Define risk-neutral pricing and explain how it is used in option pricing.
This question is time-consuming (it just took me two hours to write it) but I hope it gives practice to the notoriously difficult risk-neutral idea in Tuckman. I think if you do each step, you cannot help but confront the idea. I still have to re-read the section on risk-neutral, after years and already several reviews; and I only feel I grasp it barely. Very difficult idea - David H
AIM: Define risk-neutral pricing and explain how it is used in option pricing.
Questions:
41.1 Assume a two-step one-year binomial interest rate tree (each step is six months) used to price a one-year zero coupon bond. In six months, at the first step (when the bond has a six month maturity), the bond price will be either $978 in the up-state or $982.80 in the down-state. The REAL-WORLD probabilities that correspond are 50% (up) and 50% (down). The bond has a face value of $1,000 and, since this is Tuckman, we assume semi-annual compound frequency.
What are, respectively, the down- and up-states of the INTEREST RATES at forward six months (node 1 if today is node 0)?
* a. 3.0% and 4.0%
* b. 3.5% and 4.5%
* c. 4.0% and 5.0%
* d. 4.5% and 5.5%
41.2 Assuming the up- and down-states are symmetrically +0.5% and -0.5% from the current six month rate, what is the bond’s expected discounted expected?
* a. $968.72
* b. $969.12
* c. $960.23
* d. $961.17
41.3 If the one-year spot rate is 4.1%, what is the bond’s (expected) market price?
* a. $968.72
* b. $969.12
* c. $960.23
* d. $961.17
41.4 If p is the probability of an up jump to the up-state and (1-p) is the probability of a down jump to the down-state, what are, respectively, the up- and down-jump RISK-NEUTRAL probabilities?
* a. 70.1% (p) and 29.9% (1-p)
* b. 29.9% and 70.1%
* c. 80.1% and 19.9%
* d. 19.9% and 80.1%
41.5 Assume a call option on this bond (i.e., option to purchase the $1,000 face value bond) has a strike price $980.00. What is the expected discounted value of the call option?
* a. $0.82
* b. $1.02
* c. $1.17
* d. $1.37
41.6 For the same call option, what is the actual (market) price of the option?
* a. $0.82
* b. $1.02
* c. $1.17
* d. $1.37
41.7 What is the implied drift in the interest rate?
* a. -30 basis point
* b. -20 basis points
* c. +20 basis points
* d. +30 basis points
41.8 Which of the following best describes risk-neutral pricing?
* a. Risk-neutral probabilities as inputs into expected discounted valued
* b. Real world probabilities as inputs into expected discounted valued
* c. Risk-neutral probabilities but only in the imaginary world of risk aversion
* d. Risk-neutral probabilities replace the risk-free rate to incorporate risk aversion
Answers:
* Here in forum
[Learn] Tax argument for risk management L1.T1.41 [practice, foundation]
Posted: 03 Aug 2010 11:12 AM PDT
Time To Pay
AIM: Explain how risk management can create value moving income across time and reducing taxes
Questions:
41.1 Assume Stulz’ stylized firm (Pure Gold) that produces one lump-sum, pretax cash flow at the end of a one-year period; i.e., firm value is the discounted present value of the one future cash flow. The riskfree rate is 3%. The firm’s future cash flow is only a function of the the spot price of gold, which is a purely UNSYSTEMATIC market risk. Without hedging, the firm’s future pretax cash flow will be either $0 (low future spot price of gold) or $100,000 (high future spot price of gold). Each outcome is equally likely. The corporate tax schedule of MARGINAL tax rates includes (these are currently accurate):
* 15% of pretax income from $0 to 50,000
* 25% from $50 to $75,000
* 34% from $75 to $100,000
Assume pretax cashflow is identical to pretax income. If the firm hedges the price risk of gold, by how much does this risk management increase firm value?
* a. Zero, per the Hedging Irrelevance Theorem
* b. $3,520
* c. $4,520
* d. $5,520
41.2 What does it mean for, instead, shareholders to practice a homemade hedge in this situation. And would that create value?
* a. Short gold futures, yes
* b. Short gold futures, no
* c. Long gold futures, yes
* d. Long gold futures, no
41.3 Which of the following is the necessary assumption that makes validity Stulz’ “tax argument for risk management” (the assertion that risk management can add value where taxes are concerned)?
