P2.T5. Tuckman Chapter 7 The Science of Term Structure Models Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal copy and also violates GARP s ethical standards.
P2.T5. Tuckman Chapter 7 P1.T5. Tuckman Chapter 7 Workbook Exam Relevance (XLS not topic) Worksheet T5.29.7 Term Structure Low 29.7. Exp Disc Value (Zero) Low Low Low Low 29.7. Risk Neutral Pricing 29.7. Replicate Port (Option) 29.7. Three Steps 29.7. CMS Note: If you are unable to view the content within this document we recommend the following: MAC Users: The built-in pdf reader will not display our non-standard fonts. Please use adobe s pdf reader (http://get.adobe.com/reader/otherversions/) PC Users: We recommend you use the foxit pdf reader (http://www.foxitsoftware.com/secure_pdf_reader/) or adobe s pdf reader (http://get.adobe.com/reader/otherversions/). Mobile and Tablet users: We recommend you use the foxit pdf reader app or the adobe pdf reader app. All of these products are free. We apologize for any inconvenience. If you have any additional problems, please email Suzanne at suzanne@bionicturtle.com. 2
Chapters 7: The Science of Term Structure Models
Calculate the expected discounted value of a zero coupon bond using a binomial tree. Six-month interest rate tree T5.29.7 Exp Disc Value (Zero) Spot rates 0 s(0.5) s(1.0) Spot (0,T) 5.00% 5.15% PV of $1,000 Par 975.61 950.42 Interest Rate Binomial Tree T= 0.0 T= 0.5 T= 1.0 5.0% p 0.5 5.50% 1-p 0.5 4.50%
Calculate the expected discounted value of a zero coupon bond using a binomial tree. One-year interest rate tree T = 1.0 year and dt or Δt= 0.5 years T5.29.7 Exp Disc Value (Zero) Spot rate term structure s(0.5) s(1.0) s(1.5) Spot (0,T) 5.00% 5.15% 5.25% PV of $1,000 Par 975.61 950.42 925.21 Interest Rate Process T= 0.0 T= 0.5 T= 1.0 T= 1.5 6.0% 5.50% 5.0% 5.0% 4.50% 4.0% 5
Calculate the expected discounted value of a zero coupon bond using a binomial tree. Expected discounted value is the weighted the future nodes, weighted by price, discounted to present value: Interest Rate Binomial Tree T= 0.0 T= 0.5 T= 1.0 5.0% T5.29.7 Exp Disc Value (Zero) p 0.5 5.50% 1-p 0.5 4.50% $1, 000 1 5.5% 2 1 $973.24 1 $978.00 2 2 $951.82 1 5% 2 $1, 000 1 4.5% 2 $973.24 $978.00 Price of Zero (semi-annual compounding) T= 0.0 T= 0.5 T= 1.0 $1,000 Par $1,000 $973.24 $951.82 $1,000 $978.00 $1,000 Same, but discount factors instead T= 0.0 T= 0.5 T= 1.0 0.95182 0.97324 0.97800 6
Using replicating portfolios, construct and apply an arbitrage argument to price a call option on a zero-coupon security. The payoff of the call option (i.e., $0 or $3) can be replicated by the portfolio that is long the one-year bond and short the six month bond. Therefore, the value of the option must equal the cost of the replicating portfolio T5.29.7 Replicate Port (Option) Interest Rate Binomial Tree T= 0.0 T= 0.5 p 50% 5.50% 5.0% 1-p 50% 4.50% II. Replicating Portfolio T= 0.0 T= 0.5 Bond $973.24 Call $0.00 In this (Tuckman s) example, long $629.34 face amount of the one-year bond plus short -$612.50 face amount of the sixmonth bond replicates option payout and has an cost of $0.58 Bond Par: $1,000 Option Strike @ $975 Long B (1.0) $612.50 Short B (0.5) ($612.50) $0.00 Replicating Face(1.0) $629.34 Portfolio Face(0.5) -$612.50 Bond $978.00 Cost: $0.58 Call $3.00 Long B (1.0) $615.50 Short B (0.5) ($612.50) $3.00 7
Using replicating portfolios, construct and apply an arbitrage argument to price a call option on a zero-coupon security. Tuckman s prior example has three basic steps: 1. Specify the interest rate assumptions which includes both an interest rate tree (50% probability of an up-jump from current 5.0% to 5.5% and 50% probability of down-jump to 4.5%) and a one-year rate of 5.15%. 2. Assume the derivative instrument: in this case, a call option with a strike price of $975 (on a bond with face value of $1,000). Find the replicating portfolio (II.). This is the combination of long position in a one-year bond plus a short position in a six-month bond that produces a payoff identical to the derivative. The cost of the portfolio is $0.58, which therefore must be the price of the derivative. 3. Compare the expected discounted value of $1.46, which discounts with the true (or real-world) probabilities (p = 50% and 1-p = 50%), to the arbitrage price of $0.58, which discounts with the risk-neutral probabilities (p = 80.09% and 1-p = 19.91%). 8
