Question

# Interest rates are expressed as annualized rates for the term specified. Report your interest rate answers as fractional numbers like 0.11 for 11% per year.

Problem B. The price of a stock is currently 69. The stock price by the end of the next three-month period is expected to be up by 10 percent or down by 10 percent. The risk-free interest rate is 7.8 percent per annum with continuous compounding. What is the current value of a three-month American call option with strike price of 64 using a single-step binomial tree? How will you trade involving one call option to make arbitrage profits if the call option’s current market price is 39.5.

25. What is the value of p? What are the two values (cash flows) of the call option at maturity from top to bottom of the tree?

26. Cash flows at the top of the tree.

27. Cash flows at the bottom of the tree.

28. What is the current fair value of the call option?

29. What is the value of delta at time 0?

30. Write 1 if your answer is to take a long position in the call option and 0 if it is to take a short position in the call option in an arbitrage trading strategy at time zero.

31. Write the net cash position (if borrowed write with a negative sign) at time zero. Write the two net cash positions from top to bottom at the end of one step (maturity) in your trading strategy:

32.

33.

34. Write the net cash made as of maturity from the arbitrage trading strategy.

CAN YOU PLEASE SHOW WORK.

Answer 1

Let S0 be initial stock price, S0=69 and S1 be stock price after one period

Step 1:

We shall first determine the expected stock prices. The prices are expected to move up or down by 10%

Let us denote upward movement by u as a fraction of S0

Let upward possible stock price, S1u = S0 * (1+10%) = S0 * 1.1 = S0 * u

where u = 1.1

S1u = 69 *1.1 = 75.9

Similarly, downward possible price, S1d = S0(1-0.1) = S0 * .9 = S0*d

where d= 0.9 and hence S1d = 62.1

The stock price tree is shown below:

	S1u=75.9
S0=69
	S1d=62.1

Step 2: Calculate call expiration prices

The value of call option can be given by formula: MAX (expiry_price - strike price, 0)

The above formula says if the expiry price of a stock is greater than strike price, the value of call at expiry will be the difference between the two prices else 0. Based on this, the call expiration price tree is drawn below:

Time=0	Time=1
	Cu=MAX(75.9-64,0)
C
	Cd=MAX(62.1-64,0)

In terms of values,

Time=0	Time=1
	11.9
C
	0

Step 3:Calculate call prices

We can now calculate the call prices using risk-neutral probabilies

The call price can be calculated as the weighted average of expected call prices calculated above with the weights being risk-neutral probabilities and discounting it to time 0.

The risk neutral probability can be calculated using the following formula (its derivation beyond scope of this Qn)

p = (1+r - d) / (u -d)

where u, d are up and down factors already calculated and r is riskfree interest rate per period

p = ( 1+0.078/4 - 0.9) / (1.1 - 0.9) = 0.5975

Call prices at time = 0 is weighted average of call prices at time=1 using probability weights discounted to current time 0

C = (p * Cu + (1-p) * Cd) / (1+r)

C = (0.5975 * 11.9 + (1-0.5975)*(0)) / (1+0.078/4) = 6.9742

So, the call price is 6.9742

Arbitrage Strategy:

This involves creating a hedge portfolio. A hedge portfolio involves buying a proportion of stock and writing a call in such a way that risk free return is achieved.

The proportion of stock to be purchased can be calculated using hedge ratio.

Hedge ratio is given by, h = (Cu - Cd) / (Su-Sd) = (11.9 -0)/(75.9-62.1) = 0.86232

So, the strategy will be to purchase 0.86232 shares and write one call option.

(Since market call option price is greater than theoretical price, we sell (i.e. write) call and purchase stock)

Value of portfolio at time t=0; V = h * S0 - C

V = 0.86232 * 69 - 39.5 = 20

(Actually, stock is purchased and call written, the 20 given above is cash outflow)

On expiration, the portfolio will be worth as follows:

If stock price is 75.9,

V1u = stock value + call value = 0.86232*75.9 -(11.9) = 53.55

if the price is 62.1,

V1d = stock value + call value = 0.86232*62.1 -(0) = 53.55

Thus, the value of the position will be $53.55 irrespective of the stock price on maturity expiration date while the portoflio value was created at value of 20. Hence there is a risk free profit of (53.33-20) over a period of 3 months.

Answers:

25) The value of p is calculated as above as 0.5975

26) Cash flows in terms of stock at expiration at top of tree already drawn above

27) Cash flows of call expiration are already given above

28) Current value of the call option is 6.9742

29) Value of delta

delta, h = (Cu - Cd) / (Su-Sd) = (11.9 -0)/(75.9-62.1) = 0.86232

30) answer = 0

(we have to write a call and buy h or delta shares of stock)

31) At time zero, stock is purchased at $69 for quantity of 0.86232 and call is written, Hence there is an outflow of $59.5 and inflow of $39.5

So, Net cash position = -20

At time t=1, if the stock price is 75.9, stock will be sold and cash received will be +65.45 but there willl be a loss of 11.9 in short call which will be exercise at strike price of 64.

So, net cash position = 65.45-11.9 = +53.55

If stock price is 62.1, stock will be sold and cash received will be 53.55 and call not be exercised by the buyer.

Hence, net cash position = +53.33

Time=0	Time=1
	53.55
-20
	53.55

The above will be cash flow position at time = 1.

Net cash position at time t =1 are