**FM320/FM322**

**Exercises**

金融衍生品代写 The exercises marked by the symbol ★ will be covered in class. You should do the remaining exercises on your own.

The exercises marked by the symbol ★ will be covered in class. You should do the remaining exercises on your own.

**1 The Binomial Model and the FTAP**

#### 1.1

Below are some examples of a financial market with two dates and two states of the world. Asset prices and payoffs are given by (π, D). In each case, answer the following questions:

– Is there an arbitrage opportunity?

– If there is no arbitrage, what are the underlying state prices?

– Which securities can be priced by no-arbitrage?

– What is the no-arbitrage price of a derivative with contingent payoffs V_{u} and V_{d} in the two states?

#### 1.2 ★

Consider a two-date economy with three states of the world, and three securities traded at the initial date. The asset payoff matrix is:

The (i, j)-th element of D is the payoff of the j-th security in the i-th state of the world.

a. Are security markets complete? If not, is it possible to introduce another asset so that markets do become complete? Give an example of such an asset.

b. Characterize the set of arbitrage-free prices for the securities in D. In other words, find all the prices π for the assets such that there does not exist an arbitrage, given D. Give an example of a price vector in this set.

c. Provide an example of (positive) prices for the securities in D, such that there does exist an arbitrage. Exhibit an arbitrage in your example.

d. Suppose (D, π) does not admit arbitrage. Consider the following asset:

Show that d can be priced by no-arbitrage. Let π_{d} be the no-arbitrage price of d. Write down a formula for π_{d} in terms of π. Is it possible to express π_{d} as a risk-neutral expectation of the payoffs of d, discounted at the riskfree rate? If so, give an explicit expression for π_{d} in this form.

#### 1.3 ★ 金融衍生品代写

XYZ’s stock price is £100 and in each 3-month period will either increase by 25% (with probability 0.75) or fall by 20% (with probability 0.25). A 6-month call on XYZ stock has an exercise price of £90. The riskfree interest rate is 4% per annum, compounded quarterly.

a. Find the no-arbitrage price of the XYZ call without calculating a replicating portfolio strategy.

b. Derive the self-financing trading strategy that replicates the option. Verify that the initial cost of this strategy is equal to the no-arbitrage price of the option that you calculated in the previous part.

c. Explain intuitively why the option delta for the second 3-month period varies with the level of the stock price.

d. Calculate the expected rate of return on the option, under the risk-neutral measure, over the second 3-month period. Intuitively justify your answer.

#### 2.2 ★

Your cash position in millions of pounds, denoted by X, follows a Brownian motion with drift 0.5 per quarter and variance 4.0 per quarter. Let X_{0} denote your current cash position. What is the probability distribution of your cash position a year from now? How high does X_{0} have to be for you to have a less than 5% chance of a negative cash position after 1 year? Write your answer in terms of the standard normal cdf Φ, and then give a numerical answer using the norminv function in Excel.

### 3 The Black-Scholes Model 金融衍生品代写

#### 3.1 ★

Option strategies and the cost of these strategies in the BS model

In this question the riskfree rate is quoted as a continuously compounded rate.

##### a.

Entering a long straddle position involves buying a call and a put with the same strike price and expiration date.

(i) Represent graphically the payoff of a straddle as a function of the price of the underlying at expiration.

(ii) Suppose the current price of the underlying is £50, its annual volatility is 22%, and the riskfree rate is 5%. What is the cost of entering a straddle at strike price £50 and maturity 4 months?

##### b.

Entering a long butterfly position involves buying a call option with a low strike price, buying another call option with a high strike price, and selling two call options with an intermediate strike price.

(i) Represent graphically the payoff of a butterfly as a function of the under-lying price at expiration (suppose the intermediate strike price is exactly half way between the other two).

(ii) Suppose the current S&P500 index is at 1420, its annual volatility is 19%, its dividend yield is 0.3%, and the riskfree rate is 5%. What is the cost of entering a butterfly at strike prices 1400, 1420 and 1440 and maturity 2 months?

