---
title: "Multi-Asset Options"
book: "Derivatives and Volatility"
subject: quant
language: en
chapter: 17
exercises: 8
source: https://one-course.com/books/quant/5/en/chapter/17-multi-asset-options
---

# Chapter 17 — Multi-Asset Options

A note pays a coupon at the end of the year only if all three of its shares are then at or above 70% of their starting levels. The client sees three familiar names and a condition that looks easy to meet. The dealer sees a position in correlation that no listed market quotes. If the three shares were independent, with volatilities of 25%, 30% and 35%, the probability of paying the coupon would be 62%. If they moved together it would be 79%. The dealer who sold the coupon is short that difference. This chapter prices options on several assets, baskets, worst-ofs and best-ofs, and shows how desks read correlation from the index and single-stock option markets. It then covers [dispersion trades](#def-dv-multi-asset-options-dispersion) on the gap between the two, [local correlation](#def-dv-multi-asset-options-local), which fits the index skew, and the quanto and [composite options](#def-dv-multi-asset-options-composite) that bring a second currency into the payoff.

## 17.1 Correlation and baskets

Every model in this chapter starts from correlated lognormal assets: $dS_i/S_i=\mu_i\,dt+\sigma_i\,dW^i$ with $d\langle W^i,W^j\rangle=\rho_{ij}\,dt$. Paths are simulated by the Cholesky factor $L$ of the correlation matrix, $LL^\top=\rho$ (One Quant Book 4, chapter 25), applied to independent normals, and prices by Monte Carlo (Book 4, chapter 26). The build takes each $\sigma_i$ as a number or as a [local volatility](https://one-course.com/books/quant/5/en/chapter/9-local-volatility#def-dv-local-volatility-model) function $\sigma_i(t,S_i)$, so each asset’s own smile is respected.

**Definition 17.1 (Basket option).**

A *basket option* is an option on a weighted sum of assets, typically of their performances: $\bigl(\sum_iw_iS_{i,T}/S_{i,0}-K\bigr)^+$ for a call.

A basket’s variance is $\sum_{ij}w_iw_j\rho_{ij}\sigma_i\sigma_j$ in the lognormal approximation. It is below the weighted average of the variances whenever the correlations are below one, which is the diversification the client buys. A basket of lognormals is not lognormal, but a lognormal with the same first two moments prices it well. Match $M_1=\sum w_iF_i$ and $M_2=\sum_{ij}w_iw_jF_iF_je^{\rho_{ij}\sigma_i\sigma_jT}$, then price with Black at the volatility $\sqrt{\ln(M_2/M_1^2)/T}$. On an equally weighted basket of the three shares, one year, the approximation is within 0.0004 of the simulated at-the-money price for every correlation from 0 to 0.95.

![Three shares with volatilities of 25%, 30% and 35%, one year, zero rates, equal pairwise correlation. Left: worst-of puts and best-of calls lose value as correlation rises and basket calls gain. Right: the probability that all three end above 70% of their start, the note’s coupon condition. Data: the tutorial.](https://one-course.com/images/onecourse/chapters/quant-5/dv-multi-asset-options/fig-565dc9c5f116.svg)

***Figure 17.1.** Three shares with volatilities of 25%, 30% and 35%, one year, zero rates, equal pairwise correlation. Left: worst-of puts and best-of calls lose value as correlation rises and basket calls gain. Right: the probability that all three end above 70% of their start, the note’s coupon condition. Data: the tutorial.*

## 17.2 Worst-of and best-of

**Definition 17.2 (Worst-of option).**

A *worst-of option* pays on the worst performance among several assets, for instance $(K-\min_iS_{i,T}/S_{i,0})^+$ for a worst-of put.

**Definition 17.3 (Best-of option).**

A *best-of option* pays on the best performance, for instance $(\max_iS_{i,T}/S_{i,0}-K)^+$ for a best-of call.

Correlation enters these payoffs in the opposite direction from a basket. When the assets move independently, the worst performer is far below the average and the best far above it. When they move together, all performances are close, and the worst-of and best-of look like options on a single asset. So a worst-of put and a best-of call lose value as correlation rises, and a basket call gains ([Figure 17.1](#fig-dv-multi-asset-options-correlation)).

**Example 17.4 (The note’s three shares).**

For the three shares (volatilities 25%, 30% and 35%, one year, zero rates), the at-the-money worst-of put is worth 0.250 per unit notional with independent shares and 0.152 at a correlation of 0.95. The best-of call falls from 0.284 to 0.160, and the basket call rises from 0.071 to 0.118. The probability that all three end at or above 70% of their start rises from 0.616 to 0.787. At a correlation of 0.4 it is 0.673, and at 0.6 it is 0.707: a five-point error in correlation moves the coupon’s value by about one point of probability.

