Nikos SKANTZOS 2010. Stochastic methods in Finance. What is the fair price of an option? Consider a call option on an asset. Delta hedging: For every time interval: Buy/Sell the asset to make the position: Call – Spot * nbr Assets insensitive to variations of the Spot
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Stochastic methods in Finance
What is the fair price of an option?
Consider a call option on an asset
Delta hedging:
For every time interval:
Buy/Sell the asset to make the position:
Call – Spot * nbr Assets
insensitive to variations of the Spot
Fair price is the amount spent during delta-hedging:
Option Price =
=Δ1+Δ2+ Δ3 +Δ4+ Δ5
It is fairbecause that is how much we spent!
Δ1
Δ2
Δ3
Δ4
Δ5
no-arbitrage & delta hedging
Assume a Spot Dynamics
σ: size of fluctuations
μ: steepness of main trend
ΔWt: random variable (pos/neg)
expected spot = market forward
is Gaussian normal of zero mean, variance ~ T
= discounted average of the call-payoff over various realizations of final spot
Solution
Price = value of position at maturity – value of cash-flow at maturity
Δ=0, Delta-neutral value:
if S S+dS then portfolio value does not change
Vega=0, Vega-neutral value:
if σσ+dσ then portfolio value does not change
“Greeks” measure sensitivity of portfolio value
portfolio value
Delta neutral position,
∂Portfolio/∂S=0
S
S+dS
S
Comparison with market:
BS < MtM when in/out of the money
Plug MtM in BS formula to calculate volatility smile
Inverse calculation “implied vol”
Market observable
Main causes:
Fat tails:
Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes
the accumulated vol
Moneyness: Δ=DF1·N(d1)
Sensitivity of Delta to spot is maximum
1 underlying is needed to hedge
Linear regime: S-K
Δ<0, price gets smaller if spot increases
Spot is far from barrier and far from OTM: risk is minimum, price is maximum
Vanna: Sensitivity of Vega with respect to Spot
Volga: Sensitivity of Vega with respect to Vol
E[Σi Si] , E[ (Σi Si)2 ],
(time to maturity)
(because Wt1 is independent of Wt2-Wt1)
E[St1 ∙ St2] = S02 exp[r∙(t1+t2) + σ2 ∙min(t1,t2)]
(something the BS implied vol is not!)
and its time/strike derivatives
Maturity, Strike
Rule of thumb:
Local vol varies with index level twice as fast as implied vol varies with strike
(Derman & Kani)
Sfinal
Sinitial
Example:
Take smile quotes
Build local-vol
Use them in simulation and price vanillas
Compare resulting price of vanillas vs market quotes
(in smile terms)
EURUSD market
Lines: market quotes
Markers: LV pricer
Blue: 3 years maturity
Green: 5 years maturity
These equal to the volatility
Dupire Local Vol is therefore not a real stochastic model
(model output – market observable)2
(model ATM vol – market ATM vol)2
Processes
Affecting skew:
Affecting overall shift in vol:
Three factor model in FOREX:
spot + domestic/foreign rates
To replicate FX volatilities match
FX,mktwith FX,model
Θ(s) is a function of all model parameters: FX,d,f,ad,af
Hull-White is often coupled to another underlying
FX vol + IR vols up to a certain date have exceeded the FX-model vol.
