Nikos SKANTZOS 2010

1. Nikos SKANTZOS2010 Stochastic methods in Finance

2. 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! Fair price Δ1 Δ2 Δ3 Δ4 Δ5

3. Black-Scholes: the mother model • Black-Scholes based option-pricing on no-arbitrage & delta hedging • Previously pricing was based mainly on intuition and risk-based calculations • Fair value of securities was unknown

4. Black-Scholes: main ideas Assume a Spot Dynamics • The rule for updating the spot has two terms: • Drift : the spot follows a main trend • Vol : the spot fluctuates around the main trend • Black-Scholes assume as update rule • This process is a “lognormal” process: • “lognormal” means that drift and fluctuations are proportional to St σ: size of fluctuations μ: steepness of main trend ΔWt: random variable (pos/neg)

5. Black-Scholes: assumptions • No-arbitrage  drift  = risk-free rate • Impose no-arbitrage by requiring that expected spot = market forward • Calculations simplify if • fluctuations are normal: is Gaussian normal of zero mean, variance ~ T • Volatility (size of fluctuations)  is assumed constant • Risk-free rate is assumed constant • No-transaction costs, underlying is liquid, etc

6. Black-Scholes formula • Call option = e–rT∙ E[ max(S(T)-K,0)] = discounted average of the call-payoff over various realizations of final spot Solution

7. Interpretation of BS formula Price = value of position at maturity – value of cash-flow at maturity

8. 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 How much does the portfolio value change when spot changes? portfolio value Delta neutral position, ∂Portfolio/∂S=0 S S+dS S

9. Comparison with market: BS < MtM when in/out of the money Plug MtM in BS formula to calculate volatility smile Inverse calculation  “implied vol” Black-Scholes vs market

10. Spot probability density • Distribution of terminal spot (given initial spot) obtained from Market observable Main causes: • Spot dynamics is not lognormal • Spot fluctuations (vol) are not constant Fat tails: Market implies that the probability that the spot visits low-spot values is higher than what is implied by Black-Scholes

11. What information does the smile give ? • It represents the price of vanillas • Take the vol at a given strike • Insert it to Black-Scholes formula • Obtain the vanilla market price • It is not the volatility of the spot dynamics • It does not give any information about the spot dynamics • even if we combine smiles of various tenors • Therefore it cannot be used (directly) to price path-dependent options • The quoted BS implied-vol is an artificial volatility • “wrong quote into the wrong formula to give the right price” (R.Rebonato) • If there was an instantaneous volatility σ(t), the BS could be interpreted as the accumulated vol

12. Types of smile quotes • The smile is a static representation of the implied volatilities at a given moment of time • What if the spot changes? • Sticky delta: if spot changes, implied vol of a given “moneyness” doesn’t change • Sticky strike: if spot changes, implied vol of a given strike doesn’t change Moneyness: Δ=DF1·N(d1)

13. Spotladders: price, delta & gamma • Vanilla • Knock-out spot=1.28 strike=1.25 barrier=1.5 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

14. Spotladders: vega, vanna & volga Vanna: Sensitivity of Vega with respect to Spot • Vanilla • Knock-out spot=1.28 strike=1.25 barrier=1.35 Volga: Sensitivity of Vega with respect to Vol

15. Simple analytic techniques: “moment matching” • Average-rate option payoff with N fixing dates • Basket option with two underlyings • TV pricing can be achieved quickly via “moment matching” • Mark-to-market requires correlated stochastic processes for spots/vols (more complex)

16. “Moment matching” • To price Asian (average option) in TV we consider that • The spot process is lognormal • The sum of all spots is lognormal also • Note: a sum of lognormal variables is not lognormal. Therefore this method is an approximation (but quite accurate for practical purposes) • Central idea of moment matching • Find first and second moment of sum of lognormals: E[Σi Si] , E[ (Σi Si)2 ], • Assume sum of lognormals is lognormal (with known moments from previous step) and obtain a Black-Scholes formula with appropriate drift and vol

17. Asian options analytics (1) • Prerequisites for the analysis: statistics of random increments • Increments of spot process have 0 mean and variance T (time to maturity) • E[Wt]=0, E[Wt2]=t • If t1<t2 then E[Wt1∙Wt2] = E[Wt1∙(Wt2-Wt1)] + E[Wt12] = t1 (because Wt1 is independent of Wt2-Wt1) • More generally, E[Wt1∙Wt2] = min(t1,t2) • From this and with some algebra it follows that E[St1 ∙ St2] = S02 exp[r∙(t1+t2) + σ2 ∙min(t1,t2)]

18. Asian options analytics (2) • Asian payoff contains sum of spots • What are its mean (first moment) and variance? • Looks complex but on the right-hand side all quantities are known and can be easily calculated ! • Therefore the first and second moment of the sum of spots can be calculated

19. Asian options analytics (3) • Now assume that X follows lognormal process, with λ the (flat) vol, μthe drift • Has solution (as in standard Black-Scholes) • Take averages in above and obtain first and second moment in terms of μ,λ • Solving for drift and vol produces

