Options - HOST

Question 1

Price direction of Options (both American/European) as S, K, tau, Sigma, rf or Div Changes

Answer 1

• Call Payoff: max(S-K,0)
• Put Payoff: max(K-S,0)

1. As S goes up, Call Value goes up and Put value goes down
2. As K goes up, Call value decreases and Put value decreases
3. Impact of Tau (time to maturity) is uncertain on European Call/Put. If there is a large dividend payoff between two different maturities, a European call with shorter maturity (before the ex-div date) may be worth more.
   
   1. For European Puts, one with the shorter maturity is worth more b/c of time value
4. Increase in Sigma (Vol) - increases the value of an option
5. As risk-free rate goes up, value of call goes up and value of put goes down.
6. Divdiend lowers the call value and increases the value of a put option.

Question 2

Put-Call Parity equation and proof

Answer 2

c + K^-rt = p + S - D

• Here the European Call/Put has the same underlying, same strike and same maturity

Prove:

• Portfolio A: Call + zero-coupon bond with the face value K
• Portfolio B: Put option and long the underlying stock (Protective Put)
• Portfolio A payoff: max(St-K,0)+K= max(St,K)
• Portfolio B payoff: max (K-St,0) + St = max (St,K)
• Since both portfolio have the same payoff at T and no payoff in between, the no arbitrage argument dictates they must have the same value at t.

Question 3

Forward and Put-call Parity

Answer 3

• Re-arrange Put-call parity: c-p = St - K^-rt
• Long a call and short the put is equal to owning the stock and borrow the money today.
• This expression tells us that when K = S^rt, both Call and Put prices are the same.

Question 4

Why should you never exercise an American call on a non-dividend paying stock before maturity?

Answer 4

If you exercise the call option, you only receive the intrinsic value S-K. Call option also has the time-value of money. Hence it is better off to sell the option rather than exercising it.

• Re-arrange put-call parity: c = (S-K)+(K-K^-rt )+ p.
  
  • 1st component is the intrinsic value
  • 2nd component is the time value of money
  • 3rd component is the value of protection (price of a put).
  • 2nd and 3rd component are always positive.
• Hence the European call is always worth more than its intrinsic value. Considering that the American call is worth at least as much as the European option, it is never optimal to exercise the American Call option.
• There are two other arguments.
  
  • Compare the payoff of selling the call option
    
    • Keep the call option, short the stock and invest K amount to earn the risk-free rate
  • Jensen’s inequality, f(X) is a convex function and E[f(X)] > = f[E(X)]

Question 5

Arbitrage example: European put option on a non-dividiend paying stock.

• Put option with strike price $80 is priced at $8
• Put option with strike price $90 is priced at $9

Is there an arbitrage opportunity in these two options?

Answer 5

The price of a put option (we are comparing $8 and $9) as a function of the strike price is a convex function.

• Π*P(K) > P(ΠK).
• For this specific problem, we should have
  
  • 8/9 x P(90) > P (8/9 * 90)
  • 8/9 * 9 is not greater than P(80) (which is also 8)
  • Hence there is an arbitrage opportunity

Let’s short 9 units of 80 puts and long 8 units of 90 puts. Our cost and proceeds both equal to 27. Payoff at different scenarios:

• S_T >= 90, payoff = 0 (no put exercised).
• 90 > S_T >= 80, payoff = (90-S_T)*8 > 0
• If S_T<80, payoff = (90-S_T)*8 - (80-S_T)*9 = S_T>0

The final payoff >= 0 with the positive probability. hence this is an arbitrage opportunity.

Question 6

What are the assumptions of BS formula?

Answer 6

1. Stock pays no dividend
2. Risk-free rate is constant and known
3. The storck price follows GBM with the constant drift Mu and volatility Sigma
4. There are no transaction costs or taxes.
5. Proceeds of Short-selling can be fully invested
6. All securities are perfectly divisible.
7. There are no risk-free arbitrage opportunities

Question 7

What is the Call and Put Options Formula?