A note that pays a coupon when all shares are above a level, and returns less than the capital when the worst share has fallen through a lower level (chapter 18’s autocallables), leaves the dealer long a worst-of put and short a worst-of digital. Both lose value as correlation rises, so the dealer is short correlation, and the market for index options is where that exposure can be partly offset. The dealer’s other exposures are cross-gammas, the second derivatives in pairs of assets (One Quant Book 6, chapter 3). They make the hedge in each share depend on the moves of the others, and they are largest near the barrier where the worst performer changes.

## 17.3 Implied correlation, dispersion and correlation skew

An index is a basket of its members, so index options and single-stock options together imply a correlation.

**Definition 17.5 (Implied correlation).**

The *implied correlation* of an index with weights $w_i$ is the single correlation $\bar\rho$ that makes the members’ implied volatilities $\sigma_i$ consistent with the index’s implied volatility $\sigma_I$:

$$
\bar\rho=\frac{\sigma_I^2-\sum_iw_i^2\sigma_i^2}{\bigl(\sum_iw_i\sigma_i\bigr)^2-\sum_iw_i^2\sigma_i^2}.
$$

Computed with at-the-money volatilities, it is a single number. Computed at each strike, using the index’s volatility there and the members’ volatilities at the same moneyness, it varies with the strike: index skews are steeper than single-stock skews, so the [implied correlation](#def-dv-multi-asset-options-implied) is higher at low strikes.

**Definition 17.6 (Correlation skew).**

The *correlation skew* is the variation of [implied correlation](#def-dv-multi-asset-options-implied) with the strike: higher for index options struck below the money, where the market prices stocks falling together, than above it. It is the equity analogue of the base-correlation skew of credit tranches (One Quant Book 2, chapter 24).

**Example 17.7 (A synthetic 20-stock index).**

Take an index of twenty equally weighted stocks with flat smiles and volatilities spaced evenly from 22% to 36%, and let its own one-year smile be chapter 9’s: 21.9% at 90, 19.2% at 100, 17.0% at 110. The [implied correlation](#def-dv-multi-asset-options-implied) is 0.549 at 90, 0.408 at 100 and 0.308 at 110 ([Figure 17.2](#fig-dv-multi-asset-options-skew), left). Priced with a constant correlation of 0.408, the members produce an index smile that is flat, 19.3% at all three strikes: constant correlation cannot produce an index skew out of flat member smiles. In variance-swap terms, using the index’s strip volatility of 22.1%, the [implied correlation](#def-dv-multi-asset-options-implied) is 0.559.

[Implied correlation](#def-dv-multi-asset-options-implied) has been higher, on average, than the correlation stocks then realised. Driessen, Maenhout and Vilkov report averages of 39.5% implied against 32.5% realised for the S&P 500, and 46.0% against 35.5% for the Dow Jones 30. They read the gap as a large negative premium for correlation risk, and attribute the index [variance risk premium](https://one-course.com/books/quant/5/en/chapter/14-variance-swaps-and-volatility-derivatives#def-dv-variance-swaps-and-volatility-derivatives-vrp) of chapter 14 to it. The trade that harvests it has a name.

**Definition 17.8 (Dispersion trade).**

A *dispersion trade* sells index volatility and buys the members’ volatility, in [variance swaps](https://one-course.com/books/quant/5/en/chapter/14-variance-swaps-and-volatility-derivatives#def-dv-variance-swaps-and-volatility-derivatives-vs) or straddles, in amounts that cancel the exposure to each member’s volatility. What remains is a short position in the index’s [implied correlation](#def-dv-multi-asset-options-implied): it gains when realised correlation comes in below implied.

**Definition 17.9 (Correlation swap).**

A *correlation swap* pays the weighted average of the pairwise correlations realised between the members of a basket, $\sum_{i<j}w_iw_j\hat\rho_{ij}/\sum_{i<j}w_iw_j$, against a fixed strike. It isolates the correlation that a [dispersion trade](#def-dv-multi-asset-options-dispersion) holds only approximately.

The [dispersion trade](#def-dv-multi-asset-options-dispersion)’s weights follow from the index variance formula. At a fixed correlation, the derivative of the index variance with respect to member $i$’s variance is $w_i^2(1-\bar\rho)+\bar\rho\,w_i\sum_jw_j\sigma_j/\sigma_i$. A short index [variance swap](https://one-course.com/books/quant/5/en/chapter/14-variance-swaps-and-volatility-derivatives#def-dv-variance-swaps-and-volatility-derivatives-vs) with [variance notional](https://one-course.com/books/quant/5/en/chapter/14-variance-swaps-and-volatility-derivatives#def-dv-variance-swaps-and-volatility-derivatives-varnot) $N$, hedged with member [variance swaps](https://one-course.com/books/quant/5/en/chapter/14-variance-swaps-and-volatility-derivatives#def-dv-variance-swaps-and-volatility-derivatives-vs) of $N$ times these derivatives, is insensitive to small changes in each member’s volatility. Its P&L is then $N(\bar\rho-\hat\rho)C$, where $C=(\sum w_i\sigma_i)^2-\sum w_i^2\sigma_i^2$ is the cross term of the index variance.

