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