FI 478 Final Exam
S0= 100 -> Su=120, Sd=80; Call with K=110 has payoffs of (up, down)
(10,0)
What is NOT true about equity options
----->they typically have up to five expirations and strike prices options have large trading costs options are a zero sum game options provide high leverage
Portfolio consists of 5 calls with delta call =0.4 and 4 puts with delta put =-0.6. What's the overall delta of this portfolio?
-0.4
You buy a 10-year swap IR, paying a swap fixed rate of 3.0% and having a notional of $4m. If LIBOR turns out to be 2.5% in next quarter, then you receive from the swap dealer at that date (net $ amount)...
-5,000 Notional∗(LIBOR-Fixed)∗per quarter 4,000,000*(2.5%-3%)*(1/4)=-5,000
Stock price at option expiration is 105. Put strike price is 100. Payoff of this put at expiration....
0
S0= 100 -> Su=120, Sd=80; Rf=0; What the risk-neutral probability (of up move)?
0.5
Stock price is 100 right now, and it can increase to 120 or drop to 80. Risk free rate is zero. Compute the risk-neutral probability of 100->120?
0.5 u=120/100=1.2, d=80/100=0.8 p=(1+r-d)/(u-d)=(1+0-0.8)/(1.2-0.8)=0.5
What is the price of a put with strike K=100 and expiration T=1 year? You simulate 5 stock paths that produce the following prices at option expiration, S(T)=110, 96, 102, 99, 100. Risk-free rate rf=10%. Hint: use the Monte Carlo method.
0.9 1. Compute option payoff for each stock path: P(T)=0, 4, 0, 1, 0. 2. Average payoff E(P(T))=(0+4+0+1+0)/5=1 3. Option price P=1/(1+rf)^T=1/1.1=0.91
Eurodollar future with delivery in September 2021 trades for "99.815" The September 2021 FRA (which the price for forward on 90-day LIBOR) is ...
100-99.815=0.185%
You put a K= 100 call for C0= 3. S0 = 95. What should be St, for you to break even?
103
You want to hedge a $2 million portfolio that has a 1.2 beta against market risk with E-mini futures. Assume that S&P500 is at 4,000 and E-mini contract multiplier is $50. How many contracts do you need to sell?
12 Your effective exposure to S&P is $2m*1.2=$2.4m. Each contract's exposure is 4000*50=$0.2m. You need to sell 2.4/0.2=12 contracts
S0= 100 -> Su=120, Sd=80; Rf=0; Find price of a call (K=90). Hint use RN probability
15
Stock price is 100 right now, and it can increase to 120 or drop to 80. Risk free rate is zero. Using the risk-neutral probability from the previous question or other methods, find price of a call with strike 90.
15 Call has payoffs of 30 (120-90) and 0 -> C=(0.5*30+0.5*0)/1=15
You buy a call with K = 50 and T= 1m for C0 =2. ST increases to 55. What's your return from the call?
150%
What is the delta of a portfolio that consists of {two calls with delta 0.5, five puts with delta -0.2, and two shares of the underlying stock}?
2 Delta=Sum of position deltas=2*0.5+5*(-0.2)+2*1=2
A call with strike K = 80. Call price C0 = 5. Stock price S0= 82. What's the time value of this call?
3
So = 100. You buy a put with K = 95 for C1 = 2. St = 90. Your net profit?
3
You buy a call (K=50) for C=1 and a put (K=55) for P=6. Stock [rice drops to S(T)=45. What's your net profit (P&L)?
3
What is the time value of a call with a strike of $40 expiring in one month. If the stock price is $50 right now, and the call trades for $13.
3 13-(50-40)=Price - Intrinsic Value
You pay C(0)=$2 to buy a call with a strike of K=$50 that expires in a month. The underlying stock price was S(0)=$45 then you bought the call, but it increased to S(T)=$55 at the option expiration. What is your net profit (P&L)?
3 P&L (for a long call) =Payoff - Price = max(S(T)-K,0)-C(0)=(55-50)-2=3
Your equity portfolio has a 2% alpha and a 0.5 beta. Expected return on S&P 500 is 5%. Risk free rate is 1%. What is the expected return on this portfolio after you hedge away systematic (beta) risk?
3% After you hedge beta, portfolio beta becomes zero, and expected return = rf + alpha + beta (ER_S&P)= 1%+ 2% + 0*5%=3%
C0=5, K=90, S0=91, rf=0, T=1. Compute put (K=90,T=1) price?
4
Option payoff for 5 simulated stock paths are 0, 10, 5, 6, 4. T=1. Rf=20%. Option price?
4
S0= 100 -> Su=120, Sd=80; Call with K=110, if prices of simple securities are U(1,0)=0.4, D(0,1)=0.5?
4
So= 100, St = 110, K=105. Call price C=1. Compute net profit (P&L) of long call at T
4
A call with a strike of 50 trades for 5. The current stock price is 52. Risk free rate is zero. The stock pays a $1 dividend during the life of the option. What is the price of the put with the same strike and expiration?