**c. 金融衍生品代写**

A bear spread involves buying a put option at a given strike price and selling one at a lower strike price

(i) Represent graphically the payoff of a bear spread as a function of the underlying price at expiration.

(ii) Suppose the dollar is currently at £0.51, its volatility is 15%, the UK riskfree rate is 5.25%, the US riskfree rate is 5%. You are based in London. What is the cost of entering a bear spread on the dollar with strike prices 0.51 and 0.41 and with a maturity of one year?

#### 3.2 ★

Suppose that the Black-Scholes assumptions are satisfied. In particular, there is a stock whose price follows a GBM with drift µ and volatility σ, and there is a money market account with a constant instantaneous riskfree rate r. The stock does not pay any dividends before T. Consider a derivative on the stock with payoff g(S_{T }) at time T, for some function g, whose price at time t ≤ T is given by F(S, t). Recall that the Black-Scholes PDE is

### 4 Hedging in Black-Scholes 金融衍生品代写

#### 4.1 ★

Answer the following questions in the context of the Black-Scholes model with a non-dividend paying underlying.

a. Recall that a long straddle is a long position in one call and one put with the same strike price and maturity.

(i) What is the delta of a long straddle position?

(ii) Now fix a strike price K. What combination of a call and a put with strike price K and the same maturity achieves a delta of zero?

b. Consider a trading strategy that is continuously rebalanced in order to keep it delta-neutral. What should be the expected return on this strategy? Justify your answer.

c. Is the value of delta for an at-the-money call above, below or equal to 1/2 (prior to expiration)?

d. Characterize analytically how the delta of a call option is affected by the volatility of the underlying. Derive a condition on S/PV(K) for which delta is increasing in the volatility parameter.

e. Consider a derivative with payoff g(ST ) at time T. Suppose the volatility parameter σ is 22%, and the riskfree rate r is 5%. The current value of the stock (at t = 0) is 100, and the derivative is worth 1.37, with a delta of 0.175, and a gamma of 0.017. What is its theta?

#### 4.2 ★ 金融衍生品代写

Delta-hedging

A US bank’s position in options on the euro has a delta of 30,000 and a gamma of −80,000. Explain how these numbers can be interpreted. The exchange rate (dollars per euro) is 0.90. What position should the US bank take in the spot foreign exchange market to make the overall position delta-neutral? After a short period of time, the euro appreciates to $0.93. Assuming the bank did set up a delta-neutral position, how is the value of its overall position affected by the exchange rate movement (use an approximation)? Estimate the new delta (of the original book of options). What additional trade is necessary to keep the position delta-neutral?

### 5 The FTAP and Risk-Neutral Pricing in Continuous Time

**6 Volatility 金融衍生品代写**

**6.1 ★ **

Implied volatility

a. The riskfree rate is 5% per annum, continuously compounded.

- QQR shares trade at £75. A call option on QQR with strike price £80 and maturity 4 months trades at £1.9982.
- XXY shares trade at £150. A call option on XXY with strike price £160 and maturity 4 months trades at £3.9964.

Compute the implied volatilities of the two calls. Prove that they must be equal.

b. The LSE.com stock trades at £50, and the riskfree rate is 5% per annum, continuously compounded. Consider two European put options on this stock expiring in 2 months. The first one, with strike price £50, trades at £1.4224; the second, with strike price £45, trades at £0.3215.

Compute the implied volatilities of the two puts. Comment on your finding.

**7 Exotic Options**

#### 7.1 ★

Barrier option in a binomial tree

Consider an underlying asset whose price evolves according to the binomial tree in Figure 1. The riskfree interest rate is 10% for each period (with no compounding within a period). Price an up-and-in European call option on this underlying with barrier 55 and strike price 40.