* a. Tax carryforwards
* b. Tax shield
* c. Personal taxes
* d. Differential tax rates
Answers:
|
AIM: Define risk-neutral pricing and explain how it is used in option pricing.
Questions:
41.1 Assume a two-step one-year binomial interest rate tree (each step is six months) used to price a one-year zero coupon bond. In six months, at the first step (when the bond has a six month maturity), the bond price will be either $978 in the up-state or $982.80 in the down-state. The REAL-WORLD probabilities that correspond are 50% (up) and 50% (down). The bond has a face value of $1,000 and, since this is Tuckman, we assume semi-annual compound frequency.
What are, respectively, the down- and up-states of the INTEREST RATES at forward six months (node 1 if today is node 0)?
* a. 3.0% and 4.0%
* b. 3.5% and 4.5%
* c. 4.0% and 5.0%
* d. 4.5% and 5.5%
41.2 Assuming the up- and down-states are symmetrically +0.5% and -0.5% from the current six month rate, what is the bond’s expected discounted expected?
* a. $968.72
* b. $969.12
* c. $960.23
* d. $961.17
41.3 If the one-year spot rate is 4.1%, what is the bond’s (expected) market price?
* a. $968.72
* b. $969.12
* c. $960.23
* d. $961.17
41.4 If p is the probability of an up jump to the up-state and (1-p) is the probability of a down jump to the down-state, what are, respectively, the up- and down-jump RISK-NEUTRAL probabilities?
* a. 70.1% (p) and 29.9% (1-p)
* b. 29.9% and 70.1%
* c. 80.1% and 19.9%
* d. 19.9% and 80.1%
41.5 Assume a call option on this bond (i.e., option to purchase the $1,000 face value bond) has a strike price $980.00. What is the expected discounted value of the call option?
* a. $0.82
* b. $1.02
* c. $1.17
* d. $1.37
41.6 For the same call option, what is the actual (market) price of the option?
* a. $0.82
* b. $1.02
* c. $1.17
* d. $1.37
41.7 What is the implied drift in the interest rate?
* a. -30 basis point
* b. -20 basis points
* c. +20 basis points
* d. +30 basis points
41.8 Which of the following best describes risk-neutral pricing?
* a. Risk-neutral probabilities as inputs into expected discounted valued
* b. Real world probabilities as inputs into expected discounted valued
* c. Risk-neutral probabilities but only in the imaginary world of risk aversion
* d. Risk-neutral probabilities replace the risk-free rate to incorporate risk aversion
Answers:
* Here in forum
[Learn] Tax argument for risk management L1.T1.41 [practice, foundation]
Posted: 03 Aug 2010 11:12 AM PDT
Time To Pay
AIM: Explain how risk management can create value moving income across time and reducing taxes
Questions:
41.1 Assume Stulz’ stylized firm (Pure Gold) that produces one lump-sum, pretax cash flow at the end of a one-year period; i.e., firm value is the discounted present value of the one future cash flow. The riskfree rate is 3%. The firm’s future cash flow is only a function of the the spot price of gold, which is a purely UNSYSTEMATIC market risk. Without hedging, the firm’s future pretax cash flow will be either $0 (low future spot price of gold) or $100,000 (high future spot price of gold). Each outcome is equally likely. The corporate tax schedule of MARGINAL tax rates includes (these are currently accurate):
* 15% of pretax income from $0 to 50,000
* 25% from $50 to $75,000
* 34% from $75 to $100,000
Assume pretax cashflow is identical to pretax income. If the firm hedges the price risk of gold, by how much does this risk management increase firm value?
* a. Zero, per the Hedging Irrelevance Theorem
* b. $3,520
* c. $4,520
* d. $5,520
41.2 What does it mean for, instead, shareholders to practice a homemade hedge in this situation. And would that create value?
* a. Short gold futures, yes
* b. Short gold futures, no
* c. Long gold futures, yes
* d. Long gold futures, no
41.3 Which of the following is the necessary assumption that makes validity Stulz’ “tax argument for risk management” (the assertion that risk management can add value where taxes are concerned)?
* a. Tax carryforwards
* b. Tax shield
* c. Personal taxes
* d. Differential tax rates
Answers:
|
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