ADVANCED DERIVATIVES LECTURE NOTES. Claudio Tebaldi Università Bocconi

Transcription

1 ADVANCED DERIVAIVES LECURE NOES Claudio ebaldi Università Bocconi

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3 Contents 1 A primer on differential equations Ordinary Differential Equations and risk free securities Ito stochastic calculus Stochastic Differential Equations PDE derived from Stochastic Differential Equations Feynman-Kac formula Backward Kolmogorov equation Forward Kolmogorov equation B&S valuation approach No Arbitrage restrictions: an empirical test Pricing under the historical measure B&S complete market valuation approach B&S replication argument From theory to practice he forward measure he implied volatility surface Option strategies and PDE Static portfolio strategies he Breeden and Litzenberger formula he Local Volatility model Static replication of exotic derivatives he replication of volatility index Stochastic Volatility Volatility, Subordination and Stochastic Clocks he Heston Model Multidimensional Ito Lemma he replication argument when the market is incomplete he market price of volatility risk iii

4 iv CONENS 4.6 he Heston pricing formula A new approach to option pricing and hedging he Vega-Vanna-Volga approach he Stochastic Volatility Inspired parametrization he Bibliography. 45

5 Chapter 1 A primer on differential equations. he theory of differential equations is by far the most important branch of applied mathematics. Virtually any quantitative application involves the statement and the solution of a boundary value problem for a differential equation. While we refer to standard textbooks for a general introduction to the subject, in the following we introduce the classes of differential equations that we will use in the analysis of stochastic markets. 1.1 Ordinary Differential Equations and risk free securities Definition 1.1 An autonomous Ordinary Differential Equation (ODE) in Cauchy normal form is a differential equation which can be written in the following form: d dt f (t) = F ( f (t) ) f () = f where t I = [, t 1 ), F : U R n, f : I U R n is differentiable in I. f is said the unknown or indeterminate function, f is the initial condition. Example 1.1 he simplest ordinary differential equation is the one which gives the accrued capital in a bank account as a function of time and of the continuously compounded interest rate r: let b the cash in the account at time 1

6 2 CHAPER 1. A PRIMER ON DIFFERENIAL EQUAIONS., then by definition of r: d dt C t = rc t C = b in this case the differential equation is integrated (i.e. solved) as follows: s s 1 d C t dt C tdt = d dt ln (C t) dt = s d ln (C t ) = s s s rdt rdt ln (C s ) ln (C ) = r (s ) exp (ln (C s /C )) = exp (rs) C s = C exp (rs) rdt (1.1) hence the general integral to the equation is given by: C s = C exp (rs) and imposing the initial value condition: we get the solution at time t: C = C exp (r) = b C t = b exp (rt) he same ODE determines the price B t of a bond expiring at time and maturing an interest rate r. In fact the price today of the bond is determined imposing the condition that the reimbursement value at time is equal to its nominal reimbursement, conventionally set to 1, B = 1. Hence: B t = B exp (rt) B = B exp (r ) = 1 B t = B exp ( r ( t)) Notice that in this case, in order to restate the boundary value problem in Cauchy form, it is necessary to change time variable introducing the time to maturity τ := t, then the ODE equation in Cauchy form becomes: d B (τ) dτ = rb (τ) B () = 1

7 1.2. IO SOCHASIC CALCULUS 3 and its solution is: B (τ) = B () exp ( rτ) Since τ = t, it is easy to verify that B t coincides with B (τ): B t = B (τ) τ= t While the price evolution of a risk-free asset does not involve uncertainty and risk, in general we need to describe the price evolution of a stock which by definition is risky. his fact substantially complicates the equation which describes the price evolution which becomes stochastic. In the next section we recall the basic notions on stochastic differential equations which are required to understand the Black Merton and Scholes (B&S from now on) derivation of the fundamental PDE valuation equation. 1.2 Ito stochastic calculus B&S valuation approach makes use of a sophisticate language: the rules of this language are those of Ito stochastic calculus. Let the price of a stock at time t, S t (ω), be described by an adapted random ( process with ) continuous price paths in a filtered probability space Ω, {F t } t [, ], P. he general expression of the stochastic evolution of the process is then specified through the Ito evolution equation: ds t (ω) S t (ω) = µ t (ω) dt + σ t (ω) dw t S (ω) = s by its characteristics, i.e. two adapted random functions called the drift µ t (ω) and its volatility σ t (ω). his equation should be interpreted as the differential version of the stochastic integral equation for S t (ω): S t (ω) = s + t µ τ (ω) S τ (ω) dτ + t σ τ (ω) S τ (ω) dw τ (ω) he economic intuition behind this equation is quite clear, the trend represents the instantaneous expected return on the security, while the volatility represents the infinitesimal expected risk. he fundamental innovative content of the above equation is the continuous time representation of the risk evolution by means of a Wiener process. While this representation is extremely useful in computation, it complicates the mathematical formulation of the theory requiring an extension of conventional integration theory. Let

8 4 CHAPER 1. A PRIMER ON DIFFERENIAL EQUAIONS. us briefly explain why this complication is unavoidable. Standard Riemann- Lebesgue-Stieltjes integration can be applied only to functions with finite total variation V a,b (f) < +. Recall that the total variation V a,b (f) of a real function f on an interval [a, b] R, is defined by: V a,b (f) = sup p P N p i=1 f (t i ) f (t i 1 ) P = { p = ( t, t 1,..., t Np ) a = t < t 1 <... < t Np = b } while the quadratic variation of a real function f defined on an interval [a, b] R, is given by: V Q a,b (f) = sup p P N p i=1 f (t i ) f (t i 1 ) 2 P = { p = ( t, t 1,..., t Np ) a = t < t 1 <... < t Np = b }. and functions with finite quadratic variation have infinite first order variation. In order to quantify risk one needs a continous function with zero trend (completely unpredictable) and finite volatility over an infinitesimal period of time. In other terms any function useful to provide a dynamic representation of risk must have finite quadratic variation by construction. In particular, almost any path of the Wiener process has finite quadratic variation. hus conventional integration theory cannot be applied and an extension is required. Consider a function f C 1 (R), then for t [, ] and ω C ([, ]) with fintie quadratic variation and a nested sequence of partitions π 1 π 2.. π N, then it is possible to define a pathwise integral: t f (ω (s)) dω (s) := + lim f (ω (π n k t)) [ ω ( π n k+1 t ) ω (π n k t) ] n + k= It can be proved that this limit exists for any t and defines a function in C ([, ]). he above definition of pathwise integral of a function of quadratic variation deviates from standard integration theory for the following crucial property: assume that F C 2 (R), then for all t [, ] it is possible to show that the following change of variable formula applies: F (ω (t)) = F (ω ()) + t F (ω (s)) dω (s) t F (ω (s)) dv Q,t (ω (s)) (1.2)

9 1.3. SOCHASIC DIFFERENIAL EQUAIONS 5 While the above definition of integral is given for ordinary functions, the definition of the integral for functions with finite quadratic variation was first given by Ito using stochastic analysis under a stronger assumption. Ito required the existence of an underlying probability space and constructed integration theory for continuous stochastic processes. He showed that the change of variables formula (1.1) applies to adapted stochastic processes. In fact the central result of Ito stochastic calculus is the following: Lemma 1.1 Let G is a measurable function, twice differentiable with respect to the first argument and differentiable with respect to the second one, then the differential characteristics of the process G (S t (ω), t) in terms of the characteristics of S t (ω), dg (S t (ω), t) are given by: dg (S t (ω), t) = [G s (S t (ω), t) µ t (ω) S t (ω) + 12 ] G ss (S t (ω), t) σ 2t (ω) S t (ω) 2 dt +G s (S t (ω), t) σ t (ω) S t (ω) dw t (ω) Having introduced the notion of an Ito stochastic process, it is now possible to extend the notion of differential equation. 1.3 Stochastic Differential Equations Definition 1.2 A stochastic differential equation (SDE) in Cauchy form is a differential equation which can be written in the following form: dp (t, ω) = µ (t, P (t, ω)) dt + σ (t, P (t, ω)) dw t (ω) P () = ξ where µ : [, ] U R, σ : [, ] U R, W t is a standard Brownian motion in the probability space (Ω, F, P). A random process P (t, ω) adapted to F solves (in the strong sense) the SDE in the interval I = [, ] if: P (P (ω) = ξ) = 1, ( ) P { b (s, X s) + σ (s, X s ) } ds < + = 1 for t [, ] : P (t, ω) = ξ + t µ (s, P (s, ω)) ds + σ (s, P (s, ω)) dw s (ω)

