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The following is a review of the Derivatives principles designed to address the learning outcome statements set forth by CFA Institute. Cross-Reference to CFA Institute Assigned Reading #38.
READING 38: VALUATION OF
CONTINGENT CLAIMS
Study Session 14
EXAM FOCUS
This topic review covers the valuation of options. Candidates need to be able to calculate value of an option using the binomial tree framework and should also understand the inputs into the Black-Scholes model and how they influence the value of an option. While this topic review is somewhat quantitative, candidates need to understand the material conceptually as well. This reading has a lot of testable material.
MODULE 38.1: THE BINOMIAL MODEL
LOS 38.a: Describe and interpret the binomial option valuation model and its component terms.
LOS 38.b: Calculate the no-arbitrage values of European and American options using a two-period binomial model.
LOS 38.e: Describe how the value of a European option can be analyzed as the present value of the option’s expected payoff at expiration.
CFA®^ Program Curriculum, Volume 5, pages 386, 388, and 408
BINOMIAL MODEL
A binomial model is based on the idea that, over the next period, the value of an asset will change to one of two possible values (binomial). To construct a binomial model, we need to know the beginning asset value, the size of the two possible changes, and the probability of each of these changes occurring.
One-Period Binomial Model
Consider a share of stock currently priced at $30. Suppose also that the size of the possible price changes, and the probability of these changes occurring are as follows:
S 0 = current stock price = $ U = up move factor = (1 + % up) = S+/S = 1. D = down move factor = (1 − % down) = S–/S = 0. S+^ = stock price if an up move occurs = S 0 × U = 30 × 1.333 = $ S–^ = stock price if a down move occurs = S 0 × D = 30 × 0.75 = $22.
πU= probability of an up move πD = probability of a down move = (1 – πU)
A one-period binomial tree for this stock is shown in Figure 38.1. The beginning stock value of $30 is to the left, and to the right are the two possible paths the stock can take, based on that starting point and the size of an up- or down-move. If the stock price increases by a factor of 1.333 (a return of 33.3%), it ends up at $40.00; if it falls by a factor of 0.75 (a return of –25%), it ends up at $22.50.
Figure 38.1: One-Period Binomial Tree
The probabilities of an up-move and a down-move are calculated based on the size of the moves, as well as the risk-free rate, as:
π U =
where: Rf = periodically compounded annual risk-free rate
PROFESSOR’S NOTE
These up- and down-move probabilities are not the actual probabilities of up- or down-moves. They are the risk-neutral probabilities that are consistent with investor risk-neutrality. The distinction between actual probabilities and risk-neutral probabilities is not relevant for the exam.
We can calculate the value of an option on the stock by:
Calculating the payoff of the option at maturity in both the up-move and down-move states. Calculating the expected value of the option in one year as the probability-weighted average of the payoffs in each state. Discounting the expected value back to today at the risk-free rate.
EXAMPLE: Calculating call option value with a one-period binomial tree Calculate the value today of a one-year call option on a stock that has an exercise price of $30. Assume that the periodically compounded (as opposed to continuously compounded) risk-free rate is 7%, the current value of the stock is $30, the up-move factor is 1.333, and the down move factor is 0.75. Answer: First, we have to calculate the probabilities:
1+Rf−D U−D
1+R −D
S 0 + P 0 = C 0 + PV(X)
Note that both options are on the same underlying stock have the same exercise price, and the same maturity.
In our previous example, PV(X) = 30 / 1.07 = $28.04. We can verify the put call parity as:
S 0 + P 0 = $30 + $3.15 = $33.15.
C 0 + PV(X) = $5.14 + 28.04 = $33.19 (rounding accounts for the slight difference).
Put call parity can be used to create a synthetic instrument that replicates the desired instrument.
For example, a synthetic call can be created by creating a portfolio that combines a long position in the stock, a long position in a put and a short position in a zero coupon bond with a face value equal to the strike price (i.e., borrowing the present value of the exercise price at the risk-free rate).
C 0 = S 0 + P 0 − PV(X)
EXAMPLE: Using put-call parity A 1-year call option on the common stock of Cross Reef Inc., with an exercise price of $60 is trading for $8. The current stock price is $62. The risk-free rate is 4%. Calculate the price of the put option implied by put-call parity. Answer: According to put-call parity, to prevent arbitrage, the price of the put option must be: P 0 = C 0 − S 0 + [ ]
= $8 − $62 + = $3.
