How Stock Price Changes Put Value

Why a Put Moves Opposite to Its Underlying

A put gives its holder the right, but not the obligation, to sell the underlying stock for a specified exercise price at expiration. If the stock price rises while the exercise price, expiration date, interest rate, and other relevant conditions are held fixed, the put generally becomes less valuable. The reason begins with the payoff: at expiration, a put pays only when the stock price is below the exercise price.

The relationship is not usually dollar for dollar. A rise in the stock price can reduce the put’s value by less than that rise, and the amount depends on the possible stock prices at expiration. A put can also retain value when it is currently out of the money, because the stock price may fall before expiration.

Definition: A European put with exercise price $K$ and expiration time $T$ has expiration payoff $(K-S_T)^+=\max(K-S_T,0)$, where $S_T$ is the stock price at expiration. Its current value is the price paid today for that contingent payoff, not merely its current expiration payoff evaluated using today’s stock price.

A Model for Comparing Matched Puts

To isolate the effect of the current stock price, compare puts with the same exercise price $K$, expiration $T$, and underlying stock, while holding the interest rate and the stock’s return model fixed. For the results and examples here, assume the stock pays no dividends, trading is frictionless, and an arbitrage-free risk-neutral pricing model applies. Under risk-neutral pricing, the value of a European put is the discounted expected expiration payoff using risk-neutral probabilities—not probabilities intended as forecasts of actual outcomes.

Write $G=S_T/S_0$ for the stock’s gross price multiplier from today to expiration, where $S_0$ is today’s stock price. In the matched comparison, the risk-neutral distribution of $G$ is held fixed as $S_0$ changes. With an effective annual risk-free rate $i$, the valuation formula is:

P_0=(1+i)^{-T}\,E^Q\!\left[(K-S_0G)^+\right].

Here $E^Q$ denotes expectation under the risk-neutral distribution. The formula assumes that the rate is effective annually and that $T$ is measured in years. For a one-year option, the discount factor is $1/(1+i)$. In a one-period binomial model with up multiplier $u$ and down multiplier $d$, the risk-neutral probability of an up move is $q=(1+i-d)/(u-d)$, provided $d<1+i<u$. If that condition fails, the usual binomial model does not have a valid risk-neutral probability between zero and one.

Why a Higher Stock Price Lowers Put Value

The key is a comparison of expiration payoffs under the same possible stock-price multipliers. Fix a multiplier $G>0$ and compare two current stock prices $S_1<S_2$. The expiration stock prices in that outcome are $S_1G$ and $S_2G$. The put payoff at the higher stock price cannot exceed the payoff at the lower price:

(K-S_2G)^+\le (K-S_1G)^+.

Taking risk-neutral expectations and applying the same positive discount factor preserves the inequality. Therefore, under the stated matched-model assumptions, the higher current stock price cannot make the European put more valuable. This argument does not require a particular binomial model or a normal distribution; it uses the ordering of the put payoffs outcome by outcome.

The comparison also gives a useful bound on the size of the change. For $S_2>S_1$, the reduction in the payoff in any outcome is at most $(S_2-S_1)G$. Since the stock pays no dividends, risk-neutral pricing requires $E^Q[G]=(1+i)^T$. Discounting the expected bound gives:

0\le P_0(S_1)-P_0(S_2)\le S_2-S_1.

Thus, in this model, raising the stock price lowers or leaves unchanged the put value, but the decrease cannot exceed the increase in the stock price. The inequality is a bound, not a claim that every put’s value changes by a fixed fraction of the stock-price change. If the stock has dividends, the matched model and bound need to account for them; the non-dividend assumption used here should not be silently discarded.

Worked Example: Comparing Three Matched Puts in a One-Period Tree

A hypothetical non-dividend-paying stock has one year to a European put’s expiration. The exercise price is $K=\$100$, and the effective annual risk-free rate is $i=10\%$. In a one-period binomial model, the stock’s gross return multiplier is $u=1.4$ in the up state and $d=0.8$ in the down state. Calculate the put values when today’s stock price is $\$100$, $\$120$, and $\$80$. Keep the same tree assumptions for all three puts.

Method. First determine the risk-neutral probability, then calculate each state’s expiration price and put payoff, and finally discount the expected payoff. The risk-neutral up probability is:

q=\frac{1+i-d}{u-d} =\frac{1.10-0.80}{1.40-0.80} =\frac{0.30}{0.60} =0.5.

The down probability is $1-q=0.5$, and the one-year discount factor is $1/1.10$. These probabilities are valid because $0.8<1.1<1.4$.