Answer 7

c = Se^-yt N(d₁ ) - Ke^-rtN(d₂)

p = Ke^_-rt N(-d₂) - Se^-ytN(-d₁)

d₁ = [ln(S/K) + (r - y + σ²/2)t] / [σ*sqrt(t)]

d₂ = d₁ - σ*sqrt(t)

N(x) is the cdf of the standard normal distribution.

y = dividend rate (continuous)

if this is a currency option, y = rf (which is the foreign risk-free interest rate).

Question 8

Option’s Delta. Explain

Answer 8

European Call with Div Yield: Delta = e^-ytN(d₁)

European Put with Div Yield: Delta = -e^-yt(1-N(d₁)).

• This is true for the ATM. Longer the maturity, higher the delta. Think about the formula of d1, since time is going up, so is d1. Hence the N(d1) goes up as well. This is the probability of the option actually ending up in the money.
• For shorter maturities, the shorter the maturity, faster the delta approaches one.

Question 9

St = 100, Rf = 5%, One-Year European Call Option, Option is ATM, Vol = 0. What is the call worth and how would I hedge it?

Answer 9

• Since the volatility is zero, the stock price drifts up at the expected return on the stock with no deviation. With no deviation, the stock is riskless
• The stock must offer the expected return equal to the riskless rate
• Stock price will be $105 for sure at end of the year.
• Call will be worth $5 and the PV of this cash flow is: $5/1.05 = $4.75 (about 5 cents per dollar)
• This is a good example to explore the connection between forwards and options:
  
  • c(t) = e^-r(T-t)*max(F-X,0)
    
    • F = S(t)e^r(T-t)
• Hedging: if F>X, that is the forward price greater than the strike price, then I would need a delta of +1.

Question 10

Two standard options have the same features except one has longer maturity, which one has higher gamma?

Answer 10

• Gamma - rate of change of its delta with respect to stock price. Option gamma is called “curvature” or “convexity”
• Gamma is non-negative for standard puts/calls
  
  • Means that their delta rises with increases S
• Put-Call Parity: gamma of European call = Gamma of European Put
• Theta is Large and negative for ATM options and it increases in magintude as maturity approaches. Usually - Theta and Gamma have opposite signs.
• Large negative Theta goes in hand with large positive Gamma for ATM (short-maturity) options
• Strike price plays a crucial role in reversing this behavior: if a call is deep-in-the money, as expiration approaches, deltas do not vary much (+1 or -1) with changing S, hence the Gamma is zero.
• Looking at the formula, when the stock price goes to infinity, the gamma goes to zero. (makes sense - as the call is deep in the money and put is deep-out of the money, and delta won’t change much as the stock price change).
• Same thing occurs if the stock price goes to zero.
• Infinite gamma means the sensitivity of delta to small changes in price of underlying is infinite.

Question 11

Option Value - as expiration approaches

Answer 11

• This is time decay - called Theta
• Option value ‘decays’ toward kinked payoff as expiration approaches
• Theta is negative for all vanilla options
  
  • Except:
  • Deep in-the-money European style call
    
    • Positive Theta - if the div yield is high enough. high div yield pushes price below intrinsic value and option has to “decay upward” in value as expiration approaches
    • Deep in the money European put - because the life does not get much better (bounded by zero stock price). Since the option cannot be exerciesed immediately - the put price will be traded at discount

Question 12

The Put has limited downside potential and no upside. The Call has unlimited upside and no downside. Given the random direction of stock price movement, the disparity in potential payoff suggest that Call should be worth more than the put. Put-Call Parity says the otherwise. Explain

Answer 12

• Key is the shape of risk-neutral distribution of the final stock price.
• This distribution is Lognormal
  
  • This distribution is right-skewed
• If we start at S(t) = X and r = 0, then the skewness in the distribution of S(T) means that the final stock price is more likely to end up below the strike than above it.
  
