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Bounding the Risk-Free Rate with a European Call Spread

Article Quant Q&A · Author: MikeHeimlich
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Summary

The document shows how a pair of European calls on the same underlying, with the same maturity, can imply an upper bound on the risk-free rate. The example uses a lower-strike call priced at 12 and a higher-strike call priced at 3, with strikes of 40 and 50 and maturity of three years.

Buying the lower-strike call and selling the higher-strike call creates a call spread whose maximum payoff is the 10-point strike difference. Under the answer’s annual compounding convention, the present value of that maximum payoff is 10 divided by the three-year accumulation factor. The spread costs 9, so requiring its price not to exceed that discounted maximum payoff yields a maximum rate of about 3.6%. This is a no-arbitrage bound based on the stated prices and payoff limit, not a method for estimating a unique market rate. It depends on the options being comparable and on the assumed discounting convention.

Key ideas

  • A lower-strike call combined with a short higher-strike call forms a call spread.
  • The spread’s maximum payoff equals the difference between its strikes.
  • The spread price cannot exceed the discounted value of that maximum payoff under the stated setup.
  • The example derives a maximum annual risk-free rate of 3.6% using three-year maturity and annual compounding.
  • The result is a bound, not a unique estimate of the risk-free rate.

Tags

OptionsDerivatives pricingRisk management
Full text
# Calculating the max. risk free interest rate with two given options


# Calculating the max. risk free interest rate with two given options












I have an excercise where we have two European Call Options, which have the same underlying, same maturity $t = 3$, same interest. The only difference is their price and their strike. The price of the first Call is $C_A = 3$ with a Strike of $K_A = 50$ and the second is $C_B = 12$ with Strike $K_B = 40$. Now I have to find out the maximal riskfree interest rate $r_f$. Does anyone how to find this $r_f$ thanks in advance

## Answer by dm63 (score 3, accepted)

https://quant.stackexchange.com/a/45652

Consider the 40/50 call spread. This has a maximum payoff of 10, and hence has a maximum value of $$10/(1+r)^3$$. Where r is the annual risk free rate. But we know it is priced at 12-3 =9, so the maximal risk free rate satisfies $$(1+r)^3=10/9$$ which gives r= 3.6%

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.