When $S_0=\$100$. The up-state stock price is $\$100(1.4)=\$140$, giving put payoff $(100-140)^+=\$0$. The down-state stock price is $\$100(0.8)=\$80$, giving payoff $(100-80)^+=\$20$. The risk-neutral expected payoff is $0.5(0)+0.5(20)=\$10$. The put value is $10/1.10=\$9.0909$, or \$9.09 to the nearest cent.

When $S_0=\$120$. The up-state price is $\$120(1.4)=\$168$, with payoff $(100-168)^+=\$0$. The down-state price is $\$120(0.8)=\$96$, with payoff $(100-96)^+=\$4$. The expected payoff is $0.5(0)+0.5(4)=\$2$, so the put value is $2/1.10=\$1.8182$, or \$1.82.

When $S_0=\$80$. The up-state price is $\$80(1.4)=\$112$, with payoff $\$0$. The down-state price is $\$80(0.8)=\$64$, with payoff $\$36$. The expected payoff is $0.5(0)+0.5(36)=\$18$, so the put value is $18/1.10=\$16.3636$, or \$16.36.

The check is consistent with the price ordering: the put values are $\$16.36$, $\$9.09$, and $\$1.82$ as the stock price rises from $\$80$ to $\$100$ to $\$120$. The changes also satisfy the bound: raising the stock from $\$100$ to $\$120$ lowers the put by about $\$7.27$, less than the $\$20$ stock-price increase.

Moneyness Is Not the Same as Put Value

Moneyness describes the relationship between today’s stock price and the exercise price. A put is in the money when $S_0<K$, at the money when $S_0=K$, and out of the money when $S_0>K$. Its current intrinsic value is $(K-S_0)^+$. That amount describes the payoff if the put could be exercised immediately; it is not generally the European put’s full value.

An out-of-the-money put can have positive value because it has time remaining and the stock could fall below $K$ before expiration. Conversely, being in the money does not mean that the put’s value equals its intrinsic value: the option’s value reflects the possible expiration payoffs under the pricing model. This distinction is especially important when comparing puts on either side of $S_0=K$.

Worked Example: A Put Can Remain Valuable Above Its Exercise Price

A hypothetical non-dividend-paying stock has a one-year European put with $K=\$100$. Under the risk-neutral model, the gross multiplier $G$ is $0.74$ with probability $0.25$, $1.04$ with probability $0.50$, and $1.34$ with probability $0.25$. The effective annual rate is $i=4\%$. Verify that the probabilities are risk-neutral, then calculate the put value at $S_0=\$100$ and $S_0=\$110$.

Check the model. The risk-neutral expected multiplier is $0.25(0.74)+0.50(1.04)+0.25(1.34)=0.185+0.520+0.335=1.040$, which equals $1+i$. The model is consistent with a non-dividend-paying stock at the stated rate.

At $S_0=\$100$. The three expiration prices are $\$74$, $\$104$, and $\$134$. The corresponding put payoffs are $\$26$, $\$0$, and $\$0$. The expected payoff is $0.25(26)+0.50(0)+0.25(0)=\$6.50$. Discounting gives $6.50/1.04=\$6.25$.

At $S_0=\$110$. The expiration prices are $\$81.40$, $\$114.40$, and $\$147.40$. The payoffs are $\$18.60$, $\$0$, and $\$0$. The expected payoff is $0.25(18.60)=\$4.65$, and the put value is $4.65/1.04=\$4.4711538$, or \$4.47 to the nearest cent.

The stock-price increase of $\$10$ lowers the put value from $\$6.25$ to $\$4.47$, a decrease of about $\$1.78$. At $S_0=\$110$, the put is out of the money and its intrinsic value is zero, yet its model value is positive because the low-price outcome still produces a payoff. The calculations check both the inverse price relationship and the distinction between intrinsic value and option value.

Comparing Prices Across a Two-Period Tree

In a multi-period model, the same principle applies, but the possible expiration prices and their probabilities must reflect every path through the tree. The next example uses a two-period binomial tree to show why one should calculate terminal payoffs rather than infer a put’s value directly from its current moneyness.

Worked Example: A Two-Period Matched-Price Comparison

A hypothetical non-dividend-paying stock has a two-year European put with $K=\$100$. The effective annual rate is $i=0\%$. In each one-year period, the stock rises by a factor $u=1.5$ or falls by a factor $d=0.75$. Calculate the put values for $S_0=\$80$ and $S_0=\$100$.

