Derivative hw2 v5
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The document discusses various concepts related to derivatives including: 1) A trading strategy is proposed for a firm to sell overvalued euro forward contracts and buy euros, borrowing USD to take advantage of mispricing. 2) A call option on a stock with binomial up/down movements is valued at $0.4762 using risk-neutral probabilities. Trading opportunities are explored if the option price differs from this value. 3) Forward prices for different commodities like coffee, gold and natural gas are explained in terms of factors like storage costs, demand patterns and whether the good is perishable or not. 4) Interest rates are deduced that would eliminate arbitrage opportunities between investing in local currency or
![Introduction to Derivatives – HW01
Dong Li, Lehana Singh, Nader Hosseini-Sianaki
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4. (**) In Manor Farm, Snowball and Napoleon are trying to figure out how to manage
the financial portfolio of the community. They know that the risk-free rate available for
investment from Mr. Jones is at 10%. MF dollars, the currency at Manor Farm, are
currently trading at $2= 1 MF dollars. They know that the risk-free rate available for
investment from Mr. Jones is at 10%. That is, they can turn over 100 MF dollars to Mr.
Jones, and he will give them back 110 MF dollars in one year. Snowball and Napoleon
know that Mr. Whymper offers risk-free investments, but in US dollars ($). Snowball
and Napoleon have the following forward prices on MF dollars.
Maturity Forward price
1 year 2.40
2 years 2.88
3 years 3.46
4 years 4.15
5 years 4.98
If financial markets in and around Manor Farm are arbitrage free, transaction costs are
minor, and the such - what rate of interest (for $ deposits) do you think Mr. Whymper
should offer?
rUS = [ F / S * (1 + rMF)T ]1/T - 1
Maturity Fwd Price rUS
1 2.4 0.32
2 2.88 0.32
3 3.46 0.32
4 4.15 0.32
5 4.98 0.32
The interest rate offered by Mr. Whymper (rUS) = 32%
4](https://image.slidesharecdn.com/derivativehw2v5-111205060800-phpapp02/85/Derivative-hw2-v5-4-320.jpg)
![Introduction to Derivatives – HW01
Dong Li, Lehana Singh, Nader Hosseini-Sianaki
5. (**) Consider a binomial world in which a stock, over the next year, can go up in
value by 50% (subjective probability of 60%) or down by 33.33% (subjective probability
of 40%). The stock is currently trading at $50. The risk free return is 5%.
(a) What is the value of the call option that expires in one year with a strike price of
$55?
Risk neutral upper probability = Pu = (r-d)/(u-d) = 0.46
Risk neutral downward probability = Pd = ( u-r)/(u-d) = 0.54
Value of Call option = (0.46*(75-55))/(1.05) = 8.762
(b) Is the expected return on the call option higher or lower?
(0.6(75-55)+0.4*0)/(1+rc) = 8.76
rc = 45.28%
The expected return on the call option is higher than the stock
(c) What is the value of the put option that expires in one year with a strike price of $45?
What is its expected return?
Risk neutral upper probability = Pu = ( r-d)/(u-d) = 0.46
Risk neutral downward probability = Pd = ( u-r)/(u-d) = 0.54
Value of put option = (0.54*(45-33.335))/(1.05) = 5.9991 ~ 6
(0.6(0)+0.4*(45-33.33)/(1+rc) = 6
rc = - 22.20%
(d) Say something interesting about the expected returns of calls and puts in this
binomial world.
