# A Day in the Life of a Trader

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1 Siena, April 2014

2 Introduction 1 Examples of Market Payoffs Sticky Smile e Floating Smile 5

3 Examples of Market Payoffs Understanding risk profiles of a payoff is conditio 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 How to hedge a swap

4 Examples of Market Payoffs Understanding risk profiles of a payoff is conditio 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 How to hedge a swap

5 Examples of Market Payoffs II Flattening and Steepening: a bet on the curve shape Example: Es: what does pay mean? Cap & Floors Collars : which kinds of risk to hedge entails? Swaptions Bermudian Swaptions and callable bonds A somewhat strange contract: Wedge:

6 Example of a Structured Swap C, (Customer) pays 4% if Euribor6m < 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, pays Euribor6m and receives 4% and at the same time buys from C two caps: One plain struck at 5.5 and one digital, struck at 5.5, which pays 1.5% Risks for B: directional risk (short rates)coming from paying E6m on the swap It is in part compensated by the long directional on the cap B is also long ν why? On the other side C is short the rates on the 2 caps and is also short ν

7 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

8 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

9 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 C t+ t S S C S 2 ( S)+... (1) Define Γ = 2 C S 2 Recall that in the binomial approximation of the model from which we can deduce S = σs t (2) ( S) 2 = Σ 2 S 2 t (3)

10 Continues... The P &L of a delta hedged call and in which we suppose to know for certain the value of the future volatility, denoted by Σ, è is given by 1 2 Γ( S)2 = 1 2 Γ(Σ2 S 2 δt) (4) while the loss because of time decay is given by (Θ = C t ) Hence the evolution of the P &L is given by dp &L = d(c S) = 1 2 Γ(Σ2 S 2 t) + Θ( t) (5)

11 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 quadratic P &L S. 1 2 Γ( S)2 (7)

12 Recap These relations, besides being the fundamental issues for the writing of the classical PDE according to B&S, tell us that if we know the future volatility the P &L would be deterministic, regardless of the direction of the underlying Hence if the realized volatility is σ r instead of Σ we can write 1 2 Γ(σ2 r Σ 2 )S 2 δt (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 In B&S there exist a unique parameter for each K and T, because this is stock volatility, not the option one. To represent smiles we need a different formulation for the dynamics of the underlying and a different distribution

13 Making sense of some divergences between the Model and the Market If the B&S model 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 Hence the Lognormal distribution represent a very particular case...

14 Divergences between Market and Model II By construction in the standard model exceptional 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 use B&S BUT change volatility with the movement of the strike testifies that in the picture some is missing

15 How the market uses Black and Scholes Traders use the volatility parameter, which is the more opaque 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 &L Jumps and to provide a bit more accurate hedges ( or in the trades control) Unexpected P &L jumps are the true Traders BANE! If yoo lose money (but even if you make it) never tell your boss that you do not know HOW.

16 Inputs I Sticky Smile e Floating Smile Repetita iuvant: given two options on the market d C(t, T, S, K 1 ) e C(t, t, S, K 2 ), is there a single number σ such that and C(t, T, S, K 1 ) = B&S(t, T, r, S, K 1, σ) (9) C(t, T, S, K 2 ) = B&S(t, T, r, S, K 2, σ) (10) NEIN! (Brigo Mercurio, 2006) On the market you see σ(t, K). Unfortunately you need σ(t, S). One is then forced to do ad hoc assumptions about the function connecting σ(t, S) and σ(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)

17 Definition of two Polar cases Sticky Smile e Floating Smile 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 S K e and hence follow the underlying As usual on the market we observe an intermediate behaviour between the two extremes

18 Rebonato, Riccardo, Volatility and Correlation, second edition W iley&sons, 2004

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lambda (options leverage) INTRODUCTION The lambda (option leverage) is defined as the percentage change in an option price divided by the percentage change in an underlying price. Also called effective