# Schweser Mock, AM, Question 50, Schweser are WRONG!!

The Bi-nomial option pricing model in the Schweser answer is just plain wrong, I think?

They have used Mulroney’s probability of an up move, not the risk neutral probability.

The binomial pricing model requires there to be no arbitrage at the nodes, so risk neutral probabilities are calculated based on the quantum of up and down moves and the risk free rate, you cannot use the 60% given in the question.

If up was 1.2 and down 1/1.2 and the probability of up was 60% then the risk free must be 5.3% per period otherwise there is arbitrage.

Up multiplier = 1.2

Down multiplier = 1/1.2 = 0.83333

risk neutral prob of up = (1.012-0.8333)/(1.2-0.83333)= 0.487

risk neutral prob of down = 1-up=0.513

The model answer’s payoff’'s are correct, but the option value should be 0.513*0.513*8.61= \$2.23 discounted to today at risk free which = \$2.21

I have seen one question in CFAI mock where down move was also given and they have not used it. Are you saying we should simply discard other values in a given question except up move and calculate others?

The key of using the binomial tree to price put/call options is that they need to be risk neutral.

The process is effectively saying at each node, “what must happen here for there to be no arbitrage at the next node”. If we discount the expected values at the risk free rate they must be risk free expectations.

So we must use risk neutral probabilities, not the probability that Mulroney believes to be the chance of an up move.

This is different from valueing bonds bi-nomially, becuase we do not discount at the risk free, we discount at the epxected yield at each node.

bump, any other thoughts fromm anyone?

Could you post the whole problem?

You need to check how it’s worded…I got this one wrong on the CFA mock…

From the Vignette:

Mulroney is considering the purchase of a one year Euro style put option with exercise price = 35 on her Merrills Material stock, which is currently trading at 38. To check her dealer’s price on these puts she uses a binomial model with a semi-annual probability of an up move equal to 60% and an up factor of 1.2. She uses the yield on a 6month t-bill as the risk free rate for her model.

The question:

Assuming the risk free rate for a 6 month holding period is a constant 1.2%, the value of the 1 year put options on Merrill Materials is closest to:

A - \$3.33, B- \$1.38 , C \$1.35

Note the comment on Page 182 of CFA Volume 6: It might appear that pi and 1-pi are probabilities of the up and down movements, but they are not. In fact, the probabilities of the up and down movements are not required…

If so… My approach is:

1. Prob(up) = 0.6

2. Prob(down) = 1 - 0.6 = 0.4

3. Up factor = 1.2

4. Down factor = 1/1.2 = 0.833

5. Discount rate for 6 months = 1.2%/2 = 0.006

Binomial Tree for Stock Price (I can’t use a table to present here):

1 Period away:

Vstock (up) = 45.6

Vstock (down) = 31.667

2 Periods away:

Vstock(up,up) = 54.72

Vstock(up,down) = 38

Vstock(down,down) = 26.389

At the end of 2 semi-annual periods. We get Value of Put in the 3 nodes as

Vput(up,up) = 0

Vput(up,down) = 0

Vput(down,down) = 35 - 26.389 = 8.611

Discount, using both probabilities and discount rate of 0.006

Value of Put at Node(down) = PV (weighted average of Value of Put at Node(down,down) and Node(down,up))

= PV Factor * [Prob(down)*Node(down,down) +Prob(up)*Node(down,up)]

= 1/1.006 * (0.4 * 8.6111 + 0.6 * 0)

= 3.423

Similarly, Value of Put at Node (up) = PV(weighted average of next period’s 2 nodes) = 0

Now, we are at:

Vput (up) = 0

Vput (down) = 3.423

Do this again:

Value of Put at Node (t=0) = PV(weighted average of next period’s 2 nodes)

=PV Factor * [Prob(down)*Node(down) +Prob(up)*Node(up)]

= 1/1.006 * (0.4 * 3.423 + 0.6 * 0)

= 1.36 (Ans)

Kyh, that’s essentially the Schaefer model answer, except it’s 1.2% per period, not pa.

However, the tree you have, same as Schweser, isn’t a risk neutral tree. IF the prob of an up move is 60% and an up factor of 1.2, down prob 40% and factor 1/1.2, then the expected price one period from now is

(38*1.2*0.6)+(38/1.2 *0.4)= 27.36+12.66=40.03

if I can borrow at risk free =1.2% then I can borrow 38, buy the stock, after one period repay 38.46, and own a stock with an expected price of 40.03, and so there is arbitrage. The reason we use risk neutral probabilities is that these are the probabilities that mean the expected return = the risk free return = no arbitrage, and so we can derive a value for the put that is fair and allows no arbitrage. If the put is worth 1.35 Then arbitrage is available.

Erm… I’m not sure whether you could call the difference, between expected future stock price return and risk free rate, an arbitrage.

Sorry but that is how I would do that question.

Hope someone else can help! =)

^ Agreed - your understanding of arbitrage and risk neutral valuation is a little off. You can’t have arbitrage when you are basing your expectations on the “expected value of the stock”. Arbitrage requires a market price, and you can’t price an option to be risk free at each of the nodes because there are no market prices of the underlying available for the future nodes (without using a strategy around futures, etc).