Options2 - HOST

Question 1

Price of 100 day European Call is $100, what is the price of 200 day European Call?

Answer 1

• How does the value of call changes with time to expiration?
• c(t) = σ_A* sqrt [(T-t)/2π)]
  
  • Here σ is the standard deviation of price
• Call value increases at something like square root of term to maturity (in our case the call value increases by 40% to 50%)
• It also depends on where the call is:
  
  • Sensitivity to term to maturity increases as you move out-of-money
  • Sensitivity moves down to zero if the option is deep-in-the money
• Doubling the term-to-maturiy can easily double/triple or quadruple the value of call if it is well out-of the money.
• Deep out of the money calls equal to “lottery tickets”
  
  • The option has more time to finish ITM
  • PV of cost of exercising the option decreases
• This is the opposite of Put: deep ITM put looses value if the maturity is extended.

Question 2

BS World. One-Year ATM European Call

r = 0, Spot = $100, Std of Stock price = $10. Is the call price closer to $1, $5, or $10?

Answer 2

• Std of $10 means, if the stock price is either 90 or 110.
  
  • Call Value = .5*0 + .5*10 = $5 (Approx)
• Exact using this formula:
  
  • Call = σ* sqrt [(T-t)/ 2π] = 10*.4 = $4
  • Notice: This is different than the one we been using with the stock price.
• Call value is approximately $4

Question 3

100-day European Call option with implied vol of 20. Suppose tomorrow IV jumps to 25 but after that it will return to 20 for the remainder of its life. What extension to the life of the call would produce the same change in the PV of the call as the above-mentioned single-day increase in Vol?

Answer 3

• Applying that short form formula
  
  • Increasing Vol from .20 to .25 is the increase of Variance by more than 50% in one day
  • Option’s life is 100 days
    
    • Hence Variance increase by .5% for every day
  • Since Vol is under square root and also the time (from the short-hand formula)
• This is equal to the increasing the life of the option by .005*100 = .5 day (half a day)

Question 4

Long a Straddle with strike of $25. Underlying is at $25. It costs you $5 to buy this straddle. What price movement am I looking for?

Answer 4

• Long Straddle is Long call plus a long a put with the same strike
• If I hold the straddle till maturity then any movement more than $5 will generate profit
  
  • Or any movement before the maturity (may) generate profit

Question 5

Eurodollar futures contract with 6 months to maturity, selling @ 5%, settled @ 3M Libor (MTM every day). 6M Euro-dollar forward, same thing as above except the settled at maturity? Which one do you prefer and why?

Answer 5

• Pick the forward. Euro-dollar moves inversely to the interest rate movements.
  
  • You have to post the margins (b/c you incur losses) when the rates are high
  • You earn less on your proftis
• Euro-dollar future contract value =
  
  • $10,000 x [100 - (90/360)*Delta]
    
    • Where Delta is settlement discount rate. Put is as a number 5 (Not .05) if the discount is 5%

Question 6

For the Monte-carlo simulation, does it make sense to simulate GBM process for the call itself or the underlying?

Answer 6

• Simulate the GBM process for the underlying
• It is difficult to model the call
  
  • Why? The instantenous vol of the call changes whenver the leverage of the call changes (assuming the underlying is of constant vol)
  • And the leverage of the call changes whenever the stock price moves (and even if it doesn’t move - the theta dost its job).

Question 7

If I am long the MBS and if I am expecting a bond market rally - would I be better off with positive convexity or negative convexity?

Answer 7

• I would be better off with the positive convexity (in general)
  
  • It means that as the rates fall (the bond market rally), the value of the MBS will increase
• However - for MBS - owners of this security are exposed to “prepayment risk” and “extension risk”.
  
  • Prepayment risk: Rates fall and the homeowners refinance at the lower rate (worst time for investors to get their money back)
  • Extension risk: Rates rise and the pre-payment slows down.
  • Hence borrowers are the long the call option on the mortgage
    
    • This option has the negative convexity
• Negative Convexity - also called “Compression to par”
• We assume a parallel shift in the yield curve

Question 8

What is wrong with this hedging strategy: Buy one share if the stock price above the stirke price and sell the stock when the stock price is below the strike price?

Answer 8

• This naive strategy is called: “Stop-loss” strategy
• It might seem like it replicates the payoff of the call, but it is not.
• It is almost impossible to buy and sell at the exactly the strike price
  
  • You will buy at the slightly higher price than the strike price
  • You will sell at the slightly lower price than the strike price
• You will get eaten alive by the transaction cost
• The timing of the CF to the optin and the hedge are different. It is not a hedge.