#### 7.2 ★

Barrier option

Consider a down-and-out European call option on a non-dividend paying stock. The option barrier H, the strike price K and the spot S are such that H = K and S>K. Assume that r = 0, i.e. you can ignore the time value of money. Also assume that the stock price follows a continuous-time process with continuous sample paths, but not necessarily geometric Brownian motion (this implies that if the stock price crosses the barrier H it must actually hit that barrier).

a. Suppose you sell the knock-out call and buy the underlying. In case the barrier is hit at time τ , what is the value of your position at τ? In case the barrier is not hit, what is the value of your position at expiration? Use a no-arbitrage argument to pin down the fair price of the knock-out call. Calculate its gamma and vega.

b. In reality, when the barrier is hit, you would typically not be able to sell the stock immediately at price K but would instead sell the stock at K^{−}< K, a phenomenon referred to as “slippage”. How does this consideration affect the fair price of the knock-out call?

c. From the strategy in part (a), deduce a replicating portfolio for the barrier option. What is the effect of slippage on the value of this portfolio relative to the value of the option?

**7.3**

Lookback option

Use a three-step binomial tree to price a lookback call option on a non-dividend paying stock. The time-to-maturity is 1 year, the initial stock price is £50, the annual stock volatility is 25%, and the riskfree rate is 5% (continuously compounded).

**8 Forwards and Futures 金融衍生品代写**

**8.1**

Forward contracts – pricing and marking-to-market

Consider a 6-month forward contract to deliver 100 shares of XXY company stock. The current price per share is £25 and XXY is expected to pay a £2 dividend per share in three months. The continuously compounded interest rate is 10% per annum.

a. What forward price (per share) should be stipulated in the contract?

b. Suppose you entered a long position at an earlier time (expiring 6 months from now) stipulating a forward price of £22. What is the marked-to-market value of your position?

c. Suppose the quoted forward price is actually £24.73. Describe a strategy to take advantage of the arbitrage opportunity.

#### 8.2 ★ 金融衍生品代写

Hedging with forward contracts

For this question assume that the interest rate is constant.

a. IBZ stock pays no dividend and its current price is £100. Its one-year fair forward price is £105. What is the value of a long position in a forward contract on IBZ stock expiring in a year, with forward price £100?

b. Consider a long forward position in a non-dividend-paying stock with forward price F_{0}and expiration date T. What is the value of this forward position at t ≤ T? What is the delta and gamma of this forward position?

c. Your book of derivatives on the non-dividend-paying stock TTW.com has a delta of 1000 and a gamma of −2000.

(i) Suppose forward contracts for delivery of 100 shares are traded. How can you make your position delta-neutral by trading in the forward market?

(ii) In the options market, an option on 100 TTW.com shares has a delta of 75 and a gamma of 20. What position do you need to take in this option and in the forward market to make your overall position both delta- and gamma-neutral?

#### 8.3 ★ 金融衍生品代写

Interest rates (per annum with continuous compounding) in the UK and in the US are constant at 4% and 1%, respectively. The GBP-USD exchange rate is currently $1.60 per pound.

a. What is the fair forward exchange rate on a contract expiring in 6 months?

b. Suppose you are based in the UK and you are short in a forward contract on the US dollar for delivery of $5m in 6 months at the exchange rate of $1.55 per pound. What is the marked-to-market value of your position?

c. Suppose that over the next 6 months the US dollar can go up by 2.5p or down by 2.5p, with equal probability. What is the fair price of an at-the-money asset-or-nothing digital call option contract on the US dollar, with notional amount of $100, and with time-to-expiration of 6 months?

**9 Introduction to Fixed Income and Linear Interest Rate Derivatives 金融衍生品代写**

**9.1 ★**

Yield-to-maturity, zero rates, forward rates, bond returns

In this question all interest rates are annual rates with annual compounding. The interest rates that you are asked to calculate should also be expressed as annual rates with annual compounding.