For call option:
[(Neutral Upside Prob)*(Call Payoff)] / (1 + rf) = [(Subjective Prob) * (Call Payoff)] / (1 + rc)
This simplifies to:
(Neutral Upside Prob) / (1 + rf) = (Subjective Upside Prob) / (1 + rc)
We know (Neutral Upside Prob) < (Subjective Upside Prob), therefore rf < rc
Similarly for put option:
(Neutral Downside Prob) / (1 + rf) = (Subjective Downside Prob) / (1 + rc)
We know (Neutral Downside Prob) > (Subjective Downside Prob), therefore rf > rc
5](https://image.slidesharecdn.com/derivativehw2v5-111205060800-phpapp02/85/Derivative-hw2-v5-5-320.jpg)
Derivative hw2 v5
- 1. Introduction to Derivatives – HW01 Dong Li, Lehana Singh, Nader Hosseini-Sianaki 1. (*) Suppose that the current spot price for the euro is $0.90, and that the interest rate in the euro zone stands at 15%. Further assume that the term structure in the US is at 5%, and that the volatility of the dollar/euro exchange rate stands at 35%. Venis Telroy Inc. is a new firm in the forward trading business. They are buying and selling one-year forward contracts on the euro. They are quoting a bid of $0.86 and an ask of $0.87. (a) What would you assess as a fair value for the one-year forward price on the euro? Spot Price = $ 0.90 Interest rate in Europe = 0.15 Interest rate in US = 0.05 Forward rate (USD/Euro) = Spot rate *(1+R$)/(1+Re) = 0.8217 (b) If you could trade with Venis Telroy, can you suggest a trading strategy? The quoted bid for Euro is $0.86, but the calculated forward rate is $0.8217. This means that the forward rate for the Euro is overvalued. S0 = Spot Price of Euro today St = Spot Price of Euro in future Trading strategy: At t=0 At t= T Sell Euro Forward 0 $0.86 – St Buy Euro -0.9/1.15 St Borrow USD 0.86/1.05 -0.86 +0.3649 0 2. (*) Consider a binomial world in which a stock, over the next year, can go up in value by 20% (subjective probability of 55%) or down by 10% (subjective probability of 45%). The stock is currently trading at $10. The riskfree return is 5%. Consider a call that expires in one year, with a strike price of $11. (a) What is the value of the call option? Risk neutral value of pu = 0.5 Risk neutral value of pd = 0.5 Value of Call option = (0.5*(12-11)+0.5*(0))/(1.05) = 0.4762 (b) If the call option was trading for $0.32, can you find an arbitrage opportunity? 1
- 2. Introduction to Derivatives – HW01 Dong Li, Lehana Singh, Nader Hosseini-Sianaki At t=0 D U (St < =11) (12 > St > 11) Buy Call (a nos) -0.32 (a) 0 a*St –a* 11 Short (b nos) Asset +10*b -St*b -St*b Lend Money (c -c +1.05*c +1.5* c amount) 0 0 (a-b)St-11*a+1.5c >0 Solving for a, b and c -0.32a+10b-c=0 -St*b+1.05c=0 Min. value of St in D region = 9 Hence, c = 9b/1.05 a = 4.44625b In region D value 11<St<12 3.44625*St-11+1.05c >0 (As c is positive) (c) If the call option was trading for $0.61, can you find an arbitrage opportunity? At t=0 D U (St < =11) (12 > St > 11) Sell Call +0.61 0 11-St Long 1/3 Asset -3.33 3 4 Borrow Money +2.72 - 2.86 - 2.86 0 +0.14 +2.14 (-St+ 1.14 >0) 3. (**) In the _le hw2data.xls in the course web page you will find the future prices of several assets. How can you explain the different patterns in the forward prices for the different assets? Be as precise as you can in terms of quantifying deviations from the pricing equations discussed in class, but also focus on the qualitative nature of the deviations (i.e. try to explain why they arise). Coffee is a perishable commodity and cannot be stored away for very long. Furthermore, the shape of the graph is determined by near future demand outstripping supply. This results in near future price ending up higher than later months. This is called normal backwardation. 2
- 3. Introduction to Derivatives – HW01 Dong Li, Lehana Singh, Nader Hosseini-Sianaki Coffee Last Price 190 180 170 160 Coffee Last Price 150 140 2M 15M 1M 3M 4M 5M 6M 7M 8M 9M 10M 11M 13M 14M 1Y Price of gold for future delivery is higher than sport price. This is expected for non-perishable commodities such as gold which has a cost of carry. Such costs include warehousing fees and interest forgone on money tied up, less income from leasing out the commodity if possible. Gold can be leased out. Here, the cost of carry s such that the Forward Price formula would determine the forward price to be higher than the spot price or near future price. This is called Contango. Gold Last Price 1350 1300 1250 Gold Last Price 1200 1150 1M 3M 5M 7M 9M 11M 13M 15M 17M Similar to gold, natural gar has a higher future delivery than sport price as it is a non-perishable commodity as well. Natural gas also has a high storage cost. The ripples in the graph are due to the cyclical nature of highs and lows in demand for natural gas due to fluctuations in seasonal demand. In colder months there is higher demand for natural gas and less demand at other times of the year. 3
