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F105 > Chapter 13: Valuation of investments (2) > Flashcards

Chapter 13: Valuation of investments (2) Flashcards

1
Q

e. Introduction to fixed income option pricing

Give four reasons why interest rate derivatives are more difficult to value than equity derivatives.

A
  1. The behaviour of an individual interest rate is more complicated than that of a stock price, as interest rates vary by term.
  2. For many products, it is necessary to develop a model describing the behaviour entire yield curve, whereas Black-scholes model of share option prices is based on single share price only.
  3. Volatilities of different points on yield curve are different
  4. Interest rates are used for discounting as well as for determining payoffs from derivative.
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2
Q

e. Introduction to fixed income option pricing

State Black’s formula for the price of a call option, defining all notation used.

A

c = P(0, T)[ F_0*Q(d1) - XQ(d2))]

where:

  • Q(x) is the cumulative standard normal distribution function
  • d1 = (ln(F_0/X) +0.5VT)/squrt(VT) and d2 = d1 - squrt(VT)
  • P(0, T) is discount factor from strike date T back to time 0
  • F_0 is forward price of the underlying asset
  • squrt (V) is the volatility of forward price
  • X is option strike price

(When valuing the option, we assume that the distribution of the value of the underlying asset is normally distributed)

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3
Q

e. Introduction to fixed income option pricing

State Black’s formula for the price of a put option.

A

p = P(0 T)[XQ(-d2) - F_0Q(-d1)

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4
Q

e. Introduction to fixed income option pricing

State the two approximations involved when interest rates are stochastic.

A

Approximations

  1. The expected value of VT is assumed equal to its forward price F0.
  2. The stochastic behaviour of interest rates is not taken into account in the way the discounting is done.
    These two assumptions have exactly offsetting effects, when the model is being used to value an option on a (bond or other derivatives consider in this chapter)
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5
Q

e. Introduction to fixed income option pricing

State Forward price of coupon-bearing bond.

A

F_0 = (B_0 - I)exp(rT) = (B_0 - I)/(P(0, T)

Where:

  • B dirty price of
  • I is the present value of the coupons paid by the bond for duration T
  • r is the risk free rate of interest compounded continuously
  • T time to maturity of option

(F_0 can then be used in Black’s formula to value bond options)

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6
Q

e. Introduction to fixed income option pricing

Explain what is meant by the following in relation to a coupon bearing bond.
1. the clean price
2. the dirty price
3. accrued interest

A
  1. Clean price
    - quoted price of the bond which appears on the dealer’s screens and does not include accrued interest
  2. Dirty price
    -clean price plus accrued interest
    - it is the price at which the bond is traded.
    - represents discounted present value of future cashflows paid by bond
  3. accrued interest
    - proportion of the next coupon deemed to have accrued since the last coupon was paid
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7
Q

e. Introduction to fixed income option pricing

State a formula relating to price and yield volatilities of a bond.

A

s.d = Dy0s.d_y

  • s.d is forward price volatility
  • s.d_y is the corresponding forward yield volatility
  • D is (modified) duration of forward bond underlying option
  • y0 is initial (forward)yield on forward bond underlying option

(This is needed since yield volatilities are usually quoted, but price volatility is used in Black’s formula)

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8
Q

e. Introduction to fixed income option pricing

State a formula for the modified duration in terms of the duration for a fixed interest bond.

A

Modified D = D/(1 + y/m)

Where m is the frequency per annum with which y is compounded.

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9
Q

e. Introduction to fixed income (Interest rate caps and floors) option pricing

Describe how interest rate caps and floors work.

A
  • OTC derivatives that can be purchased from investment banks.

Cap:
- in return for initial premium it provides a payment each time floating rate R_K rises above fixed cap rate R_X
- Can be used to cap interest rate paid, e.g., on a mortgage
- it can be viewed as a pf of put options on a zero-coupon bonds

Floor:
- buyer receives a payment each time floating interest rate falls below fixed floor rate
- Can be used to provide insurance against a fall in interest rates on a floating rate note.