**Example 17.10 (Correlation fifteen points below implied).**

Sell one-year index variance with a [vega notional](https://one-course.com/books/quant/5/en/chapter/14-variance-swaps-and-volatility-derivatives#def-dv-variance-swaps-and-volatility-derivatives-veganot) of 100 000 at the strip’s 22.12%, a [variance notional](https://one-course.com/books/quant/5/en/chapter/14-variance-swaps-and-volatility-derivatives#def-dv-variance-swaps-and-volatility-derivatives-varnot) of 2 261 per variance point, and buy the members’ variance in the hedge amounts. If each member realises its implied volatility and correlation realises 0.409, fifteen points below the implied 0.559, the P&L is $2\,261\times0.15\times798=270\,600$, with $C=798$ variance points. Over 4 000 simulated years of daily returns the P&L averages 271 000, with a standard deviation of 34 000 from the noise in realised volatilities and correlation. Nine times in ten it lies between 215 000 and 326 000 ([Figure 17.3](#fig-dv-multi-asset-options-dispersion)).

![A synthetic index of twenty stocks with flat smiles, whose own smile is chapter 9’s. Left: the implied correlation at three strikes. Right: the local correlation, a function of the index level, that reproduces the index smile at the three strikes. It is capped at one below the money and falls as the index rises to about 10% above its start; the upturn beyond is an artefact of a quadratic fitted to three strikes, which constrain nothing there. Data: the tutorial.](https://one-course.com/images/onecourse/chapters/quant-5/dv-multi-asset-options/fig-465d9f3f138f.svg)

***Figure 17.2.** A synthetic index of twenty stocks with flat smiles, whose own smile is chapter 9’s. Left: the [implied correlation](#def-dv-multi-asset-options-implied) at three strikes. Right: the [local correlation](#def-dv-multi-asset-options-local), a function of the index level, that reproduces the index smile at the three strikes. It is capped at one below the money and falls as the index rises to about 10% above its start; the upturn beyond is an artefact of a quadratic fitted to three strikes, which constrain nothing there. Data: the tutorial.*

![The P&L of a one-year dispersion trade (short index variance, vega notional 100 000, long member variance in the hedge amounts) over 4 000 simulated years in which correlation realises fifteen points below implied. The dashed line is the formula. Data: the tutorial.](https://one-course.com/images/onecourse/chapters/quant-5/dv-multi-asset-options/fig-38bdd6178c2a.svg)

***Figure 17.3.** The P&L of a one-year [dispersion trade](#def-dv-multi-asset-options-dispersion) (short index variance, [vega notional](https://one-course.com/books/quant/5/en/chapter/14-variance-swaps-and-volatility-derivatives#def-dv-variance-swaps-and-volatility-derivatives-veganot) 100 000, long member variance in the hedge amounts) over 4 000 simulated years in which correlation realises fifteen points below implied. The dashed line is the formula. Data: the tutorial.*

Dispersion is not free money. The trade is short correlation, and the [correlation skew](#def-dv-multi-asset-options-skew) is the market’s statement that correlation rises when the index falls. In a sell-off the short index leg then loses more than the member legs gain. The limits-to-arbitrage reading of Driessen, Maenhout and Vilkov is that realistic trading frictions eat the premium: the trade involves dozens of option positions, rebalanced as weights and volatilities drift.

## 17.4 Local correlation

[Example 17.7](#ex-dv-multi-asset-options-index) showed that members with flat smiles and a constant correlation cannot produce the index’s skew. The index skew is steeper than the members’ skews because correlation rises when the index falls. Langnau argued that a Gaussian copula (One Quant Book 4, chapter 15) of the single-stock distributions fails to explain the steepness of the index skew. He extended multi-asset [local volatility](https://one-course.com/books/quant/5/en/chapter/9-local-volatility#def-dv-local-volatility-model) by making correlation a function of the market state.

**Definition 17.11 (Local correlation).**

A *local correlation* model gives each asset its own [local volatility](https://one-course.com/books/quant/5/en/chapter/9-local-volatility#def-dv-local-volatility-model), fitted to its own options, and makes the correlation between them a function of time and of the market state, typically the index level: $\rho_{ij}(t,I_t)$. The function is chosen so that the model’s index options reprice the index smile, which it does by construction in the same way that [local volatility](https://one-course.com/books/quant/5/en/chapter/9-local-volatility#def-dv-local-volatility-model) fits one asset’s smile.

Exact calibration uses the [Markovian projection](https://one-course.com/books/quant/5/en/chapter/9-local-volatility#def-dv-local-volatility-projection) of chapter 9. The index’s local variance must equal the conditional expectation of the basket’s instantaneous variance given the index level, and that equation is solved for the correlation at each index level. The build uses the simplest version that shows the mechanism. The correlation is common to all pairs and quadratic in the index’s [log-moneyness](https://one-course.com/books/quant/5/en/chapter/7-implied-volatility-and-its-surface#def-dv-implied-volatility-and-its-surface-surface) $x$, $\rho(x)=a-bx+cx^2$ clipped to $[0,1]$, and its three coefficients are fitted to the index volatilities at the three strikes by Newton steps on a simulation with common random numbers.

