Chapter 13: Valuation of investments (2) Flashcards
Give 4 reasons why interest rate derivatives are more difficult to value than equity derivatives
Behaviour of an individual interest rate more complicated than that of a stock price, as interest rates vary by term
For many products, it is necessary to develop a model describing behaviour of entire yield curve, whereas Black Scholes model of share option prices is based on single share price only
Volatilities of different points on yield curve are different
Interest rates are used for discounting as well as for determining payoffs from derivatives
State the Black’s formula for the price of a call option, defining all the notation used
c = P(0,T)[F0Ŷ(d1) - XŶ(d2)]
Where
Ŷ(x) is the cumulative standard normal distribution function
d1 = [In(F0/x) + (σ2T/2)]/[σT1/2 and d2 = d1-σT1/2
F0 is forward price of underlying asset
σ is the volatility of forward price
X is option strike price
Explain what are meant by each of the following in relation to a coupon bearing bond
The clean price
The dirty price
Accrued interest
Clean Price
Equals dirty prices less accrued interest
Is quoted price
Dirty price
Include accrued interest
Is price paid for bond
Represents discounted present value of future cashflows paid by bond
Accrued interest
Accrued interest is proportion of next coupon deemed to have accrued since last coupon was paid
State the formula relating the price and yield volatilities of a bond
Price and yield volatilities of bond
σ = Dy0σy
Where
σ is forward price volatility
σy is corresponding forward yield volatility
D is modified duration of forward bond underlying option
y0 is initial (forward) yield on forward bond underlying option
State a formula for the modified duration in terms of the duration for a fixed interest bond
D = Duration/(1+y/m)
Where m is the frequency per annum with which y is compounded
Describe how interest rate caps and floors work
Over the counter derivatives that can be purchased from investment bank
In return for initial premium, interest rate cap provides payment each time floating interest rate Rk rises above fixed cap rate, Rx
In contrast, buyer of interest rate floor recieves payment each time floating interest rate falls below fixed floor rate
Can be used to hedge against movements in short term interest rates, or speculate on such movements
Explain how each pay off is determined for a caplet
In each sub-period of interest rate cap, interest payment is made under relevant caplet if floating interest rate in that sub - period exceeds cap interest rate. Otherwise no payment made in that sub - period
Interest payment made at end of sub - period
Interest payment based on interest rate that applies over sub - period at start of sub period
Actual monetary payment based on payment in interest rate terms, multiplied by both cap principal and length of sub - period
Outline in words how to value an interest rate cap and an interest rate floor
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 constituents caplets
Likewise, floor isi valued as sum of values constituent floorlets, where each floorlet valued (using black formula) as put option on floating rate, with strike price equal to floor interest rate
State the put call parity relationship between swaps, caps and floors
Cap price = floor price +value of swap
Where
cap interest rate and floor interest rate are same
terms, principals, frequency of payments
swap is agreement to recieve floating and pay fixed
Explain what is meant by an interest rate collar
Consists of long position in interest rate cap and short position in floor
design to guaranteee that interest rate on underlying floating rate note always lies between two levels
Usually constructed so that price of long position in cap initially equal to price of short position in floor, so that cost of entering into collar is zero
Assuming that you hold a swaption, explain, with reference to the swap rate, how you would decide whether or not to exercise your option to enter into the swap
Calculate net present value of swap to you on strike date
If its positive, so that you expect to recieve more than you pay out, then you enter into it
If it is 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 have zero NPV at outset. It is equivalent to par yield
Describe how to value a securitised bond
Use deterministic discounted cashflow approach
Discount rate reflects overall riskiness of bond and should be similar to yield offered by equally risky bonds (similar to credit rating)
Simulate and discount possible cashflows, allowing for
Probablility, timinig of any defaults, and likely recoveries
ranking and structure of different bond tranches
Discount rates should allow for risks not captured in the cashflows
State what the price of a plain vanilla credit default swap should be if it purchased via a single premium and an annual premium
Explain what the value of a total return swap should be equal to
Price of single premium credit default swap
+Equals expected default loss on reference bond
Price of annual premium credit default swap
+Equals 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
Equal why the equally of a company can be considered as a call option on the company’s assets
Suppose company has amount D of zero coupon bond outstanding that matures at time T and let Vt = value of company’s assets at time t.
If VT D, company will repay debt and value of equity = Vt - D
So, value of firm’s equity at time T is Et = max(Vt - D,0)
This is equivalent to payoff from a call option on company’s assets with a strike price equal to the amount of debt, D