Method and probabilities. The one-period risk-neutral up probability is $q=(1-0.75)/(1.5-0.75)=0.25/0.75=1/3$. The down probability is $2/3$. The two-year discount factor is $1/(1+0)^2=1$. Over two periods, the probability of two up moves is $(1/3)^2=1/9$, the probability of one up and one down is $2(1/3)(2/3)=4/9$, and the probability of two down moves is $(2/3)^2=4/9$.

At $S_0=\$80$. The three terminal stock prices are $80(1.5)^2=\$180$, $80(1.5)(0.75)=\$90$, and $80(0.75)^2=\$45$. The put payoffs are $\$0$, $\$10$, and $\$55$. The expected payoff, and therefore the put value, is:

\frac{1}{9}(0)+\frac{4}{9}(10)+\frac{4}{9}(55) =\frac{40+220}{9} =\frac{260}{9} =\$28.89.

At $S_0=\$100$. The terminal prices are $\$225$, $\$112.50$, and $\$56.25$. The payoffs are $\$0$, $\$0$, and $\$43.75$. Thus:

\frac{1}{9}(0)+\frac{4}{9}(0)+\frac{4}{9}(43.75) =\frac{175}{9} =\$19.44.

The put value falls by $\$28.8889-\$19.4444=\$9.4445$, approximately $\$9.44$ using the unrounded values. This is less than the $\$20$ increase in the stock price. The result also checks the general ordering: the put is more valuable at the lower current stock price.

Common Exam Trap: Using Intrinsic Value as the Option Price

A frequent error is to use $(K-S_0)^+$ as if it were the price of a European put. That is only the current intrinsic value. For instance, in the first worked example, the put with $K=\$100$ and $S_0=\$120$ is out of the money. The incorrect calculation is:

(100-120)^+=0 \quad\Longrightarrow\quad \text{“put value”}=\$0.

The correction is to calculate and discount the possible expiration payoffs. In that example, the down-state payoff is $\$4$, the up-state payoff is $\$0$, and the risk-neutral probabilities are $0.5$ each. The value is $[0.5(4)+0.5(0)]/1.10=\$1.82$, not zero. The put is out of the money today but can still pay at expiration.

A second trap is to assume the put’s value must fall by exactly the amount the stock rises. The bound derived above shows that the decrease is no more than the stock-price increase under the stated assumptions, but the examples demonstrate that the actual decrease depends on the distribution of expiration outcomes. To compare matched puts reliably, keep the contract terms and model assumptions fixed, then value the payoffs.

Key takeaway: For a European put with fixed terms and a fixed non-dividend stock return model, increasing today’s stock price cannot increase the put’s value. The put’s moneyness may change as the stock crosses the exercise price, but neither moneyness nor intrinsic value alone gives the full option value.

Check Your Understanding

For each problem, use the stated matched model and give dollar values to the nearest cent where needed.

A one-year put has $K=\$100$ and $i=0\%$. Its risk-neutral gross multiplier is $0.75$ or $1.25$, each with probability $0.5$. Find its value at $S_0=\$80$ and $S_0=\$100$, and state the direction of the change. (Answer: $\$20.00$ at $S_0=\$80$, $\$12.50$ at $S_0=\$100$; the value falls by $\$7.50$.)
A one-year put has $K=\$110$, $i=10\%$, and binomial multipliers $u=1.4$, $d=0.8$. Find the risk-neutral probabilities and the put values at $S_0=\$90$ and $S_0=\$105$. (Answer: $q=0.5$, down probability $=0.5$; values $\$17.27$ and $\$11.82$, respectively.)
A one-year put has $K=\$100$, $i=0\%$, and risk-neutral multipliers $0.6,1.0,1.4$ with probabilities $0.25,0.50,0.25$. Find its values at $S_0=\$90$ and $S_0=\$110$. (Answer: $\$16.50$ and $\$8.50$, respectively.)
A two-year put has $K=\$100$, $i=0\%$, and a two-period binomial model with $u=1.5$, $d=0.75$. Calculate the put value at $S_0=\$80$, using the probabilities for the three distinct terminal-price paths. (Answer: $\$28.89$.)
In a matched, non-dividend-paying model, a stock price rises from $\$95$ to $\$108$. The put exercise price is $\$100$. State the put’s moneyness at each stock price and the largest possible decrease in value under the stated assumptions. (Answer: in the money at $\$95$, out of the money at $\$108$; the value decrease is no more than $\$13$.)

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