Both can be used to hedge against movements in ST interest rates or to speculate on such movements.

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10
Q

e. Introduction to fixed income (Interest rate caps) option pricing

Explain how each payoff is determined for a caplet.

A
  • In each subperiod of interest rate cap, interest payment is made under relevant caplet if the floating rate in that subperiod exceeds the cap rate. Otherwise no payment is made in that subperiod.
  • Interest payment made at end of sub-period
  • Interest payment based on interest rate that applies over subperiod at start of subperiod
  • Actual monetary payment based on payment in interest rate terms, multiplied by both cap principal and length of sub-period (or tenor)

(Payments are likewise made under a floor if the floating rate is less than the floor interest rate)

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11
Q

e. Introduction to fixed income (Interest rate caps and floors) option pricing

State the formulae for the payoff from a caplet and a floorlet.

A

Caplet:

Ltenormax(R_X - R_K,0) –> each payment is like a call option on LIBOR

Floorlet:

Ltenormax(R_K - R_X, 0)) –> each payment is like a put option on LIBOR

where:

  • L is the principal
  • tenor = t_k+1 - t_k (time between resets)
  • R_K is floating rate (compounding frequency = tenor)
  • R_X is the cap rate (compounding frequency = tenor)
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12
Q

e. Introduction to fixed income (Interest rate caps and floors) option pricing

Outline in words how to value an interest rate cap and an interest rate floor.

A
  • Each interest rate caplet valued using Black’s formula.
  • This values cap as call option on floating interest rate, with strike price equal to cap interest rate
  • Value of interest cap is then sum of values of constituent caplets.
  • Likewise, floor valued as sum values of constituent floorlets, where each floorlet valued (Using Black’s formula) as put option on floating rate, with strike rate equal to floor interest rate.
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13
Q

e. Introduction to fixed income (Interest rate caps and floors) option pricing

State the formulae for valuing an interest rate caplet.

A

c = LtenorP(0, T_k+1)[F_K * Q(d1) - R_X * Q(d2)

Where:

  • d1 = (ln(F_K/R_X) + 0.5v_K * t_k)/sqrt(v_kt_k) and d2 = d1 - sqrt(v_k*t_k)
  • F_K is the forward rate between t_k and t_k + 1
  • squrt (v_k) is volatility of forward rate

squrt(v_k) can be spot volatilities or flat volatilities. spot v curves are humped, flat also humped but less so. flat are cumulative averages of spot v.

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14
Q

e. Introduction to fixed income (Interest rate caps and floors) option pricing

State the formulae for valuing an interest rate floorlet.

A

c = LtenorP(0, t_k+1)[R_X * Q(-d2) - F_K * Q(d1)]

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15
Q

e. Introduction to fixed income (Interest rate caps and floors) option pricing

State the put-call parity relationship between swaps, caps, floors.

A

cap price = floor price + value of a swap

Where:
- cap interest rate and floor interest rate are same
- terms, principals, frequency of payments etc are same
- swap is the agreement to receive floating and pay fixed

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16
Q

e. Introduction to fixed income (Interest rate caps and floors) option pricing

Explain what is meant by an interest rate collar.

A
  • Consists of long position in interest rate cap and short position in floor.
  • Designed to guarantee that interest rate on underlying floating rate note always lies between two levels
  • Usually constructed so that the price of long position in cap initially equal to the price of short position in floor, so that cost of entering collar is zero.
17
Q

e. Introduction to fixed income (Interest rate caps and floors) option pricing

Outline the likely circumstances in which a company might choose to purchase an interest rate collar.

A
  • An interest rate cap can be used to cap, or fix a maximum level, on a floating interest rate
  • this might be useful if the company has issued a floating rate bond.
  • due to changes in circumstances it now wishes to cap the interest rate that it committed to pay
  • Although it could achieve this by purchasing an interest rate cap, these cost money
  • One way of reducing costs of the cap might by simultaneously selling an interest rate floor.
  • Which is effectively achieved by purchasing an interest rate collar, which is equivalent to buying a cap and selling a floor.
18
Q

e. Introduction to fixed income (Swaption) option pricing

Assuming that you hold a swaption, explain, with refence to the swap rate, how you would decide whether to exercise your option into swap.