  • with r = 0, the median of the risk-neutral distribution of S(T) conditional on S(t) is
    
    • S(t) e^{(r-.5*Sigma^2)(T-t)}< S(t)
    • The option is stuck at the money so the median is below the strike
• Call has bigger potential payoff but lower probability of achieving them.
• Put has smaller potential payoff but higher probability of achieiving them.

Question 13

For standard European call option, graph of the “Delta” of a call option. What does this delta mean (in terms of hedging)?

Answer 13

• Stock price is bounded by 0 to infinity
• Delta is bounde by 0 to 1
• At S(T) = 0, Delta = 0 and S(T) = infinity, Delta = 1
• At S(t) = X, Delta is slightly above 1/2
• The graph looks very much like the Normal CDF
  
  • Delta = N(d1)
• Delta tells us how many stocks we need to own in order to hedge our short call option position.

Question 14

No dividend. Standard European call stuck at ATM with one year maturity. If r = 6%. Is the option’s delta greater than 0.5? What does it depend on?

Answer 14

• Let’s think about. Envision the BS formula for European Call
  
  • c(t) = S*N(d1) - e^-r(T-t)XN(d2)
  • d1 = [ln(St/X) + (r+ .5*sig²)(T-t)] / sig*(sqrt(T-t))
• If S(t) = X (ATM option), then d1 is slightly higher than zero, d1 > 0 and N(d1) >0.5. N(.) is an increasing function of its input.
• Hence the Option’s delta is greater han 0.5 for ATM option.

Question 15

BS world with continuous dividend. Standard European call struck ATM. If r = 0.06, and div rate p = .03. Is option’s delta greater than 0.5?

Answer 15

• This is a bit tricky. Since the new formula is
• Delta = e^-p(T-t)N(d₁)
• It also depends on the Volatility of the stock.
• In our case, since the Div rate p = 0.03, the Delta will be greater than 0.5.

Question 16

I am long a call MITCO. I am delta hedged. CEO dies and the stock plunges. How do I adjust my delta hedge?

Answer 16

• I am long the call, so my hedge includes shorting the stock (by Delta amount)
• Stock plunges, the call value drops too.
• I need less stocks to hedge my positions, so I would need to reduce the short position of my Stock
• Borrow money and buy-back stocks to cover my short position.

Question 17

Explain very carefully N(d1) and N(d2)

Answer 17

• N(d1) measures how the call price changes per unit change in the price of the underlying
• N(d1) - another interpretation - replicating portfolio. Number of unit of stocks you must hold in a continuously rebalanced portfolio that replicates the payoff to the call.
• e^-r(T-t)XN(d2) - Value of the borrowing (or short position in bonds) required in a continuously rebalanced portfolio that replicates the payoff a call option.
  
  • Intuition: Value of the borrowing in the replicating portfolio is always less than or equal to the value of the replicating portfolio’s long position in the stock. aka - call has non-negative value
• S(t)N(d1) - Discounted value of the expected BENEFIT of owning the option
• e^-r(T-t)XN(d2) - Discounted value of the expectd Cost of the owning the option
• N(d2) is the risk-neutral probability that the call option finishes in the money. X is the cost if it does.

Question 18

European Digital Option (Cash or nothing) pays a constant H if the stock price is above Strike Price X and zero otherwise. What is the price of this option?

