**Review of ∆ and Γ hedging**

**∆ hedging**

Recall the Delta of a portfolio Π_{t} = Π(_{t}, S_{t}) is the rate of change of its value with respect to the price of the underlying asset S_{t}:

∆_{t} = ∆_{t}(Π) := ∂_{S}Π(*t*, S_{t}).

∆-hedging involves creating a delta-neutral position: ∆_{t}(Π) = 0.

Since the delta of the underlying is simply 1, can ∆-neutralize an initial position Π^{0} by adding “−∆_{t}(Π^{0})” in the underlying asset.

**Γ hedging**

Recall the Gamma of a portfolio Π_{t} = Π(*t*, S_{t}) is the rate of change of the delta ∆_{t} = ∆(*t*, S_{t}) with respect to the underlying:

Γ-hedging involves creating a gamma-neutral position: Γ_{t}(Π) = 0.

Since gamma of the underlying is 0, can Γ-neutralize a position Π^{0} by adding “−∆_{t}(Π^{0})” in the underlying asset, giving new position Π^{1}.

However, moving from Π^{0} to Π^{1} can destroy delta neutrality (if any).

Trick: find yet another instrument with delta “−∆_{t}(Π^{1})” but with zero gamma to add to the portfolio to arrive at Π^{2} that is ∆ and Γ neutral.

WARNING: Not always feasible to maintain Γ neutrality.

**PnL of ∆-hedged position**

**Theta as a proxy for Gamma**

Recall value f = f(t, S) of a call satisfies BSM PDE

_{t}:= f(t, S

_{t}) − ∆

_{t}S

_{t}is exactly the ∆-hedged position, so we can treat the righthandside as 0 (or by taking r = 0 on small time-scale). This shows Theta can be proxy for Gamma:

This relationship between the Gamma and Theta of an option is fundamental for making money with options.

**Infinitesimal rebalancing PnL governed by Γ**

Consider the infinitesimal PnL of a portfolio consisting only of a call:

where σ_{real} is realized volatility.

In contrast, Greeks Θ_{t}, Γ_{t}, ∆_{t} computed at implied vol σ_{imp}

Putting everything together, the hedged position

Π(t, St) := f(t, St) − ∆tSt,

exhibits the “infinitesimal rebalancing PnL” (see Sinclair [4])

**Infinitesimal PnL governed by Gamma and Vega**

Can take step further with movement σ_{imp} → σ′_{imp} of implied vol.

Writing ∆-hedged position Π_{t} = Π(t, S_{t}, σ_{imp}),

where V_{t} = ∂_{σ}*f*(t, S_{t}, σ_{imp}) is the vega of the call.

Can take σ′_{imp} ≈ σ_{real} (market will match realized vol to some degree)

Thus, a goal of options trading: identify “volatility mismatches” so

- Long Gamma/Vega makes money if σ
_{real}> σ_{imp}⋆ - Short Gamma/Vega makes money if σ
_{imp}> σ_{real}⋆

**Summary**: can make money with hedged position from discrepancies in either realized or implied volatilities.

**Correlation of spot and vol: hedging vega with delta**

**Relationship between smile and spot correlation with vol**

Negative correlation of prices and vol corresponds to heavy left tail and thin right tail of implied distribution of S_{T}; this is because

- if S ↓ accompanied by σ ↑, then greater declines more likely
- if S ↑ accompanied by σ ↓, then greater increases less likely

Accordingly, the smile will have a negative skew (slope).

Positive correlation of prices and vol corresponds to thin left tail and heavy right tail of implied distribution of S_{T}; this is because

- if S ↓ and σ ↓ together, then greater declines become less likely
- if S ↑ and σ ↑ together, then greater increases more likely

Accordingly, the smile will have a positive skew (slope).

**Illustration of negative correlation of spot and vol (Hull, [3])**

**Correlation of vol and spot in cryptocurrencies**

In crypto, strong linear relationship between spot and vol observed in options with longer tenors (e.g., fix 182 days to maturity)

Calculate deltas that fixed strike has yesterday for each tenor Predicted strike: one with same delta as fixed strike had yesterday y-axis is difference between predicted and true vols wrt strikes. Darker colors = more recent…appears to exhibit regime switching!

**Correlation of vol and spot in cryptocurrencies**

Such structure forms with tenors as small as 14 (or even 7) days:

**Instruments used for Hedging in crypto derivatives**

Note transactions have to go through the blockchain. Consequently, it is VERY expensive to transact. Potential solution: create a Futures Market.

- Typically a complex, exchange dependent settlement procedure:
- Perpetual Future is VERY unique to the crypto market.

- There is opportunity for crypto carry trade!!
- There is term structure that is not necessarily consistent

**Inconsistent term structure**

**Key Takeaway**

There is a particular interaction between vega and delta in crypto. This interaction is atypical, so money can be made!

**Vega hedging**

**Model Driven Vega Hedging**

Vega hedging: closing vega risk at one strike using a different one.

– Standard in equities, which employ stochastic vol models

As people say (see Gatheral [2]), “options are hedged with options.”

More precisely, recall prototypical two factor stochastic vol model:

Issue: two sources of randomness to deal with for BSM argument

Set up portfolio Π = V − ∆S − ∆^{1}V^{1} of two options V, V^{1}, so that

main option V = V(t, S, v) hedged with auxiliary option V^{1} = V^{1} (t, S, v).

1. Delta hedge: ∂_{S}V − ∆^{1}∂_{S}V^{1} − ∆ = 0 (eliminate dS noise)

2. Vega hedge: ∂_{v}V − ∆^{1}∂_{v}V^{1} = 0 (eliminate dv noise)

Application of Ito’s confirms riskless position: dΠ = rΠ dt.