A
  • Calculate the net present value of the swap to you on strike date
  • if it’s positive, so that expect to receive more than you pay, then you enter into it.
  • if it’s negative. let option expire worthless
  • In practice, this would be done by comparing swap rate quoted on strike date with fixed rate previously agreed
  • Recall that swap rate is fixed interest rate that would make swap zero NPV at outset . it is equivalent to par yield.
19
Q

e. Introduction to fixed income (Swaption) option pricing

State a formula for the value of a swaption, which gives you the right to enter into a pay fixed rate (R_X) and receive floating rate swap.

A

V = LA/m [F_0*Q(d1) - R_XQ(d2)]

Where:
- d1 = (ln(F_0/R_X) + 0.5*vT)/squrt(vT)
L is the principal
- m is the number of swap payments per time period
- F_0 is the forward swap rate and R_X is fixed interest rate (R_X is assumed to be lognormally distrbuted - lasts n years starting in T years)
- A = sum from 1 to mn of P(0, t_i)
- p (0, t_i) is the discount factor for cashflow time t_i
- squrt (v) is the volatility of the swap rate
- T is the strike date of the swaption

20
Q

e. evaluating securitisation (including CBO’s and MBSs)

Describe how to value a securitised bond.

A
  1. Use deterministic discounted cashflow approach

Discount rate reflects overall riskiness of bond and should be similar to yield offered by equally risky bonds (similar credit rating)

  1. Simulate and discount possible cashflows, allowing for:
    - probability, timing of any defaults and likely recoveries
    - probability, timing and likely extent of any prepayments
    - extent of any recourse to originating company and any/or any guarantees
    - potential treasury management issues
    - ranking and structure of different bond tranches.

Discount rate(s) should allow for risks not captured in cashflows.

21
Q

e. evaluating securitisation (including CBO’s and MBSs)

Describe other considerations when evaluating securitisation (2)

A
  1. Evaluation concentrates more on the predictability and sustainability of cashflow than the asset’s loan to value ratio
  2. differences in legal systems (making transfers of assets located in different jurisdictions a relatively complicated process)
  3. borrowerr’s auditors called upon not only to undertake audits but provide various certificates concerning assets (& their quality), cashflows, tax position (& compliance) and contingent liabilities.
  4. complexity of securitisation –> degree of reporting required
  5. Volatility of cashflow and Analysis of credit risk
22
Q

e. evaluating securitisation (including CBO’s and MBSs)

Describe other considerations when evaluating securitisation (3)

A
  1. Lease terms
  2. Rental prospects
  3. Degree of potential competition
    - if cashflows depend on the profit stream generated by the underlying assets
  4. Barriers to competitive entry
  5. Debt service to EBTIDA (Earnings before Tax Interest Depreciation & Amortisation)
    - Aims to ensure that earnings are sufficient to cover interest payments
23
Q

e. evaluation of a credit derivative

State what the price fo a ‘plain vanilla’ credit default swap should be if purchased via a single premium and an annual premium.

Explain what the value of a total return swap should be equal to.

A

Single premium:
Equals to expected default loss on reference bond.

Annual premium:
Equals to credit spread on reference bond

value of total return swap:
Equals difference between values of assets generating returns on each side of swap.

These ignore taxes, transaction costs, bank’s profit margin and default risk of bank.

24
Q

Explain why the equity of a company can be considered as a call option on the company’s assets.

A
  • Suppose company has amount D of Zero-coupon bonds outstanding that mature at time T and let V_t = value of company’s assets at time t.
  • if V_T < D company will default on its debt at T, the value of equity is zero.
  • if V_T > D, company will repay debt and value of equity = V_T - D
  • So, value of firm’s equity at T is E_T = max (V_T - D, 0)
  • This is equivalent to payoff from call option on company’s assets with strike price equal to the amount of debt, D

F105 (21 decks)

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