Two-Period Binomial Model
Valuing an option using a two-period binomial model requires more steps, but uses the same method:
Calculate the stock values at the end of two periods (there are three possible outcomes, because an up-then-down move gets you to the same place as a down-then-up move). Calculate the three possible option payoffs at the end of two periods. Calculate the expected option payoff at the end of two periods (t = 2) using the up- and down-move probabilities. Discount the expected option payoff (t = 2) back one period at the risk-free rate to find the option values at the end of the first period (t = 1). Calculate the expected option value at the end of one period (t = 1) using up- and down-move probabilities. Discount the expected option value at the end of one period (t = 1) back one period at the risk-free rate to find the option value today (t=0).
Let’s look at an example to illustrate the steps involved.
X (1+Rf ) T $
EXAMPLE: Valuing a call option on a stock with a two-period model
Suppose you own a stock currently priced at $50 and that a two-period European call option on the stock is available with a strike price of $45. The up-move factor is 1.25 and the down-move factor is 0.80. The risk- free rate per period is 7%. Compute the value of the call option using a two-period binomial model. Answer: First, compute the probability of an up-move and a down-move, and then compute the theoretical value of the stock at the end of each period:
π U = = = 0. πD = 1 − 0.60 = 0.
The two-period binomial tree for the stock is shown in the following figure. Two-Period Binomial Tree for Stock Price
We know the value of the option at expiration in each state is equal to the stock price minus the exercise price (or zero, if that difference is negative): C++^ = max(0, $78.13 − $45.00) = $33. C−+^ = max(0, $50.00 − $45.00) = $5. C+−^ = max(0, $50.00 − $45.00) = $5. C−−^ = max(0, $32.00 − $45.00) = $
We will approach this problem by using the single-period binomial model for each period. Using this method, we can compute the value of the call option in the up-state in period one as follows:
C+^ = =
= = = $20.
The value of the call in the down-state at t=1 is computed as:
C−^ = =
= = = $2.
Now we know the value of the option in both the up-state (C+) and the down-state (C–) one period from now. To get the value of the option today, we simply apply our methodology one more time. Therefore, bringing (C+) and (C–) back one more period to the present, the value of the call option today is:
C = =
= = = $12.
1+Rf−D U−D
1+0.07−0. 1.25−0.
E(call option value) 1+Rf
( π U ×C++)+( π D ×C+−) 1+Rf (0.6×$33.13)+(0.4×$5.00)
$21.
E(call option value) 1+Rf
( π U ×C−+)+( π D ×C−−) 1+Rf (0.6×$5.00)+(0.4×$0.00)
$3.
E(call option value) 1+Rf
( π U ×C+)+( π D ×C−) 1+Rf (0.6×$20.45)+(0.4×$2.80)
$13.
The value of the option for the down node at t=1 is calculated as: P−^ = = = $9. The value of the option at time=0 is calculated as: P = = = $4. For the down move at t = 1, the exercise value of the put option is $10, calculated as Max(0, X-S) or Max(0, 50-40). Clearly, for this node, early exercise results in higher value ($10 as opposed to $9.44).
Had the option in the previous example been an American-style put option, the value would be $5.24 as shown in Figure 38.2.
Figure 38.2: Valuing an American-Style Put Option The value of the put option at time 0 can be calculated as the present value of the expected value of the option at time t=1.
P 0 = = = $5.
American-style call options on dividend-paying stocks can be evaluated similarly: determine at each node whether the exercise value is greater than the intrinsic value and, if so, use that higher value. For dividend-paying stocks, the stock price falls when the stock goes ex- dividend, and it may make sense to exercise the call option before such a decline in price.
MODULE QUIZ 38.
( π U×P+−+ π D ×P−−) (1+Rf)
(0.46×0+0.54×18) (1.03)
( π U ×P++ π D ×P−) (1+Rf)
(0.46×0+0.54×9.44) (1.03)
( π U×P++ π D×P−) (1+Rf)
(0.46×0+0.54×10.0) (1.03)
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To best evaluate your performance, enter your quiz answers online.
