Page 14 #12.03d - PQ set - Hull, Chapter 13: Binomial Trees (Hull Text Q&A)

[Dr. Jayanthi Sankaran]

Hi David,

As referenced above:

Given the following data: Stock price = $10, Strike price = $10, volatility = 20%, riskfree rate = 4%, and Term = 1.0 year

#12.03 (d) What is the delta of a call option?

Using the Black Scholes Merton model, Hull gets 0.6179 as the value of the call option.
However, using the Binomial risk-neutral method, I get 0.55. Hull states that the ATM call option has a delta of 0.5 to 0.6.
My question is when do I use the former and when do I use the latter, given that there is no information that the time steps have been increased in such a way that the Binomial Trees method converges to the BSM

Thanks!
Jayanthi

 

[David Harper CFA FRM]

Hi @Jayanthi Sankaran How did you get 0.55? under the binomial, wouldn't the delta oscillate depending on the nodes (?) ... we aren't given the number of steps ... this questions gives volatility such we should infer delta from the BSM N(d1). Thanks!

append: oh I see what you did! you solved for 10*exp(0.02)*Δ - [10*exp(0.02) - 10] = 10*exp(-0.20). I also get 0.548834 via that approach. Clever! Okay, but that's a really crude approximation, based on a single step, of the more accurate (continuous) N(d1) given by BSM. Hull shows that to setup the foundation of the binomial (and the BSM actually, as the BSM represents the convergence of the binomial). So it's still better to use BSM. Thanks,

 

[Dr. Jayanthi Sankaran]

Hi David,

This is my crude approximation: I solved for u = exp(.2) = 1.2214, d = 1/u = 0.8187, a = exp(.04) = 1.0408.
p = (1.0408 - 0.8187)/(1.2214 - 0.8187) = 0.5515, 1- p = 0.4485
Value of portfolio for up jump = 12.21Δ - 2.21
Value of portfolio for down jump = 8.19Δ
Equating 12.21Δ - 2.21 = 8.19Δ
Δ = 0.55 shares

BTW, how do you get 10*exp(0.02)*Δ - [10*exp(0.02) - 10] = 10*exp(-0.20)? The risk-free rate = 4% and the volatility = 20%....

Thanks!
Jayanthi

 

[Arka Bose]

Hi Jayanthi,

What he did is basically the same thing you did, exp(0.2) is the up move factor and exp(-0.2)is the down move. You must have got this from Hull.

However, as David said, it is delta hedge for one step only.
If we do this hedging continiously, we will be close to our black scholes N(d1).

I would like to add the basic idea about that shortcut you are doing .
Since you are writing a call option, you would like to hedge that with a portfolio that replicates a LONG CALL.

Thus, your position in that portfolio would be Δshares + (-Borrowing)
If u look at the portfolio, you can see that the borrowing (of the strike amount) is constant, but there is increase or decrease in share price, which is denoted by Δ

So, in this case, the equation is, in the case of upmove:

12.21Δ - 10 = 2.21 and in case of downmove,
8.19Δ - 10 = 0 (since downmove, thus no value of call)

If you solve, your equation will we shortened to 12.21Δ - 2.21 = 8.19Δ

But see how it takes care of only 1 node? Doing this continuously will make the Δ close to the value of delta in BSM.

This portfolio Delta hedging is the reason why in Hulls book (of course in David's notes) you will find they valued Call option through Dynamic Delta hedging in Black Scholes Chapter Greeks (Table 17.2 i think) you will see that total of the coloumn which says cost of share purchased (for delta hedging) , if discounted at the risk free rate, will give the cost of the call option approximately.

Another note:
Just my technique of how to create portfolios for delta hedging? (because I have seen people memorizing them)

The most easiest way to know what is the correct portfolio for delta hedging or for synthetic call purposes is algebric manupulation of the put call parity equation.

we know, C+ K exp(-r) = S + P

So, if you think K exp(-r) part as the value of the Bond (i.e if negative, its a borrowing, if positive, its an investment), then easily we can infer this,

C= S - Kexp(-r) [ See the same portfolio is mentioned in the delta hedging I said, i.e Shares + Borrowing]
Similarly, P = Investment + Shares.


P.S I know its an information overload, sorry

 

[Dr. Jayanthi Sankaran]

Hi @arkabose,

Thanks for your detailed explanation and the trouble you have taken to explain it in a systematic way. I appreciate it! At the outset, I must mention that I got a little thrown off by this question from the Hull text, since it has been grouped under David's Binomial Trees PQ set! I have yet to do the BSM and Greek Chapters. Nevertheless, I do have some background in the BSM and the PCP.

Also, I guess it is a typo, in David's answer above,

10*exp(0.02)*Δ - [10*exp(0.02) - 10] = 10*exp(-0.20)? It should be 10*exp(0.2)Δ - [10*exp(0.2) - 10] = 10*exp(-0.20)?

Thanks again!
Jayanthi

 

[Arka Bose]

yep, that has to be a typo