Detailed Analysis of Binomial Tree and Option Pricing for MMAF514

This document presents a comprehensive solution to an assignment focused on binomial tree option pricing. It begins by calculating the value of a call option using no-arbitrage valuation and demonstrates how to profit from arbitrage when a call option is mispriced. The solution then calculates the call option value using risk-neutral valuation. Following this, the document calculates the value of a put option using no-arbitrage valuation and illustrates arbitrage opportunities arising from mispriced put options. The put option value is also calculated using risk-neutral valuation. Finally, the solution demonstrates that the values of call and put options satisfy put-call parity. The assignment also includes the analysis of a firm's debt and equity, and the calculation of the yield to maturity and the value of a put option related to the firm's debt structure.

1) Binomial Tree and option pricing
(a) Value of call option using no arbitrage valuation
SO=100 u=1.16 d=0.86 K=110 rf=0.05 t=0.5
(i)Su= SO * u = 100 * 1.16 = $116
Sd = so* d = 100 * 0.86 = $86
(ii)Payoff - Su-K= 116 -110 = 6
-Sd- 0
(iii) Find the ∆: 116∆-6 = 86 ∆
∆ =(6/(116-86) = 0.2
Riskless portfolio is long 0.2 shares short 6 call option
iv) The value of the portfolio in 6 months is
116∆-6 = 116(0.2) -6 = 17.2
v) Today profile PV = 17.2EXP(-0.05*0.5) = 16.77533
vi) Value of each share = SO *∆ = 100 *0.2 = 20
vii_ Value of call option = value of share – today’s profile PV = 20 – 16.7753 = 3.22467
(b) Arbitrage profits when a trader quotes a call price for $2 K=110
At t=0
Call option price too low
Buy 5 option for $ 10
Short the stock to realize $100
Invest ($100-10) => 90EXP(0.05*0.5) = $92.27
At t=6
When Price is 86= Pay off for option =0
Pay off for Investment at Rf= $ 92.27
Payment to be made for short share= $ 86
Net Profit= $ 92.27- $ 86= $ 6.27
When Price is 116 = Pay off for option = 5* 6 = $30
Pay off for Investment at Rf= $ 92.27
Payment to be made for short share= $ 116
Net Profit= $ 92.27 + $ 30- $ 116= $ 6.27

● If ST>110
Exercise option to buy stock at $110
Use stock to close out short position
Received cash $100.48 from investment
Net gain = ?? 100.48- 110 = $-9.52
● If ST<110
Buy stock at ST
Use stock to close out short position
Received investment $100.48
Net gain=?? 100.48-ST (>..)
(c) Calculate the value of the call option using risk-neutral valuation
o P=erT-d/(u-d) = EXP(0.05X0.5)-0.86/(1.16-0.86) = 0.5510504
o Fu= sd-k = 116-110 =6
o Value of option: EXP(-0.05X0.5)[(P-6)+(1-P)*0]
=0.9753099*3.3063
= 3.22467
(d) Value of put option using no arbitrage valuation
SO=100 u=1.16 d=0.86 K=110 rf=0.05 t=0.5
(i)Su= SO * u = 100 * 1.16 = $116
Sd = so* d = 100 * 0.86 = $86
(ii)Payoff - 0
-K- Sd = 110-86 = 24
(iii) Find the ∆: 116∆ = 86 ∆ +24
∆ =(24/(116-86) = 0.8
Riskless portfolio is long 0.8 shares short 24 put options
iv) The value of the portfolio in 6 months is
86 ∆ +24= $92.8
v) Today profile PV = 92.8EXP(-0.05*0.5) = 90.5087
vi) Value of each share = SO *∆ = 100 *0.8 = 80
vii_ Value of call option = today’s profile PV -value of share = 90.5087 – 80 = 10.5087
(e) Arbitrage profits when a trader quotes a put price of $12 (K=110)
Is it related to lower bound of Put price?? (p>=max Ke-rT-So,0)
At t=0
Sell 1.25 put option for $15
Sell the stock for $100

Invest ($100+$15) => 115 EXP(0.05*0.5) = $117.9112
At t=6
When Price is 86= Pay off for option =24*1.25= $ 30
Pay off for share= $ 86
Receipt from investment= $ 117.9112
Net Profit= -$ 30 - $ 86+ $117.9112 = $ 1.9112
When Price is 116 = Pay off for option = 0
Pay off for share= $ 116
Receipt from investment= $ 117.9112
Net Profit= - $ 116+ $117.9112 = $ 1.9112
● If ST<110
Exercise put option to sell stock for $110
Use stock to close out short position
Pay back loan $114.835
Net gain = ?? 114.835- 110 = $-4.835
● If ST<110
No exercise put option
Sell stock at the market (ST)
Pay back loan $114.85
Net gain=?? 100.48-ST (>..)
(f) Calculate the value of the put option using risk-neutral valuation
o P=erT-d/(u-d) = EXP(0.05X0.5)-0.86/(1.16-0.86) = 0.5510504
o Fu= k -sd = 110-86 =24
o Value of option: EXP(-0.05X0.5)[(P*0)+(1-P)*24]
=0.9753099*10.77476
= 10.5087
(g) Call and put options in(C ) (F) satisfy the put-call parity.
Call C + Kexp –r*t = P + SO
C= P+ SO- Kexp –r*t
=10.5087+100 – 107.2840903
=3.2246
Put P= C + Kexp –r*t - So

=3.2246+107.2840903 – 100
=10.5087

Answer 2
Part A
Debt of Firm: $ 5 Million
Repayment Time : 4 Months
Present Value of Debt : $5 Million/ EXP(0.03*0.33333)= $ 4.950249 Million
Equity Value = 3* 1 Million = 3 Million
Present Value of Firm : $4.95 Million + $ 3 Million - Value of Put Option= $7.95 Million - Value of Put
Option
Value of Put Option:
Let the Firm Value be 7.793767 Million
Su= 10.91127
Sd= 4.67626
Pay off = 5-4.67626= $0.32374 Million
∆: 10.91127∆ = 4.67626∆ +$0.32374
∆ =($0.32374 /(10.91127-4.67626)) = 0.051923
The value of the firm in 4 months is
4.67626 ∆ +$0.32374= $.566544
Today profile PV = $.566544 EXP(.03*0.333) = $ 0.560907
Value of firm = 7.793767 *∆ = 7.793767 * 0.051923 = $ 0.404674
Value of PUt option = today’s profile PV -value of firm = $ 0.560907 – $ 0.404674= $ .156233
Value of firm = 7.793767 Million
Part B
VALUE OF Debt= $ 4.95 Million Million
Future Value: 5 Million
Time : 0.333
Using formula of rate in excel

Rate = 3.62%
Part C
Value of Firm = 7.793767 Million
Value of Debt=$ 4.794 Million
Value of Equity = $ 3 Million
Value of Debt at Maturity= $ 5 Million
Expected Price per share=3* Exp(0.03*0.3333)= 3.03
No of shares= $ 5/ 3.03= 1.650083 Million