Answer to Question #220039 in Financial Math for Favor

Question

3.1 The Cox-Ross-Rubinstein (CRR) model is a Binomial tree in which the up and down
factors are given as
u = e^(σ sqrt(h))
, d = e^(−σ sqrt(h))
,
where σ denotes the volatility parameter and h stands for the length of a single period
in a tree.
3.1.1 What is the ratio Su/Sd? [2]
3.1.2 What is the (as simplified as possible) expression for the risk-neutral probability
of the stock price going up in a single step? [2]
3.2 Find the current price of a one-year, R110-strike American put option on a non-
dividend-paying stock whose current price is S(0) = 100. Assume that the continuously compounded interest rate equals r = 0.06. Use a two-period Binomial tree with
u = 1.23, and d = 0.86 to calculate the price VP(0) of the put option.

Answer 1

3.1.1

Lets suppose, the Stock price of Stock is S, it can give an upmove and move to Su with an probability of Pu and can give an down move to Sd with an Probability of 1-Pu.

To Calculate Su, we need to multiply S(Stock price) with upmove factor & to calculate Sd, we need to multiply S(stock price ) with downmove factor .

Therefore; 

$Su = S X u = S X e^{σ \sqrt{h}} = S ×e^{σ\sqrt{h}}\\ Sd = S × d = S ×e^{(−σ \sqrt{h})} = S × e^{\frac{1}{σ\sqrt{h}}}\\ \frac{ Su}{Sd} = \frac{S × e^{σ\sqrt{h}}}{S×e^{\frac{1}{σ\sqrt{h}}}}\\ \frac{Su}{Sd} = \frac{ e^{σ\sqrt{h}}}{e^{\frac{1}{σ\sqrt{h}}}}$

Using, mathetical equation, $\frac{A^n}{ A^m} = A^{(n-m)}$

$\frac{ Su}{Sd} = e^{(σ\sqrt{h} − \frac{1}{σ\sqrt{h}})}\\ \frac{Su}{Sd} = e\frac{σ^{2}h −1}{σ\sqrt{h}}$ is the required Su/ Sd ratio.

3.1.2

Formula for Risk neutral probability for an upmove $u =\frac{ 1 + Rf −d} {u−d}$

Where, R_f is risk free rate.

Up move factor, $u = e^{(σ \sqrt{h})} = S ×e^ {σ\sqrt {h}}$

Down move factor $d = e^{(−σ \sqrt{h})} = S × e^{ \frac{1}{ σ\sqrt{h}}}$

Putting all values in formula, we get ;

$P_u=\frac{1+R_f-e^{\frac{1}{σ\sqrt{h}}}}{e^{σ\sqrt{h}}-e^{\frac{1}{σ\sqrt{h}}}}$

is our required expression for risk-neutral probability of the stock price going up.

3.2

A put choice is an agreement giving the proprietor the right, yet not the commitment, to sell–or undercut a predetermined measure of a basic security at still up in the air cost inside a predefined time span. Not really set in stone value that purchaser of the put alternative can sell at is known as the strike cost. 

Put choices are exchanged on different basic resources, including stocks, monetary standards, bonds, wares, prospects, and files. A put choice can be appeared differently in relation to a call choice, which gives the holder the option to purchase the basic at a predetermined cost, either prior to the termination date of the alternatives contract.

100 tree diagram is 123 and 86

$P = \frac{1+ r-d}{ U - d}$

$P =\frac{ 1 + 0.06 - 0.86}{ 1.23 - 0.86}$

$P =\frac{1.06- 0.86}{ 0.37}$

$=\frac{0.2}{0.37}$

= 0.5404540541

= 54.0540541%

OP of put today 

$=\frac{11.0278702}{e0.06*1}$

e^0.06= 1.0618

$=\frac{11.0278702}{1.0618}$

= 10.3860145