**Example 17.12 (Fitting the index skew).**

For the synthetic index the fit gives $\rho(x)=0.384-3.62x+16.3x^2$, which reprices the index at 21.93%, 19.19% and 16.99% against 21.93%, 19.19% and 16.98% ([Figure 17.2](#fig-dv-multi-asset-options-skew), right). Correlation is 0.91 when the index is 10% down in log terms, capped at one below that, and 0.19 when it is 10% up. The members’ marginal laws are unchanged, lognormal at their own volatilities. Only their dependence moves with the market.

[Local correlation](#def-dv-multi-asset-options-local) then prices every product on the members consistently with both option markets. That includes worst-ofs, baskets struck away from the money, and dispersion structures. Like [local volatility](https://one-course.com/books/quant/5/en/chapter/9-local-volatility#def-dv-local-volatility-model), it fixes the dynamics of correlation to a function of the state, and so it forecasts a forward [correlation skew](#def-dv-multi-asset-options-skew) that flattens, the multi-asset counterpart of the [forward smile](https://one-course.com/books/quant/5/en/chapter/9-local-volatility#def-dv-local-volatility-forward) of chapter 16. Worst-of products whose risk sits in scenarios where the index has already fallen inherit that assumption.

## 17.5 Quanto and composite options

A payoff can involve a second currency even when it references one asset. The two standard cases differ in whether the exchange rate is fixed in advance or converted at maturity.

**Definition 17.13 (Quanto option).**

A *quanto option* pays a payoff computed on a foreign asset in the foreign currency’s units but settled in the domestic currency at a fixed exchange rate: for instance $(S_T-K)^+$ dollars per index point of a yen index.

**Definition 17.14 (Quanto adjustment).**

The *quanto adjustment* is the change in the asset’s drift when it is priced under the domestic measure with a fixed conversion: $-\rho\sigma_S\sigma_X$, with $\sigma_X$ the volatility of the exchange rate (domestic per foreign unit) and $\rho$ its correlation with the asset. The quanto forward is $Fe^{-\rho\sigma_S\sigma_XT}$.

The adjustment follows from Girsanov’s theorem (One Quant Book 4, chapter 5). Under the foreign measure the asset’s drift is the foreign carry. A domestic investor who holds the asset converted at the floating rate earns the domestic rate on $S_tX_t$, which fixes the drift of $SX$. Dividing out the exchange rate’s own drift and the covariance term $\rho\sigma_S\sigma_X$ leaves the asset’s drift under the domestic measure. The hedger of a quanto holds the foreign asset and must keep re-hedging its foreign-currency value. When the asset and the currency are correlated, those re-hedges systematically gain or lose, and the adjustment prices that.

**Example 17.15 (An index forward paid in another currency).**

A foreign index with a forward of 100, volatility 20%, paid one year out at one domestic unit per point. With an exchange-rate volatility of 10% and correlations of $-0.6$, $-0.3$, $0.3$ and $0.6$, the quanto forward is 101.21, 100.60, 99.40 and 98.81. A simulation of 200 000 paths under the domestic measure with the adjusted drift agrees to 0.005 ([Figure 17.4](#fig-dv-multi-asset-options-quanto)). At a correlation of $-0.3$ the at-the-money quanto call is worth 8.29 against 7.97 for the same call settled in the foreign currency.

**Definition 17.16 (Composite option).**

A *composite option* is an option on the foreign asset’s value converted into the domestic currency at the prevailing rate, $(S_TX_T-K)^+$ with a domestic strike. Its underlying $SX$ has volatility $\sqrt{\sigma_S^2+\sigma_X^2+2\rho\sigma_S\sigma_X}$.

A quanto removes the currency risk from the payoff and leaves the correlation in the drift. A composite keeps the currency risk and puts the correlation in the volatility: at $\rho=-0.3$, the composite volatility is $\sqrt{0.04+0.01-0.012}=19.5\%$. Both are two-asset options, and both carry the same correlation position, which the desk must mark and hedge. Spread options between two assets, the other common two-asset product, appear in One Quant Book 6, chapter 16.

![The one-year forward of a foreign index (forward 100, 20% volatility) paid at a fixed exchange rate whose volatility is 10%, by the correlation between the index and the rate. Data: the tutorial.](https://one-course.com/images/onecourse/chapters/quant-5/dv-multi-asset-options/fig-15c770802777.svg)

***Figure 17.4.** The one-year forward of a foreign index (forward 100, 20% volatility) paid at a fixed exchange rate whose volatility is 10%, by the correlation between the index and the rate. Data: the tutorial.*

## 17.6 Tutorial: correlation in practice

**Goal.** Price baskets, worst-ofs and best-ofs against correlation; read the [implied correlation](#def-dv-multi-asset-options-implied) of an index at three strikes; fit a [local correlation](#def-dv-multi-asset-options-local) to the index skew; simulate a [dispersion trade](#def-dv-multi-asset-options-dispersion); measure the [quanto adjustment](#def-dv-multi-asset-options-quantoadj). **End state:** the four figures and the numbers of the weekend problem.

