Yield Binomial
Bond Option Pricing Using
the Yield Binomial
Methodology
AGENDA
• Background
• South African Complexity with option model
• Problems with Black and Scholes Approach
• Binomial Methodology
Background
• American Bond Options - some traders use Black & Scholes model
• Adjust for early exercise by forcing the answer to equal at least intrinsic
South African Complexity with Option Model
• Overseas bond options have a fixed strike price throughout the option
• South African bond options trade with a strike yield
• Thus the strike price changes throughout the life of the option
South African Complexity with Option Model
• Difference between Clean Strike prices and strike yield:
Problems with Black and Scholes approach
• Tends to under-price out of the money option
• Mispricing is the worst for short-dated bonds
• Adjusting the Black & Scholes value with the intrinsic value results in discontinuity in value.
• This also results in a discontinuity in the Greeks.
Example (1)
• Put option on R150• Settlement date: 26 Sept 2002• Maturity date: 1 Apr 200• Riskfree rate until option maturity:
10% (continuous)• Strike yield: 11.5% (semi-annual)• The YTM (semi-annual) ranges from
11.5% to 12.97% • Nominal: R100
Example (2)
• We are interested in the point where the bond option premium falls below intrinsic
0
0.5
1
1.5
2
2.5
3
3.5
11.50% 11.70% 11.90% 12.10% 12.30% 12.50% 12.70% 12.90%
Intrinsic valueBond option premium
Example (3)
• The premium falls below intrinsic at a YTM of ± 11.84%
• We are also interested in the behaviour of delta around a YTM of 11.84%
Example (4)
• To this end, we use a numerical delta, calculated as follows:
• Delta = UBOP(i+1) – UBOP(i)
AIP(i+1) – AIP(i)
• UBOP stands for used bond option premium, and is equal to the intrinsic whenever the option premium falls below intrinsic
• AIP is the all-in price of the bond at the option’s settlement date
Example (5)
• Delta makes a jump at the 11.84% mark
Numerical delta
-1.01
-0.81
-0.61
-0.41
-0.21
-0.0111.50% 11.70% 11.90% 12.10% 12.30% 12.50% 12.70% 12.90% 13.10%
Example (6)
• If we were to extend the data points in the first graph, it would look more or less as follows:
Example (7)
• The Black and Scholes model will use:– The bond option premium if it is larger than intrinsic– Intrinsic, wherever the option premium falls below it
• This is illustrated by the red dots:
What is different about the yield binomial model?
• Normal binomial model uses a binomial price tree
• Yield binomial uses yields instead of prices
Normal binomial model
Using Risk Neutral argument we get:
• a = exp(rt)
• u = exp[.sqrt(t)]
• d = 1/u
• p = a - d
u - d
S21= S0S0
S11=S0u
S10=S0d
p
1-p
p
p
1-p
1-p
S22=S11u
S20=S10d
Time 0 Time 1 Time 2
Normal binomial model
S0
Normal binomial model
• From an initial spot price S0, the spot price at time 1 may jump up with prob p, or down with prob 1-p.
• In the event of an upward jump, the S1 = S0u
• In the event of a downward jump, the S1 = S0d
• The probability p stays the same throughout the whole tree.
Yield binomial model
p2
1-p2
Y0
Time 0
Y11=FY1u
Y10=FY1d
Time 1
FP1
Y22=FY2u2
Y20=FY2d2
Time 2
FP2
Y21=FY2FY1
Yield binomial model
• At each time step the forward yield FYi is calculated
• Then the yields at each node are calculated
• Take first time step:– Y11 and Y10 is calculated by
– Y11 = FY1 * u and
– Y10 = FY1 * d
Yield binomial model
p1
1-p1
p2
p2
1-p2
1-p2
Y0
Time 0
P11=P(Y11)
P10=P(Y10)
Time 1
FP1
P22=P(Y22)
P20=P(Y20)
Time 2
FP2
P21=P(Y21)FP1 =P(FY1)
Yield binomial model
• In this model, a forward price FPi is calculated at time step i from the yields just calculated
• At each node i,j, a bond price BPi,j is calculated from the yield tree
• Cumulative probabilities CPi,j:
CP0,0 = 1
CPi,j = CPi,j.(1-pi) if j=0
= CPi-1,j-1.pi + CPi-1,j.(1-pi) if 1j i-1
= CPi-1,i-1.pi if j=i
Yield binomial model
The relationships between the forward prices FPi, bond prices Bpi,j and probabilities pi are given by:
FP1 = p1.BP1,1 + (1-p1).BP1,0
FP2 = CP2,2.BP2,2 + CP2,1.BP2,1 + CP2,0.BP2,0
FPi = sum(cumprob(I,j) *price(I,j) from j =0 to i
p(i) = price(i) – sum(cumprob(i-1,j) * price(i,j)/
sum(cumprob(i-1,j)* price(I,j+1) –price(i,j))
Binomial Methodology…
• Option Tree:- Calculate the pay-off at each node at the end of
the tree.- Work backwards through the tree.
- Opt. Price = Dics * [Prob. Up(i) * Option Price Up +
Prob. Down(i) * Option Price Down]
Binomial Methodology
• Checks on the model:
– Put call parity must hold – Volatility in tree must equal the input volatility
Binomial Methodology in summary
Option Inputs:
• Strike yield
• Type of option (A/E)
• Is the option a Call or a Put?
Greeks
• Numerical estimates• Alternative method for Delta and Gamma:
– Tweak the spot yield up and down.
– Calculate the option value for these new spot yields.
– Fit a second degree polynomial on these three points.
– The first ad second derivatives provide the delta and gamma.
Binomial Methodology in Summary
• Calculated parameters - Yield and Bond Tree:
- Time to option expiry in years
- Time step in years
- Forward yield and prices at each level in tree using carry model
- Up and down variables
Benefits of Binomial
• Caters for early exercise
• Smooth delta
• Flexibility with volatility assumptions
Binomial Model
• Number of time steps?
• Not a huge value in having more than 50 steps
• Useful to average n and n+1 times steps
Yield Binomial