20. Asian options analytics (4) • Since we wrote Asian payoff as max(XT-K,0) • We can quote the Black-Scholes formula • With • And μ, λ are written in terms of E[X], E[X2] which we have calculated as sums over all the fixing dates • The “averaging” reduces volatility: we expect lower price than vanilla • Basket is based on similar ideas

21. Smile-dynamics models • Large number of alternative models: • Volatility becomes itself stochastic • Spot process is not lognormal • Random variables are not Gaussian • Random path has memory (“non-markovian”) • The time increment is a random variable (Levy processes) • And many many more… • A successful model must allow quick and exact pricing of vanillas to reproduce smile • Wilmott: “maths is like the equipment in mountain climbing: too much of it and you will be pulled down by its weight, too few and you won’t make it to the top”

22. Dupire Local Vol • Comes from a need to price path-dependent options while reproducing the vanilla mkt prices • Underlying follows still lognormal process, but… • Vol depends on underlying at each time and time itself • It is therefore indirectly stochastic • Local vol is a time- and spot-dependent vol (something the BS implied vol is not!) • No-arbitrage fixes drift μ to risk-free rate

23. Local Vol • Technology invented independently by: • B. Dupire Risk (1994) v.7 pp.18-20 • E. Derman and I. Kani Fin Anal J (1996) v.53 pp.25-36 • They expressed local vol in terms of market-quoted vanillas and its time/strike derivatives • Or, equivalently, in terms of BS implied-vols:

24. Dupire Local Vol • Contains derivatives of mkt quotes with respect to: Maturity, Strike • The denominator can cause numerical problems • CKK<0 (smile is locally concave), σ2<0, σ is imaginary • The Local-vol can be seen as an instantaneous volatility • depends on where is the spot at each time step • Can be used to price path-dependent options

25. Rule of thumb: Local vol varies with index level twice as fast as implied vol varies with strike (Derman & Kani) Local Vol rule of thumb Sfinal Sinitial

26. 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) Local-Vol and vanillas • By design the local-vol model reproduces automatically vanillas • No further calibration necessary, only market quotes needed EURUSD market Lines: market quotes Markers: LV pricer Blue: 3 years maturity Green: 5 years maturity

27. Analytic Local-Vol (2) • Estimating the numerical derivatives of the Dupire Local-Vol can be time-consuming • Alternative: assume a form for the local-vol σ(St,t) • Do that, for example, by: • From historical market data calculate log-returns These equal to the volatility • Make a scatter plot of all these • Pass a regression • The regression will give an idea of the historically realised local-vol function

28. Analytic Local-Vol (2) • A popular choice is • Ft the forward at time t • Three calibration parameters • σ0 : controlling ATM vol • α: controlling skew (RR) • β: controlling overall shift (BF) • Calibration is on vanilla prices • Solve Dupire forward PDE with initial condition C=(S0-K)+

29. Stochastic models • Stochastic models introduce one extra source of randomness, for example • Interest rate dynamics • Vol dynamics • Jumps in vol, spot, other underlying • Combinations of the above Dupire Local Vol is therefore not a real stochastic model • Main problem: Calibration • minimize (model output – market observable)2 • Example (model ATM vol – market ATM vol)2 • Parameter space should not be • too small: model cannot reproduce all market-quotes across tenors • too large: more than one solution exists to calibration

30. Processes • Lognormal for spot • Mean-reverting for variance • Correlated Brownian motions Heston model • Coupled dynamics of underlying and volatility • Interpretation of model parameters • μ : drift of underlying • κ : speed of mean-reversion • ρ : correlation of Brownian motions • ε : volatility of variance • Analytic solution exists for vanillas ! • S L Heston "A Closed form solution for options with stochastic volatility"Rev Fin Stud (1993) v.6 pp.327-343

31. Effect of Heston parameters on smile Affecting skew: • Correlation ρ • Vol of variance ε Affecting overall shift in vol: • Speed of mean-reversion κ • Long-run variance v∞

32. Local-vol vs Stochastic-vol • Dupire and Heston reproduce vanillas perfectly • But can differ dramatically when pricing exotics! • Rule of thumb: • skewed smiles: use Local Vol • convex smiles: use Heston

33. Hull-White model • It models mean-reverting underlyings such as • Interest rates • Electricity, oil, gas, etc • 3 parameters to calibrate • obtained from historical data: • rmean (describes long-term mean) • obtained from calibration: • a: speed of mean reversion • σ : volatility • Has analytic solution for the bond price P = E[ e-∫r(t)dt ]

34. 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 Three-factor model in FOREX Hull-White is often coupled to another underlying • Common calibration issue: "Variance squeeze“: FX vol + IR vols up to a certain date have exceeded the FX-model vol. • Solution (among other possibilities): Time-dependent parameters (piecewise constant) parameter time