Time-dependent parameters (piecewise constant)
parameter
time
δ = benefit of direct access – cost of carry
Not observable but related to physical ownership of asset
(although δt can be pos/neg)
no need for stochastic interest rates
Miltersen extension:
Time-dependent parameters
This model adds a new element to the stochastic models: jumps in spot
Motivated by real historic data
Disadvantages
Risk cannot be eliminated by delta-hedging as in BS
Hedging strategy is not clear
Advantages
Extra term to the Black-Scholes process:
Then,
Therefore, Y: size of the jump
Jump size & jump times:
Random variables
Vega Vanna Volga
are responsible the smile impact
zero out the Vega,Vanna,Volga of exotic option at hand
(easy BS computations from the market-quoted volatilities)
+ Smile impact of portfolio of vanillas
KATM
KATM
K25ΔP
K25ΔC
KATM
K25ΔP
K25ΔC
ATM Straddle
25ΔRisk-Reversal
25ΔButterfly
RR carries mainly Vanna BF carries mainly Volga
Vanna-Volga market price is
XVV = XBS
+ wATM ∙ (ATMmkt-ATMBS)
+ wRR ∙ (RRmkt-RRBS)
+ wBF ∙ (BFmkt-BFBS)
Other market practices exist
Further weighting to correct price when spot is near barrier
It reproduces vanilla smile accurately
"Vanna-Volga methods in FX derivatives: from theory to market practise“
Int J Theor Appl Fin (to appear)
Local Vol Heston Vanna-Volga Multifactor Local-Stoch Vol
Pros
Cons
and a liquid exotic market (OT)
W. Schoutens, E. Simons, and J. Tistaert,
"A Perfect calibration! Now what?“ Wilmott Magazine, March 2004
0%-100% (away-close to barrier)
OT tables depend on
Monte Carlo
Advantages:
Easy to implement
Easy for multi-factor processes
Easy for complex payoffs
Disadvantages
Not accurate enough
CPU inefficient
Greeks not stable/accurate
American exercise: difficult
Depends on quality of
random number generator
PDE
Disadvantages
Hard to implement
Hard for multi-factor processes
Hard for complex payoffs
Advantages
Very accurate
CPU efficient
Greeks stable/accurate
American exercise: very easy
Independent of random numbers
Monte Carlo
Based on discounted average payoff over realizations of spot:
Outline of Monte Carlo simulation
Partial Differential Equation (PDE)
Based on alternative formulation of option price problem
Idea is to rewrite it in discrete terms, e.g. with t+=t+Δt, S+=S+ΔS
Apply payoff at maturity and solve PDE backwards till today
Spot
S0
K
time
today
maturity
Simulations require at each time step a random number
Statistics: for example, normal-Gaussian (for lognormal process) mean=0 variance=1
This means that if we sum all random numbers we should get 0 and st.dev.=1
In practise, we draw uniform random numbers in [0,1] and convert them to Normal-Gaussian random numbers using the normal inverse cumulative function
A typical simulation requires 105 paths & 102 steps: 107 random numbers
Deviations away from the required statistics produce unwanted bias in option price
Random numbers do not fill in the space uniformly as they should !
This effect is more pronounced as the number of dimensions (=number of steps * number of paths) increases
there is a formula behind taking as input the computer clock
LDRN fill the [0,1] space homogenously.
Gaussian cumulative function
homogenous numbers form [0,1]
1
Gaussian probability function
0
Non-homogenous numbers in (-∞ ∞)
“Peak” implies that more points should be sampled from here
Higher density of points here
, 2)
1.000
0.900
0.800
0.700
0.600
0.500
0.400
0.300
0.200
0.100
0.000
1.000
0.000
0.200
0.400
0.600
0.800
Plot pairs of columns
Draw (n x m) table of Sobol’ numbers
(1,2)(10,20)
Nbr Steps
Nbr Paths
(13,40)(20,881)
Non-uniform filling for large dimensions!
Payoff at maturity is alive if
Barrier A has not been hit
Barrier B has been hit
=probability of not hitting
Prob(A is !hit) =
Prob(A is !hit in [t1,t2])∙
Prob(A is !hit in [t2,t3])
Prob(A is !hit AND B is !hit) =
=Prob(A is !hit in [t1,t2])∙
Prob(A AND B are !hit in [t2,t3]) ∙
Prob(B is !hit in [t3,t5])
S0
K
today
today
t
maturity
Naïve Monte Carlo is clearly impractical
Cashflows
Spot Paths
Npaths
Nsteps
Out-of-the-money
In-the-money
Cashflows
Spot Paths
Out-of-the-money
In-the-money
* CF=(Sthis path(T)-K)+
Cashflows
Spot Paths
Y1(T-Δt)
Y3(T-Δt)=0
Y5(T-Δt)=0
Y6(T-Δt)
Ypath(T-Δt) = DF(T-Δt,T) ∙ CF(T)
E(S) = a0+a1∙S +a2∙S2
E(S)
Y
S
(the option can be exercised only once)
(the option can be exercised only once)
Main idea:
Note: