Recovering An Implied Financing Rate From Option Quotes

Why Recover a Financing Rate From Option Quotes?

A call and put on the same asset can reveal more than the relationship between their prices. Under the no-dividend European option model, their price difference also reflects the present value of the strike. If the asset price, strike, and time to expiration are known, matched call and put quotes therefore imply a discount factor—and, after choosing a rate convention, an implied financing rate.

This is a conditional inference, not a direct observation of a borrowing rate. It relies on the option prices and asset price fitting the same no-arbitrage model. A discrepancy from a stated market rate calls for checking the assumptions and inputs; it does not, by itself, establish that an executable arbitrage is available.

Model and assumptions: Consider European calls and puts on the same non-dividend-paying asset, with the same strike $K$ and expiration $T$. Let $S_0$ be the asset price now, $C$ the call price, $P$ the put price, and $D(0,T)$ the discount factor for one unit of currency paid at time $T$. Assume the prices are synchronized, the contracts match in all relevant terms, and the market has no arbitrage with a common financing rate. Under these assumptions, $C-P=S_0-KD(0,T)$.

Isolating the Discount Factor

Rearrange parity to put the unknown discount factor on one side:

$$ C-P=S_0-KD(0,T) $$

D(0,T)=\frac{S_0-C+P}{K}

The numerator combines the current asset value with the option-price difference. The result is a discount factor for the strike payment at expiration. It is not itself an annual rate. To turn it into a rate, use the same convention as the rate you intend to compare it with.

Rate conversion: If $i$ is an annual effective rate and $T$ is measured in years, then $D(0,T)=(1+i)^{-T}$, so $i=D(0,T)^{-1/T}-1$. If $\delta$ is a constant force of interest, then $D(0,T)=e^{-\delta T}$, so $\delta=-\ln D(0,T)/T$. These conversions require $D(0,T)>0$ and $T>0$.

A discount factor below 1 gives a positive implied rate under either convention. A factor above 1 gives a negative implied rate; that is not automatically an error, since negative rates are possible. A nonpositive factor cannot be converted into a real rate using these formulas and is inconsistent with the positive discount factor in the model.

Why the Parity Rearrangement Works

The parity equation follows by comparing two portfolios with the same expiration payoff. One portfolio holds a call and an amount of cash that grows to $K$ at expiration. The other holds a put and one share. In a frictionless market, equal future payoffs must have equal current values.

Value the strike cash today.
Invest $KD(0,T)$ now at the model's financing rate. It is worth $K$ at time $T$.

Compare expiration payoffs.
The call plus $K$ pays $\max(S_T-K,0)+K=\max(S_T,K)$. The put plus one share pays $\max(K-S_T,0)+S_T=\max(K,S_T)$.

Equate current values.
The payoffs agree in every case, so $C+KD(0,T)=P+S_0$. Rearranging gives $D(0,T)=(S_0-C+P)/K$.

This derivation also identifies the limits of the calculation. The equality depends on the European exercise feature, the absence of dividends in this version of parity, and matching contracts. For American options, equality need not hold. If the asset pays dividends, the no-dividend equation must be adjusted rather than applied unchanged.

Recovering and Checking the Rate

A reliable calculation separates the problem into two tasks: first infer the discount factor from the option quotes, then convert that factor to the rate convention being tested. If a stated market rate is given, independently calculate its discount factor and compare it with the quote-implied factor. Comparing the rates is also useful, but only after both rates have been put on the same annual and compounding basis.

Worked Example: Two-Year Quotes and an Effective Annual Rate

A non-dividend-paying share is priced at $S_0=\$90$. Matched European options have strike $K=\$100$, expire in $T=2$ years, and are quoted at $C=\$12.00$ and $P=\$12.25$. Recover the implied annual effective rate and compare it with a stated market rate of 5% effective annually. Assume synchronized, exact prices and the model above.

Method. First infer $D(0,T)$ from parity, then convert it to an annual effective rate. The quotes have call-minus-put value $\$12.00-\$12.25=-\$0.25$.

Step 1: Infer the discount factor.

D(0,2)=\frac{90-12.00+12.25}{100} =\frac{90.25}{100} =0.9025

Step 2: Convert to an annual effective rate. Since $T=2$, $D(0,2)=(1+i)^{-2}$. Thus:

i=(0.9025)^{-1/2}-1 =\frac{1}{0.95}-1 =0.05263158\ldots

The quote-implied annual effective rate is $5.2632\%$, rounded to four decimal places as a percentage. The intermediate check is $1/(1.05263158\ldots)^2=0.9025$.

Step 3: Compare with the stated rate. At 5% effective annually, the two-year discount factor would be:

D_{\text{market}}(0,2)=\frac{1}{(1.05)^2} =\frac{1}{1.1025} =0.90702948\ldots

That rate implies a call-minus-put value of $90-100(0.90702948\ldots)=-\$0.7029478\ldots$, whereas the observed difference is $-\$0.25$. The implied rate is about 0.2632 percentage points above the stated rate, so the quotes do not exactly fit the stated-rate version of the model. Check: using the inferred factor gives $90-100(0.9025)=-\$0.25=C-P$. This method avoids treating the two-year discount factor as if it were an annual rate.

Worked Example: A Fractional Expiration Under Effective Annual Compounding

A non-dividend-paying asset has spot price $S_0=\$150$. A matched European call and put have strike $K=\$160$, expiration in $T=0.5$ years, and prices $C=\$18.40$ and $P=\$25.20$. Find the implied annual effective rate. Then assess whether it matches a stated 4% annual effective market rate.

Method. Infer the six-month discount factor and use the half-year exponent in the effective-rate formula. The half-year term is essential: an annual effective rate is not applied as though the options expired after a full year.

