Chapter 14 p.37 - Calibrating binomial models

[el_george]

Hi
I noticed that the formula of the core reading for u,d in this case for a dividend paying stock should be wrong. The formulas given do not meet the assumption u*d=1and doing the math I end up with
u=exp(σ δt^0.5-v δt)
d=exp(v δt-σ δt^0.5)
and not in what is in the core reading.
Any suggestions?

 

[el_george]

Correction to the above formulas:
u=exp(σ δt^0.5+v δt)
d=exp(-σ δt^0.5-v δt)

 

[Steve Hales]

The "ud=1" assumption only applies when there are zero dividends.
When implementing dividends in the binomial model, the share price needs to be reduced to account for their payment at each node. So the update formulae are actually:
S(t) * u * exp(-v*dt) and S(t) * d * exp(-v*dt)
Once this is allowed for then "an up followed by a down" leads to the same share price as "a down followed by an up" as required.

 

[Sunil Chaudhary]

The "ud=1" assumption only applies when there are zero dividends.
When implementing dividends in the binomial model, the share price needs to be reduced to account for their payment at each node. So the update formulae are actually:
S(t) * u * exp(-v*dt) and S(t) * d * exp(-v*dt)
Once this is allowed for then "an up followed by a down" leads to the same share price as "a down followed by an up" as required.

Hi Steve,

Can you please let me know how or from where you got "S(t) * u * exp(-v*dt) and S(t) * d * exp(-v*dt)"

Also, I am unable to understand the formula for u and d in case of dividend paying stocks. How can I derive the expressions mentioned in the core reading as
u=exp(σ δt^0.5+v δt)
d=exp(-σ δt^0.5+v δt)

Thanks in advance.
Sunil

 

[Steve Hales]

Hi Steve,

Can you please let me know how or from where you got "S(t) * u * exp(-v*dt) and S(t) * d * exp(-v*dt)"

Also, I am unable to understand the formula for u and d in case of dividend paying stocks. How can I derive the expressions mentioned in the core reading as
u=exp(σ δt^0.5+v δt)
d=exp(-σ δt^0.5+v δt)

Thanks in advance.
Sunil

Consider an up-step in the binomial tree. The share price moves from S(t) to S(t)*u, but then the dividend is paid and the share price needs reducing by exp(-v*dt) in recognition of this. (Remember that v is the force of dividend payment.) So the ultimate change in the share price is S(t)*u*exp(-v*dt), ie the multiplicative increase is then reduced by the dividend payment.

The formulae you've mentioned are derived in Section 6 of the binomial model chapter of the Course Notes. Let me know if anything is unclear.
Regards

 

[Sunil Chaudhary]

Consider an up-step in the binomial tree. The share price moves from S(t) to S(t)*u, but then the dividend is paid and the share price needs reducing by exp(-v*dt) in recognition of this. (Remember that v is the force of dividend payment.) So the ultimate change in the share price is S(t)*u*exp(-v*dt), ie the multiplicative increase is then reduced by the dividend payment.

The formulae you've mentioned are derived in Section 6 of the binomial model chapter of the Course Notes. Let me know if anything is unclear.
Regards

Hi Steve,

Thanks for your response.
I have noted and understood that how u and d are derived under section 6 'calibrating binomial models'.
However, I am unable to get how u and d are derived for continuously payable dividend rate v.

Can you please suggest a detailed source which I can refer to or general understanding how to
u=exp(σ δt^0.5)
d=exp(-σ δt^0.5)
are modified for dividend rate.

Thanks
Sunil

 

[Steve Hales]

Ah, I see what you mean.
Given that the dividends are certain, the usual approach is to deduct their present value from the starting share price. This way the new initial share price is allowed to evolve according to the binomial model with the same u and d but without any deductions for the future dividends being paid. So the two following arrangements result in the same outcomes:

1. Initial share price S0 either goes up by a factor of u = exp(sigma*dt^0.5 + v*dt) or down by a factor of d = exp(-sigma*dt^0.5 + v*dt) but needs to then be multiplied by a factor of exp(-v*dt) to allow for the dividend payment.

2. Initial share price S0*exp(-v*dt) goes up by a factor of u = exp(sigma*dt^0.5 + v*dt) or down by by a factor of d = exp(-sigma*dt^0.5 + v*dt) but without the need to adjust for dividends because they're taken care of in the initial share price.

Under both arrangements you'll see that after the dividends have been accounted for the actual share price has either gone up to S0*exp(sigma*dt^0.5) or down to S0*exp(-sigma*dt^0.5). But this only works if u = exp(sigma*dt^0.5 + v*dt) and d = exp(-sigma*dt^0.5 + v*dt).

 

[Sunil Chaudhary]

Ah, I see what you mean.
Given that the dividends are certain, the usual approach is to deduct their present value from the starting share price. This way the new initial share price is allowed to evolve according to the binomial model with the same u and d but without any deductions for the future dividends being paid. So the two following arrangements result in the same outcomes:

1. Initial share price S0 either goes up by a factor of u = exp(sigma*dt^0.5 + v*dt) or down by a factor of d = exp(-sigma*dt^0.5 + v*dt) but needs to then be multiplied by a factor of exp(-v*dt) to allow for the dividend payment.

2. Initial share price S0*exp(-v*dt) goes up by a factor of u = exp(sigma*dt^0.5 + v*dt) or down by by a factor of d = exp(-sigma*dt^0.5 + v*dt) but without the need to adjust for dividends because they're taken care of in the initial share price.

Under both arrangements you'll see that after the dividends have been accounted for the actual share price has either gone up to S0*exp(sigma*dt^0.5) or down to S0*exp(-sigma*dt^0.5). But this only works if u = exp(sigma*dt^0.5 + v*dt) and d = exp(-sigma*dt^0.5 + v*dt).

Many Thanks Steve.