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PSTAT  170  S21  QUIZ  1

INTRODUCTION TO MATHEMATICAL  FINANCE

JUNE 24,  2021

Problem 1. You have purchased one call option expiring in one year with a strike price of $40. The current price of the underlying is $30, the interest rate is zero, and the premium for the call option is $2.63.

(1) Draw the payof and P&L diagrams for the call option at expiration.

(2) What is the P&L on the option at expiration if the underlying is $57.50 (i.e. S1  = 57.5)?

Problem 2. The Second National Bank is ofering a two-year equity-linked CD prod- uct based on the S&P 500 index.  The current level of the S&P 500 index is 4,000. The customer will deposit $1,000 at time 0 and is guaranteed the return of their ini- tial investment at time 2.  The customer will participate in any upside appreciation in the S&P 500 index over the two-year period at a fraction R <  1 of the actual appreciation in the S&P 500 index over the two-year period.  Mathematically, the value of the customer’s investment at the end of the two-year period is

1, 000   [1 + R ( S24,000 1)+]

The continuously compounded interest rate is T = 0.04 per year.

If the current price of a call option on the S&P 500 index expiring at time 2 with strike price 4, 000 is $664.57, what is the fair participation rate that the Second National Bank should be ofering (i. e. what is R)?

Problem 3. You have been asked to create a synthetic short position in a forward contract that permits you to sell 10 units of the underlying one year from now at a price of $50 per unit.

(1) Describe the positions you need to take in call and put options to achieve the synthetic short forward position.

(2) If the underlying is selling for $48 today (i.e. S0  = 48), what is the cost of your synthetic short position?

Problem 4. Suppose European call and put prices for a given maturity and under- lying are given by the values in the following table.

Strike price K

41          45          52

Call price

11.45      10.07      8.07

Put price

7

9.80       12.10

Determine if there are any arbitrage opportunities and, if so, explain a spread position that produces an arbitrage proit.

Problem 5. A inancial expert is giving the following advice:

“It is rarely optimal to exercise an American call option on a non-dividend paying stock early. However, when the underlying stock hits an all time high and then does not go up much over the next month, early exercise is optimal because the chance that the stock goes up in price any more is very small. In this type of situation, you can do nothing but lose by not exercising.”

Do you agree with the advice of the inancial expert? Why?

Problem 6. A non-dividend paying stock currently sells for 40 and follows a single- period binomial model. One period from now, the stock will sell for either 70 or 20. A call option written on this stock expires in one period, has a strike price of 50, and currently sells for 8.

(1) What is the continuously compounded interest rate r for the period?

(2) Compute  and B, the stock and bond positions that replicate the payofs for the call option.


Solutions

Problem 1 . (1) The payof from the call option is the function

Payof = (S1  — 40)+

and the proit and loss (P&L) from the long call option investment is the function P&L = (S1  — 40)+ — 2.63

Payof

0

40                                                                         S

1

P & L

(2) The P&L on the option investment at expiration if the underlying is $57.50 is

(57.50 — 40)+  — 2.63 = $14.87.

Problem 2. The deposit of $1,000 must be used to provide for the guaranteed amount at time 2 and any remaining amount can be used to obtain upside equity exposure using at the money call options.

The bank needs to set aside

1000e 0.04(2)  = 923.12

in order to have the guaranteed amount of $1,000 available at time 2.

The remaining amount of the deposit is 1000 — 923.12 = 76.88 and this amount is used to purchase at the money call options expiring at time 2. The payof that can be generated from using the $76.88 to purchase call options on the S&P 500 index expiring at time 2 with strike price 4, 000 is equal to the number of call options that can be bought for $76.88 multiplied by the payof from one call option. Since one call option costs $664.57, the amount of $76.88 buys 76.88/664.57 calls.  Therefore, the payof that can be generated from using the $76.88 to purchase call options on the S&P 500 index expiring at time 2 with strike price 4, 000 is equal to

76.8866457 (S2   4, 000)+ .

We may rewrite this payof as

76.8866457 (S2  4, 000)+  = 76.8866457 4, 000 ( S24,000  1)+

= 1, 000 76.8866457 4 ( S24,000  1)+  = 1, 000 [0.4627 ( S24,000  1)+]

Therefore, the value of the customer’s investment at the end of the two-year period is

Guaranteed Amount + Equity Exposure = 1, 000 + 1, 000 [0.4627 ( S24,000 1)+] = 1, 000   [1 + 0.4627 ( S24,000 1)+]

Problem 3. (1) Start from the basic relationship between call and put option payofs to show

(ST  — K)+ + K = (K — ST )+ + ST

K — ST  = (K — ST )+ — (ST  — K)+ .

