Binomial model

[Deepak Chitnis]

Hi @David Harper CFA FRM CIPM In question set R30.P1.T4.Hull question no.1.4(page 4) there is question like follows:

And there is also explanation of this question on page page 7 as follows:

But I think there is something missing there, when I calculated the prices as follows:
1st price=e^-4%($3.63*0.321)=$1.119541
and option price=e^-4%(1.119541*0.321)=$0.345281. That means answer needs to be a.0.35. Correct me if I am wrong.
Thank you

 

[Nicole Seaman]

Hello @Deepak Chitnis

Here is the original thread to this question: https://forum.bionicturtle.com/threads/p1-t4-1-one-and-two-step-binomial-valuation-models.4744/. You may find your answer among the many discussions that are posted here. If your question is not answered, you can ask your question on that specific thread. In the future, please post all questions related to practice question sets in their individual forum links that are provided in each question set instead of creating a new post each time.

This helps others who may have the same question to look up the answer more quickly, and it also helps David and others who answer your question to reference the original forum post.

Thank you,

Nicole

 

[Deepak Chitnis]

Thank you @Nicole Manley, will post my query there. In the mean time will you tell me how to delete a thread that I have added as new?
Thank you

 

[Dr. Jayanthi Sankaran]

Hi Deepak,

Your question is a classic case of the valuation of American puts since they can be exercised prior to maturity. On the first node, Price =e^-4%($3.63*0.321)=$1.119541 = $1.12
However, the intrinsic value at node 1 is $20 - $18.10 = $1.90

Hence the put can be exercised at node 1 because the intrinsic value > PV i.e $1.90 > $1.12
and the price at node 1 = $1.90

At node 0 the price = 0.321*($1.90)*e^-.04 = $0.585 = $0.59

This is what makes American puts, in general, to be more valuable than their corresponding European puts

Hope that helps!
Thanks
Jayanthi

 

[Deepak Chitnis]

Hi @Jayanthi Sankaran thank you for the clarification. Just one quick question, is it same applicable for American call option?
Thank you

 

[Nicole Seaman]

Hello @Deepak Chitnis

If you have created a thread that you no longer want to show up in the forum, please tag me in that thread and I can delete it for you.

Thank you,

Nicole

 

[Dr. Jayanthi Sankaran]

Yes. that holds true for all American options be it calls or puts!

Thanks!
Jayanthi

 

[Deepak Chitnis]

Thank you so much @Jayanthi Sankaran, will remember it. Thanks a lot.

 

[Deepak Chitnis]

Hi @Nicole Manley, I will tag you so. Thanks for all the help

 

[Deepak Chitnis]

Hi @Jayanthi Sankaran just a one question for you, look if you can explain me or otherwise I will ask David! Question is form GARP's book:
Mark, a risk manager for bank XYZ, is considering writing a 6 month american put option on a non-dividend paying stock ABC. The current price is USD 50, and the strike price of the option is USD 52. In order to find the no-arbitrage price of the option mark uses two-step binomial tree model. The stock price can go up or down by 20% each period. Marks view is that the stock price has an 80% probability of going up each period and a 20% probability of going down the annual risk free rate is 12% with continuous compounding. The no-arbitrage price of the option is closest to:
A. USD 2
B.USD 2.93
C. USD 5.22
D. 5.86
I have trying to solve a question but I have many confusions see if you can explain it to me.
U=1.2, D=0.8, P=57.61%, 1-P=42.39%.
stock=50, Stock up=60, Stock up up=72, stock up down=48 and similarly stock down=48, stock down down=32, stock down up=48. now the value of the put option last node that is 32=20 and then the value of the stock down that is 48 will be 12 I get that because this is american option, but my confusion starts now that is when valuing the call option we used the node stock up down=48 that is e^-12%*0.25($4*42.39%) now suppose if it will be american style call option do we need to solve the put option like same way means e^-12%80.25($4*57.61%)? and second if it is the european option we need to do same like get the option values from the stock up-down value?(only for no arbitrage price of option) Sorry for trouble. Please help if you can.
Thank you

 

[Dr. Jayanthi Sankaran]

Hi Deepak,

I would like to split up the answers into three parts:

Part 1 - Valuing the six-month American put with S(0) = $50, K = $52, u = 1.2, d = 0.8, r(f) = 12% per annum.
(1) There is no need to compute p and (1 - p) because they are already given, p = 0.8 and 1 - p = 0.2
(2) Value of the option at node 2: P(ud) = $52 - $48 = $4 and at node P(dd) = $52 - $32 = $20
(3) Value of the option at node 1: P(u) = $4*0.2*e^(-0.12*0.25) = $0.7764, the intrinsic value = $0
The reason we adjust the annual 12% risk free rate by multiplying it by 0.25 is because we assume that there are two periods of 3 months each. This is because the American put is a six-month put.
(4) Value of the option at node 1: P(d) = [(0.8*$4) + (0.2*$20)] = $7.2*e^-(0.12*0.25) = $6.987.
(5) However, intrinsic value of the option at node 1: P(d) = $12 > $6.987. In other words, it is valuable to exercise the option at node 1
(6) Value of option at node 0: P(0) = [($0.7764*0.8) + $12*0.2)]*e^(-0.12*0.25) = $2.93
The answer is B

Part II - Valuing the six-month American call with the same data as above. You should get an answer of P(0) = $12.054

and finally,

Part III - European puts and calls are valued the same way, except that we do not compare the intrinsic value with the present value at each node because they cannot be exercised. Hence, even if the intrinsic value > Present value through backward induction we cannot use it because the option cannot be exercised. You must review Hull's example in pages 113 - 114 for your questions.