4 C-P=S-PV(K)-PV(Div) -> P=5-52+50+1=4
You want to buy a six-month S&P 500 forward. S&P is at 4,000 right now. Risk free rate is 5%. S&P's annual dividend yield is 2%. What is a fair forward price?
4060 F0=S0*(1+r-d)^T=4000*(1+5%-2%)^0.5=4059.56 See slides 20-23 +r because we have to pay interest if we borrow to pay the spot in "cash and carry" -d because the spot index will pay us dividends, which reduce our costs
You pay P(0)=$2 to buy a put with a strike of K=$50 that expires in a month. What is the underlying price at which you break even?
48 P&L (S*(t)) = 0 for the break-even price S*(T), thus 0=max(K-S*(T),0)-P(0)=(50-S*)-2 -> S*=48
Stock price is 100, strike is 100, N(d1)= 0.55, N(d2)= 0.5, rf=0. Option price?
5
Suppose rf = 2.5%, the index value is S0 = 4,000, and futures price for 1-year delivery is F0 = 4,000 × 1.025 = 4,100. You hold $1 million of market index. How many futures contracts should you sell to hedge?
5 E-mini contract multiplier is $50. Thus, each contract gives you 4000*50=$200,000 exposure. $1m/$0.2m=5 contracts.
You invest $50 into call options. The call has a strike K=$100. Its current price is C(0)=$5. How much money will you make (net profit), if the stock price increases to $110 at the option expiration.
50 P&L per option = (110-100)-5=$5; with $50 you buy 10 calls, so that you total P&L is 10*5=$50
Stock price can only move up or down until option expiration (two outcomes). Prices of up (down) securities that pay $1 only in up (down) scenario are $0.4 ($0.5). Risk free rate is zero. What is the price of call with payoff 4 (10) in up (down) scenarios?
6.5 E(C(T))=4*Down+10*Up=4*0.4+10*0.5=6.6 Risk free rate is zero, so no need to compute present value in this case, C=E(C(T))/(1+rf)^T =E(C(T))/1=6
In binomial model, good/bad state prices: G(1,0)=0.4, B(0,1) =0.5. A call pays 10 & 6 in good/bad states. Call price?
7
Risk-neutral probability of good (bad) state is 0.4 (0.6). A call pays 10 (5) in good (bad) state. Rf=0. Call price?
7
A call option on Tesla stock is ... and ...
American, settled by delivery American = can exercise at any time; European = can exercise only at expiration Cash settled = only net payoff (e.g., S-K); Settled by delivery = a long call gets the underlying asset in exchange for cash $K
C0 = 5, P0 = 7, K=90, S0=90, rf=0, T=1. How take advantage of the arbitrage?
Buy call, sell put, sell stock
E-mini S&P future that expires in six months trades at 4010. S&P 500 index is at 4000 right now. Risk-free rate is 1%. Compute the dividend yield (% per year) implied by the futures price.
F*(T) = S0 (1 + r - d)^T -> solve for d -> d=1+r-(F/S0)^(1/T)=1+1%-(4010/4000)^2=0.5%
How to compute FRAs for long T (>10Y)?
Forward Treasury curve + TED spread
A firm can issue one of the listed products and convert them into the floating rate using IR swaps (LIBOR for 3.7% fixed). What is the lowest floating rate that the firm can get? Fixed rate note: 4% Simple FRN: L + 0.5% Inverse floater: 7.6% - L (e.g., if we can pay only L + 0.1% that would be great, but can we obtain such a low rate?)
L + 0.2% Fixed rate note: 4% + swap = 4% + (L - 3.7%) = L + 0.3% Simple FRN: L + 0.5% - already a floating rate Inverse floater: 7.6% - L + 2 swaps = (7.6% - L)+2*(L - 3.7%) = L + 0.2% -> the lower rate!
Why trading on insider info in the options market is a bad idea?
Large option trades are easy to spot
You sell a forward on gold with a forward price F(0)=$1750. Spot gold is traded at S(0)=$1725. Gold price at expiration is S(T)=$1800 at expiration. Your payoff is (you receive) ...
The buyer's payoff is S(T)-F=1800-1750=50. You are on the other side and must pay this 50. The seller's payoff is F-S(T)=-50. S(0) is needed to compute F0 but is irrelevant for forward payoff
An option can be perfectly replicated in a binomial model?
True With the stock and bond we can replicate solve for G(0,1) and B(1,0) securities and thus replicate any option (can be represented with G and B)
Volatility futures that expire in three months have a price of 22.5%. Spot volatility right now is 18%. Which of these is a better forecast of spot volatility in three months.
Volatility futures price Cash-and-carry arbitrage is not possible because spot volatility is not tradable and cannot be stored. Thus, futures help predict future value of the spot in this case.
What does the classic PC-parity (S=C-P+PV(K)) NOT require?
