1. Yield Binomial

2. Bond Option Pricing Using the Yield Binomial Methodology

3. AGENDA Background South African Complexity with option model Problems with Black and Scholes Approach Binomial Methodology

4. Background American Bond Options - some traders use Black & Scholes model Adjust for early exercise by forcing the answer to equal at least intrinsic

5. 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

6. South African Complexity with Option Model Difference between Clean Strike prices and strike yield:

7. 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.

8. 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

9. Example (2) We are interested in the point where the bond option premium falls below intrinsic

10. 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%

11. 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

12. Example (5) Delta makes a jump at the 11.84% mark

13. Example (6) If we were to extend the data points in the first graph, it would look more or less as follows:

14. 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:

15. What is different about the yield binomial model? Normal binomial model uses a binomial price tree Yield binomial uses yields instead of prices

16. Normal binomial model Using Risk Neutral argument we get: a = exp(r  t) u = exp[ .sqrt(  t)] d = 1/u p = a - d u - d

17. S 21 = S 0 S0S0 S 11 =S 0 u S 10 =S 0 d p 1-p p p S 22 =S 11 u S 20 =S 10 d Time 0Time 1 Time 2 Normal binomial model S0S0

18. From an initial spot price S 0, 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 S 1 = S 0 u In the event of a downward jump, the S 1 = S 0 d The probability p stays the same throughout the whole tree.

19. Yield binomial model p2p2 1-p 2 Y0Y0 Time 0 Y 11 =FY 1 u Y 10 =FY 1 d Time 1 FP 1 Y 22 =FY 2 u 2 Y 20 =FY 2 d 2 Time 2 FP 2 Y 21 =FY 2 FY 1

20. Yield binomial model At each time step the forward yield FY i is calculated Then the yields at each node are calculated Take first time step: –Y 11 and Y 10 is calculated by –Y 11 = FY 1 * u and –Y 10 = FY 1 * d

21. Yield binomial model p1p1 1-p 1 p2p2 p2p2 1-p 2 Y0Y0 Time 0 P 11 =P(Y 11 ) P 10 =P(Y 10 ) Time 1 FP 1 P 22 =P(Y 22 ) P 20 =P(Y 20 ) Time 2 FP 2 P 21 =P(Y 21 ) FP 1 =P(FY 1 )

22. Yield binomial model In this model, a forward price FP i 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 CP i,j : CP 0,0 = 1 CP i,j = CP i,j.(1-p i ) if j=0 = CP i-1,j-1.p i + CP i-1,j.(1-p i ) if 1  j  i-1 = CP i-1,i-1.p i if j=i

23. Yield binomial model The relationships between the forward prices FP i, bond prices Bp i,j and probabilities p i are given by: FP 1 = p 1.BP 1,1 + (1-p 1 ).BP 1,0 FP 2 = CP 2,2.BP 2,2 + CP 2,1.BP 2,1 + CP 2,0.BP 2,0 FP i = 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))

24. 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]

25. Binomial Methodology Checks on the model: –Put call parity must hold –Volatility in tree must equal the input volatility

26. Binomial Methodology in summary Option Inputs: Strike yield Type of option (A/E) Is the option a Call or a Put?

27. 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.

28. 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

29. Benefits of Binomial Caters for early exercise Smooth delta Flexibility with volatility assumptions

30. 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

31. Yield Binomial