In particular

50 — S1  = (50 — S1 )+ — (S1  — 50)+

which shows that the payof from a short position in a forward contract on the under- lying expiring one year from now at a forward price of $50 is equal to the payof from a combined portfolio that is long one put option on the underlying expiring one year from now with a strike of $50 and short one call option on the underlying expiring one year from now with a strike of $50. Therefore, the payof from a short position in a forward contract that permits you to sell 10 units of the underlying one year from now at a price of $50 per unit can be written as

10  (50 — S1 ) = 10  (50 — S1 )+  — 10  (S1  — 50)+

so that the synthetic short forward contract position we are asked to create can be obtained by going long  (i.e. buy)  10 put options on the underlying expiring one year from now with a strike of $50 and shorting (i. e. selling) ten call options on the underlying expiring one year from now with a strike of $50.

(2) The payof from the short forward contract position is 10  (50 — S1 )

at time 1. Therefore, the cost of this payof is equal to the present value of 10  50 less the cost of 10 units of the underlying:

10  50e r  — 10  S0      [ 10  PV0,1 (50) — 10  S0]

= 10  50e r  — 10  48.

If you assumed the interest rate was zero you would have determined that the cost was 10  (50 — 48) = 20.

Note that in general, a payof of K — ST  at time T will cost e rT K — S0  so that if

K = erT S0  then the cost is zero.

Problem 4 . One must check the various arbitrage conditions from the text. Generally, we let K1  < K2  < K3  and we are considering calls and puts with the same underlying and same expiration - the only thing that is varying is the strike price.

The simple arbitrage conditions in price1  are

C(K1 )  C(K2 )   and   P(K1 )  P(K2 ).

1See (9.15) and (9.16) on page 281.

The arbitrage conditions in price change2  are

C(K1 ) — C(K2 )  K2  — K1      and   P(K2 ) — P(K1 )  K2  — K1 .

The arbitrage conditions in convexity3 are

C(K1 ) — C(K2 )     C(K2 ) — C(K3 )             P(K2 ) — P(K1 )     P(K3 ) — P(K2 )

K2  — K1                           K3  — K2                                       K2  — K1                          K3  — K2

We can conveniently check these conditions using an Excel table and comparing val- ues.  We discover that the convexity condition for put option prices is violated, re- sulting in:

P(45) — P(41)     P(52) — P(45)

45 — 41                52 — 45

As we saw in class notes and the text, when convexity condition is violated it means that the cost of the associated asymmetric butterly spread is negative. However, the payof from the the associated asymmetric butterly spread is always nonnegative. This is an arbitrage. To create the spread position we need to create the associated asymmetric butterly spread.

The associated asymmetric butterly spread is computed by irst determining α where

α = K3   K2  = 52 45 =  7

The relative option positions that are used to construct the associated asymmetric butterly spread are:

. Long α units of the K1  = 41 strike put options

. Short 1 unit of the K2  = 45 strike put option

. Long (1 — α) units of the K3  = 52 strike put options

The cost of this butterly spread is

αP(41) P(45) + (1 α)P(52) = (1 α) (P(52) P(45)) α (P(45) P(41))

= 15241 [(45 41) (P(52) P(45)) (52 45) (P(45) P(41))]

= (45  41)(52  45)5241 [15245 (P(52)  P(45))  14541 (P(45)  P(41))]

< 0

by (*).

2See (9.17) and (9.18) on page 282.

3See (9.19) and (9.20) on page 282.

Typically, whenever possible, we rescale the positions by the denominator of α so as to have integer positions in the options comprising the butterly spread. In this case the natural version of the butterly spread to work from is obtained by scaling all positions by 11 = 52 — 41 so that the payof from this rescaled butterly spread is

Payof = 7(41 — S)+  — 11(45 — S)+ + 4(52 — S)+

This payof function is easily veriied as non-negative by checking the cases 0  S < 41,    41  S < 45,    45  S < 52,    52  S

or by drawing the payof function. If we check the cases we have 

' ' '

Payof =

' ' '

(

0

4(52 — S)

— 11(45 — S) + 4(52 — S)

7(41 S) 11(45 S) + 4(52 S)

52  S

45  S < 52

41  S < 45

0  S < 41

which can be simpliied as 

' ' '

Payof =

' ' '

(

0

4(52 — S)

7(S — 41)

0

52  S

45  S < 52

41  S < 45

 S < 41

and is seen to be non-negative.  Therefore the associated butterly spread pays us money up front and has a non-negative cash low - that is arbitrage.