**Valuation equation**

Argument of previous slide leads to valuation equation for

V = V(t, S, v) analogous to BSM PDE:

for some function f, usually written as

where ϕ = ϕ(*t, S, V*) is “market price of volatility risk”

(and we recall *α*, *β* are drift and volatility factor for variance).

**Delta hedging with stochastic vol**

The delta from a stochastic volatility model implies a type of

correlation between vol and spot!!

Recall a few well known facts:

- If ρ = 0, then any stochastic vol model implies “sticky delta”.
- If model is mainly local vol, then delta is mostly “sticky strike.”

Crypto lives in the middle of these:

- Short term vols are mostly sticky delta
- Low delta strikes are mostly sticky strike
- Interestingly tends to be the case that ρ > 0
- recall ρ controls the skew of the surface
- thus can have positive volatility skew as argued before

In general impossible to fit stochastic vol model well to market.

Accordingly, we resort to some tricks…

**Example of vol. surface in BTC**

There is a particular interaction between vega and delta in crypto. This interaction is atypical, so money can be made!

**Statistical Vega Hedging**

Do a dimensionality reduction of the term structure on a surface to

ideally project all the risks at different tenors, to a single one.

At Arbelos, we use 3 PCA’s to do so:

**Optimal execution and nonlinear price impact**

**Optimization problem for optimal execution**

Rebalancing to preserve ∆ and Γ neutrality requires making trades

Such trades can have adverse effect on market microstructure

Must trade in careful way (say, liquidate for concreteness)

Following Ch.8 of Cartea, Jaimungal, Penalva [1], agent optimizes

over admissible controls *A*, subject to dynamical constraint

where:

**Optimal execution**

Approximation with *T* >> 0 is to take premium or depth δ to be

Note this formula assumes a linear penalty and linear price impact.

**Illustration of Price Impact (Ch.6, Cartea, Jaimungal, Penalva [1])**

Price impact can sometimes be reasonably modelled linearly:

**Issues with price impact in crypto**

But graphs of imbalance clearly shows nonlinear price impact!

**References**

Á. Cartea, S. Jaimungal, and J. Penalva.

Algorithmic and high-frequency trading.

Cambridge University Press, 2015.

J. Gatheral.

The volatility surface: a practitioner’s guide.

John Wiley & Sons, 2011.

J. C. Hull.

Options, futures, and other derivatives.

Pearson Education, 2022.

E. Sinclair.

Positional Option Trading: An Advanced Guide.

John Wiley & Sons, 2020.

**Case study of an exotic structure: One Touch**

**Terminology**

Consider problem of pricing a contingent claim with payoff

i.e., an American Binary Call with barrier H or one-touch.

In Black-Scholes, “dynamic hedging” refers to continuously trading.

By contrast, “static hedging” refers to discretely many trades.

– For our purposes, usually consists of at most two trades.

**Symmetric Case**

Suppose the price process S_{t} is Brownian motion: dS_{t} = σ dB_{t}.

Then the distribution of S_{T} will be symmetric in the sense that

- upon hitting H, binaries with payoff 1{S
_{T}≥ H} are worth $0.5.

But, upon hitting H, the one-touch will “knock-in” and pay $1.

Hence, at H, a one-touch is worth the same as 2 binaries.

Static replication strategy for one-touch in symmetric case:

- Buy two binary calls struck at H at inception t = 0.
- If price ever touches barrier H, dissolve portfolio to pocket $1.

Summary:

**Driftless Case**

Suppose S_{t} exhibits driftless log-normal dynamics with r − q = 0:

dS_{t} = σ S_{t} dB_{t}

Distribution of S_{T} has weaker positive skew that must be corrected!

Symmetry to exploit: at S = H, d_{2} = −d_{1} since log(S/H) = 0.

Then value of a call with strike K = H at S_{t} = H is given by

S_{t}N(d_{1}) − KN(d_{2}) = H · [N(−d_{2}) − N(d_{2})] = H · [1 − 2N(d_{2})].

But we know N(d_{2}) is the value of a binary call struck at H.

Rearranging, this implies two binaries and 1/H calls all struck at H is exactly equal to $1 upon hitting the barrier H.

Thus, static replication strategy for one-touch in driftless case:

- Buy two binary calls and 1/H calls struck at H.
- If price ever touches barrier, dissolve portfolio to pocket $1.

**Issue with General case**

Suppose S_{t} exhibits more general log-normal dynamics:

Distribution of S_{T} now has even more complicated skew!

Symmetry to exploit now much more complicated.

For example, *put-call symmetry* now involves exponent

Produces replication terms nonlinear in underlying (see next slide).

– This symmetry is now much less intuitive to work with.

– No finite portfolio of vanillas provides a perfect static hedge.

**Solutions for General case**

There is an elegant solution.

– But solution requires passing by some inelegant formulas.

One can check the analytic formula for a one-touch struck at H is

TRICK: Simply let T ↓ 0 here to find infinitesimal European payoff:

Reduces exactly to previous driftless case formula when r = q = 0.

**Finite static hedge for General case**

Unfortunately, as indicated, this last payoff features nonlinearities.

Solution: Settle for first order approximation of this formula:

PROS: Formula practically provides an approximate static hedging

strategy that is readily implemented and remembered.

CONS: Simple replicating portfolio will not give as accurate of a price

as PDE methods based on LSV models (forthcoming work).

**Vega and skew in binaries**

Just saw “one touches are roughly twice a binary plus a correction.”

To price a binary call, recall

Recall particular interaction between vega and delta in crypto

Aim to exploit this interaction here (forthcoming work).