Deep In-the-Money Option Time Value Depends on Its Benchmark

# Time Value of Deep ITM Put vs Call


# Time Value of Deep ITM Put vs Call












According to my understanding of the discussion in page 259 of “Options as a Strategic Investment 4/E” by Lawrence McMillan, a deep in-the-money put has less time value than a deep in-the-money call of equal moneyness.

I would like to prove or verify this statement using the Black–Scholes framework (assuming no dividends), or any rigorous method.

Could someone provide guidance or hints on how to approach this proof, or clarify whether this statement is correct under standard assumptions?

## Answer by pandashark (score 2)

https://quant.stackexchange.com/a/85568

The claim is true when the risk-free rate is positive, which is the regime McMillan was writing about. But the full picture is more nuanced — there are two competing effects, and the result depends on whether you measure "time value" relative to spot intrinsic or forward intrinsic.

### Setup

Define "equal moneyness" as both options being in-the-money by the same dollar amount $d$:

- ITM call at strike $K_c = S - d$, intrinsic value $= d$

- ITM put at strike $K_p = S + d$, intrinsic value $= d$

Time values (using spot intrinsic $\max(S-K, 0)$ and $\max(K-S, 0)$):

$$\text{TV}_C = C(K_c) - d, \qquad \text{TV}_P = P(K_p) - d$$

### Decomposition via Put-Call Parity

European put-call parity (no dividends): $C(K) - P(K) = S - Ke^{-rT}$.

Apply at each strike and substitute $d = S - K_c$ and $d = K_p - S$:

$$\text{TV}_C = P(K_c) + K_c(1 - e^{-rT})$$

$$\text{TV}_P = C(K_p) - K_p(1 - e^{-rT})$$

Since $K_c + K_p = 2S$:

$$\boxed{\text{TV}_C - \text{TV}_P = \underbrace{\bigl[P(K_c) - C(K_p)\bigr]}_{\text{skew term}} + \underbrace{2S(1 - e^{-rT})}_{\text{discount term}}}$$

where $P(K_c)$ is an OTM put at strike $S - d$ and $C(K_p)$ is an OTM call at strike $S + d$.

### The Two Competing Effects

Discount term $2S(1 - e^{-rT}) > 0$ for $r > 0$. This reflects the time value of money: holding an ITM call defers paying the strike (earning interest on cash), while holding an ITM put defers receiving the strike (forgoing interest). The call benefits from positive rates; the put is penalized.

Skew term $P(K_c) - C(K_p) < 0$ for $r \geq 0$. At the same absolute distance $d$ from spot, the OTM call is worth more than the OTM put. This is because equal dollar distance translates to unequal log-moneyness — the OTM put is further from ATM in log-space since $\lvert\ln(S/K_c)\rvert > \lvert\ln(S/K_p)\rvert$ (equivalently, $-\ln(1-d/S) > \ln(1+d/S)$, which follows from $\ln(1-x^2) < 0$). This log-distance asymmetry, combined with the log-normal distribution's heavier right tail, makes the OTM call strictly more valuable.

For deep ITM options, both OTM counterparts approach zero, so the skew term vanishes while the discount term persists — making the claim stronger the deeper in-the-money you go.

### Numerical Verification (QuantLib, Black-Scholes)

$S = 100$, $\sigma = 30\%$, $T = 1$ year:

| $d$ | $K_c$ | $K_p$ | $r$ | $\text{TV}_C$ | $\text{TV}_P$ | $\text{TV}_C - \text{TV}_P$ |
| 20 | 80 | 120 | 0% | 3.53 | 5.44 | -1.91 |
| 20 | 80 | 120 | 2% | 4.70 | 3.62 | +1.08 |
| 20 | 80 | 120 | 5% | 6.46 | 1.05 | +5.41 |
| 20 | 80 | 120 | 10% | 9.43 | -2.81 | +12.24 |
| 30 | 70 | 130 | 5% | 4.40 | -1.67 | +6.06 |
| 40 | 60 | 140 | 5% | 3.20 | -3.71 | +6.90 |

Decomposition at $r = 5\%$, $T = 1$:

| $d$ | OTM Put $P(K_c)$ | OTM Call $C(K_p)$ | Skew term | Discount term | $\text{TV}_C - \text{TV}_P$ |
| 5 | 7.17 | 11.98 | -4.81 | +9.75 | +4.95 |
| 20 | 2.56 | 6.90 | -4.34 | +9.75 | +5.41 |
| 40 | 0.27 | 3.12 | -2.85 | +9.75 | +6.90 |
| 50 | 0.04 | 2.06 | -2.01 | +9.75 | +7.74 |

As $d$ increases, the skew term vanishes (both OTM options expire worthless) while the discount term is fixed — confirming the claim grows stronger with depth.

### The Deeper Story: Spot Intrinsic vs Forward Intrinsic

The decomposition above reveals something important. The discount term $2S(1 - e^{-rT})$ is not optionality — it is a deterministic discounting effect. It equals the difference between spot intrinsic and the present value of forward intrinsic summed across both options.

If we instead measure time value relative to forward intrinsic — defined as $\max(F - K, 0)\,e^{-rT}$ where $F = Se^{rT}$ is the forward price — then the discount term vanishes entirely. What remains is purely the skew term, which is always negative (for $r \geq 0$). Under forward intrinsic:

$$\text{TV}_C^{\text{fwd}} - \text{TV}_P^{\text{fwd}} = P(K_c) - C(K_p) < 0$$

(The inequality $C(K_p) > P(K_c)$ follows from the log-distance asymmetry discussed above.)

So measured by genuine optionality, the deep ITM put always has more time value than the call, regardless of the interest rate. The apparent advantage of the ITM call under spot intrinsic comes entirely from the interest earned on deferred strike payment, not from greater optionality.

### When Does McMillan's Claim Hold?

McMillan's observation uses spot intrinsic (the standard in equity markets). Under this convention:

- $r > 0$ with moderate vol: the discount term dominates. $\text{TV}_C > \text{TV}_P$. This is the regime McMillan was writing about (US rates were 5%+ in the 1980s-90s).

- $r = 0$: only the skew term survives. $\text{TV}_C < \text{TV}_P$ — the claim reverses.

- Small $r$ with high vol: the skew term can dominate even with $r > 0$. At $r = 1\%$, $\sigma = 60\%$, $d = 30$: $\text{TV}_C = 8.77$ vs $\text{TV}_P = 13.59$ (QuantLib-computed).

For $\sigma = 30\%$, the crossover rate (where the two terms balance) is around $r \approx 1\%$. For $\sigma = 60\%$, it is around $r \approx 3\text{--}4.5\%$ depending on depth. In short, the claim holds comfortably for any rate environment above 2-3%, which covers most of financial history.

### Caveats







Computed with QuantLib using `AnalyticEuropeanEngine`.