- 4. Introduction to Derivatives – HW01 Dong Li, Lehana Singh, Nader Hosseini-Sianaki Natural Gas Price 8 7 6 5 4 3 Natural Gas Price 2 1 0 1M 19M 46M 10M 28M 37M 55M 64M 73M 82M 91M 100M 109M 118M 127M 136M 145M 4. (**) In Manor Farm, Snowball and Napoleon are trying to figure out how to manage the financial portfolio of the community. They know that the risk-free rate available for investment from Mr. Jones is at 10%. MF dollars, the currency at Manor Farm, are currently trading at $2= 1 MF dollars. They know that the risk-free rate available for investment from Mr. Jones is at 10%. That is, they can turn over 100 MF dollars to Mr. Jones, and he will give them back 110 MF dollars in one year. Snowball and Napoleon know that Mr. Whymper offers risk-free investments, but in US dollars ($). Snowball and Napoleon have the following forward prices on MF dollars. Maturity Forward price 1 year 2.40 2 years 2.88 3 years 3.46 4 years 4.15 5 years 4.98 If financial markets in and around Manor Farm are arbitrage free, transaction costs are minor, and the such - what rate of interest (for $ deposits) do you think Mr. Whymper should offer? rUS = [ F / S * (1 + rMF)T ]1/T - 1 Maturity Fwd Price rUS 1 2.4 0.32 2 2.88 0.32 3 3.46 0.32 4 4.15 0.32 5 4.98 0.32 The interest rate offered by Mr. Whymper (rUS) = 32% 4
- 5. Introduction to Derivatives – HW01 Dong Li, Lehana Singh, Nader Hosseini-Sianaki 5. (**) Consider a binomial world in which a stock, over the next year, can go up in value by 50% (subjective probability of 60%) or down by 33.33% (subjective probability of 40%). The stock is currently trading at $50. The risk free return is 5%. (a) What is the value of the call option that expires in one year with a strike price of $55? Risk neutral upper probability = Pu = (r-d)/(u-d) = 0.46 Risk neutral downward probability = Pd = ( u-r)/(u-d) = 0.54 Value of Call option = (0.46*(75-55))/(1.05) = 8.762 (b) Is the expected return on the call option higher or lower? (0.6(75-55)+0.4*0)/(1+rc) = 8.76 rc = 45.28% The expected return on the call option is higher than the stock (c) What is the value of the put option that expires in one year with a strike price of $45? What is its expected return? Risk neutral upper probability = Pu = ( r-d)/(u-d) = 0.46 Risk neutral downward probability = Pd = ( u-r)/(u-d) = 0.54 Value of put option = (0.54*(45-33.335))/(1.05) = 5.9991 ~ 6 (0.6(0)+0.4*(45-33.33)/(1+rc) = 6 rc = - 22.20% (d) Say something interesting about the expected returns of calls and puts in this binomial world. For call option: [(Neutral Upside Prob)*(Call Payoff)] / (1 + rf) = [(Subjective Prob) * (Call Payoff)] / (1 + rc) This simplifies to: (Neutral Upside Prob) / (1 + rf) = (Subjective Upside Prob) / (1 + rc) We know (Neutral Upside Prob) < (Subjective Upside Prob), therefore rf < rc Similarly for put option: (Neutral Downside Prob) / (1 + rf) = (Subjective Downside Prob) / (1 + rc) We know (Neutral Downside Prob) > (Subjective Downside Prob), therefore rf > rc 5
- 6. Introduction to Derivatives – HW01 Dong Li, Lehana Singh, Nader Hosseini-Sianaki 6. (***) In the Iniesta case (discussed in lecture 5), in part 3 we discussed the situation that would arise when Barcelona's stock could take on the values (150; 100; 60), there was a risk-free asset yielding 10%, and a call option on Barcelona stock with a known price. This problem studies what happens when there is no third security, i.e. what can we say about the prices of securities in a trinomial world when we only have two securities to try to replicate the payoffs of the derivative. In particular, what can you say about the price of a put option on Barcelona's stock with a strike price of 110? Assume you can only trade on Barcelona's stock (currently trading at $100) and risk free bonds (with 10% risk-free return), i.e. ignore the information on the call option on Barcelona's stock given in the case. Hint: the answer is that the put option, absent arbitrage opportunities, should be priced in the range (a; b). You need to figure out what a and b are, and find an argument to justify the bound. 6