A Day in the Life of a Trader
Antonio Mannolini, Ph.DIntroduction
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
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
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
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:
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
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
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
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
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
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)
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.
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
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
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)
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)
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
References
Rebonato, Riccardo,
Volatility and Correlation, second edition