35. Two-factor model in commodities • Commodity models introduce the “convenience yield” (termed δ) δ = benefit of direct access – cost of carry Not observable but related to physical ownership of asset • No-arbitrage implies Forward: F(t,T) = St ∙ E [ e∫(r(t)-δ(t))dt ] • δt is taken as a correction to the drift of the spot price process • What is the process for St, rt, δt ? • Problem: • δt is unobserved • Spot is not easy to observe • for electricity it does not exist • For oil, the future is taken as a proxy • Commodity models based on assumptions on δ

36. Gibson-Scwartz model • Classic commodities model • Spot is lognormal (as in Black-Scholes) • Convenience yield is mean-reverting • Very similar to interest rate modeling (although δt can be pos/neg) • Fluctuation of δ is in practise an order of magnitude higher than that of r  no need for stochastic interest rates • Analysis based on combining techniques • Calculate implied convenience yield from observed future prices Miltersen extension: Time-dependent parameters

37. 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 Merton jump model Advantages • Can produce smile • Adds a realistic element to dynamics • Has exact solution for vanillas

38. Merton jump model Extra term to the Black-Scholes process: • If jump does not occur • If jump occurs Then, Therefore, Y: size of the jump • Model has two extra parameters: • size of the jump, Y • frequency of the jump, λ Jump size & jump times: Random variables

39. Merton model solution • Merton assumed that • The jump size Y is lognormally-distributed, • Can be sampled as Y=eη+γ∙g; g is normal ~N(0,1) and η,γ are real • Jump times: Poisson-distributed with mean λ, Prob(n jumps)=e-λT(λT)n /n! • Jump times: independent from jump sizes • The model has solution a weighted sum of Black-Scholes formulas • σn , rn , λ’ are functions of σ,r and the jump-statistics given by η, γ

40. Merton model properties • The model is able to produce a smile effect

41. Vanna-Volga method • Which model can reproduce market dynamics? • Market psychology is not subject to rigorous math models… • Brute force approach: Capture main features by a mixture model combining jumps, stochastic vols, local vols, etc • But… • Difficult to implement • Hard to calibrate • Computationally inefficient • Vanna-Volga is an alternative pricing “recipie” • Easy to implement • No calibration needed • Computationally efficient • But… • It is not a rigorous model • Has no dynamics

42. Vanna-Volga main idea • The vol-sensitivities Vega Vanna Volga are responsible the smile impact • Practical (trader’s) recipie: • Construct portfolio of 3 vanilla-instruments which zero out the Vega,Vanna,Volga of exotic option at hand • Calculate the smile impact of this portfolio (easy BS computations from the market-quoted volatilities) • Market price of exotic = Black-Scholes price of exotic + Smile impact of portfolio of vanillas

43. Vanna-Volga hedging portfolio • Select three liquid instruments: • At-The-Money Straddle (ATM) =½ Call(KATM) + ½ Put(KATM) • 25Δ-Risk-Reversal (RR) = Call(Δ=¼) - Put(Δ=-¼) • 25Δ-Butterfly (BF) = ½ Call(Δ=¼) + ½ Put(Δ=-¼) – ATM KATM KATM K25ΔP K25ΔC KATM K25ΔP K25ΔC ATM Straddle 25ΔRisk-Reversal 25ΔButterfly RR carries mainly Vanna BF carries mainly Volga

44. Vanna-Volga weights • Price of hedging portfolio P = wATM ∙ ATM + wRR ∙ RR + wBF ∙ BF • What are the appropriate weights wATM ,, wRR,wBF? • Exotic option at hand X and portfolio of vanillas P are calculated using Black-Scholes • vol-sensitivities of portfolio P = vol-sensitivities of exotic X: • solve for the weights:

45. 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 price

46. Vanna-Volga vs market-price • Can be made to fit the market price of exotics • More info in: • F Bossens, G Rayee, N Skantzos and G Delstra "Vanna-Volga methods in FX derivatives: from theory to market practise“ Int J Theor Appl Fin (to appear)

47. Models that go the extra mile • Local Stochastic Vol model • Jump-vol model • Bates model

48. Local stochastic vol model • Model that results in both a skew (local vol) and a convexity (stochastic vol) • For σ(St,t) = 1 the model degenerates to a purely stochastic model • For ξ=0 the model degenerates to a local-volatility model • Calibration: hard • Several calibration approaches exist, for example: • Construct σ(St,t) that fits a vanilla market, • Use remaining stochastic parameters to fit e.g. a liquid exotic-option market

49. Jump vol model • Consider two implied volatility surfaces • Bumped up from the original • Bumped down from the original • These generate two local vol surfaces σ1(St,t) and σ2(St,t) • Spot dynamics • Calibrate to vanilla prices using the bumping parameter and the probability p

50. Bates model • Stochastic vol model with jumps • Has exact solution for vanillas • Analysis similar to Heston based on deriving the Fourier characteristic function • More info: • D S Bates “Jumps and Stochastic Volatility: Exchange rate processes implicit in Deutsche Mark Options“ Rev Fin Stud (1996) v.9 pp.69-107