1. **[Local correlation](#def-dv-multi-asset-options-local) paths.** Equicorrelated shocks whose common correlation is read from the index level at every step: `def local_correlation_paths (s0, vols, weights, rho_fn, times, n_paths: int = 100_000 , seed: int = 17 , steps_per_date: int = 21 ) -> np.ndarray: """Equicorrelated lognormal assets (zero rates) whose common correlation rho_fn(t, x) depends on the index's log-moneyness x = ln(I_t / I_0), I = sum w_i S_i: each shock is sqrt(rho) Z + sqrt(1 - rho) Z_i.""" s0, vols, w = (np.asarray(a, float ) for a in (s0, vols, weights)) m = len (s0) i0 = float (w @ s0) rng = np.random.default_rng(seed) half = n_paths // 2 x = np.tile(np.log(s0), (2 * half, 1 )) out = [np.exp(x)] now = 0.0 for target in np.asarray(times, float ): dt = (target - now) / steps_per_date for j in range (steps_per_date): idx = np.exp(x) @ w rho = np.clip(rho_fn(now + (j + 0.5 ) * dt, np.log(idx / i0)), 0.0 , 0.999 )[:, None ] zc = rng.standard_normal((half, 1 )) zi = rng.standard_normal((half, m)) zc, zi = np.vstack([zc, -zc]), np.vstack([zi, -zi]) z = np.sqrt(rho) * zc + np.sqrt(1 - rho) * zi x = x - 0.5 * vols * vols * dt + vols * math.sqrt(dt) * z out.append(np.exp(x)) now = target return np.stack(out, axis=1 )` **Listing 17.1.** The local-correlation path generator. code/firm/multiasset/firm_multiasset.py
2. **[Implied correlation](#def-dv-multi-asset-options-implied) and the index variance**, the two directions of the same formula: `def implied_correlation (index_vol: float , weights, vols) -> float : """The single correlation that makes the index variance equal sum_ij w_i w_j rho_ij sigma_i sigma_j.""" w, v = np.asarray(weights, float ), np.asarray(vols, float ) own = float (np.sum((w * v) ** 2 )) cross = float (np.sum(w * v)) ** 2 - own return (index_vol ** 2 - own) / cross def index_variance (weights, vols, rho: float ) -> float : w, v = np.asarray(weights, float ), np.asarray(vols, float ) own = float (np.sum((w * v) ** 2 )) return own + rho * (float (np.sum(w * v)) ** 2 - own)` **Listing 17.2.** Implied correlation from index and member volatilities. code/firm/multiasset/firm_multiasset.py
3. **The quanto forward and the composite volatility**: `def quanto_forward (fwd: float , sigma_s: float , sigma_x: float , rho: float , t: float ) -> float : """Forward of a foreign asset paid in domestic currency at a fixed rate: F e^(-rho sigma_S sigma_X T), rho the correlation between the asset and the exchange rate quoted as domestic per foreign.""" return fwd * math.exp(-rho * sigma_s * sigma_x * t) def composite_vol (sigma_s: float , sigma_x: float , rho: float ) -> float : """Volatility of the foreign asset converted at the prevailing rate into domestic currency.""" return math.sqrt(sigma_s ** 2 + sigma_x ** 2 + 2 * rho * sigma_s * sigma_x)` **Listing 17.3.** Quanto and composite adjustments. code/firm/multiasset/firm_multiasset.py
4. **Run** `dv_multiasset.three_share_prices()` , `correlation_by_strike()` , `calibrate_local_correlation()` , `dispersion()` , `quanto_example()` and `fig_multiasset.py` .

**What to change next.** Give the members their own skews (a [local volatility](https://one-course.com/books/quant/5/en/chapter/9-local-volatility#def-dv-local-volatility-model) each) and refit the [local correlation](#def-dv-multi-asset-options-local); price the three-share worst-of digital under the fitted [local correlation](#def-dv-multi-asset-options-local); run the [dispersion trade](#def-dv-multi-asset-options-dispersion) with correlation realising at implied but member volatilities five points below.

## 17.7 Build: multi-asset paths and payoffs

**Purpose.** The miniature firm’s multi-asset engine: correlated paths for baskets, worst-ofs and quantos, the implied- and realised-correlation analytics of its dispersion book, and the generator that chapter 18’s autocallables run on.

