Questions:
Problem Solving
1) The following call and put options are on the same stock and have the same expiration
date. Given the following strike prices and call/put premiums of the option contracts:
Strike Price $ 130 140 150 160 170
Call Price $ 26.6 20.4 15.3 11.2 8.1
Put Price $ 5.9 9.6 14.5 20.4 27.2
Answer the following questions and use a payoff/profit diagram to illustrate your answer.
Option prices are per share, but option contracts are for 100 shares.
a. Calculate the maximum loss of the following trading strategy: write 2 calls with a strike
price of $130, buy 4 calls with a strike price of $150 , and buy 3 puts with a strike price
of $170.
b. Calculate the breakeven points of the above options trading strategy.
c. If the time to expiry of the option is 3 months, the risk free rate is 2% and no dividends
are paid, what is current price of the stock and the implied volatility of the stock using
the Black-Scholes model? (Use the BKM spreadsheet from Blackboard.) Copy the
relevant section of the spreadsheet into your submission as below
Sd deviation
(annual) 0.2783 d1 -0.0225
Maturity (in
years) 0.5 d2 -0.2193
Risk-free rate
(annuatandarl) 0.05 N(d1) 0.4910
Stock Price 100 N(d2) 0.4132
Exercise price 105 B/S call value 6.7864
Dividend yield
(annual) 0 B/S put value 9.1939
2) The stock of Dusty Coal Ltd is selling for $53. The mine is not fully producing, no
dividends are anticipated and there is doubt regarding the level of demand from Asia.
Due to these uncertainties, you estimate that in any three months the stock may rise by
11% or fall be 8% both are equally likely. T-bill yield is 5% per annum.
a. Construct and a two period binomial tree to show the distribution of possible prices for
Dusty Coal stock in 6 months time. Fully explain the diagram.
b. Using your tree calculate the value of a $60 strike 6 month call option.
c. Using the tree, calculate the value of a $60 strike 6 month put option.
d. Do your values satisfy put-call parity? Explain.
Hint: Work backwards to get the call price after 3 months using the hedge ration for
second three months. That gives the payoffs for the first three months etc..
Conceptual Question
3) Today is 31st May. The yield on T-bills is 3% per annum. The futures price for June 30th
delivery of Gold is $1593.60 for December 30th delivery the price is $1600.00.
a. Does this pricing present an arbitrage opportunity? Provide a full explanation of your
reasoning.
b. How might you construct a portfolio to exploit any such arbitrage opportunity that
existed?
c. What are the risks in any such portfolio? (100 words max)
SOLUTION
1) The following call and put options are on the same stock and have the same expiration date. Given the following strike prices and call/put premiums of the option contracts:
Strike Price $ 130 140 150 160 170
Call Price $ 26.6 20.4 15.3 11.2 8.1
Put Price $ 5.9 9.6 14.5 20.4 27.2
Answer the following questions and use a payoff/profit diagram to illustrate your answer. Option prices are per share, but option contracts are for 100 shares.
a. Calculate the maximum loss of the following trading strategy: write 2 calls with a strike price of $130, buy 4 calls with a strike price of $150, and buy 3 puts with a strike price of $170.
Ans: For the above strategy we first need to calculate the option premiums for each option contract as shown below:
- Write 2 calls with a strike price of $130: As we are writing the option there will be an inflow equal to the call price multiplied by the number of contracts. i.e. Inflow = 26.6*2 = $53.20
- Buy 4 calls with a strike price of $150: As we are buying the option there will be an outflow in the form of option premium which can be calculated in the similar way as above, Outflow = 15.3*4 = $61.20
- Buy 3 puts with a strike price of $170: Similar to the above call option there will be an outflow in the case of buying a put option also, Outflow = 27.2*3 = $81.60
Therefore the net initial cash flow is the difference between the inflows and outflows, and it was found that there was a net outflow i.e. initial investment = $89.60
Therefore, the maximum loss that one can incur in the above trading strategy is equal to the initial investment ($89.60)
b. Calculate the breakeven points of the above options trading strategy.
Ans: The breakeven was calculated using the payoff table as shown below:
St | payoff | profit | |
St< 130 |
115.00 |
165.00 |
75.40 |
130<St<150 |
136.08 |
89.60 |
0.00 |
150<St<170 |
170.00 |
0.00 |
(89.60) |
St>170 |
214.80 |
89.60 |
0.00 |
The breakeven points were achieved by using the Solver tool in excel. The two break even points are highlighted in the table above with the price at $136.08 and $214.80.
c. If the time to expiry of the option is 3 months, the risk free rate is 2% and no dividends are paid, what is current price of the stock and the implied volatility of the stock using the Black-Scholes model? (Use the BKM spreadsheet from Blackboard.)
2) The stock of Dusty Coal Ltd is selling for $53. The mine is not fully producing, no dividends are anticipated and there is doubt regarding the level of demand from Asia. Due to these uncertainties, you estimate that in any three months the stock may rise by 11% or fall by 8% both are equally likely. T-bill yield is 5% per annum.
a) Construct and a two period binomial tree to show the distribution of possible prices for Dusty Coal stock in 6 months time. Fully explain the diagram.
Ans: For simplicity, the option is assumed to be a European option.
S_{o }denotes the stock price at T=0 i.e. now which is given in the question, the terms uS_{o }and dS_{o }are the stock prices after 3 months where u denotes the 11% increase whereas d denotes 8% decrease in the stock price. Similarly for the second period, the upper node uuS_{o} is the 11% increase in the price from uS_{o. }It is important to notice here that the value udS_{o} can be calculated either as 8% decrease in uS_{o} or 11% increase in dS_{o}. The value ddS_{o }is 8% decrease in the value of dS_{o}. Thus, the above binomial tree displays the movement in price during 2 3-month periods.
b) Using your tree calculate the value of a $60 strike 6 month call option.
