Practice-problems - Fall Term Lecturer: Paymon Khorrami PDF

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Fall Term
Lecturer: Paymon Khorrami...

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Mathematics for Finance Practice Problems with Solutions Paymon Khorrami∗ December 3, 2019 Useful Mathematical Formulas. • Inverse of a 2 × 2 matrix: −1    1 a b d −b . = c d ad − bc −c a • Formula for geometric series: ∞ X

ρn =

n=0

1 , 1−ρ

if 0 < ρ < 1.

• Variance formula for the sum of two random variables X and Y , with a, b constants: Var(aX + bY ) = a2 Var(X) + b2 Var(Y ) + 2abCov(X, Y ). • Standard normal cumulative distribution function (cdf): Z x y2 1 Φ(x) := √ e− 2 dy, 2π −∞ i.e., the probability that a standard normal random variable realizes a value below x. Recall that “standard normal” means a normal random variable with mean 0 and variance 1. • Law of iterated expectations: Et [Et+s [X]] = Et [X]. • Itˆo’s lemma (where Bt is a one-dimensional Brownian motion, dXt = µ(t, Xt )dt + σ(t, Xt )dBt , and f (t, x) is a twice-differentiable function): df (t, Xt ) =

∗

h ∂f ∂t

+ µ(t, Xt )

∂f ∂f 1 ∂ 2f i + σ(t, Xt )2 2 dt + σ(t, Xt ) dBt . ∂x 2 ∂x ∂x

Please email me at [email protected] if there are any typos or mistakes.

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Questions. Question 1. Interest rate risk. The yield curve is upward-sloping on average, meaning long-term yields are typically higher than short-term yields. One explanation is that investors expect short-term yields to rise in the future. This is the view under the expectation hypothesis, for example. Another view is that long-term bonds are somehow riskier. (a) Suppose you have a two-year investment horizon. Which is riskier: buy and hold a two-year zero, or roll over two one-year zeros? Why? (b) Suppose you have a one-year investment horizon. Which is riskier: buy a two-year zero and sell after one year, or buy a one-year zero? Why? (1)

(c) Suppose investors tend to have one-year investment horizons. Suppose y t and ( 8%, with probability 21 (1) yt+1 = 12%, with probability 21 (2)

(1) so that E[yt+1] = 10%. Do you expect yt Justify your answer.

(2)

= 10%, yt

(2)

> 10%, or y t

= 10%

< 10%?

Question 2. Coupon bonds. (a) Suppose you want to price a coupon bond which pays a semiannual coupon of C = $5 and which matures 2 years from now. The face value of the bond is F = $100. Now suppose that I give you the prices of zero-coupon bonds that mature 0.5, 1, 1.5 and 2 years from now. Let these zero-coupon bond prices be denoted Zt,t+0.5 , Zt,t+1 , Zt,t+1.5 , and Zt,t+2 , respectively. Write down the formula for the coupon bond price. (b) Now suppose the yield curve is flat, with net (annualized) yield of 5%. What is the value of this coupon bond? (c) Now assume the bond pays annual coupons of c = $5 instead. What is the bond price now? (d) Now assume the bond pays annual coupons of C = $10 and has an infinite maturity (i.e., it continues to pay this coupon every year, starting next year, forever). What is the bond price now? Question 3. Insurance and the mean-variance frontier. Insurance is a risky asset that offers a negative expected return. For example, you pay more than 1% of your house value per year in order to insure against a fire that occurs with 1% probability. 2

(a) Suppose you pay exactly a 1% premium for this insurance. What is the expected return on buying this insurance? (b) Suppose the risk-free rate is 5%. Would a mean-variance efficient investor buy this insurance product? Question 4. One-period model I. Consider the following asset market with payoffs A and price vector S (the columns of A are the assets, and the rows represent states):     1 3 0 1    and S = 2 . A= 1 2 6 1 1 12 6 (a) Is this market complete? If there is any redundant asset, find it, and point it out. (b) Find the state prices in this market. (c) Is there any arbitrage in this market? If so, find an investment strategy that delivers the arbitrage. If not, explain why not. (d) Suppose we introduce another asset with payoffs   7  b = 1 . 0 Can this asset be perfectly hedged?

(e) What are the valid no-arbitrage prices for this asset (or unique price if there is only 1)? Question 5. One-period model II. Consider the following asset market with payoffs A and price vector S (the columns of A are the assets, and the rows represent states):     7 1 5 4 A = 2 2 0 and S = 1 . 7 1 5 2 (a) Is this market complete? If there is any redundant asset, find it, and point it out. (b) Find the state prices in this market. 3

(c) Is there any arbitrage in this market? If so, find an investment strategy that delivers the arbitrage. If not, explain why not. (d) Suppose we introduce a risk-free asset:   1  b = 1 . 1

Can this asset be perfectly hedged?

(e) What are the valid no-arbitrage prices for the riskless asset (or unique price if there is only 1)? Question 6. One-period model III. Consider the following asset market with payoffs A and price vector S (the columns of A are the assets, and the rows represent states):     3 1 0 4    A= 2 0 4 and S = 1 . 1 1 0 2 (a) Is this market complete? If there is any redundant asset, find it, and point it out. (b) Find the state prices in this market. (c) Is there any arbitrage in this market? If so, find an investment strategy that delivers the arbitrage. If not, explain why not.  ′ (d) Introduce a riskless asset with payoff b = 1 1 1 . What are the valid noarbitrage prices for the riskless asset (or unique price if there is only 1)? (e) Introduce a put option on the first asset with strike price 2.5. What are the valid no-arbitrage prices for this put option (or unique price if there is only 1)? Question 7. Option payoffs. Describe a set of options and any other assets you could trade to generate the following terminal payoffs (in solid black). State the type (call or put) for each option traded, the amounts of each (positive for long, negative for short), and the strike prices of each. Suppose the current stock price is 100. (a) Bear spread:

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payoff at time T

40

0

80

120

stock price at time T

i. Do you want the stock price to rise or fall if you take this position? ii. Is the option premium positive or negative for this strategy (positive means you have to pay to take the position; negative means you are paid )? (b) Long Straddle: payoff at time T

20

0 80

100

120

stock price at time T

i. Do you want the stock price to rise or fall if you take this position? ii. Is the option premium positive or negative for this strategy (positive means you have to pay to take the position; negative means you are paid )? Question 8. Binomial tree I.

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There is a market with a bond that has constant riskless rate 25% and a stock with S0 = 1 which can either double or halve each period, i.e., St+1 = 2St or 0.5St with probabilities p and 1− p, respectively. Let’s consider a model of 3 periods, i.e., we have times t = 0, 1, 2, 3. (a) Draw the stock price binomial tree and write the stock prices at each node. (b) Compute the time-0 prices of European call and put options with strike prices K = 1. Use any method you like. (c) Now we will increase the spread in returns. Suppose instead of doubling or halving, the stock triples or thirds each period, i.e., St+1 = 3St or (1/3)St each period. Compute the time-0 prices of European call and put options with strike prices K = 1. Use any method you like. Question 9. Binomial tree II. There is a market with a bond that has constant riskless rate 10%. A risk-neutral CEO is evaluating a biomedical project which can either succeed, partially succeed, or fail. If the project succeeds, it will pay off 100. It it partially succeeds, it will pay off 50. If the project fails, it will pay nothing. Each period, there are drug trials about whether the project will succeed or fail, and a positive trial arrives iid with probability p = 0.5. Let’s consider a model of 3 trials, i.e., we have times t = 0, 1, 2, 3. Only at the end (t = 3) does the CEO (and everyone else) find out for sure how successful the project will be. Assume that the project requires 3 positive trials to be fully successful, 2 out of 3 positive to be partially successful, and otherwise the project is a failure. (a) Draw a binomial tree representing the CEO’s valuation of the project at each time period and each “node” (nodes are outcomes of the signals). (b) At time t = 0, an investment firm gives the CEO a loan with a face value of 50 due at time t = 3, but which may also be paid back early (this is known as pre-payment or “callable debt”). If the CEO does not pay back by time t = 3, then the lender gets control of the firm’s assets. When will the CEO pay back the loan? Explain. (c) At t = 0, if the debt is priced by the CEO, how much must the CEO have borrowed to from the investment firm? Question 10. Trinomial tree (hard, but if you understand this, then binomial trees will be easy). There is a market with a bond that has constant riskless rate 0% and a stock with S0 = 1 which can either double, stay unchanged, or halve each period, i.e., St+1 = 2St , St+1 = St , or St+1 = 0.5St with probabilities p1 , p2 , and p3 = 1 − p1 − p2 , respectively. Let’s consider a model of 2 periods, i.e., we have times t = 0, 1, 2. 6