**Interface.** `correlated_paths(s0, vols, corr, times, r, q, n_paths, seed, quanto, steps_per_date)` with `vols[i]` a number or a function $\sigma_i(t,S)$; `local_correlation_paths(s0, vols, weights, rho_fn, times, …)`; `equicorrelation`; `basket_payoff`, `worst_of_payoff`, `best_of_payoff`, `worst_of_digital`; `basket_moment_match`; `implied_correlation`, `index_variance`, `realised_correlation`; `quanto_forward`, `composite_vol`.

**Rules.** Correlation matrices are checked positive definite before use (the Cholesky factor fails otherwise); the quanto correlation is quoted against the rate in domestic per foreign units; every Monte Carlo number carries a standard error.

**Acceptance tests.** `code/firm/multiasset/tests/`: simulated assets are martingales with the specified correlation; worst-of and best-of reduce to the vanilla at correlation one; moment matching against simulation; [implied correlation](#def-dv-multi-asset-options-implied) inverts the index variance; [local correlation](#def-dv-multi-asset-options-local) reduces to the constant case; the quanto forward matches a simulation.

**Stretch.** Per-member local volatilities and a pairwise [local correlation](#def-dv-multi-asset-options-local) calibrated by [Markovian projection](https://one-course.com/books/quant/5/en/chapter/9-local-volatility#def-dv-local-volatility-projection); the cross-gamma report of One Quant Book 6.

Sources and further reading

- J. Driessen, P. Maenhout and G. Vilkov, “The price of correlation risk: evidence from equity options”, SSRN 673425 (2005); “Option-implied correlations and the price of correlation risk”, SSRN 2359380 (2013).
- A. Langnau, “Introduction into ‘local correlation modelling”’, arXiv 0909.3441 (2009).

## 17.8 Exercises

**Exercise 17.1 ★.**

Two assets with volatilities 20% and 30%, equally weighted, correlation 0.5: what is the basket’s volatility in the lognormal approximation?

**Solution of Exercise 17.1.**

$\sigma_B^2=0.25\times0.04+0.25\times0.09+2\times0.25\times0.5\times0.2\times0.3=0.0475$: 21.8%, below the average volatility of 25%.

**Exercise 17.2 ★.**

Why does a best-of call lose value as correlation rises while a basket call gains?

**Solution of Exercise 17.2.**

A best-of call pays on the largest performance, which is far above the average when the assets move independently and close to it when they move together: correlation shrinks its upside. A basket call pays on the average, whose volatility rises with correlation: less diversification, more option value.

**Exercise 17.3 ★.**

An index of two equally weighted members with volatilities 25% and 35% trades at 24%. Compute its [implied correlation](#def-dv-multi-asset-options-implied).

**Solution of Exercise 17.3.**

$\sum w_i^2\sigma_i^2=0.25(0.0625+0.1225)=0.04625$; $(\sum w_i\sigma_i)^2=0.09$, so the cross term is $0.04375$; $\bar\rho=(0.0576-0.04625)/0.04375=0.26$.

**Exercise 17.4 ★★.**

Derive the quanto forward $Fe^{-\rho\sigma_S\sigma_XT}$ from the requirement that $S_TX_T$, converted at the floating rate, earns the domestic rate.

**Solution of Exercise 17.4.**

Let $X$ be domestic per foreign. The domestic value of the foreign asset, $SX$, is a domestic traded asset, so under the domestic measure it drifts at $r_d-q$. The exchange rate drifts at $r_d-r_f$. By Itô, the drift of $S=(SX)/X$ is $(r_d-q)-(r_d-r_f)+\sigma_X^2-\operatorname{cov}(\ln SX,\ln X)/dt$, and $\operatorname{cov}(\ln SX,\ln X)/dt=\rho\sigma_S\sigma_X+\sigma_X^2$. The drift is $r_f-q-\rho\sigma_S\sigma_X$, so the quanto forward is the foreign forward times $e^{-\rho\sigma_S\sigma_XT}$.

**Exercise 17.5 ★★.**

Explain why a dealer who sells worst-of notes is short correlation, and name one listed instrument that offsets part of the position.

**Solution of Exercise 17.5.**

The note embeds a worst-of put sold by the client (the capital-at-risk leg) and a worst-of digital coupon bought by the client; the dealer is long the put and short the digital, and both lose value for the dealer as correlation rises. Index options, or index variance against single-stock variance (a reverse dispersion), carry the opposite correlation exposure and offset part of it.

**Exercise 17.6 ★★.**

Using the synthetic index, compute the dispersion P&L if realised correlation comes in at implied but every member’s volatility realises one point above its implied level (hedge amounts unchanged).

**Solution of Exercise 17.6.**

About $+28$: the hedge amounts cancel the first-order exposure to each member’s variance, and what is left is a second-order term, small against the 270 600 that a fifteen-point correlation gap produces.

**Exercise 17.7 ★★★.**

*Coding.* Price the three-share worst-of digital (all above 70%) under the synthetic index’s fitted [local correlation](#def-dv-multi-asset-options-local) function, applied to the three shares’ own basket. Compare with the constant-correlation prices of the chapter.

