FRM 2008 Practice Question

[al66440]

Hi David, Can you show me how to get to 5.34% on this question?
Thanks,
Alex
16. Jeff is an arbitrage trader, and he wants to calculate the implied dividend yield on a stock while looking at
the over-the-counter price of a 5-year put and call (both European-style) on that same stock. He has the
following data:
• Initial stock price = USD 85
• Strike price = USD 90
• Continuous risk-free rate = 5%
• Underlying stock volatility = unknown
• Call price = USD 10
• Put price = USD 15
What is the continuous implied dividend yield of that stock?
a. 2.48%
b. 4.69%
c. 5.34%
d. 7.71%
Answer: c
We can use the Put-Call parity here to easily solve for the continuous dividend yield.
We have C - P = S0e-q*T - Ke-r*T, so 10 - 15 = 85e-q*5 - 90e-0.05*5. Solving for q, we get 5.34%.
a. Incorrect. C and P where inverted in the formula.
b. Incorrect. C and P where inverted in the formula, and S and K where also inverted in the formula.
c. Correct. The above formula was used correctly, C - P = S0
e-q*T - Ke-r*T.
d. Incorrect. S and K where inverted in the formula.
Reference:
John Hull, Options, Futures, and Other Derivatives, 6th ed. (New York: Prentice Hall, 2006).,
Chapter 13 – The Black-Scholes-Merton Model

 

[David Harper CFA FRM]

Hi Alex,

Sure, that's a tough b/c it takes the put-call parity and adds the dividend yield, so you've got to remember that the dividend yield effectively reduces the stock price. So replace (S) with "discounted" stock price S*EXP[-qt]. For what it's worth, my own memorization sequence starts with the minimum value, which is my favorite:

MV of call option = S - K*exp[-rt]. So c > S-K*exp[-rt].

I like this b/c it's "inside" the Black-Scholes, so for put call parity, only need to include the "p"
c > S-K*exp[-rt], and
c - p = S-K*exp[-rt]. See how these are sort of a pair?

and now just, if dividend, remember to reduce/discount the S:
(c-p) = S*EXP[-qt] - K*EXP[-rt]
i.e., same put call parity but stock is "discounted"/reduced by dividend yield. Why? Option hold forgoes dividends, or if you like, dividends takeaway from capital appreciation that option holder would have enjoyed.

Okay, then solve that for q, which is a very useful study

(c-p) = S*EXP[-qt] - K*EXP[-rt]
(c-p) + K*EXP[-rt] = S*EXP[-qt]
[(c-p) + K*EXP[-rt]]/S = EXP[-qt]
ln[(c-p) + K*EXP[-rt]]/S = ln(EXP[-qt])
ln[(c-p) + K*EXP[-rt]]/S = [-qt]
ln[(c-p) + K*EXP[-rt]]/S * -(1/t) = q

note 4th step performs very common step: need to unlock q in the exponent by taking natural log of both sides, ln() an exp() are inverse functions: ln(exp(x)) = x. This liberates the q. So I do get 5.34%...

David