Volatility is constant
How should you trade in the stock to delta hedge a long position of four puts with delta of -0.5 (each)?
buy 2 shares Because the total put delta is 4*(-0.5)=-2, need to buy two shares to make it zero
An insurance company holds a $100m bond portfolio with average duration of 10 years, and thus receives a fixed coupon. The management is afraid of interest rate risk. How should the company use IR swap to hedge the risk.
buy a swap Fixed + Swap = Fixed + (Libor - Fixed) = LIBOR -> and LIBOR will change with the level of interest rate
A pension fund is long fixed-rate bonds. To convert from fix to floating rate...
buy a swap (S=L-F)
A straddle...
buy call, buy put
How can Southwest Airlines hedge its jet fuel costs with crude oil futures?
buy futures Oil and jet fuel prices are highly correlated. Southwest should buy a forward/futures because it want to fix a price of its fuel for future delivery. Southwest should buy a forward/futures because it want to fix the price of future fuel purchases at today's prices.
A put trades for $5, but the put call parity implies that this put should trade for $6. How to profit from this arbitrage opportunity?
buy put, sell call, buy stock C-P=S-PV(K) -> P=C-S+PV(K) if P<C-S+PV(K) , the put is cheap -> buy put, sell the synthetic put: -(C-S+PV(K)) -> sell call, buy stock, sell bond
If stock price decreases, then
call decreases and put increases
If volatility increases, then
call increases and put increases
You buy a put to bet that the stock price will...
decrease
You are short an IR swap (i.e., you pay LIBOR in exchange for fixed rate). If interest rates increase, the value (P&L, cashflows) on your swap ...
decrease Short Swap = Fixed - Libor (i.e., receive fixed, pay LIBOR). If rates go up, so does LIBOR, but the fixed amount ("coupon") is unchanged. Thus, "unchanged" - "increase" = "decrease"
If you buy a put, you bet that stock price will ... and volatility will ...
decrease, increase Put price will increase if stock price drops or/and if volatility increases
Which market is the largest?
derivatives
According to the BlackScholes model, implied volatility surface should be..,
flat
Corporations like Google primarily use derivatives to ...
hedge
If gold storage costs increase (e.g., because of limited storage capacity), then gold futures price will ...
increase F*(T) = S0 (1 + r + c)^T. F will increase if c increases.
Your option portfolio has a delta of 5. Stock price increases by 0.02. The value of your portfolio will....
increase by 0.1
A protective put consists of long index and ...
long OTM put long OTM put protects that index portfolio from market crashes by limiting loses.
A call with a larger strike has a ... price (compared to another put with the same expiration)
lower Call's payoff (and thus price) negatively depends on the strike: S(T)-K
What is the main way that corporation increase value by hedging with derivatives?
lower the discount rate Hedging makes the stock less risky -> lower beta -> lower required rate of return
What is the cash flows (how much do you pay) at the time you buy a futures or forward contract
nothing forward is like a bet ("I bet that Lions will win the next SB") so no money changes hands at the time of a trade (except for small collateral-margin to insure the parties do not disappear ). All payments occur at contract expiration.
Which equity portfolio risks can be eliminated by hedging with index futures?
only systematic/beta We can make beta zero with hedging but cannot hedge against stock-specific events that affect our portfolio.
What is NOT required by the put-call parity?
options must be cash settled
Derivatives are popular tool for hedging and speculation because they ...
provide high leverage
What is the relation between risk-free rate (rf) and the expected return of the stock market (ERm) in the risk neutral economy (under risk neutral probability)?
rf = ERm The expected returns reflect that the risk-neutral investor only cares about return and is indifferent between investing in the stock versus the risk-free bond
A covered call = buy stock+
sell call
The oil forward is traded at $65 while its fair price (or parity value) is $64. To take advantage of the arbitrage, you should ...
sell forward and buy spot oil
What is NOT a way to earn risk premium in S&P 500 index options?
sell out-of-the-money puts sell at-the-money straddles --->buy at-the-money calls
If P>C-S+PV(K), you can make arbitrage profits by...
sell put, buy call, sell stock
A long IR swap (you receive LIBOR and pay fixed rate) can be represented as ...
short fixed rate bond + long floating rate bond Long Swap = get LIBOR + pay fixed rate ... getting LIBOR = coupons from the floating rate bond, paying fixed rate = being short fixed rate bond and paying coupons. Swap can be be views as a sequence of FRA (but not a single FRA
If you sell a straddle (sell ATM call + sell ATM put), you bet that the stock price ...
stays close to the current level
If an option portfolio is properly delta-hedged, and stock price decreases a bit, then its value ...
stays the same The idea behind delta-hedging is that P&L from options is offset by P&L from the stock hedge
By buying a put you bet that ... (select all that apply)
stock price will increase --->stock price will decrease --->stock volatility will increase stock volatility will decrease Put price negatively depends on the stock price and positively depends on volatility
If you sell a put, your max possible loss is
strike price
You sell a call with a price C(0)=$10 that expires in one month and has a strike K=$100. What is your maximum loss (P&L)?
unlimited For a long call the gain are unlimited (e.g., Tesla price going to infinity). These potentially unlimited *gains* comes from the other side that is short the call. Thus, the *losses* on the short call are potentially unlimited
You buy a straddle if you bet that
volatility will increase