Question 9

Stochastic Calculus: Integrate w(t)dt from 0 to T. Here w(t) is a standard brownian motion.

Answer 9

• w(t) has continuous paths with probability 1
• Sn = sum from i to n
  
  • (t_n - t_i) [w(t_i) - w(t_i-1)]
• Sn is just the weighted sum of increments of a standard brownian motion
• These increments are independently Normally distributed
• Finite sum of a constant-weighted independent Normal is also Normal
• I_T(w) is distributed N(0, T³/ 3]

Question 10

Integration of w(t)dw(t) from 0 to T. Here w(t) is a standard brownian motion.

Answer 10

• It requires Ito’s Lemma
• Apply Ito’s lemma to F(t,w) = w²(t) / 2
  
  • dF = F_tdt + F_wdw + .5F_ww(dw)²
  • = w(t)dw(t) + .5dt
• Integrate to find F(T) - F(0)
  
  • Answer: [w²(T) - T] / 2

Question 11

IBM is trading at $75. What does it cost to construct a derivative security that pays exactly one dollar when IBM hits $100 for the first time?

(Ignore: dividends, assume riskless rate of 0, all assets are infinitely divisible, ignore any short-sale restrictions and ignore any taxes or transactions costs).

Answer 11

• If there is a security that promises one dollar when IBM hits $100 for the firs time
• If this secruity is more than $.75, then I should issue 100 of them, use these proceeds to buy the $75 IBM stock.
• I am perfectly hedged. By no arbitrage - this security cannot sell for more than 75 cents

If security costs less than 75 cents, I should short-sell the IBM stock and buy 100 of such securities. For this argument to work, we are assuming that we can roll-over our short position indefinitely. Hence the value of this security should be 75 cents

Question 12

Why do you get a “smile” effect when you plot implied volatility of options against the strike price?

Answer 12

• BS assumes constant vol
• Option prices are determined by supply and demand
  
  • If the volatility is changing with both level of the underlying and time to maturity, then the distribution of future stock prices is no longer LogNormal
• As vol changes trhue time, you are likely to get periods of little activity and preiod of intense activity
  
  • These periods produced peakedness and fait tails (Respectively - together Leptokurtosis)
• Fat tails are likely to lead to some sort of smile effect - because they increase the chance of payoffs away from the money.

Question 13

Is the price of a double-barrier, knock out option (one with both down and out and up and out) just the price of an up-out option plus the price of a down-out option?

Answer 13

• Barrier Option relationship (remember)
  
  • Down and out call Option + Down and In Call option = Standard Call
• Answer: No
  
  • Pair of up/out and down/out is worth more than the double barrier knock-out option
  • The most obvious reason: If the price moves one way - the double knock out is knocked out but a portfolio of a down-out + up-out still contains one LIVE option

Question 14

Gold price follows a Gaussian process. Current price =$400, Riskless rate = 0, Vol = $60 per annum. What is the value today of a digital cash-or-nothing option that pays $1 million in six months if the price of gold is at or above $430?

Answer 14

• Volatility grows with the sqrare-root of time
• $60 per annum volatility translates into:
  
  • $60 * sqrt(1/2) = $42 (approx)
• What is the probability of:
  
  • P (G(T) > 430)
  • With r = 0, in the risk-neutral world, the distribution is centered at G(t) = 400 and standard deviation at $42.
• P(G(T) > 430) = P(G(T) - G(t) > 30)
  
  • = P [(G(T) - G(t) / 42 ) > 30/42]
  • = 1 - N(3/4)
    
    • first term is roughly standard normal
    • N(0) = .50
    • N(1) = 84
    • N(3/4) = .75
• We conclude that there is roughly 25% chance that the digital option finishes in-the-money. Discounted expected payoff = $250,000

Question 15

“L” denotes the 3M US dollar LIBOR rate. Consider an interest rate swap arrangement where Party A pays L to party B. Party B pays 24% - 2 x L to party A. Can you reverse engineer this deal and express it in simple terms?

Answer 15

• Subtract L from both parties
• then B pays 24% - 3 x L
• divide the above by 3%
  
  • 8% - L
• Hence this swap deal is the three swaps, where there is a swap of Libor for 8% fixed rate