Answer 18

• Price = e^-r(T-t)*H*N(d2)
  
  • 2nd term from the BS equation
  • N(d2) is the probability that the options ends up in the money. This is simply the discounted (risk-neutral) expected payoff to the option
  • H is the payoff if the option ends up in the money. H is sometimes called a “bet”
• First term from the BS equation: S(t)N(d1)
  
  • Value of the Long position in a digital “asset or nothing” option
• Long position in the “Asset or nothing” option combined with a short position in “Cash or Nothing” option replicates the payoff of the Euroepan Call

Question 19

European Digital Option (Cash or nothing). How does this option price vary with volatility? Explain

Answer 19

• Cash or nothing: e^-r(T-t)XN(d2)
• N(d2) is the probability of the option ends up in the money.
• Since this is a digital option:
  
  • If you are in-the money, Volatility is your enemy
  • If you are out-of-the moeny, Volatility is your friend
• dc/dsigma^2 is positive iff S(t) < Xe^{-(r+.5*sig^2)(T-t)}

Question 20

We find that Monthly Variance > 4* Weekly Variance

and Monthly Variance > 20 * Daily Variance

And Weekly Variance > 5 * Daily Variance

Monthly volatility estimated from the weekly and daily time series is significantly smaller than the monthly time series. How do we price the option?

Answer 20

• Our observation is that the sample variances are not linear in time and that the differences are statistically significant.
• This observation is consistent with the empirical finding that some financial stock indices are positively auto-correlated.
• We can adjust the original BS formula to take in account for the new diffusion process that captures the predictability
  
  • here the autocorrelation can be described using a more complicated drift in the diffusion
• Presence of auto-correlation in stock returns is only one example of a real-world divergence from the BS assumptions (in addition to the effect of restrictions of short sales proceeds).

Question 21

“Intrinsic Value” = Max (St-K,0), let’s call’s value be C(t).

C(t) - Max(S(t)-X,0). Explain carefully how this graph behaves and what does it mean

Answer 21

• The difference of C(t) - Max(S(t)-X,0) is the value of not exercising
• When the option is deep out of the money, this value is zero
• When the Option is deep in the money, and if you do not exercise
  
  • X (1 - exp^(-r(T-t))
• Without dividends, it is never optimal to EXERCISE the plain vanilla American call.
  
  • This does not mean that you should continue to hold on to the option
  • Rather, if you want to exit the position, you should capture the Time-value of the option by SELLING the option

Question 22

It is 10 months since you sold one-year European Call option to a customer. You have been delta-hedging to the written call. Option is well-in the money (e.x. delta is close to 90). Suppose that you watch the underlying stock is falling in price in last two months, what happens to the delta of the replicating portfolio?

Answer 22

• Naive (also incorrect) answer is sell the stock and reduce the delta exposure.
• Time Value is also moving against you. If the Option stays in the money (even if the prices are falling), we will need to deliver the shares. (means the Option’s delta will jump to +1
  
  • It all depends on how fast the option’s delta is falling compared to the time value of the moeny.

Question 23

Standard European Call option: Explain three graphs

1. Call Price at maturity T
2. Call Price at maturity t
3. Call price at maturity t versus the future price F

Answer 23

• At maturity, the option’s price is not asymptotic ( that is following a curve arbitrarily closely)
  
  • At maturity - Call value rises with slope 1 from the point S(T) = X
• The plot of call price versus future price is a smooth curve that is asymptotic to the line C = 0 when the F is very small.
  
  • From F = X, when the futures price is large, it is asymptotic to a line rising with slope e^-r(T-t)

Question 24

2 Europen call everything is the same, except one call matures in one year and other option matures in 4 Years. One year vol = 15%, What value of Vol do I put into the BS formula to value the four-year option?

Answer 24

• BS assumes arithmetic brownian motion in log price
• This assumption yields a geometric brownian motion in priceand arithmetic brownian motion in continuously compounded returns
• Vol of continuously compounded return Sigma^2 gorws linearly with time for ABM
• Answer: sqrt(4)* Vol = .30 or 30%

Question 25

What is the value of a three-month ATM call option on a $100 stock when the implied vol is 40?

Answer 25

• Arithmetic Brownian motion approximation
  
  • c(t) = σS * sqrt( [T-t] / 2π )
    
    • sqrt(1/2π) = .4
    • Sqrt(3/12) = .5
    • Vol = .4
    • S = 100
  • Call = 100*.4*.4*.5 = $8