- An analyst has calculated the value of a 2-year European call option to be $0.80. The strike price of the option is 100.00, and the underlying asset is a 7% annual coupon bond with three years to maturity. The two-period binomial tree for the European option is shown in the following figure.
The value of the comparable 2-year American call option (exercisable after 1 year) with a strike price of 100.00 is closest to: A. $1.56. B. $2.12. C. $3.80.
MODULE 38.4: HEDGE RATIO
LOS 38.c: Identify an arbitrage opportunity involving options and describe the related arbitrage.
CFA®^ Program Curriculum, Volume 5, page 391
ARBITRAGE WITH A ONE-PERIOD BINOMIAL MODEL
Let’s revisit our original example of a single period binomial model. Recall that the call option has a strike price equal to the stock’s current price of $30, U = 1.333, D = 0.75, and the risk-free rate is 7%. We calculated that the probability of an up-move is 55% and that of a down-move 45%.
If the market price of the one-period $30 call option were to be different from the $5.14 value calculated before, there would be an arbitrage opportunity. This arbitrage will involve the call option and shares of the stock. If the option is overpriced in the market, we would sell the option and buy a fractional share of the stock for each option we sold. If the call option is underpriced in the market, we could purchase the option and short a fractional share of stock for each option purchased.
The fractional share of stock needed in the arbitrage trade (commonly referred to as the hedge ratio or delta), is calculated in the one-period model as:
C+−C−
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- In a one-period binomial model, the hedge ratio is 0.35. To construct a riskless arbitrage involving 1,000 call options if the option is “overpriced,” what is the appropriate portfolio? Calls Stock A. Buy 1,000 options Short 350 shares B. Buy 1,000 options Short 2,857 shares C. Sell 1,000 options Buy 350 shares
- A synthetic European put option is created by: A. buying the discount bond, buying the call option, and short-selling the stock. B. buying the call option, short-selling the discount bond, and short-selling the stock. C. short-selling the stock, buying the discount bond, and selling the call option.
MODULE 38.5: INTEREST RATE OPTIONS
LOS 38.d: Calculate and interpret the value of an interest rate option using a two-period binomial model.
CFA®^ Program Curriculum, Volume 5, page 409
WARM-UP: BINOMIAL INTEREST RATE TREES
We can use an estimate of the volatility of an interest rate to create a set of possible rate paths for interest rates in the future called a binomial interest rate tree. The diagram in Figure 38.3 depicts a two-period binomial interest rate tree.
Figure 38.3: Two-Period Binomial Interest Rate Tree
The interest rate at each node is a one-period forward rate. Beyond the root of the tree, there is more than one one-period forward rate for each nodal period (i.e., at year 1, we have two 1- year forward rates, i 1,U and i 1,D). The interest rates are selected so that the (risk-neutral)
probabilities of up- and down-moves are both equal to 0.5.
PROFESSOR’S NOTE You will not have to construct a tree for the exam—just be able to use the given tree to value an interest rate option.
Interest Rate Options
An interest rate call option has a positive payoff when the reference rate is greater than the exercise rate:
call payoff = notional principal × [Max (0, reference rate − exercise rate)]
Interest rate call options increase in value when rates increase.
An interest rate put option has a positive payoff when the reference rate is less than the exercise rate:
put payoff = notional principal × [Max (0, exercise rate − reference rate)]
Interest rate put option values increase in value when rates decrease. Valuing interest rate options using the binomial tree is similar to valuing stock options; the value at each node is the present value of the expected value of the option. While LIBOR-based contracts pay interest in arrears, to keep things simple, we assume that options cash settle at maturity.
EXAMPLE: Interest rate call option valuation Given the two-period interest rate tree below, what is the value of a two-period European interest rate call option with an exercise rate of 5.50% and a notional rincipal of $1 million? (Assume that options cash settle at time T = 2.)