The price of a 1-year zero-coupon bond with face value £100 is £95.24. A 2-year bond with face value £100, annual coupon rate 8%, and annual coupon payment, has yield-to-maturity y = 6.92%.

a. Find y(0, 1) and y(0, 2), the zero rates for maturities 1 year and 2 years respectively.

b. A 2-year bond with face value £100, annual coupon rate 6.93%, and annual coupon payment has just been introduced. At what price should this bond trade?

c. Compute the one-year forward rate in one year.

d. Suppose the one-year zero rate prevailing after one year is 9.04%. What return would you have realized by investing in a 2-year zero-coupon bond for one year? What return would you have realized over 2 years by saving at the one-year rate and rolling over?

e. Suppose instead that the one-year zero rate prevailing after one year is 5%. What return would you have realized by investing in a 2-year zero-coupon bond for one year?

**9.2 ★ 金融衍生品代写**

In this question, the instantaneous riskfree rate r follows an arbitrary stochastic process. Consider a forward contract, entered into at date 0, to buy a non-dividend paying underlying at date T.

a. Find the marked-to-market value V_{t}of this contract at time t<T in terms of the spot price at date t, S_{t}, and the forward price at date 0, F_{0}.

b. Now suppose that S_{t}= S_{0}, i.e. the price of the underlying at date t is the same as its price at date 0. Find a condition on zero-coupon prices under which V_{t}= 0. Interpret.

**9.3 ★ 金融衍生品代写**

Marked-to-market value of a swap

You enter into a short position in a one-year swap with notional amount £100,000 and semi-annual coupon payments (i.e. there is one payment after six months and another one after a year). The yield curve is flat at 10%. Three months later, the yield curve is flat at 12%. Yields are quoted as annual rates compounded semiannually.

a. Calculate the marked-to-market value of your position (at the 3-month point) by valuing the fixed and floating rate bonds implicit in the swap.

b. Calculate the marked-to-market value of your position (at the 3-month point) by valuing the FRAs implicit in the swap.

**10 Interest Rate Options 金融衍生品代写**

**10.1**

The one-year interest rate (quoted as an annual rate with annual compounding) is currently 12%. A two-year zero-coupon bond with face value 100 trades at 81.25. Consider the binomial model where we assume that the value of this bond in a year will be either 95 or 90 with equal probability.

a. Without calculating a replicating portfolio, determine the no-arbitrage price of a European call option on the two-year bond, struck at 92 and expiring in a year.

b. Suppose the quoted price of the option differs from the one you found in part (a). Solve for an arbitrage strategy that you can implement in order to profit from the mispricing.

**10.2 ★ 金融衍生品代写**

All interest rates in this question are quoted as annual rates with annual compounding, and all zero-coupon bonds have face value £100. Consider the evolution of the one-year LIBOR rate. The current LIBOR (at t = 0) is L_{0} = 1.5%. The LIBOR rate in one year’s time (at t = 1) is either L_{u} = 1% or L_{d} = 2%, with equal probability.

The price at t = 0 of a two-year zero-coupon bond is P(0, 2) = £97.16.

**a. **

Price a European call option on the two-year zero-coupon bond with maturity at t = 1 and strike K = £98.52.

##### b. 金融衍生品代写

Consider a long position in a swap with annual payments based on the oneyear LIBOR rate, with swap rate 1.5%, notional £10,000, and two remaining payments (at t = 1 and t = 2). Calculate the value of this swap position at t = 0 using each of the following methods:

(i) By valuing the fixed and floating coupon bonds implicit in the swap.

(ii) By valuing the FRAs implicit in the swap.

(iii) By combining the long position in the 1.5% swap with a short position in the same swap but at the fair swap rate, and valuing the riskless cash flows that result from this portfolio.

(iv) Using risk-neutral valuation.

(v) By calculating a dynamic self-financing strategy, using zero-coupon bonds, that replicates the swap.

**c.**

Calculate the price at t = 0 of a floor with payments at t = 1 and t = 2 based on the one-year LIBOR rate, with strike 1.5%, and notional £10,000.

**d. **

Consider a receiver swaption with strike rate 1.5%. The underlying of the swaption is a two-year swap with annual payments and notional £10,000. The swaption expires at t = 0. What is its value at t = 0?

**Formula Sheet**

In the final exam you will be given the formula sheet on the next page. This does not mean that you will need to use all of these formulas.

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