1.1 “Free” Option

1. The option is not really free because we may end up at a loss at and above the strike price. (See Figure S.1.)
2. The replicating portfolio would include selling x digital calls struck at K at price p and buying a vanilla call struck at K for the premium of m. In order for the portfolio to have zero cost we must have .
3. The cost of one digital call using the Black-Scholes model with the given parameters is $0.5398. The premium of the vanilla call from the Black-Scholes model is $7.97. Solving for x we get .

1.2 Autocallable

Answer: .

1.3 Geometric Asian Option

1. From Ito-Doeblin we have where α = r − q − ½σ². Substituting into the definition of A_T we get:
   
2. which yields the required expression for A_T after simplifications.
3. From Ito-Doeblin, we get , which yields the required result after integration of both sides over [0, T]. The distribution of is thus normal with zero mean and variance .
4. Substituting and we need to show that ln A_T is normally distributed with mean:
   
5. and variance . Indeed the mean matches the expression from question (a), and using question (b) we find that the variance of is as required.

1.4 Change of Measure

We have with . Since we have after substitution:

But and thus . Expanding the second squared bracket and cancelling terms we are left with as required.

1.6 Siegel's Paradox

1. Straightforward application of Ito-Doeblin.
2. The “risk-neutral” dynamics of 1/X from question (a) correspond to the dollar risk-neutral measure, in which 1/X is non tradable (number of euros per dollar). The paradox is resolved by introducing the euro risk-neutral measure where 1/X follows the process:
   

In the dollar risk-neutral measure, 1/X is the price of the dollar-euro exchange rate quanto dollar. Following the notations of Section 1.2.4 and defining S = 1/X, we have ρ = −1 and η = σ; thus the drift of S quanto dollar under the dollar risk-neutral measure is r_€ − r_$ − ρση = r_€ − r_$ + σ² as required.

2.1 No Call or Put Spread Arbitrage Condition

We know the upper bound is:

and we know that so after rearranging terms we get:

We know the lower bound is:

and we know that so we get

From put-call parity: , and thus:

Putting them together we get:

Furthermore and

Thus:

as required.

2.2 No Butterfly Spread Arbitrage Condition

1. Identities:
   • We have:
   

   Differentiating with respect to K:

   

   as required.

   Differentiating with respect to σ:

   

   Using the chain rule:

   

   But:

   

   Substituting and :

   

   Using the fact that , we get:

   
   • Differentiating with respect to σ:
   

   We have:

   

   and:

   

   Substituting back into :

   

   Factoring by and recognizing d₁ we obtain as required:

   
2. First-order derivative: . Second-order derivative:
   
3. which yields the required result after noting that f_xy = f_yx by Schwarz's theorem.
4. Applying the second-order chain rule:
   

   Substituting the identities from (a) we get the required result after further straightforward algebra.

5. After simplifications is equivalent to:
   

2.3 Sticky True Delta Rule

1. Applying the chain rule: , whence the required result after substituting K = 1 and T = 1.
2. If then and thus yielding the required result after appropriate substitutions.
3. c. Because Δ′ > 0 (call delta goes up as S goes up) and b < 0 (volatility goes down as S and Δ go up) the sticky true delta rule would produce a lower delta than Black-Scholes.

3.1 Overhedging Concave Payoffs

Assume f(K₁) = 0 for simplicity. From left to right: start with f′(K₁) calls struck at K₁ so as to be tangential to the payoff; add calls struck at K₂ such that the portfolio matches the payoff f(K₃) at K₃; then add calls struck at K₃ so as to be tangential to the payoff after K₃; and so on (see Figure S.2).

3.2 Perfect Hedging with Puts and Calls

From Section 3.2:

Splitting the integral at F and using terminal put-call parity we get:

But , and:

After substitutions and simplifications we get:

Thus:

because .

3.3 Implied Distribution and Exotic Pricing

1. • Answer: ≈ .1043
   • Answer: ≈ 1.0789
   • Answer: ≈ .2693
   • Answer: ≈ .0211. Vanilla overhedge with strikes 0.5, 0.9, 1, 1.1, 1.3, 1.7: Quantities are 0, 0, 0.0100, 0.1200, 0.6700, and 1.5200 (see Figure S.3).
2. 1. (i Answer: approximately $0.8965.
   2. (ii Answer: approximately $0.8626.

Figure S.3

3.5 Path-Dependent Payoff

1. For example: forward start option, Asian option.
2. Pseudo-code:
   1. Loop for i = 1 to N:
      1. Generate and calculate
      2. Generate and independently and calculate
         
      3. Calculate
   2. Return
3. We only know the marginal distributions of but we are missing the conditional implied distribution of .

3.6 Delta

From we get and thus:

atm_vol = SVI(1);
atm_vol1 = SVI(1/(1+epsilon));
price = 0;
for i=1:n
  normal = randn;
  x = exp(atm_vol*normal*sqrt(T) - 0.5*atm_vol∧2*T);
  x1 = (1+epsilon)*exp(atm_vol1*normal*sqrt(T) − 0.5*atm_vol1∧2*T);
  price = (i-1)/i*price + Payoff(x)*ImpDist(x) …
          / lognpdf(x,-0.5*atm_vol∧2*T,atm_vol*sqrt(T)) / i;
  price1 = (i-1)/i*price1 + Payoff(x1)*ImpDist(x1) …
          / lognpdf(x1,-0.5*atm_vol1∧2*T,atm_vol1*sqrt(T)) / i;
end
delta = price − price1

4.1 From Implied to Local Volatility

1. We rewrite Equation (4.1) by solving for after using the result from Problem 2.4.2(c) for .
   

   Take and derive with respect to T:

   

   We are given and so:

   

   We then plug in and into Equation (4.1):

   

   By dividing both numerator and denominator by , we obtain the desired Equation (4.2):

   
2. Replace with so we now have:
   

   Now using the Hint:

   

   Thus = and by integration we find that:

   

   as required.