Answer: Given the exercise rate of 5.50%, the call option has a positive payoff for nodes C++^ and C+–. The value of the option at node C++^ can be calculated as: [Max (0, 0.107383 – 0.055)] × $1,000,000 = $52,
P 0 = e–rTXN(–d 2 ) – S 0 N(–d 1 )
where: d 1 =
d 2 = d 1 − σ √T C 0 and P 0 = values of call and put option T = time to option expiration r = continuously compounded risk-free rate S 0 = current asset price X = exercise price σ = annual volatility of asset returns N(*) = cumulative standard normal probability N(–x) = 1 – N(x)
While the BSM model formula looks complicated, its interpretations are not:
- The BSM value can be thought of as the present value of the expected option payoff at expiration. For a call, that means C 0 = PV {S 0 erTN(d 1 ) − XN(d 2 )}. Similarly, for a put option, P 0 = PV {XN(–d 2 ) − S 0 erTN(–d 1 )}.
- Calls can be thought of as a leveraged stock investment where N(d 1 ) units of stock are purchased using e–rTXN(d 2 ) of borrowed funds. (A short position in bonds can also be interpreted as borrowing funds.) A portfolio that replicates a put option consists of a long position in N(–d 2 ) bonds and a short position in N(–d 1 ) stocks.
- N(d 2 ) is interpreted as the risk-neutral probability that a call option will expire in the money. Similarly, N(–d 2 ) or 1 − N(d 2 ) is the risk-neutral probability that a put option will expire in the money.
EXAMPLE: BSM model Stock of XZ Inc., is currently trading at $50. Suppose that the return volatility is 25% and the continuously compounded risk-free rate is 3%. Calls and puts with a strike price of $45 and expiring in 6 months (T=0.5) are trading at $7.00 and $1.00 respectively. If N(d 1 ) = 0.779 and N(d 2 ) = 0.723, calculate the value of the replicating portfolios and any arbitrage profits on both options. Answer: The replicating portfolio for the call can be constructed as long 0.779 shares (0.779 × $50 = $38.95), and borrow 45 × e-0.03(0.5)^ × (0.723) = $32.05. net cost = $38.95 − $32.05 = $6. Because the market price of the call is $7.00, the profitable arbitrage transaction entails writing a call at $7.00 and buying the replicating portfolio for $6.90 to yield an arbitrage profit of $0.10 per call. For the put option valuation, note that N(–d 1 ) = 1 − N(d 1 ) = 1 − 0.779 = 0.221 and N(–d 2 ) = 1 − 0.723 = 0.277. The replicating portfolio for the put option can be constructed as a long bond position of 45 × e-0.03(0.5)^ × (0.277) = $12.28 and a short position in 0.221 shares resulting in short proceeds of $50 × 0.221 or $11.05. net cost = $12.28 – $11.05 = $1. Because the market price of the put option is $1.00, arbitrage profits can be earned by selling the replicating portfolio and buying puts, for an arbitrage profit of $0.23 per put.
ln(S/X) +(r+ σ^2 /2)T σ √T
LOS 38.h: Describe how the Black–Scholes–Merton model is used to value European options on equities and currencies.
CFA®^ Program Curriculum, Volume 5, page 419
Options on Dividend Paying Stocks
So far we have assumed that the underlying stock does not pay dividends. If it does, we can adjust the model using a lowercase delta (δ) to represent the dividend yield, as follows:
C 0 = S 0 e–δTN(d 1 ) − e–rTXN(d 2 )
P 0 = e–rTXN(–d 2 ) − S 0 e–δTN(–d 1 )
where: δ = continuously compounded dividend yield d 1 =
d 2 = d 1 − σ √T
Note that S 0 e–δT^ is the stock price, reduced by the present value of any dividends expected to
be paid during the option’s life.
The put-call parity relation must also be modified if the stock pays dividends:
P 0 + S 0 e–δT^ = C 0 + e–rTX
Options on Currencies
We can also use the Black-Scholes-Merton model to value foreign exchange options. Here, the underlying is the spot exchange rate instead of a stock price.
The value of an option on a currency can be thought of as being made up of two components, a bond component and a foreign exchange component. The value can be calculated as:
C 0 = S 0 e–r(B)TN(d 1 ) − e–r(P)TXN(d 2 )
and
P 0 = e–r(P)TXN(–d 2 ) − S 0 e–r(B)TN(–d 1 )
where: r(P) = continuously compounded price currency interest rate r(B) = continuously compounded base currency interest rate
For currencies, the carry benefit is not a dividend but rather interest earned on a deposit of the foreign currency. The spot exchange rate, S 0 , is discounted at the base or foreign currency
interest rate, and the bond component, e–r(P)TX, is the exercise exchange rate discounted at the price (or domestic currency) interest rate.
LOS 38.i: Describe how the Black model is used to value European options on futures.
ln(S/X)+(r− δ + σ^2 /2)T σ √T
C 0 = (AP)e−r(actual/365)^ [FRA(M×N) N(d 1 ) − XN(d 2 )] × NP
where: AP = accrual period = [ ]
NP = notional principal on the FRA
Equivalencies in Interest Rate Derivative Contracts
Combinations of interest rate options can be used to replicate other contracts, for example:
- A long interest rate call and a short interest rate put (with exercise rate = current FRA rate) can be used to replicate a long FRA (i.e., a forward contract to receive a floating rate and pay a fixed rate).
- Similarly, if the exercise rate = the current FRA rate, a short interest rate call and long interest rate put can be combined to replicate a short FRA position (i.e., a pay-floating, receive-fixed forward contract).
- A series of interest rate call options with different maturities and the same exercise price can be combined to form an interest rate cap. (Each of the call options in an interest rate cap is known as a caplet.) A floating rate loan can be hedged using a long interest rate cap.
- Similarly, an interest rate floor is a portfolio of interest rate put options, and each of these puts is known as a floorlet. Floors can be used to hedge a long position in a floating rate bond.
- If the exercise rate on a cap and floor is same, a long cap and short floor can be used to replicate a payer swap.
- Similarly, a short cap and long floor can replicate a receiver swap.
- If the exercise rate on a floor and a cap are set equal to a market swap fixed rate, the value of the cap will be equal to the value of the floor.
Swaptions
A swaption is an option that gives the holder the right to enter into an interest rate swap. A payer swaption is the right to enter into a specific swap at some date in the future at a predetermined rate as the fixed-rate payer. As interest rates increase, the right to take the pay- fixed side of a swap (a payer swaption) becomes more valuable. The holder of a payer swaption would exercise it and enter into the swap if the market rate is greater than the exercise rate at expiration.
A receiver swaption is the right to enter into a specific swap at some date in the future as the fixed-rate receiver (i.e., the floating-rate payer) at the rate specified in the swaption. As interest rates decrease, the right to enter the receive-fixed side of a swap (a receiver swaption) becomes more valuable. The holder of a receiver swaption would exercise if market rates are less than the exercise rate at expiration.
A swaption is equivalent to a an option on a series of cash flows (annuity), one for each settlement date of the underlying swap, equal to the difference between the exercise rate on the swaption and the market swap fixed rate.
actual 365
If PVA represents the present value of such an annuity, the value of a payer swaption using the Black model can be calculated as:
pay = (AP) PVA [SFR N(d 1 ) − X N(d 2 )] NP
where: pay = value of the payer swaption AP = 1/# of settlement periods per year in the underlying swap SFR = current market swap fixed rate X = exercise rate specified in the payer swaption NP = notional principal of the underlying swap d 1 =
d 2 = d 1 − σ √T
The value of a payer swaption is essentially the present value of the expected option payoff:
pay = PV (E(payoff))
Similarly, the value of a receiver swaption (which we’ll call “REC”) can be calculated as:
REC = (AP) PVA [X N(–d 2 ) − SFR N(–d 1 )] NP
Equivalencies
A receiver swap can be replicated using a long receiver swaption and a short payer swaption with the same exercise rates. Conversely, a payer swap can be replicated using a long payer swaption and short receiver swaption with the same exercise rates. If the exercise rate is set such that the values of the payer and receiver swaptions are equal, then the exercise rate must be equal to the market swap fixed rate.
A long callable bond can be replicated using a long option-free bond plus a short receiver swaption.
MODULE QUIZ 38.5, 38. To best evaluate your performance, enter your quiz answers online.
- Compare the call and put prices on a stock that doesn’t pay a dividend (NODIV) with comparable call and put prices on another stock (DIV) that is the same in all respects except it pays a dividend. Which of the following statements is most accurate? The price of: A. a DIV call will be less than the price of NODIV call. B. a NODIV call will equal the price of NODIV put. C. a NODIV put will be greater than the price of DIV put.