**Solution of Exercise 17.7.**

0.736, between the constant-correlation prices at 0.6 (0.707) and 0.8 (0.748), and well above the 0.673 at the index’s at-the-money [implied correlation](#def-dv-multi-asset-options-implied) of about 0.4: under [local correlation](#def-dv-multi-asset-options-local) the shares move together in exactly the falling scenarios that decide the digital.

**Exercise 17.8 ★★★.**

*Find the flaw.* “[Implied correlation](#def-dv-multi-asset-options-implied) has exceeded realised correlation on average, so a dispersion book that sells index variance and buys member variance is a riskless carry trade.”

**Solution of Exercise 17.8.**

It is a premium, not a carry: the book is short correlation and short index volatility, and it loses heavily in a sell-off, when correlation rises towards one and index volatility spikes. The published evidence also says that realistic trading frictions make the premium hard to capture. The average gap is compensation for that risk.

## 17.9 Problem: The Index and Its Members

**Problem 17.1.**

Weekend problem — implied correlation and a dispersion trade

A synthetic index holds twenty stocks in equal weights, with flat smiles and volatilities evenly spaced from 22% to 36%. The index’s own one-year smile is chapter 9’s. A dispersion desk considers selling index variance against member variance.

**Part I — The members.**

1. Compute $\sum w_i^2\sigma_i^2$ and the cross term $C$ .
2. What index volatility would a correlation of zero imply? Of one?
3. Why is the index volatility below the average member volatility?
4. Give the index’s volatility at 90, 100 and 110.
5. Give the index volatility that a constant correlation of 0.408 produces at 90 and 110.

**Part II — [Implied correlation](#def-dv-multi-asset-options-implied).**

6. Give the [implied correlation](#def-dv-multi-asset-options-implied) at 90, 100 and 110.
7. Give the variance-swap [implied correlation](#def-dv-multi-asset-options-implied) .
8. Explain the [correlation skew](#def-dv-multi-asset-options-skew) in one sentence.
9. Give the [local correlation](#def-dv-multi-asset-options-local) function that reprices the three strikes.
10. What [local correlation](#def-dv-multi-asset-options-local) does it assign 10% down and 10% up?

**Part III — The [dispersion trade](#def-dv-multi-asset-options-dispersion).**

11. For a [vega notional](https://one-course.com/books/quant/5/en/chapter/14-variance-swaps-and-volatility-derivatives#def-dv-variance-swaps-and-volatility-derivatives-veganot) of 100 000 on the index, give the [variance notional](https://one-course.com/books/quant/5/en/chapter/14-variance-swaps-and-volatility-derivatives#def-dv-variance-swaps-and-volatility-derivatives-varnot) .
12. Give the member hedge amounts’ formula and explain what they cancel.
13. Give the P&L if correlation realises fifteen points below implied.
14. Give the simulated mean, standard deviation and 5–95% range.
15. What would a [correlation swap](#def-dv-multi-asset-options-corrswap) struck at the implied 0.559 pay per unit notional?

**Part IV — Judgement.**

16. When does the [dispersion trade](#def-dv-multi-asset-options-dispersion) lose, and how much could it lose?
17. What does the published evidence say about implied against realised correlation?
18. Why might the premium persist?
19. State the *named result* : the [implied correlation](#def-dv-multi-asset-options-implied) of the synthetic index at three strikes, and the P&L of the [dispersion trade](#def-dv-multi-asset-options-dispersion) when realised correlation comes in fifteen points below implied.
20. In one sentence: what does an index option price that its members’ options do not?