Step 1: Calculate the quote-implied factor.

D(0,0.5)=\frac{150-18.40+25.20}{160} =\frac{156.80}{160} =0.98

Step 2: Recover the annual effective rate. Solve $0.98=(1+i)^{-0.5}$:

i=(0.98)^{-1/0.5}-1 =(0.98)^{-2}-1 =\frac{1}{0.9604}-1 =0.0412328\ldots

The implied annual effective rate is $4.1233\%$, rounded to four decimal places as a percentage.

Step 3: Test the stated 4% rate. Its six-month discount factor is:

D_{\text{market}}(0,0.5) =(1.04)^{-0.5} =\frac{1}{\sqrt{1.04}} =0.98058068\ldots

The corresponding parity difference is $150-160(0.98058068\ldots)=-\$6.892908\ldots$. The observed difference is $18.40-25.20=-\$6.80$. They differ by about $\$0.0929$, so the exact quotes imply a slightly higher effective rate than 4%. Check: the inferred factor gives $150-160(0.98)=-\$6.80=C-P$. The common mistake avoided here is using $i=1/D-1$, which would treat a half-year discount factor as a one-year factor.

Worked Example: Recovering a Force of Interest

A hypothetical non-dividend-paying asset has $S_0=\$80$. Matched European options with strike $K=\$75$ expire in $T=1.5$ years and have prices $C=\$11.00$ and $P=\$3.00$. Recover the constant force of interest implied by the quotes and compare it with a stated force of 2.5% per year.

Method. Use parity to obtain the 1.5-year discount factor, then take its logarithm using the force-of-interest conversion. Do not compare this force directly with an effective annual rate; they are different conventions.

Step 1: Infer the discount factor.

D(0,1.5)=\frac{80-11+3}{75} =\frac{72}{75} =0.96

Step 2: Convert to a force of interest.

\delta=-\frac{\ln(0.96)}{1.5} =\frac{0.04082199\ldots}{1.5} =0.02721466\ldots

The implied force is $2.7215\%$ per year, rounded to four decimal places as a percentage.

Step 3: Compare on the same convention. A force of 2.5% gives:

D_{\text{market}}(0,1.5) =e^{-0.025(1.5)} =e^{-0.0375} =0.96319442\ldots

It would imply $C-P=80-75(0.96319442\ldots)=\$7.76042\ldots$, compared with the quoted difference of $\$8.00$. The exact quotes imply a higher force than the stated 2.5%. Check: the inferred factor gives $80-75(0.96)=\$8.00=C-P$. This comparison avoids mixing a force of interest with an effective annual rate.

Common Traps and Interpretation

The calculation is short, but its interpretation depends on keeping the model, time period, and rate convention straight. In an exam setting, show the inferred discount factor before reporting the rate; this makes sign and exponent errors easier to detect.

Reversing the signs. The correct rearrangement is $D(0,T)=(S_0-C+P)/K$. Using $(C-P-S_0)/K$ reverses the sign and can produce a negative factor, which is not a valid discount factor in this model.
Calling the factor a rate. A value such as $0.98$ is the value today of one unit paid at expiration, not an annual rate. Convert it using the actual time to expiration.
Using the wrong time unit. If expiration is six months away, use $T=0.5$ years when the rate is annual. Do not use $T=6$ or silently treat the term as one year.
Comparing unlike conventions. Convert the stated market rate to a discount factor using its own convention, or convert both factors to the same rate convention before comparing.
Applying equality outside the model. The equation used here assumes European options on a non-dividend-paying asset with matching terms. American options or known dividends require a different analysis.

An implied financing rate is best read as the rate that makes the matched quotes satisfy the assumed parity equation. If it differs from the stated market rate, verify the asset price, strike, expiration, exercise style, dividend assumption, and quote precision. This tutorial assumes exact, synchronized prices; it does not determine whether displayed quotes can be traded at those prices.

Key takeaway: For matched European options on a non-dividend-paying asset, infer $D(0,T)$ as $(S_0-C+P)/K$, then convert that factor using the stated time and rate convention. The implied rate matches the market rate only when the quote-implied and market discount factors agree under the same model.

Check Your Understanding

For each problem, assume synchronized, exact quotes and the no-dividend European model unless stated otherwise. Use rates per year and give implied rates to four decimal places as percentages.

A non-dividend-paying asset has $S_0=\$100$. Matched options with $K=\$105$ and $T=1$ year have $C=\$8$ and $P=\$11$. Find the implied annual effective rate. (Answer: $D(0,1)=103/105=0.9809524$; $i=1.9417\%$.)
A non-dividend-paying asset has $S_0=\$70$. Matched options with $K=\$75$ and $T=2$ years have $C=\$9$ and $P=\$12.50$. Find the implied annual effective rate. (Answer: $D(0,2)=0.98$; $i=(0.98)^{-1/2}-1=1.0153\%$.)
A non-dividend-paying asset has $S_0=\$48$. Matched options with $K=\$50$, $T=0.5$ years, $C=\$4$, and $P=\$5.50$ are quoted. Find the implied force of interest and compare it with a stated force of 2% per year. (Answer: $D(0,0.5)=0.99$; $\delta=-\ln(0.99)/0.5=2.0101\%$, slightly above 2%.)
A non-dividend-paying asset has $S_0=\$100$. Matched options with $K=\$100$ expire in one year and are quoted at $C=\$12$, $P=\$7$. Recover the implied annual effective rate. If the stated market rate is 5% effective annually, state whether the quotes match it. (Answer: $D(0,1)=0.95$; $i=1/0.95-1=5.2632\%$, so the quotes do not exactly match the stated rate.)

What You'll Learn