Discounting The Strike Under Different Rate Conventions

Why the Strike Must Be Discounted Correctly

In European put-call parity, the strike price is paid or received at the option’s expiration, not at the valuation date. Its value today is therefore the present value of a future payment. The arithmetic depends on how the interest rate is quoted: an effective annual rate, a nominal rate convertible several times per year, and a force of interest do not use the same accumulation formula.

The key is to match the rate convention to the time interval. Express expiration as a number of years, use the accumulation rule implied by the rate, and discount the strike by dividing by that accumulation factor. This is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: If a certain amount $K$ is payable at time $T$, its present value is $PV(K)=K/A(0,T)$, where $A(0,T)$ is the accumulation factor from time 0 to time $T$ under the stated interest-rate convention. The rate and time units must refer to the same convention.

Present Value Under Three Rate Conventions

Let $K$ be the strike price and $T$ the time to expiration in years. Under the standard conventions below, the present value is obtained using the corresponding accumulation factor.

Rate quote	Accumulation factor to time $T$	Present value of $K$
Effective annual rate $i$	$(1+i)^T$	$K(1+i)^{-T}$
Nominal annual rate $i^{(m)}$, convertible $m$ times per year	$(1+i^{(m)}/m)^{mT}$	$K(1+i^{(m)}/m)^{-mT}$
Force of interest $\delta$, constant over the term	$e^{\delta T}$	$Ke^{-\delta T}$

For the nominal-rate row, the formula directly applies when expiration is at a conversion date, so $mT$ is an integer number of conversion periods. If expiration falls between conversion dates, the contract or problem must specify how to handle that partial period; do not silently assume a fractional-period rule. For the effective annual and constant-force formulas, $T$ may be a fractional number of years.

Rate-convention check: A nominal annual rate $i^{(m)}$ convertible $m$ times per year gives a rate of $i^{(m)}/m$ per conversion period. It is not itself an effective annual rate. The corresponding effective annual rate is $(1+i^{(m)}/m)^m-1$.

Why the Discount Formulas Hold

Present value is the amount that, accumulated under the stated model, grows to the future payment. This gives a short derivation for each convention.

Effective annual rate.
At an effective annual rate $i$, one dollar grows to $1+i$ dollars in one year. Over $T$ years, the accumulation factor is $(1+i)^T$. If $P$ is the present value of $K$, then $P(1+i)^T=K$, so $P=K(1+i)^{-T}$.

Nominal rate convertible $m$ times per year.
The rate per conversion period is $i^{(m)}/m$. Each period multiplies the balance by $1+i^{(m)}/m$. After $mT$ complete periods, $P(1+i^{(m)}/m)^{mT}=K$, giving the nominal-rate present-value formula.

Constant force of interest.
A constant force $\delta$ gives an accumulation factor $e^{\delta T}$. Thus $Pe^{\delta T}=K$, and solving for $P$ yields $P=Ke^{-\delta T}$.

The derivation also explains the central discipline: the future amount must equal the present amount multiplied by the accumulation factor for the model actually specified. The same strike and expiration can have different present values under different rate conventions.

Worked Example: Effective Annual Rate and a Two-Year Strike

A European option has strike $K=\$15{,}000$ and expires in 2 years. The risk-free rate is 5% effective annually. Find the present value of the strike and the amount by which it is below the strike.

Method. Use the effective annual accumulation factor $(1+i)^T$, with $i=0.05$ and $T=2$. This avoids treating the annual effective rate as if it were a nominal rate with an unstated conversion frequency.

The accumulation factor is

$$ (1.05)^2=1.1025. $$

Therefore,

PV(K)=\frac{\$15{,}000}{1.1025}=\$13{,}605.44.

The present value is $\$15{,}000-\$13{,}605.44=\$1{,}394.56$ below the strike. As a check, $\$13{,}605.44(1.1025)$ is approximately $\$15{,}000$, with the small difference attributable to rounding to cents. Money is rounded to two decimal places.

Worked Example: Nominal Rate Convertible Quarterly

A hypothetical European option has strike $K=\$10{,}000$, expires in 18 months, and is valued using a risk-free nominal annual rate of 8% convertible quarterly. Find $PV(K)$.

Method. Convert the annual nominal quote into a quarterly rate, then count the quarters to expiration. The time is $T=18/12=1.5$ years, so the number of conversion periods is $mT=4(1.5)=6$. Expiration coincides with a quarterly conversion date.

The quarterly rate is $0.08/4=0.02$, or 2%. The accumulation factor is

$$ (1.02)^6=1.126162419264. $$

Thus,

PV(K)=\frac{\$10{,}000}{1.126162419264}=\$8{,}879.71.

The check is that $\$8{,}879.71(1.02)^6$ is approximately $\$10{,}000$. The common mistake is to use $(1.08)^{1.5}$: 8% is a nominal rate convertible quarterly, not an effective annual rate. The correct nominal-rate calculation uses a 2% rate for each of six quarters. Money is rounded to two decimal places.

Worked Example: Continuous Compounding to a Fractional-Year Expiration

A hypothetical option has strike $K=\$12{,}000$, expires in 18 months, and is discounted at a constant force of interest $\delta=0.06$ per year. Find the present value of the strike.