Note that the class notes gave general formulas for this type of problem.  In this

solution, direct calculations were used. Both approaches are acceptable.

Problem 5. We need to carefully read and interpret the“advice”that is given. The complete text of the advice given by the expert is:

“It is rarely optimal to exercise an American call option on a non-dividend paying stock early. However, when the underlying stock hits an all time high and then does not go up much over the next month, early exercise is optimal because the chance that the stock goes up in price any more is very small. In this type of situation, you can do nothing but lose by not exercising.”

We proved that it is never optimal to exercise an American call option on a non- dividend paying stock early so the statement“[i]t is rarely optimal to exercise an Amer- ican call option on a non-dividend paying stock early”is wrong.  We talked about the fact that this result does not say that the value of an American call option on a non- dividend paying stock will not decline.  What the result says is that the payof one obtains from exercising an American call option on a non-dividend paying stock is less than or equal to the market price of the American call option on a non-dividend paying stock. Therefore, one should sell the American call option on a non-dividend paying stock and not exercise it.

The statement“when the underlying stock hits an all time high and then does not go up much over the next month, ... the chance that the stock goes up in price any more is very small”may be true but is irrelevant to whether one should exericse the American call option.

If the expert had said:

“[w]hen the underlying stock hits an all time high and then does not go up much over the next month, the chance that the stock goes up in price any more is very small.  In this type of situation, a call option is not likely to go up much in value. Therefore, you may want to sell your position to realize the gains”

then this statement may make sense to someone long a call option.

The answer should be that we do not agree with the advice of the expert because the statement about exercising an American call option on a non-dividend paying stock early is incorrect and the expert is confusing the idea that a American call option investment is unlikely to gain in value and should therefore be sold with exercising an American call option.

Problem 6. The most direct approach to determine r is to work from the discounted expectation formula for the binomial model:

C = e rh  [p* Cu + (1 p* )Cd]

as presented in (10.6) page 300 of McDonald. In words, every security’s price must be the discounted risk-neutral expectation of its payofs. [This is not the fastest approach if we also need to compute the hedge portfolio.  We will present a second approach that is faster when the hedging portfolio is needed.]

The call option has the payofs max(0,S1  — 50), which means 20 in the upstate (i. e. . stock at time 1 is 70) and 0 in the downstate (i. e. . stock at time 1 is 20). We have the pair of equations, the irst for the stock and the second for the call option:

40 = e r  [70p* + 20(1 p* )]

8 = e r  [20p* + 0(1 p* )]  e rp*  = 8/20 = 0.4

From the second equation: 8 = e r  [20p* + 0(1 p* )]  rp*  = 8/20 = 0.4. The irst equation may now be used as:

40 = e r  [70p* + 20(1 p* )]

= e r  [20 + 50p* ] = 20e r + 50e rp*

= 20e r  + 50(0.4)  ⇐⇒ e r  = 40 50(0.4)20

⇐⇒ e r  = 1  ⇐⇒ r = 0.

Therefore r = 0.

To obtain  and B we setup the usual pair of equations for the call option replicating portfolio:

20 = 70 + Ber    70 + B

0 = 20 + Ber    20 + B

[er  = 1 since we have shown that T = 0.] Subtract the second equation from the irst

to obtain 50 = 20 =⇒  = 0.4 Therefore, B =   8. Therefore,

(,B) = (0.4,   8).

The faster approach is to write the replicating equation for the call option: 20 = 70 + Ber

0 = 20 + Ber

and upon subtracting the second equation from the irst equation we ind 50  = 20  =⇒  = 0.4. The second equation shows B =   e r 20 =   8e r . Even though we do not know T we can write the price of the call option as

8 = Price of Call Option = Cost of Replicating Portfolio = S0 + B = 0.4(40) + (  8e r )

=⇒ 8 = 16  8e r    ⇐⇒ e r  = 1  ⇐⇒ T = 0.

Therefore T = 0 and we have already computed  and B as a function of T so that (,B) = (0.4,   8e r ) = (0.4,   8e0 ) = (0.4,   8).