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Problem Set 51
Due date: Oct 3, 2022
Question 1: European and American Puts (4/10) You wish to price an European
put on a stock which currently trades for $100. The put expires in nine months, and has a
strike of $100. The nine-month interest rate (annualized continuously compounded) is 5%.
The estimated volatility of the stock is 25%. The stock pays no dividends.
Important hint: To solve this question, you could use the excel worksheet
on moodle titled “BinomialTreeAmerican.xls”. Doing your own calculations is
also fine, but the algebra is very messy.
(i) What is the Black-Scholes-Merton price of the European put?
(ii) What is the price of the European put according to a standard 3-step binomial tree?
(iii) Suppose the standard 3-step binomial tree is the true description of stock price
movements in the real world. If the European put is trading for $6, is there an
arbitrage? If not, explain why not. If so, explain in detail what your strategy is.
(iv) What is the price of the American put according to a 3-step binomial tree?
(v) Under what circumstances, if any, do you exercise the put before maturity?
Question 2: Implied Volatility and Put-Call Parity (3/10)
.
Suppose S = 100 and there are both a 9-month European call and a 9-month European
put with K = 100. The continuously compounded risk-free rate is 5%, and there are no
payouts.
(i) The call currently trades at a price of 14.087. What is the Black-Scholes implied
volatility?
(ii) The put trades at an implied volatility of 36.85%. Is there an arbitrage opportunity
here? If so, how would you take advantage of it and what are the cash flows?
1
Note: optional questions are for your practice only. They are not counted toward your grades.
1
Question 3: Greeks for Black-Scholes-Merton Model (3/10)
Scholes-Merton Model,
Consider the Black-
(i) What are the Delta (∆) and the Gamma (Γ) of an European call option? You need to
show how to derive the formula.
(ii) What is the Vega of an European call option? You need to show how to derive the
formula.
2
Stock price
100
Strike Price
100
Volatility
0.36
Risk free rate
0.05
T
0.75
d1
0.27616588
d2
-0.0356033
N(d1)
0.60878967
N(d2)
0.48579935
Call price
14.0870445
(ii)
Yes. There is an arbitrage opportunity here. I can buy synthetic put and short put to take advantage of it.
ake advantage of it.

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Explanation & Answer:

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