10 6 CHAPER 1. A PRIMER ON DIFFERENIAL EQUAIONS. If σ (t, P (t, ω)) is zero (no source of uncertainty) then the SDE reduces to: dp (t) = µ (t, P (t)) dt which is equivalent to the ODE formally obtained "dividing by dt": dp (t) dt = µ (t, P (t)) P () = p 1.4 PDE derived from Stochastic Differential Equations In the formulation of the stochastic market model, the price evolution is exogenous and determined by a differential equation whose solution is a random process. As explained above, the mathematical meaning of Ito differentials is far from intuitive, due to the highly irregular character of (nowhere differentiable) price paths. In applications, in order to connect SDE with observed quantities, it is often useful possible to associate a partial differential operator to each SDE as follows: Definition 1.3 Consider a SDE defined by: dx t = µ (X t, t) dt + σ (X t, t) dw t with X t R. hen the associated differential operator is given by: A X f (x, t) = 1 2 σ2 (x, t) 2 f (x, t) f (x, t) + µ (x, t) x 2 x his operator defines a partial differential equation of the parabolic type with terminal value condition if the problem to be solved is given by: f (t, x) t + A X f (t, x) r (x, t) f (x, t) = f (, x) = F (x) while it defines a partial differential equation of the parabolic type with initial value condition if the problem to be solved is given by: f (t, x) t + A X f (t, x) r (x, t) f (x, t) = (1.3) f (, x) = F (x)

11 1.5. FEYNMAN-KAC FORMULA 7 Observe that the partial differential equations is linear, i.e. the linear combination of two solutions is still a solution and classical solutions are differentiable with respect to time and twice differentiable with respect to the x variable. 1.5 Feynman-Kac formula he key computational step that is necessary to connect stochastic market models to market prices is the computation of (conditional) expectations: E M [f (S t (ω)) F t ] where M is a measure that absolutely continuous with respect to P 1. he known methods to compute conditional expectations fall in three classes: Analytical methods. hey rely on the methods of analysis and algebra to express conditional expectations in closed form expressions dependent from parameters known at the instant of the valuation. Numerical integration of partial differential equations. Computation is reduced to the numerical integration of partial differential equations. Simulation Methods. Within this approach statistical information on probability distributions and on expectations is extracted from effi cient statistical sampling procedures made available by high speed, high capacity computing facilities. here exists a fundamental relation between conditional expectations of regular functions of the SDE solution and the solution of the associated parabolic PDE. It is given by the Feynman-Kac formula which (in a simplified formulation) states: heorem 1.1 Suppose µ (x, t), σ (x, t), r (x, t) continuous functions in their arguments and f (x, t) twice differentiable in x and of polynomial growth in x uniformly in t. hen the solution to the boundary value problem f (t, x) t + A X f (t, x) r (x, t) f (x, t) = (1.4) f (, x) = F (x) 1 Recall that absolute continuity grants the existence of a -adapted random variable m (ω) such that: dm (ω) = m (ω)dp (ω).

12 8 CHAPER 1. A PRIMER ON DIFFERENIAL EQUAIONS. admits the following stochastic representation in terms of the expectation: f (x, t) = E M F (X ( )) exp r (X s, s) ds F t Proof. See Duffi e (21). his theorem bridges the problem of the computation of conditional expectations with the problem of solving a PDE Backward Kolmogorov equation As a first application of the Feynman-Kac representation, one can compute the probability of the solution at time, X hits a target set B given that X t = x: g B (x, t) = M (X B X t = x) = E M t,x [I B (X ( )) X t = x] where B is a generic measurable event. hen this quantity solves the PDE equation: = [ t g B (x, t) + µ (x, t) g B (x, t) + 1 ] x 2 σ2 (x, t) 2 gb (x, t) (1.5) x { 2 1 if x B lim t g B (x, t) = if x / B, Choosing B ε = ( ε + y, y + ε) and letting ε one gets the evolution of the transition probability: 1 p (y, ; x, t) = lim ε 2ε P (X ( ε + y, y + ε) X t = x) his quantity represents the probability density of a transition from the state x at time t to the state y at time. It is determined by the PDE equation: = p (y, ; x, t) p (y, ; x, t) + µ (x, t) t x lim p (y, ; x, t) = δ (x y) t t σ2 (x, t) 2 p (y, ; x, t) x 2 (1.6) where δ is a symbolic notation corresponding to the notion of a density which is unless x = y. 2 his equation is known as Backward Kolmogorov equation. 2 While δ (x y) shares many properties of conventional density functions, its mathematical definition is far more complex. In fact the δ differs from the null function, the function equal to zero everywhere, only in a single point, a set of zero measure, hence as a measurable function a delta function is indistinguishable from the null function.