(a) Draw the stock price trinomial tree and write the stock prices at each node. (b) What are the probabilities of the final states, i.e., write expressions for P0 [S2 = x] for every x that can happen at t = 2. (c) Now add to the market a European call option with strike price K = 1 and maturity T = 2. Without doing any mathematics, explain the difficulty with finding no-arbitrage prices of this call option in this market. (d) Now suppose the European call option with price tree:        C1u = 1            C0 = 0.5 → C1n = 0.25              C1d = 0    

strike price K = 1 has the following  uu   C2 = 3 → C2un = 1   C ud = 0  2 nu   C2 = 1 → C2nn = 0   C nd = 0  2 du   C2 = 0 → C2dn = 0   C dd = 0 2

Are there any arbitrages in this market? If yes, find one. If not, explain why not.

(e) Now consider adding a European put option with strike price K = 1 and maturity T = 2. Provide the set of prices for this put that introduces no additional arbitrages relative to what already exists as of part (d). In other words, you may have already found arbitrages in the previous question, but price the put such that no additional arbitrages are introduced (if you found no arbitrages to this point, then find the no-arbitrage set of prices for this put). Question 11. Continuous-time integration I. Let Bt be a standard Brownian motion. In the notes (e.g., slides 36-37 of ContinuousTime.pdf), I used a direct argument to show that Z

T 0

1 1 Bt dBt = BT2 − T, 2 2

i.e, by taking a time-partition and making it finer and finer. This answer differentiates stochastic calculus from normal calculus, since Z T Z T 1 g (t)g ′ (t)dt = g(T )2 . g(t)dg(t) = 2 0 0 Please derive the first equation using Itˆo’s formula. 7

Question 12. Continuous-time integration II. Let us complicate the previous example a bit, with Bt still a standard BM. (a) Use Itˆo’s formula to show that Z Z T tdBt + 0

T

Bt dt = T BT . 0

(b) Show that the process Xt = Bt3 − 3tBt is a local martingale. (c) Write BT3 as a stochastic integral, i.e., write Z T 3 BT = vt dBt , 0

for some adapted stochastic process vt (“adapted” means vt is part of the time-t information set). Hint: rearrange the previous answer and integrate. Question 13. Local martingales. In everything that follows, Bt is a standard BM. Show that the following are local martingales (i.e., have no drift). Hint: Remember sines and cosines? No? Well here is what you need to know for this problem: d cos(x) d sin(x) = − sin(x). = cos(x) and dx dx (a) et/2 sin(Bt ). (b) et/2 cos(Bt ). (c) exp(−αBt − 12 α2 t) for α 6= 0. (d) B1,t B2,t , where B1 and B2 are both independent BMs. Question 14. Distance of N -dimensional BM (harder than exam questions). (1) are Consider an N -dimensional BM Bt = (Bt , Bt(2) , . . . , B t(N ) )′ , where each of B (i) t (i) independent BMs starting from B0 = 0. The distance of this vector from zero is given by the standard Euclidean norm kBt k, where N 1/2 X (i) 2 (B ) . kBt k = t i=1

