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A Day in the Life of a Trader

Antonio Mannolini, Ph.D

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Introduction

1 Examples of Market Payoffs

2 Structured Products and the Management of their Risks

3 A Basic review of Black and Scholes

4 Little Recap

Sticky Smile e Floating Smile

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Examples of Market Payoffs

Understanding risk profiles of a payoff isconditio sine qua non

for a mathematical representation of financial derivatives

Long Call: Delta-Gamma-Theta-Vega Long Put:Delta-Gamma-Theta-Vega A volatility basic structure: straddle Call Spread (Cap spread for IRD traders)

Back to the swaps: they can be used to hedge directional risk, curve shape and basis risk

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Examples of Market Payoffs

Understanding risk profiles of a payoff isconditio sine qua non

for a mathematical representation of financial derivatives Long Call: Delta-Gamma-Theta-Vega

Long Put:Delta-Gamma-Theta-Vega A volatility basic structure: straddle Call Spread (Cap spread for IRD traders)

Back to the swaps: they can be used to hedge directional risk, curve shape and basis risk

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Examples of Market Payoffs II

Flattening and Steepening: a bet on the curve shape Example: Es: what doespay10−30 mean?

Cap & Floors

Collars : which kinds of risk to hedge entails? Swaptions

Bermudian Swaptions and callable bonds A somewhat strange contract: Wedge:

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Example of a Structured Swap

C, (Customer) pays 4% ifEuribor6m <5.5% else E6m. Please how can you replicate this using simple instruments?

Answer: it can be represented as a swap in which B, the bank, paysEuribor6mand receives 4% and at the same timebuysfrom

Ctwo caps: One plain struck at 5.5 and onedigital, struck at 5.5, which pays 1.5%

Risks forB: directional risk (short rates)coming from paying

E6mon the swap It is in part compensated by the long directional on the cap

Bis also longν why?

On the other sideCis short the rates on the 2 caps and is also short ν

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How to hedge a structured swap:

The Bank must hedge the delta risk via futures (which however entail basis risk) or via swaps with opposite sign on the market The vega risk can be hedged via straddles or strangles

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Caps & Floors written on CMS

Payout which depends on the difference between two swap rates It is a bet on the shape of the yield curve

It can be a flattening beto or a steepening bet Correlation is important

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Mathematical formulation I

Let us expand in a Taylor series the price of a call option

C(S+∆S, t+∆t) =C(S, t)+∂C

∂t∆t+ ∂C

∂S∆S+0.5∗ ∂2C

∂S2(∆S)+. . .

(1) Define Γ = ∂∂S2C2

Recall that in the binomial approximation of the model

∆S=σS√∆t (2)

from which we can deduce

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Continues...

TheP&L of a delta hedged call and in which we suppose to know for certain the value of the future volatility, denoted by Σ, `

e is given by

1 2Γ(∆S)

2= 1

2Γ(Σ

2S2δt) (4)

while the loss because of time decay is given by (Θ = ∂C∂t) Hence the evolution of theP&L is given by

dP&L=d(C−∆S) = 1 2Γ(Σ

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Summing up

Let us check two basic facts of the derivation precedente:

∆S=σS√∆t (6)

over time this gives a lognormal distribution, but period by period the model does not admit exceptional moves.

While the position in the stock is linear, the one long call benefits from any market movements (positive gamma ), which yields a quadraticP&L∆S.

1 2Γ(∆S)

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Recap

These relations, besides being the fundamental issues for the writing of the classical PDE according toB&S, tell us that if we know the future volatility theP&Lwould be deterministic, regardless of the direction of the underlying

Hence if the realized volatility isσrinstead of Σ we can write

1 2Γ(σ

2

r−Σ

2)S2δt (8)

The true bet on a long call is that realized volatility will be higher than the one used for calculating the buying price In trader’s slang, we are long Γ

InB&S there exist a unique parameter for eachK andT, because this is stock volatility, not the option one.

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Making sense of some divergences between the Model

and the Market

If theB&Smodel were a perfect representation of market reality, implied volatility would represent the degree of uncertainty of the

underlying

Indeed there are factors neglected by the model, features of the option market itself. If you buy put deep OTM you buy a sort of insurance against market crashes.

If there is panic around these will be very costly. These factors are not represented in the model

An option can always be replicated as a combination of other options

A constant or time dependent but deterministic volatility is the representation of a benchmark, idealized market

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Divergences between Market and Model II

By construction in the standard modelexceptional movesare not admitted when we collapse the time interval.

Only with time diffusion generates a desired probability distribution

hence to get some more realistic features of the market we need to change the probabilistic representation of the world

This is the reason why constant or deterministic time dependent volatility ( hence lognormal distribution of the Asset) must be abandoned

The same fact that dealers useB&S BUT change volatility with the movement of the strike testifies that in the picture some is missing

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How the market uses Black and Scholes

Traders use the volatility parameter, which is the moreopaque

parameter in the formula to express what the model does not explain ( institutional features of the option market and motivations to hold options)

On the market you observe the implied volatility surface σ(T, K). It represent a snapshot of the markets, in the same manner in which the Yield Curve is a snapshot of the fixed income world. But in order to hedge an Option Books you need to express a view on the movie ( you are a trader!)

A model should express a view on the future evolution (not a prediction, which is impossible) of the volatility surface in order to minimize P&LJumps and to provide a bit more accurate hedges ( or in the trades control)

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Inputs I

Repetita iuvant: given two options on the market dC(t, T, S, K1) eC(t, t, S, K2), is there a single numberσsuch that

C(t, T, S, K1) =B&S(t, T, r, S, K1, σ) (9) and

C(t, T, S, K2) =B&S(t, T, r, S, K2, σ) (10) NEIN! (Brigo Mercurio, 2006)

On the market you seeσ(T, K). Unfortunately you need σ(T, S). One is then forced to do ad hoc assumptions about the function connectingσ(T, S) andσ(T, K) and the estimate the parameters To see a possible (not happy, indeed) ending of this story please read Rebonato, Volatility and Correlation (2004)

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Definition of two Polar cases

Sticky strike Smile: σ(T, K) depends only on the strike and not on the level of the underlying or the moneyness

Floating smile: σ(T, K) depends only on KS e and hence follow the underlying

As usual on the market we observe an intermediate behaviour between the two extremes

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References

Rebonato, Riccardo,

Volatility and Correlation, second edition

References

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