Method. Convert 18 months to $T=1.5$ years and use $Ke^{-\delta T}$. A force of interest is not a nominal rate divided into a chosen number of periods.

The exponent is $\delta T=0.06(1.5)=0.09$, and $e^{0.09}\approx1.094174284$. Therefore,

PV(K)=\$12{,}000e^{-0.09} =\frac{\$12{,}000}{1.094174284} =\$10{,}967.17.

Accumulating the rounded present value gives approximately $\$10{,}967.17(1.094174284)=\$12{,}000$. The formula applies because the force is stated to be constant over the term. Money is rounded to two decimal places.

Comparing Conventions Without Confusing Them

A numerical comparison makes clear why the rate label matters. Suppose the strike is $\$10{,}000$, expiration is exactly one year away, and each of the following is considered separately: 4% effective annually, 4% nominal convertible monthly, or a constant force of 4%. These are three different models, even though each quote contains the number 4%.

For 4% effective annually, $PV(K)=\$10{,}000/1.04=\$9{,}615.38$. For 4% nominal convertible monthly, the monthly rate is $0.04/12$, so the accumulation factor is $(1+0.04/12)^{12}\approx1.040741542$, giving $PV(K)\approx\$9{,}608.53$. For a constant force of 0.04, $PV(K)=\$10{,}000e^{-0.04}\approx\$9{,}607.89$. The present values differ because the accumulation factors differ.

A useful comparison is to convert each quote to an effective annual rate before comparing. The nominal monthly quote corresponds to $(1+0.04/12)^{12}-1\approx0.0407415$, or about 4.0742% effective annually. A constant force of 0.04 corresponds to an effective annual rate of $e^{0.04}-1\approx0.0408108$, or about 4.0811%. Neither is exactly 4% effective annually.

Using the Discounted Strike in Parity

For European call and put prices $C$ and $P$ on a non-dividend-paying stock, put-call parity is $C-P=S_0-PV(K)$, where $S_0$ is the current stock price. The parity identity uses the present value of the strike, not the undiscounted strike $K$. The earlier material on put-call parity establishes the identity; the task here is to supply its strike term using the correct rate convention.

For example, if the strike is $\$10{,}000$, expiration is 18 months away, and the annual rate is 8% nominal convertible quarterly, the strike term is $\$8{,}879.71$, as calculated above. Substituting $\$10{,}000$ instead would treat a future payment as if it were payable today and would change the parity calculation by $\$1{,}120.29$.

Common Exam Traps

Using the wrong annual-rate interpretation. The expression $(1+i)^T$ requires $i$ to be effective annually. A nominal convertible rate requires its per-period rate and number of periods.
Mixing months and years. If the quoted rate is annual, 18 months corresponds to $T=1.5$ years, not $T=18$.
Using $m$ instead of $mT$ periods. For quarterly conversion over 18 months, there are $4(1.5)=6$ periods, not four.
Discounting with the wrong sign. Under a positive force, $PV(K)=Ke^{-\delta T}$. The positive exponent $e^{\delta T}$ is the accumulation factor, not the discount factor.
Assuming an unstated partial period. If expiration does not coincide with a nominal conversion date, check whether the problem defines a rule for the remaining fraction before applying a fractional exponent.
Using $K$ instead of $PV(K)$ in parity. The strike is due at expiration, so it must be discounted to the valuation date.

Key takeaway: First identify whether the quote is effective annual $i$, nominal convertible $i^{(m)}$, or a constant force $\delta$. Then express expiration in years, count conversion periods when needed, and calculate $PV(K)$ from the matching accumulation factor.

Check Your Understanding

For each problem, identify the rate convention before calculating. Round money to the nearest cent.

A European option has strike $\$8{,}000$, expires in 3 years, and the rate is 3.5% effective annually. Find $PV(K)$. (Answer: $\$7{,}215.54$)
A strike of $\$9{,}000$ is payable in 2 years. The nominal annual rate is 6% convertible semiannually. Find the present value and state the number of conversion periods. (Answer: 4 periods; $PV(K)=\$7{,}996.38$)
A hypothetical option has strike $\$6{,}000$, expires in 2.5 years, and the force of interest is constant at 5% per year. Find $PV(K)$. (Answer: $\$5{,}294.98$)
A strike of $\$4{,}000$ is due in 15 months under an 8% nominal annual rate convertible quarterly. Calculate the number of periods and $PV(K)$. (Answer: 5 periods; $PV(K)=\$3{,}622.92$)
For a $\$10{,}000$ strike due in one year, compare 4% effective annually with a constant force of 0.04. Which produces the lower present value, and by how much? (Answer: The constant-force model gives the lower value: $\$9{,}607.89$, which is $\$7.49$ less than $\$9{,}615.38$.)

Rate quote	Accumulation factor to time \(T\)	Present value of \(K\)
Effective annual rate \(i\)	\((1+i)^T\)	\(K(1+i)^{-T}\)
Nominal annual rate \(i^{(m)}\), convertible \(m\) times per year	\((1+i^{(m)}/m)^{mT}\)	\(K(1+i^{(m)}/m)^{-mT}\)
Force of interest \(\delta\), constant over the term	\(e^{\delta T}\)	\(Ke^{-\delta T}\)

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