Thanks!
Jayanthi

 

[Deepak Chitnis]

Hi @Jayanthi Sankaran, Thank you for your reply, I think you should try to solve the question one more time, because whether it is american or european option we need to calculate P and 1-P. So you need to calculate the like e^-12%*0.25($4*42.39)=1.65(call option second node). And then e^-12%*0.25($1.65*57.61%+$12*42.39%)=$5.8589, so the answer must be D, but thank you for your other clarification. Sorry for trouble.
Thank you

 

[Dr. Jayanthi Sankaran]

Hi Deepak,

My answer to your question was based on your data: "Marks view is that the stock price has an 80% probability of going up each period and a 20% probability of going down". David has often said that GARP gives you extra data to confuse you. Hope that answers your query.

Thanks!
Jayanthi

 

[Deepak Chitnis]

Hi @Jayanthi Sankaran, this type of questions are for confuse us. Even if you got official books you can look for explanation. We still need yo calculate the P and 1-p separately.
Thank you,

 

[Dr. Jayanthi Sankaran]

I am going to let David explain this!

 

[Deepak Chitnis]

Hi @Jayanthi Sankaran, yea sure it will br great, but if you look the practice question set once there is also a question like this, David mentioned there that we need to calculate the p and 1-p. But it will be great if David explain this. I will tag him in the post. Sorry for the trouble. Thank you

 

[Deepak Chitnis]

Hi @David Harper CFA FRM CIPM, Please explain the question. Question is from back of the book.
Mark, a risk manager for bank XYZ, is considering writing a 6 month american put option on a non-dividend paying stock ABC. The current price is USD 50, and the strike price of the option is USD 52. In order to find the no-arbitrage price of the option mark uses two-step binomial tree model. The stock price can go up or down by 20% each period. Marks view is that the stock price has an 80% probability of going up each period and a 20% probability of going down the annual risk free rate is 12% with continuous compounding. The no-arbitrage price of the option is closest to:
A. USD 2
B.USD 2.93
C. USD 5.22
D. 5.86
I have calculated the U=1.2, D=0.8, P=57.61%, 1-P=42.39%,stock=50, Stock up=60, Stock up up=72, stock up down=48 and similarly stock down=48, stock down down=32, stock down up=48. now the value of the put option last node that is 32=20 and then the value of the stock down that is 48 will be 12 I get that because this is american option, but my confusion starts now that is when valuing the call option we used the node stock up down=48 that is e^-12%*0.25($4*42.39%) now suppose if it will be american style call option do we need to solve the put option like same way means e^-12%80.25($4*57.61%)? and second if it is the european option we need to do same like get the option values from the stock up-down value? Finally I got the answer like e^-12%*0.25($4*42.39)=1.65(call option second node). And thene^-12%*0.25($1.65*57.61%+$12*42.39%)=$5.8589,that means answer needs to be D. In the question probability of going up and down is given, but we still need to calculate the P and 1-P, correct me if I am wrong. Please help in this question.
Thank you

 

[David Harper CFA FRM]

Hi @Deepak Chitnis Yes, see below, I get the same answer as you! At node[1,0] the value is $12 = max($12 intrinsic, $10.46 discounted expected value). In my opinion, the question is flawed. If you hardcode p = 80% and d = 20%, then it looks like you get $2.932. But especially the use of "no-arbitrage" suggests risk-neutral framework such that, just as you say, we should solve for (p) as a function of (u) and (d), as usual. Further, it is highly unusual to hardcode the (p); our source is Hull and has been for years, and I don't think he has a single question that doesn't solve for (p) as a function of (u) and (d). I hope that helps!

 

[Deepak Chitnis]

Hi @David Harper CFA FRM CIPM, Thank you for the clarification and the help! David , just one out of the syllabus question, now a days in a real life, does brokers or analyst use the binomial tree and after the disaster of LTCM does Black-scholes merton model still used? Is there any else option pricing models?
Thank you

 

[Dr. Jayanthi Sankaran]

Hi Deepak,

(1) I did calculate the value of the American put and got the same value as David and you: $5.856. As David said the question is indeed flawed - so that was good learning for me! The data on p and 1 - p was meant to confuse us - I agree

(2) As far as I know, the Black-Scholes-Merton model is extensively used in the trading community to price and hedge derivatives. The most important property of the BSM is that it is independent of risk preferences. This considerable simplifies the analysis of derivatives. The BSM with modifications is highly robust even when valuing exotics!

(3) In the limit as the number of time steps is increased say to 500 steps, the binomial converges to the Black Scholes but this is computationally intensive.

(4) Hedge funds use trading strategies such as long/short, convertible arbitrage, distressed securities, emerging markets, global macro and merger arbitrage. In the case of LTCM the convertible arbitrage strategy was used. As far as I know, BSM is not used in implementing these type of strategies.

(5) Black's Model is used for valuing American calls on dividend paying stocks.

(6) There are a host of option pricing models on interest rates such as Black-Derman-Toy Model, Black Karasinski Model to value swaptions, caps and floors, captions and floortions, Ho and Lee Model, Hull and White interest models and so on....

Thanks!
Jayanthi