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# Implied Interest Rates from Option Prices

#### jyothi1965

##### New Member
Consider the following call option with 6 months till expiry, the strike price is $50, the current stock price is$55, and the value of the option is $5. What does this imply about the level of 6-month interest rates. a. Interest rates are positively sloped around the 6-month period. b. Interest rates are negatively sloped around the 6-month period. c. Interest rates are at zero for the 6-month period. d. Cannot be determined from the information given David, the above is a question from FRM 2001. My understanding is that options have been used to determine implied volatilities and this is the first time I am seeing options used to determine the implied interest rates. The answer is supposed to be C. The logic is that you if you buy the call and short the stock (for six months!!!!) , you can earn interest on 50$. At expiry, you can excercise the option to buy at 50$and square up and keep the interest. If the stock is less than$50, you would have made money by shorting in addition to interest income

(the option of what happens if stock is above 50 is not relevant as the option holder can buy at 50)

Since this creates interest free artibtrage income and therefore interest rates have to be zero ([email protected]@?)

How on earth can one conclude that interest rates are zero?

Jyothi

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Jyothi,

Agreed, this is a metaphysical question!

But yet another example of why (after years of client contact on this), I learned to teach option pricing from a lower bound perspective rather than a no-arbitrage deduction. If you keep in mind the minimum value (the lower bound), you will quickly see an option like this implies: zero rate and zero volatility.

Minimum Value (lower bound) on a call: c >= S - K[exp(-rt)]. In words, "the call must be worth at least stock price minus discounted strike price." This is a lower bound because it assumes zero volatility. Again, because i think it is worth dwelling on: a European call must be worth at least the stock price minus the discounted strike price; this lower bound assumes no volatility.

(In my opinion, this is the starting point for learning the closed-form OPM. It is worth visualizing why this formula holds: the call must be worth at least the discounted present value of the future gain on a stock that increases at the risk free rate. BTW, this was also the expensing method used by private companies prior to FAS 123R to expense their employee stock option- i.e., they could assume no volatility because they had no public volatility)

This lower bound is then the teaching/learning foundation that leads to put-call parity (add the put term) and the Black-Scholes itself [wrap in the N()s, as they serve to "plus up the minimum value for volatility]!

But, back to the question, we see a call here with no time value: $5 =$55 - \$50[exp(-rt)]. So, based on that we can infer two things (assuming there is no maturity. the other alternative is an option with no time value because it is at expiration): zero volatility and zero riskless rate.

Thanks,
David

#### jyothi1965

##### New Member
David

The way that you have explained makes the answer pretty obvious, no question on that. Obviously when exp (-rt) = 1, it follows that "r"" is zero.

But do we infer that riskless rates and interest rates are the same (please see the question above). that is stretching it too far. and then Prof Rene Stulz throws in a couple of twists (upward sloping and downward sloping, etc, etc)- it would appear that the question has been designed to trip, come what may. This is the disappointing part of FRM.

But anyway thanks as always

Jyothi