- Which of the following is not an assumption underlying the Black-Scholes-Merton options pricing model? A. The underlying asset does not generate cash flows. B. The price of the underlying is lognormally distributed. C. The option can only be exercised at maturity.
Use the following information to answer Questions 3 and 4. Stock ABC trades for $60 and has 1-year call and put options written on it with an exercise price of $60. ABC pays no dividends. The annual standard deviation estimate is 10%, and the continuously compounded risk-free rate is 5%. The value of the call is $4.09.
ln(SFR/X)+( σ^2 /2)T σ √T
However, because we’re now using a linear relationship to estimate a non-linear change, the following relationships are only approximations:
∆C ≈ e–δTN(d 1 ) × ∆S
∆P ≈ –e–δTN(–d 1 ) × ∆S
where: ∆C and ∆P = change in call and put price
The approximations are close for small changes in stock price, but the approximation becomes less accurate as the ∆S becomes larger.
EXAMPLE: Calculating change in option price e–δTN(d 1 ) from the BSM model is 0.58. Calculate the approximate change in the price of a call option on the stock if the stock price increases by $0.75. Answer: ∆C ≈ 0.58 × $0.75 = $0.
Interpreting Delta
The payoff diagrams for a European call option before- and at-expiration are shown in Figure 38.5. The “at expiration” line represents the call option’s intrinsic value, which is equal to:
Zero when the call option is out-of-the-money. The stock price minus the exercise price when the option is in-the-money.
Before expiration, the option also has time value, so the prior-to-expiration curve lies above the at-expiration diagram by the amount of the time value.
Figure 38.5: European Call Option Payoff Diagrams
The slope of the “prior-to-expiration” curve is the change in call price per unit change in stock price. Sound familiar? It should—that’s also the definition of delta. That means delta is
the slope of the prior-to-expiration curve. The delta of the put option is also the slope of the prior-to-expiration put curve.
Take a closer look at Figure 38.5. When the call option is deep out-of-the-money, the slope of the at-expiration curve is close to zero, which means the call delta will be close to zero. For deep out-of-the-money call options (i.e., when the stock price is low), the option price does not change much for a given change in the underlying stock. When the call option is in-the- money (i.e., when the stock price is high), the slope of the at-expiration curve is close to 45 degrees, which means the call delta is close to one. In this case, the call option price will change approximately one-for-one for a given change in the underlying stock price.
The bottom line is that a call option’s delta will increase from 0 to e– δT as stock price increases. For a non-dividend paying stock, the delta will increase from 0 to 1 as the stock price increases.
For a put option, the put delta is close to zero when the put is out-of-the-money (i.e., when the stock price is high). When the put option is in-the-money (i.e., when the stock price is close to zero), the put delta is close to – e– δT.
The bottom line is that a put option’s delta will increase from – e– δT to 0 as stock price increases. For a non-dividend paying stock, the put delta increases from –1 to 0 as the stock price increases.
Now, let’s consider what happens to delta as the option approaches maturity, assuming that the underlying stock price doesn’t change. The effects on call and put options are different and will depend on whether the options are in- or out-of-the-money.
Remember that a call option delta is between 0 and e– δT. Assuming that the underlying stock price doesn’t change, if the call option is:
Out-of-the-money (the stock price is less than exercise price), the call delta moves closer to 0 as time passes. In-the-money (the stock price is greater than exercise price), the call delta moves closer to e– δT as time passes.
Remember that a put option delta is between – e– δT and 0. If the put option is:
Out-of-the-money (the stock price is greater than exercise price), the put delta moves closer to 0 as time passes. In-the-money (the stock price is less than exercise price), the put delta moves closer to
Gamma measures the rate of change in delta as the underlying stock price changes. Gamma captures the curvature of the option-value-versus-stock-price relationship. Long positions in calls and puts have positive gammas. For example, a gamma of 0.04 implies that a $1. increase in the price of the underlying stock will cause a call option’s delta to increase by 0.04, making the call option more sensitive to changes in the stock price.
Gamma is highest for at-the-money options. Deep in-the-money or deep out-of-money options have low gamma. Gamma changes with stock price and with time to expiration. To lower (increase) the overall gamma of a portfolio, one should short (go long) options.
Recall that delta provides an approximation for change in option value in response to a change in the price of underlying. Including gamma in our equation would improve the