Explain why a call option on a zero-coupon security cannot be properly priced using expected discounted values. T5.29.7 Replicate Port (Option) The true value ($0.58) is less than the discounted expected value ($1.46) Investors dislike the risk of the call option: risk-aversion insists on paying less than expected discounted value. The risk penalty implicit in the call option price is inherited from the risk penalty of the one-year zero, that is, from the property that the price of the one-year zero is less than its expected discounted value III. Discounting at true and risk-neutral probabilities T= 0.0 T= 0.5 Exp. Discount Value: $1.46 Risk Neutral Price: $0.58 50.00% $0.00 80.09% 50.00% $3.00 19.91% 9
T5.29.7 Risk Neutral Pricing Explain the role of up state and down state probabilities in the option valuation. A remarkable feature of arbitrage pricing is that the up/down probabilities never enter into the calculation of the arbitrage (risk-neutral) price 10
Define risk neutral pricing and explain how it is used in option pricing. Risk-neutral pricing modifies an assumed interest rate process so that any contingent claim can be priced without having to construct and price its replicating portfolio. Since the original interest rate process has to be modified only once, and since this modification requires no more effort than pricing a single contingent claim by arbitrage, risk-neutral pricing is an extremely efficient way to price many contingent claims under the same assumed rate process. 11
T5.29.7 Risk Neutral Pricing Define risk neutral pricing and explain how it is used in option pricing. Spot rates 0 s(0.5) s(1.0) Spot (0,T) 5.00% 5.15% PV of $1,000 Par 975.61 950.42 Interest Rate Binomial Tree T= 0.0 T= 0.5 p 0.5 5.50% 5.0% 1-p 0.5 4.50% "Real-world" probabilities T= 0.0 T= 0.5 T= 1.0 T= 0.0 T= 0.5 T= 1.0 Par: $1,000 $1,000 50% $973.24 $973.24 $950.42 $1,000 $951.82 50% $978.00 $978.00 $1,000 "Risk neutral" probabilities T= 0.0 T= 0.5 T= 1.0 80.1% $973.24 $950.42 19.9% $978.00 www.bionicturtle.com 12
T5.29.7 Risk Neutral Pricing Define risk neutral pricing and explain how it is used in option pricing. "Real-world" probabilities T= 0.0 T= 0.5 The expected discounted value $951.82 = [(50%)(973.24)+(50%)(978)]/(1+5%/2) 50% $973.24 $951.82 50% $978.00 "Risk neutral" probabilities T= 0.0 T= 0.5 The market price $950.42 = $1,000 / (1 + 5.15%/2)^2 80.1% $973.24 $950.42 19.9% $978.00 13
Define risk neutral pricing and explain how it is used in option pricing. Step 1: Given trees for the underlying securities, the price of a security that is priced by arbitrage does not depend on investors risk preferences. 14
Define risk neutral pricing and explain how it is used in option pricing. Step 2: Imagine an economy identical to the true economy with respect to current bond prices and the possible value of the six-month rate over time but different in that the investors in the imaginary economy are risk neutral. Unlike investors in the true economy, investors in the imaginary economy do not penalize securities for risk and, therefore, price securities by expected discounted value. It follows that, under the probabilities in the imaginary economy, the expected discounted value of the one-year zero equals its market price. But these probabilities satisfy equation (7.8), namely the risk-neutral probabilities of.8024 and.1976. 15
Define risk neutral pricing and explain how it is used in option pricing. Step 3: The price of the option in the imaginary economy, like any other security in that economy, is computed by expected discounted value. Since the probability of the up state in that economy is.8024, the price of the option in that economy is given by equation (7.9) and is, therefore, $.58. 16
Define risk neutral pricing and explain how it is used in option pricing. Step 4: Step 1 implies that given the prices of the six-month and one-year zeros, as well as possible values of the six-month rate, the price of an option does not depend on investor risk preferences. It follows that since the real and imaginary economies have the same bond prices and the same possible values for the six-month rate, the option price must be the same in both economies. In particular, the option price in the real economy must equal $.58, the option price in the imaginary economy. More generally, the price of a derivative in the real economy may be computed by expected discounted value under the risk-neutral probabilities 17
