Price Continuous Compounding of a European Option

Problem 1

In these problems, $n$ refers to the number of binomial periods. Assume all rates are continuously compounded unless the problem explicitly states otherwise.
Let $S=\$ 100, K=\$ 105, r=8 \%, T=0.5,$ and $\delta=0 .$ Let $u=1.3, d=0.8$ and $n=1$
a. What are the premium, $\Delta$, and $B$ for a European call?
b. What are the premium, $\Delta$, and $B$ for a European put?

Problem 2

In these problems, $n$ refers to the number of binomial periods. Assume all rates are continuously compounded unless the problem explicitly states otherwise.
Let $S=\$ 100, K=\$ 95, r=8 \%, T=0.5,$ and $\delta=0 .$ Let $u=1.3, d=0.8$ and $n=1$
a. Verify that the price of a European call is $\$ 16.196$
b. Suppose you observe a call price of $\$ 17 .$ What is the arbitrage?
c. Suppose you observe a call price of $\$ 15.50 .$ What is the arbitrage?

Problem 3

In these problems, $n$ refers to the number of binomial periods. Assume all rates are continuously compounded unless the problem explicitly states otherwise.
Let $S=\$ 100, K=\$ 95, r=8 \%, T=0.5,$ and $\delta=0 .$ Let $u=1.3, d=0.8$
and $n=1$
a. Verify that the price of a European put is $\$ 7.471$
b. Suppose you observe a put price of $\$ 8 .$ What is the arbitrage?
c. Suppose you observe a put price of $\$ 6 .$ What is the arbitrage?

Problem 4

In these problems, $n$ refers to the number of binomial periods. Assume all rates are continuously compounded unless the problem explicitly states otherwise.
Let $S=\$ 100, K=\$ 95, \sigma=30 \%, r=8 \%, T=1,$ and $\delta=0 .$ Let $u=1.3$
$d=0.8,$ and $n=2 .$ Construct the binomial tree for a call option. At each node provide the premium, $\Delta,$ and $B$

Problem 5

In these problems, $n$ refers to the number of binomial periods. Assume all rates are continuously compounded unless the problem explicitly states otherwise.
Repeat the option price calculation in the previous question for stock prices of $\$ 80,590, \$ 110, \$ 120,$ and $\$ 130,$ keeping everything else fixed. What happens to the initial option $\Delta$ as the stock price increases?

Problem 6

In these problems, $n$ refers to the number of binomial periods. Assume all rates are continuously compounded unless the problem explicitly states otherwise.
Let $S=\$ 100, K=\$ 95, \sigma=30 \%, r=8 \%, T=1,$ and $\delta=0 .$ Let $u=1.3$
$d=0.8,$ and $n=2 .$ Construct the binomial tree for a European put option. At each node provide the premium, $\Delta,$ and $B$

Problem 7

In these problems, $n$ refers to the number of binomial periods. Assume all rates are continuously compounded unless the problem explicitly states otherwise.
Repeat the option price calculation in the previous question for stock prices of $\$ 80, \$ 90, \$ 110, \$ 120,$ and $\$ 130,$ keeping everything else fixed. What happens to the inital put $\Delta$ as the stock price increases?

Problem 8

In these problems, $n$ refers to the number of binomial periods. Assume all rates are continuously compounded unless the problem explicitly states otherwise.
Let $S=\$ 100, K=\$ 95, \sigma=30 \%, r=8 \%, T=1,$ and $\delta=0 .$ Let $u=1.3$
$d=0.8,$ and $n=2 .$ Construct the binomial tree for an American put option. At each node provide the premium, $\Delta,$ and $B$

Problem 9

In these problems, $n$ refers to the number of binomial periods. Assume all rates are continuously compounded unless the problem explicitly states otherwise.
Suppose $S_{0}=\$ 100, K=\$ 50, r=7.696 \%$ (continuously compounded), $\delta=0$ and $T=1$
a. Suppose that for $h=1,$ we have $u=1.2$ and $d=1.05 .$ What is the binomial option price for a call option that lives one period? Is there any problem with having $d > 1 ?$
b. Suppose now that $u=1.4$ and $d=0.6 .$ Before computing the option price, what is your guess about how it will change from your previous answer? Does it change? How do you account for the result? Interpret your answer using put-call parity.
c. Now let $u=1.4$ and $d=0.4 .$ How do you think the call option price will change from (a)? How does it change? How do you account for this? Use put-call parity to explain your answer.