3. To establish the hint, note that by linearity of σ^*, and substitute . Differentiating with respect to K we obtain:
   

   Near the money both denominators are close to 1; furthermore gives . After substitution and simplification:

   

   whose leading term is .

4.2 Market Price of Volatility Risk

1. The delta-hedged portfolio Π is long one option and short units of S. Thus:
   
2. which yields the required result after cancelling terms.
3. We have:
   

   But which yields the required result after substitution and simplifications.

4. Conditional expectation: . Conditional standard deviation:
   

   Thus , that is, at time t the risk-neutral expected return on the delta-hedged portfolio is the risk-free interest rate plus a positive or negative risk premium, which is proportional to the risk of the portfolio, with proportionality coefficient λ. This result applies to any option and λ is independent from the particular option chosen, which is why it is called the market price of volatility risk.

4.3 Local Volatility Pricing

1. Figure S.4 shows a graph of the corresponding local volatility surface.
2. Using Monte Carlo simulations:
   • “Capped quadratic” option: ; answer: 0.76949
   • Asian at-the-money-call: ; answer: 0.1242
   • Barrier call: if S always traded above 80 using 252 daily observations, 0 otherwise. Answer: 12.87577

5.1 Delta-Hedging P&L Simulation

1. Positive Path: See Figure S.5

   Negative Path: See Figure S.5

2. Average P&L: −$11,663.80 (see Figure S.6)

5.2 Volatility Trading with Options

1. (i) If σ is constant then . For a vanilla call Γ > 0 and thus the cumulative P&L is always positive.

   (ii) We have . By iterated expectations:

   
2. The aggregate cumulative P&L is:
   

   As a sum of two independent normal distributions the aggregate distribution is normal with parameters:

   • Mean: q₁m₁ + q₂m₂
   • Standard deviation:

   This result sheds light on the diversification effect obtained by delta-hedging several options within an option book: the expected income is the sum of individual delta-hedging P&Ls, but it is less risky than delta-hedging a single position.

5.4 Generalized Variance Swaps

1. From Ito-Doeblin: . But ; rearranging terms we get: . Integrating over [0, T] yields the desired result.

   Thus generalized variance may be replicated by a combination of cash, two derivative contracts paying off g(S_T) at maturity, and dynamically trading S to maintain a short position of 2g′(S_t) at all times. Taking expectations we find that the fair value is:

   

   since (dW_t) = 0.

2. From Problem 3.4.2 we get: whence the required formula after substituting and annualizing.
3. For the corridor varswap

5.5 Call on Realized Variance

1. Solving for v_t we have: Thus v_t is lognormally distributed with mean and standard deviation . From Black's formula we have where . At the money this simplifies to which yields the required formula because N(x) − N(−x) = 2N(x) − 1.
2. For x ≈ 0 we have .

6.1 Lower Bound for Average Correlation

1. Substitute x = e in .
2. Because we have . Define the Lagrangian . The first-order condition yields that is, . The constraint e^Tx = 1 then gives the value of λ and we get the required result after substitution and simplification.
3. (i) By spectral decomposition: . Furthermore, Parseval's identity states that for any x, and thus . Scaling by appropriate constants we obtain the required result.

   (ii) Rewrite and make the approximation that for i = 1,…, n − 1:

   

   which is justified by the fact that α_n ≈ 1 since v_n and e are assumed to form a tight angle. Proceed similarly for H.

6.2 Geometric Basket Call

1. We have:
   
2. where is the annualized risk-neutral drift of S⁽ⁱ⁾. As a sum of normal variables ln b_T is normally distributed with mean and variance:
   
3. where . Using Black's formula: with . Further simplifications are possible.

6.4 Continuously Monitored Correlation

After substitution, we have by Cauchy-Schwarz:

where .

7.1

1. is positive because straddles always have a positive payoff and thus price. It must be less than 1 because of the triangle inequality:
2. Use the proxy

7.2

1. From the Black-Scholes PDE . Applying Ito-Doeblin to Θ: . Taking expectations we get:
   
2. which vanishes because Greeks must also satisfy the Black-Scholes PDE.

7.3

At time 0 the portfolio value is , and the vega of each component is . Hence the portfolio vega is .

8.1 Implied Correlation

Implied correlation is constant iff , i.e., iff , whence the required result after simplifications.

8.2 Dynamic Local Correlation I

When D = I and U = ee^T Langnau's alpha is:

where . Define and divide both the numerator and denominator of the above expression by B to get:

It is easy to verify that .

9.2

1. The process clearly remains within [0,1] because it is continuous and its drift and volatility coefficients vanish at 0 and 1. Let us show that the bound 0 is nonattracting:
   

   Thus since diverges. A similar analysis shows that the bound 1 is also nonattracting.

2. We want to find f(x), g(x) such that:
   

   Thus we must solve:

   

   Taking ratios we get ; dividing both numerator and denominator by g² on the left-hand side we obtain . Solving for p after substituting we find , that is, . Substituting in and simplifying we get , and thus .

3. This model is not suitable for several reasons: it has only one parameter ω, which leaves little freedom for parameterization, and the volatility of basket variance f is lower than the volatility of constituent variance g, which is contrary to empirical observation.