**Solution of Problem 17.1.**

**1.** $\sum w_i^2\sigma_i^2=0.00430$ (43.0 variance points); $C=0.0798$ (798). **2.** 6.6% at zero correlation; 29.0% at one, the weighted average volatility. **3.** Diversification: with correlations below one, members’ moves partly cancel. **4.** 21.9%, 19.2% and 17.0%. **5.** 19.3% at both: a flat index smile. **6.** 0.549, 0.408 and 0.308. **7.** 0.559, from the index’s strip volatility of 22.1%. **8.** Index options below the money price the members falling together, which is a higher correlation. **9.** $\rho(x)=0.384-3.62x+16.3x^2$, clipped to $[0,1]$. **10.** 0.91 at $x=-0.1$ and 0.19 at $x=+0.1$. **11.** $100\,000/(2\times22.12)=2\,261$ per variance point. **12.** $N\bigl(w_i^2(1-\bar\rho)+\bar\rho w_i\sum_jw_j\sigma_j/\sigma_i\bigr)$: the index variance’s sensitivity to each member’s variance at the [implied correlation](#def-dv-multi-asset-options-implied); they cancel the first-order exposure to each member’s volatility. **13.** $2\,261\times0.15\times798=270\,600$. **14.** 271 000, 34 000, from 215 000 to 326 000. **15.** The realised average correlation, 0.409, minus 0.559: $-0.15$ per unit notional to the long side, $+0.15$ to the short. **16.** When correlation realises above implied, typically in a sell-off; the loss is $N(\hat\rho-\bar\rho)C$ plus the residual from volatility moves, and correlation can go to one: about $2\,261\times0.441\times798\approx800\,000$ from correlation alone if members realise their implied volatilities. **17.** Implied has averaged above realised: 39.5% against 32.5% for the S&P 500, 46.0% against 35.5% for the Dow Jones 30, in Driessen, Maenhout and Vilkov’s sample. **18.** Investors pay to hedge correlation spikes (they are long index puts), and the trade that sells it needs many option positions and bears crash risk; frictions limit arbitrage. **19.** [Implied correlation](#def-dv-multi-asset-options-implied) 0.549 at 90, 0.408 at 100 and 0.308 at 110 (0.559 in variance terms); the [dispersion trade](#def-dv-multi-asset-options-dispersion) makes 270 600 on a [vega notional](https://one-course.com/books/quant/5/en/chapter/14-variance-swaps-and-volatility-derivatives#def-dv-variance-swaps-and-volatility-derivatives-veganot) of 100 000 when realised correlation is fifteen points below implied (simulated 271 000 $\pm$ 34 000). **20.** The joint behaviour of its members, above all how their correlation changes when the market falls.

## 17.10 Interview questions

**Interview question 17.1 ★ trader, researcher.**

How does correlation affect the prices of a basket call, a worst-of put and a best-of call?

**Solution of Interview question 17.1.**

A basket call’s value rises with correlation (less diversification, more volatility of the average); a worst-of put and a best-of call fall with it (the extremes of the performances move closer to the average as correlation rises).

*What the interviewer is looking for: the three signs and the mechanism.*

**Interview question 17.2 ★ researcher.**

Define [implied correlation](#def-dv-multi-asset-options-implied). Why is it higher for low strikes?

**Solution of Interview question 17.2.**

The single correlation that reconciles the index’s implied variance with its members’: $(\sigma_I^2-\sum w_i^2\sigma_i^2)/(\text{cross
term})$. Index skews are steeper than single-stock skews, so the index’s volatility at low strikes implies a higher correlation: the market prices stocks falling together.

*What the interviewer is looking for: the formula and the [correlation skew](#def-dv-multi-asset-options-skew).*

**Interview question 17.3 ★★ trader.**

Explain a [dispersion trade](#def-dv-multi-asset-options-dispersion): the legs, the weights and what you are exposed to.

**Solution of Interview question 17.3.**

Short index variance (or straddles), long members’ variance in amounts that cancel each member’s volatility exposure at the [implied correlation](#def-dv-multi-asset-options-implied). The residual is short correlation: P&L $\approx N(\bar\rho-\hat\rho)C$. Exposed to correlation spikes in sell-offs, to the volatility of the weights and hedge ratios, and to transaction costs across many legs.

*What the interviewer is looking for: the legs, the weights and the residual exposure.*

**Interview question 17.4 ★★ researcher, developer.**

Derive the [quanto adjustment](#def-dv-multi-asset-options-quantoadj). What is the sign for a Japanese index paid in dollars if the index tends to fall when the yen strengthens?

**Solution of Interview question 17.4.**

Under the domestic measure the foreign asset’s drift is $r_f-q-\rho\sigma_S\sigma_X$, with $X$ in domestic per foreign units (Girsanov on the change of numeraire). For a yen index paid in dollars, $X$ is dollars per yen; if the index falls when the yen strengthens, $\rho<0$, and the quanto forward is above the yen forward.

*What the interviewer is looking for: the drift and the sign convention.*

**Interview question 17.5 ★★ developer.**

Your correlation matrix for 30 assets fails the Cholesky factorisation. Why, and what do you do?

**Solution of Interview question 17.5.**

The matrix is not positive definite: pairwise estimates from different windows or data, or hand-set entries, are inconsistent. Project it onto the nearest correlation matrix (clip negative eigenvalues and rescale the diagonal, or an alternating-projections method), or build it from a factor model, and check the change in prices.

*What the interviewer is looking for: positive definiteness and a principled repair.*

**Interview question 17.6 ★★★ risk, trader.**

A book of worst-of autocallables is short correlation. How would you measure, hedge and reserve for that exposure?

**Solution of Interview question 17.6.**

Measure: the book’s value change for a shift of all pairwise correlations (and for a [correlation skew](#def-dv-multi-asset-options-skew) shift), by bucket of maturity and moneyness of the worst performer. Hedge: with index options or index variance against single-stock variance, which carry correlation exposure of the opposite sign, knowing the hedge is imperfect (index correlation is not the basket’s). Reserve: for the unhedged correlation at a stressed level and for the model (local against constant correlation).

*What the interviewer is looking for: a correlation sensitivity, a proxy hedge and a reserve.*