Problem 10

In these problems, $n$ refers to the number of binomial periods. Assume all rates are continuously compounded unless the problem explicitly states otherwise.
Let $S=\$ 100, K=\$ 95, r=8 \%$ (continuously compounded), $\sigma=30 \%$ $8=0, T=1$ year, and $n=3$
a. Verify that the binomial option price for an American call option is $\$ 18.283 .$ Verify that there is never early exercise; hence, a European call would have the same price.
b. Show that the binomial option price for a European put option is $\$ 5.979$ Verify that put-call parity is satisfied.
c. Verify that the price of an American put is $\$ 6.678$

Problem 11

In these problems, $n$ refers to the number of binomial periods. Assume all rates are continuously compounded unless the problem explicitly states otherwise.
Repeat the previous problem assuming that the stock pays a continuous dividend of $8 \%$ per year (continuously compounded). Calculate the prices of the American and European puts and calls. Which options are early-exercised?

Problem 12

In these problems, $n$ refers to the number of binomial periods. Assume all rates are continuously compounded unless the problem explicitly states otherwise.
Let $S=\$ 40, K=\$ 40, r=8 \%$ (continuously compounded), $\sigma=30 \%, \delta=0$ $T=0.5$ year, and $n=2$
a. Construct the binomial tree for the stock. What are $u$ and $d ?$
b. Show that the call price is $\$ 4.110$.
c. Compute the prices of American and European puts.

Problem 13

Use the same data as in the previous problem, only suppose that the call price is $\$ 5$ instead of $\$ 4.110$
a. At time $0,$ assume you write the option and form the replicating portfolio to offset the written option. What is the replicating portfolio and what are the net cash flows from selling the overpriced call and buying the synthetic equivalent?
b. What are the cash flows in the next binomial period (3 months later) if the call at that time is fairly priced and you liquidate the position? What would you do if the option continues to be overpriced the next period?
c. What would you do if the option is underpriced the next period?

Problem 14

In these problems, $n$ refers to the number of binomial periods. Assume all rates are continuously compounded unless the problem explicitly states otherwise.
Suppose that the exchange rate is $\$ 0.92 / € .$ Let $r_{5}=4 \%,$ and $r_{e}=3 \%, u=1.2$ $d=0.9, T=0.75, n=3,$ and $K=\$ 0.85$
a. What is the price of a 9 -month European call?
b. What is the price of a 9 -month American call?

Problem 15

Use the same inputs as in the previous problem, except that $K=\$ 1.00$.
a. What is the price of a 9 -month European put?
b. What is the price of a 9 -month American put?

Problem 16

Suppose that the exchange rate is 1 dollar for 120 yen. The dollar interest rate is $5 \%$ (continuously compounded) and the yen rate is $1 \%$ (continuously compounded). Consider an at-the-money American dollar call that is yendenominated (i.e., the call permits you to buy 1 dollar for 120 yen). The option has 1 year to expiration and the exchange rate volatility is $10 \%$. Let $n=3$
a. What is the price of a European call? An American call?
b. What is the price of a European put? An American put?
c. How do you account for the pattern of early exercise across the two options?

Problem 17

An option has a gold futures contract as the underlying asset. The current 1 year gold futures price is $\$ 300 / 0 z$, the strike price is $\$ 290,$ the risk-free rate is $6 \%,$ volatility is $10 \%,$ and time to expiration is 1 year. Suppose $n=1 .$ What is the price of a call option on gold? What is the replicating portfolio for the call option? Evaluate the statement: "Replicating a call option always entails borrowing to buy.the underlying asset."

Problem 18

Suppose the S\&P 500 futures price is $1000, \sigma=30 \%, r=5 \%, \delta=5 \%, T=1$ and $n=3$
a. What are the prices of European calls and puts for $K=\$ 1000 ?$ Why do you find the prices to be equal?
b. What are the prices of American calls and puts for $K=\$ 1000 ?$
c. What are the time- 0 replicating portfolios for the European call and put?

Problem 19

For a stock index, $S=\$ 100, \sigma=30 \%, r=5 \%, \delta=3 \%,$ and $T=3 .$ Let $n=3$
a. What is the price of a European call option with a strike of $\$ 95 ?$
b. What is the price of a European put option with a strike of $\$ 95 ?$
c. Now let $S=\$ 95, K=\$ 100, \sigma=30 \%, r=3 \%,$ and $\delta=5 \% .$ (You have exchanged values for the stock price and strike price and for the interest rate and dividend yield.) Value both options again. What do you notice?

Problem 20

Repeat the previous problem calculating prices for American options instead of European. What happens?

Problem 21

Suppose that $u < e^{(r-s) h},$ Show that there is an arbitrage opportunity. Now suppose that $d > e^{(r-5) !}$. Show again that there is an arbitrage opportunity.

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Source: https://www.numerade.com/books/chapter/binomial-option-pricing-i/