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(a) Apply the multi-dimensional Itˆo’s formula to Yt := kBt k to find an SDE for Xt . (b) (very hard) Argue that N Z t (i) X Bs Zt := dBs(i) Y s 0 i=1

is a standard BM. Hint: to make this argument, you have to show that Z satisfies all the properties of a BM (see slide 20 of ContinuousTime.pdf). (c) Combine the first two parts to show that Yt follows the SDE dYt =

N −1 dt + dZt , 2Yt

where Z is a standard BM. Question 15. Quotient rule. If Xt and Yt are to Itˆo processes, i.e., dXt = µx,t dt + σx,t dBt dYt = µy,t dt + σy,t dBt , where Bt is a one-dimensional BM, show that hσ σy,t 2 i σy,t i d(Xt /Yt ) h µx,t µy,t σx,tσy,t x,t +( ) dt + = dBt − − − Xt Y t Yt Xt /Yt Xt Xt Yt Yt Hint: use the product rule from the homework.

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Answers. Question 1. Interest rate risk. The yield curve is upward-sloping on average, meaning long-term yields are typically higher than short-term yields. One explanation is that investors expect short-term yields to rise in the future. This is the view under the expectation hypothesis, for example. Another view is that long-term bonds are somehow riskier. (a) Suppose you have a two-year investment horizon. Why? Answer. Rolling over one-year zeros is riskier since the investor cannot know the rate he or she will get in the future. Buying two-year zero allows the investor to the 2-year interest rate, so you exactly know your cash flows by the end of your investment horizon. (b) Suppose you have a one-year investment horizon. Which is riskier: buy a two-year zero and sell after one year, or buy a one-year zero? Why? Answer. Buying the two-year zero is risky now, because you don’t know in advance what price you will get when you sell after one period. This is Buying a one-year zero means you know exactly your cash flows by the end of your investment horizon. (1)

(c) Suppose investors tend to have one-year investment horizons. Suppose y t and ( 8%, with probability 21 (1) yt+1 = 12%, with probability 21 (2)

(1) so that E[yt+1] = 10%. Do you expect yt Justify your answer.

(2)

= 10%, yt

(2)

> 10%, or y t

= 10%

< 10%?

(2) Answer. I expect y t > yt(1) = 10%. Investors have one-year horizons, so let’s (1) assume are OK with the risk embodied in the future one-year rate yt+1. Thus, to incentivize investors to invest in the two-year bond, it must offer a higher rate than the one-year rate, which would be the rate it would offer under the expectations hypothesis.

Question 2. Coupon bonds. (a) Suppose you want to price a coupon bond which pays a and which matures 2 years from now. The face value of the bond is F = $100. Now suppose that I give you the prices of zero-coupon bonds that mature 0.5, 1, 1.5 and 2 years from now. Let these zero-coupon bond prices be denoted Zt,t+0.5 , Zt,t+1 , Zt,t+1.5 , and Zt,t+2 , respectively. Write down the formula for the coupon bond price. 10

Answer. Literally use the formula from class: P = 5Zt,t+0.5 + 5Zt,t+1 + 5Zt,t+1.5 + 105Zt,t+2 . (b) Now suppose the yield curve is flat, with net (annualized) yield of 5%. What is the value of this coupon bond? Answer. First find out the values of the ZCBs, which are Zt,t+0.5 = 1.05−0.5 = 0.9759 Zt,t+1 = 1.05−1 = 0.9524 Zt,t+1.5 = 1.05−1.5 = 0.9294 Zt,t+2 = 1.05−2 = 0.9070 Then plug this into the formula from before: P = 5 × 0.9759 + 5 × 0.9524 + 5 × 0.9294 + 105 × 0.9070 = 109.5235. (c) Now assume the bond pays annual coupons of c = $5 instead. What is the bond price now? Answer. Similar method as before: P = 5 × Zt,t+1 + 105 × Zt,t+2 = 5 × 0.9524 + 105 × 0.9070 = 99.997. If you put this answer, that is fine, even though there is rounding error. Alternatively, you could have realized that this bond has the same coupon rate as the interest rate, meaning that its price should be $100, i.e., it is priced at par. You realize this by writing the pricing formula for an N -period coupon bond, with c := C/F the coupon rate: N

