4
From put-call parity:
Differentiate once with respect to \(S\):
Differentiate again:
That is:
The second derivative of \(Ke^{-rT}\) (a constant with respect to \(S\)) is zero, and the second derivative of \(-S\) is also zero. Therefore the gamma of the put equals the gamma of the call for options with the same strike and maturity. This also follows intuitively: gamma measures the curvature of the option price with respect to \(S\), and since put and call prices differ by a linear function of \(S\), their curvatures are identical.
Exercise 5. A market maker observes the following call prices for three strikes with the same maturity: \(C(90) = 15.20\), \(C(100) = 9.50\), \(C(110) = 5.80\). Check whether the convexity condition \(C(100) \leq \frac{C(90) + C(110)}{2}\) holds. If violated, describe the arbitrage strategy.
Solution to Exercise 5
Check convexity: \(\frac{C(90) + C(110)}{2} = \frac{15.20 + 5.80}{2} = 10.50\).
Compare with \(C(100) = 9.50\).
Since \(9.50 \leq 10.50\), the convexity condition \(C(100) \leq \frac{C(90) + C(110)}{2}\) holds. ✓
There is no arbitrage opportunity. The butterfly spread (buy \(C(90)\), sell \(2 \times C(100)\), buy \(C(110)\)) costs \(15.20 - 2(9.50) + 5.80 = 2.00 > 0\), which is consistent with its non-negative payoff. If the condition had been violated (say \(C(100) = 11.00 > 10.50\)), one would sell the butterfly (sell \(C(90)\), buy \(2 \times C(100)\), sell \(C(110)\)) to collect a positive upfront cash flow with a non-positive future liability.
Exercise 6. Explain why an American call on a non-dividend-paying stock is never exercised early. Use the lower bound \(C \geq S - Ke^{-r(T-t)} > S - K\) (for \(r > 0\) and \(T - t > 0\)) to argue that the option is always worth more alive than dead.
Solution to Exercise 6
For a European call on a non-dividend-paying stock, the no-arbitrage lower bound is:
When \(r > 0\) and \(T - t > 0\): \(e^{-r(T-t)} < 1\), so \(Ke^{-r(T-t)} < K\), which gives:
If the American call were exercised early, the holder would receive \(S - K\) (the intrinsic value). But the option is worth at least \(S - Ke^{-r(T-t)} > S - K\), so exercising destroys value equal to at least \(K(1 - e^{-r(T-t)}) > 0\).
This excess value has two components:
- Time value of money: By not exercising, the holder defers paying \(K\), earning interest \(K(1 - e^{-r(T-t)})\) on the deferred payment.
- Insurance value: The option protects against the stock falling below \(K\); early exercise forfeits this downside protection.
Since the option alive is always worth more than the exercise value \(S - K\), early exercise is never optimal for an American call on a non-dividend-paying stock.
Exercise 7. The Black-Scholes theta for a call is
Show that both terms are negative, so \(\Theta < 0\) in general. Under what limiting conditions (deep ITM, near expiration) might the interest rate term dominate the volatility term? Can theta ever be positive for a European call on a non-dividend-paying stock?
Solution to Exercise 7
The theta formula is:
Term 1: \(S > 0\), \(\mathcal{N}'(d_1) = \phi(d_1) > 0\), \(\sigma > 0\), \(\sqrt{T-t} > 0\), so Term 1 is strictly positive, making \(-\text{Term 1} < 0\).
Term 2: \(r > 0\), \(K > 0\), \(e^{-r(T-t)} > 0\), \(\mathcal{N}(d_2) > 0\), so Term 2 is strictly positive, making \(-\text{Term 2} < 0\).
Both terms are negative, so \(\Theta < 0\) in general when \(r > 0\).
When the interest rate term dominates: For deep ITM calls (\(S \gg K\)), \(d_1 \to +\infty\), so \(\mathcal{N}'(d_1) = \phi(d_1) \to 0\) (the volatility term vanishes since the normal density decays rapidly). Meanwhile \(\mathcal{N}(d_2) \to 1\), so Term 2 \(\to rKe^{-r(T-t)}\), which remains bounded and positive. In this regime, theta is dominated by the interest rate term.
Can theta be positive? For a European call on a non-dividend-paying stock with \(r > 0\), theta is always negative (both terms have the same sign). However, if we consider the edge case \(r = 0\), Term 2 vanishes and \(\Theta = -\frac{S\phi(d_1)\sigma}{2\sqrt{T-t}} < 0\) still.
In practice, theta can become very slightly positive for deep ITM European calls on dividend-paying stocks (or for European puts), but for calls on non-dividend-paying stocks, \(\Theta \leq 0\) always holds (with equality only in degenerate limits).