L12 - The Black-Derman-Toy binomial model Flashcards
What is this model used for?
- This model prices bonds and bond options. It creates a binomial tree depending on the interest rate and volatility term structure –> in each case there are two scenarios –> interest rates go up or go down
- Volatility is constant in time
- No mean reversion
- Returns are supposed to distribute according to a lognormal process
- it returns are log-normal prices are normally distributed
What was the problem with all options model in relation to interest rates?
- They were all created around the fact that you couldn’t get negative interest rates
How do prices and volatility change with maturity?
- volatility of the term structure decreases as maturity increases
- prices are actually based on duration so the volatility of prices increases as maturity increases
How can you transform the rate tree into the price tree?
- interest rate is now = 10%
- next period: –>50 % probability of each happening
- up its going to be 14.32%
- down its going to be 9.79%
- we know that the bond is has a par value of 100 and is maturing in two years
- to find the 1 period prices we discount the final value of the bond by both the up and down rates to get the respective prices
Other than using the tree method how can you find the current value of the bond?
- AS the bond matures in two years you can just discount the par value by the 2-year spot rate
What do black-derman-toy try to infer about the pairs of rate in the binomal tree?
- That you can estimate the future price and therefore the fair value of a bond based on a pair of future rates
- however, there are many rates(prices) that could lead to the same fair value
- thus they say there are many different pairs of rates that could be used to lead to the same fair value
- thus we need to use volatility to define what the true pair rates are!
How can we estimate the volatility structure from the binomial tree?
can figure out what the 2 year volatility is based on the log-normal ratio of the two future period 1 rates
How do we create the next step of the rates tree?
- In one year two there will be two possibilities
- Interest rate will be 14.32% (up) with han expected volatility: σ = Ln(ruu/rud)/2
- The interest rate will be 9.79% with an expected volatility σ = Ln(rud/rdd)/2
Since the middle volatilities are equal it means that :
- r(u,u)/r(u,d) = r(u,d)/r(d,d)
- r(u,d)/d(d,u) = r(u,d)^2 ={ r(uu)/r(dd)]0.5
(up/up) and (down/down) have a 25% chance of occurring but 50% chance of the middle rate occurring
Step 1: of bond valuation with the Binomial model:
- Let us suppose we have a Government bond with a coupon of 10 per cento, nominal value (NV) 100 and maturity 3 years
- It is like a ZCB portfolio composed as follows:
- 1) 1 ZCB 1 year NV=10;
- 2) 1 ZCB 2 years NV=10;
- 3) 1 ZCB 3 years NV=110.
- 1st –> discount the 1 year ZCB using the 1 year rate
- 2nd –> discount the 2 year bond with the 2 year rate and then using the formula to find out the PV
- 3rd…
How long is an option written on a bond?
- it is always shorter than the maturity of the bond
- if we write an option that will expire at the same time as the bond as we already know the price at maturity there will be no uncertainty in the bond
How can we value a European Call option on a bond?
STEP 1
Option will expire in 2 years with the strike price of K=95
- By the price tree, we know that the bond could assume the following prices: 102,11; 106,69 or 110,22
- The reason we get rid of the coupon is that the option gives you the right to buy the principal of the bond
- Payoff = call price - spot price –> if negative we do not exercise
Pricing an European call option?
STEP 2:
- Once you have figured out if you are going to exercise the option at expiry you then can estimate the call options prices in each of the previous periods:
- UU(0) –> not exercised
- M(1.69) –> exercised
- DD(5.22) –> exercised
- NOTE: if the rates go up the price of the bond goes down and the call option moves OTM and vice versa
Valuing a European Put Option?
STEP 1
European Put Option valuation?
STEP 2
How can you value an American option on a bond?
- Binomial trees are particularly useful to estimate the value for an American option
- Since the American option can be exercised in every period before the expiry, its value for each node is the highest between the holding value and the exercising value