P =F

X 1 1 + c N (1 + y) (1 + y)j j=1

1 1 i 1 (1+y)N +1 1+y − c =F + c 1 1 (1 + y)N 1 − 1+y 1 − 1+y 1 i h 1 − (1+y) N 1 + c =F (1 + y)N y i hc  c 1 + 1− . =F y y (1 + y)N

h

Now, if c = y, which is the case in this question, then we have P = F . (d) Now assume the bond pays annual coupons of C = $10 and has an infinite maturity (i.e., it continues to pay this coupon every year, starting next year, forever). What is the bond price now? 11

Answer. Take the previous formula and take N → ∞ to find P =F

C c = y y

Thus, P = $10/0.05 = $200. Question 3. Insurance and the mean-variance frontier. For example, you pay more than 1% of your house value per year in order to insure against a fire that occurs with 1% probability. (a) Suppose you pay exactly a return on buying this insurance?

What is the expected

Answer. The expected return to insurance is 0%. To see this, first get the return. If the insurance does not pay off, the cash flow is 0; if the insurance pays off, it will deliver H, your house value. The . Thus, divide to get the return: ( 0, if the house survives R= 1/0.01, if the house burns down. The expectation of this is E(R) = 0.01 × (1/0.01) + 0.99 × 0 = 1. So the expected net return is E(R) − 1 = 0. (b) Suppose the risk-free rate is 5%. Would a mean-variance efficient investor buy this insurance product? Answer. Since Rf = 1.05, the insurance asset must be on the negative portion of the frontier. Thus, a mean-variance investor would not choose it (i.e., it is too expensive, thus earning a low expected return). Question 4. One-period model I. Consider the following asset market with payoffs A and price vector S (the columns of A are the assets, and the rows represent states):     1 3 0 1 A = 1 2 6  and S = 2 . 1 1 12 6 (a) Is this market complete? If there is any redundant asset, find it, and point it out.

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Answer. Clearly assets 1 and 2 are linearly independent. Let us check asset 3. 0 = x1 + 3x2 6 = x1 + 2x2 12 = x1 + x2 . Solving the last two of these equations for x1 and x2 we find x1 = 18 and x2 = −6. Plug these back into the first equation and you will see that it holds. Therefore, asset 3 can be replicated from 18 units of asset 1 and -6 units of asset 2. Since there are only 2 linearly independent assets but 3 states, the market is incomplete. (b) Find the state prices in this market. Answer. Set up the state-price equation A′ ψ = S: 1 = ψ1 + ψ2 + ψ3 2 = 3ψ1 + 2ψ2 + ψ3 6 = 6ψ2 + 12ψ3 . The third equation gives ψ2 = 1 − 2ψ3 . The first equation then gives ψ1 = ψ3 . Plugging everything into the second equation gives 3ψ1 + 2 − 4ψ1 + ψ1 = 2 which holds as an (i.e., regardless of the value of ψ1 ; as you can see, the ψ1 will be cancelled out in the left-hand-side). Thus, the set of valid state price vectors are   α ψ =  1 − 2α  , for any α ∈ R. α Because the market is incomplete, there is

(c) Is there any arbitrage in this market? If so, find an investment strategy that delivers the arbitrage. If not, explain why not. Answer. By picking α ∈ (0, 1/2), we have ψ ≫ 0. Therefore, there exists a strictly positive state price vector, which implies no arbitrage by our arbitrage theorem. (d) Suppose we introduce another asset with payoffs   7  b = 1 . 0 Can this asset be perfectly hedged?

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Answer. Try to hedge this asset (but only using the first two assets, since we know the third one is redundant): 7 = x1 + 3x2 1 = x1 + 2x2 0 = x1 + x2 . Solv...