Ans: Using the above tree and the probability of 0.5 as given and the formula given below, the option prices at each node were calculated:
Figure 1: Option pricing through binomial tree (Parameswaran, S)
Where r= 1+ riskfree rate (5% p.a). The value of C_{0} i.e. the option price at T=0 can be calculated by the same formula by substituting the C_{u} and C_{d }values in place of C_{uu} and C_{dd}. The table below shows the option price at T=0 as well as the other nodes.
Cuu |
$5.30 |
Cu |
$2.62 |
Cud |
0 |
||
Cdu |
0 |
Cd |
0 |
Cdd |
0 |
C_{0} |
$1.29 |
The important thing to consider here is the value of C_{uu} and C_{dd} which are calculated as Max(0, X-St) i.e., the value of the option will be maximum of 0 and the excess of the spot price over the strike price (which is the only case when the option will be exercised).
c) Using the tree, calculate the value of a $60 strike 6 month put option.
Ans: Similarly, the value of P_{uu} and P_{dd} are calculated as Max(0, St-X) i.e., the value of the option will be maximum of 0 and the excess of the strike price over the spot price (which is the only case when the option will be exercised).
Using the same formula as above, the following values were obtained:
Puu | 0 | Pu | $2.90 |
Pud | $5.88 | ||
Pdu | $5.88 | Pd | $10.38 |
Pdd | $15.14 | P_{0} | $6.56 |
d) Do your values satisfy put-call parity? Explain.
Ans: The put-call parity is given as:
Figure 2: Put-call parity Equation (Parameswaran, S)
Where C_{E,t }& P_{E,t }are the call and put option premiums at time t=0, the subscript E denotes that it is a European option. St is the spot price of the stock on t=0 and X is the strike price.
Substituting the values given we find the following:
C_{E,t }= $1.29
P_{E,t} = $6.56
St = $53
X = $60, PV(X)= $58.55
We can clearly see that the LHS (-$5.27) and RHS (-$5.55) are not equal, therefore, the put-call parity condition is not satisfied. This implies that there is a arbitrage opportunity for the trader to book profits.
3) Today is 31st May. The yield on T-bills is 3% per annum. The futures price for June 30th delivery of Gold is $1593.60 for December 30th delivery the price is $1600.00.
a) Does this pricing present an arbitrage opportunity? Provide a full explanation of your reasoning.
Ans: In order to check for an arbitrage opportunity we need to compare the Futures price as well as the Spot price of the underlying asset. We can say that arbitrage exists only if the Futures price is different from the future value of the spot price (S) i.e. S*(1+r), where r is the yield on T-bills or any other return on investment. Therefore, in this case there exist two possibilities for each of the delivery date i.e. 30th June and 30th Dec.
A) The Futures price is less than the future value of the spot price, in which case the futures contract is underpriced and the trader can sort out a trading strategy in order to book profits through this arbitrage opportunity. This type of arbitrage is called Reverse Cash and Carry arbitrage (Kolb & Overdahl)
B) The futures price is greater than the future value of the spot price, in which case the contract is overpriced. This type of arbitrage is called Cash and Carry arbitrage (Kolb & Overdahl)
In the context of the question, we can only be sure of the arbitrage opportunity once we have information regarding the spot price. However, for Cash and Carry arbitrage we need the borrowing rate of interest, but the question only provides the T-Bills rate which is necessary for detecting the Reverse Cash and Carry arbitrage. Therefore, we can conclude that there is a possibility of Reverse Cash and Carry arbitrage opportunity in this case.
b) How might you construct a portfolio to exploit any such arbitrage opportunity that existed?
Ans: As explained above, we have ruled out the possibility of the Cash and Carry arbitrage which leaves out Reverse Cash and Carry arbitrage, so in order to exploit the opportunity the following needs to be done (Kolb & Overdahl):
- The arbitrageur needs to short sell the gold at the current spot price
- The proceeds from the sale can be invested in T-Bills at 3% p.a.
- The short position in gold should be covered by going long in the futures contract
At the expiry, the difference between the value of the T-Bills and the futures price is the profit that the arbitrageur makes.
c) What are the risks in any such portfolio? (100 words max)
Ans: The above portfolio is a combination of a short position in the asset and long position in futures, which gives rise to a synthetic short position in a T-Bill i.e. a riskless asset (Bond, P). This means that the combination of a short and long in two risky assets gives rise to a risk free investment, thus, no risk is involved in the portfolio. Moreover, since futures transactions occur in stock exchanges, the counterparty risk is also eliminated.
References
Kolb, R.W. and Overdahl, J.A. ‘Understanding Futures Markets, Chapter 3, Futures Prices’. 6th Ed: Blackwell Publishing. [online] Available at: http://www.blackwellpublishing.com/ufm/chapter3.ppt [Accessed on 29 May 2012]
Bond, P. ‘Forward Contracts on Financial Assets’ [online] Available at: http://finance.wharton.upenn.edu/~pbond/teaching/Forwards.pdf [Accessed on 29 May 2012]
Figure 1: Option pricing through binomial tree (Parameswaran, S), ‘Futures and Options: Concepts and Applications- The Binomial Option Pricing Model’: Tata Mcgraw Hill Education Private Limited
Figure 2: Put-call parity Equation (Parameswaran, S), ‘Futures and Options: Concepts and Applications- Arbitrage Restrictions’: Tata Mcgraw Hill Education Private Limited
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