Relate the difference between true and risk neutral probabilities to interest rate drift. Under true probabilities there is a 50% chance that the six-month rate rises from 5% to 5.50% and a 50% chance that it falls from 5% to 4.50%. Hence the expected change in the six-month rate, or the drift of the sixmonth rate, is zero. Under risk-neutral probabilities there is an 80.24% chance of a 50 basis point increase in the six-month rate and a 19.76% chance of a 50 basis point decline. Hence the drift of the six-month rate under these probabilities is 30.24 basis points. 18
T5.29.7 Three Steps Explain how the principles of arbitrage pricing of derivatives on fixed income securities can be extended over multiple periods. Spot rate term structure s(0.5) s(1.0) s(1.5) Spot (0,T) 5.00% 5.15% 5.25% PV of $1,000 Par 975.61 950.42 925.21 Interest Rate Process T= 0.0 T= 0.5 T= 1.0 T= 1.5 6.0% 5.50% 5.0% 5.0% 4.50% 4.0% Risk-neutral probabilities p 0.8024 1-p 0.1976 q 0.6489 q-1 0.3511 Binomial Price Tree Par: $1,000.00 $1,000.00 $970.87 $946.51 $1,000.00 $925.21 $975.61 $955.78 $1,000.00 $980.39 $1,000.00 19
Describe the rationale behind the use of non recombining trees in option pricing. If ud du, tree is non-recombining Nothing wrong with non-recombining tree, from economic perspective For example, to justify this particular tree, could argue that when short rates are 5% or higher they tend to change in increments of 50 basis points. But when rates fall below 5%, the size of the change starts to decrease. In particular, at a rate of 4.50% the short rate may change by only 45 basis points. A volatility process that depends on the level of rates exhibits state-dependent volatility. www.bionicturtle.com 20
Describe the rationale behind the use of non recombining trees in option pricing. But practitioners tend to avoid non-recombining trees because they are difficult or impossible to implement After six months there are two possible rates, after one year there are four, and after N semiannual periods there are 2N possibilities. For example, a tree with semiannual steps large enough to price 10-year securities will, in its rightmost column alone, have over 500,000 nodes, while a tree used to price 20-year securities will in its rightmost column have over 500 billion nodes. 21
T5.29.7 CMS Calculate the value of a constant maturity Treasure swap, given an interest rate tree and the risk neutral probabilities. Pricing a constant maturity Treasury (CMT) swap Pays notional/2 * (sovereign Treasury yield - 5%) every six months Notional 1,000,000 Fixed rate 5.0% Interest Rate Process T= 0.0 T= 0.5 T= 1.0 q 64.89% 6.0% p 80.24% 5.50% 5.0% q-1 35.11% 5.0% p-1 19.76% 4.50% 4.0% Expected discounted value T= 0.0 T= 0.5 T= 1.0 $5,000 Payoff: $2,500 PV: $5,658 PV= $3,616 $0 Payoff: -$2,500 PV: -$4,217 -$5,000 22
Describe the advantages and disadvantages of reducing the size of the time steps on the pricing of derivatives on fixed income securities. If cash flows do not occur at given intervals, reducing time step will get closer (more proximate) to actual cash flows More attention must be paid to numerical issues like round-off error Six-month rate trees can too coarse. Reducing the step size (i.e., increasing the granularity) fills the tree with enough rates to price with greater accuracy Decreasing the time step increases computation time
Explain why the Black Scholes Merton model to value equity derivatives is not appropriate to value derivatives on fixed income securities. The price of a bond must converge to its face value at maturity But the stochastic process underlying the stock price assumption in the Black-Scholes is not so constrained. Due of the maturity constraint, the volatility of a bond s price must eventually get smaller as the bond approaches maturity. But the Black-Scholes makes the simpler assumption that stock volatility is constant: not appropriate for bonds. 24
Describe the impact of embedded options on the value of fixed income securities. European call option on a 5-year bond paying 6% coupon (each step = six months) Price 0 = $105.17 r 0 = 5.0% Option 0 = $4.14 25 Price u = $102.34 r u = 5.5% Option u = $2.09 Price d = $106.43 r d = 4.5% Option d = $5.67 Price uu = $100.00 r uu = 6.0% Option uu = $0.00 Price ud = $103.59 r ud = 5.0% Option ud = $3.59 Price dd = $107.33 r dd = 4.0% Option dd = $7.33
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