13 1.5. FEYNMAN-KAC FORMULA Forward Kolmogorov equation he forward Kolmogorov is the equation that describes the evolution of the transition density with respect to the first time argument, assuming the second one fixed. It is given by: (y, t + τ) p (y, t + τ; x, t) p (y, t + τ; x, t) = µ + τ y σ 2 (y, t + τ) p (y, t + τ; x, t) 2 y 2 lim p (y, t + τ; x, t) = δ (x y) τ + Notice that the knowledge of the transition density simplifies the computation of the conditional expectations for any function F (X ) thanks to the following: Proposition 1.1 he solution to the parabolic PDE terminal condition is a convolution integral 3 between the transition probability p (y, ; x, t), which solves the backward Kolmogorov equation and the final condition F (x): V (x, t) = obtained from the solution to the PDE. p (y, ; x, t) F (y) dy he formal verification of this statement is simply obtained by observing that the integrand is composed by the forward transition density multiplied by a term F (y) which is independent from t and x. hen, exchanging the integral with the PDE defining the forward transition density operator, one gets: = V (x, t) V (x, t) + µ (x, t) + 1 t x 2 σ2 (x, t) 2 V (x, t) x 2 + t p (y, ; x, t) F (y) dy + = µ (x, t) p (y, ; x, t) F (y) dy x σ 2 (x, t) 2 p (y, ; x, t) F x 2 (y) dy 3 he convolution integral between two functions f, g with domain the real line R is defined as: f g (x) = f (x, y) g (y) dy

14 1 CHAPER 1. A PRIMER ON DIFFERENIAL EQUAIONS. he next example shows that, using the above results, it is possible to compute the conditional expectation for a generic payoff F (S ) for a security whose price follows a lognormal diffusion. Example 1.2 Consider the evolution of a lognormal stock price: ds t S t = (µ q) dt + σdw t S = s and a contract with payoff F (S ). Define the process: X t = log (S t ) Application of the Ito rule implies: dx t = (µ q 12 ) σ2 dt + σ 2 dw t that is the SDE corresponding to the log return evolution. hen, the function V (x, t) := f (e x, t) verifies the associated PDE with constant coeffi cients: ( V (x, t) + µ q 1 ) V (x, t) t 2 σ V (x, t) x 2 σ2 x 2 = (1.7) x = log (s) V (x, ) = F (e x ) he verification that the normal distribution density ( ) 1 N ( µ, σ) (x, t) = 2πσ2 ( t) exp (x µ ( t))2 2σ 2 ( t) with mean µ ( t) = ( µ q 1σ2) ( t) and variance σ 2 ( t) solves 2 the backward Kolmogorov equation associated to the process X t goes as follows: first one computes each differential term in the PDE (long but strightfrward), then collecting these contributions one gets: N ( µ, σ) [ +N ( µ, σ) [ +N ( µ, σ) = N ( µ, σ) [ (x µ ( t)) µ σ 2 ( t) (x µ ( t)) µ σ 2 ( t) 1 2 ( t) (x µ ( t))2 2σ 2 ( t) 2 + ] + (x µ ( t))2 2σ 2 ( t) 2 ] ] 1 2 ( t)

15 1.5. FEYNMAN-KAC FORMULA 11 Hence it is possible to conclude that: p (y, ; x, t) = N ( µ, σ) (y x, t) thus: V (x, t) = N ( µ, σ) (x y, t) F (e y ) dy and Using the previous formula, computation of the conditional expectations E M [ I {S >K} S t ] E M [ S I {S >K} S t ] is straightforward. he final payoffs are given by I {log(k/s )>} and S I {log(k/s )>}. he computation goes as follows: E [ M I {S >K} S t = s ] = = = log(k/s ) d µ 2 N ( µ, σ) (y log(s), t) I {y>log(k/s )}dy N ( µ, σ) (y log(s), t) dy 1 2π exp ( (y ) 2 2 ) dy = N (d µ 2) where d µ 2 = ln(s/k)+(µ q 1 2 σ2 )( t) σ. he computation of the second expectation is left as an exercise for the t reader.

16 12 CHAPER 1. A PRIMER ON DIFFERENIAL EQUAIONS.

17 Chapter 2 B&S valuation approach 2.1 No Arbitrage restrictions: an empirical test. By the law of one price, if the payoff of two portfolios are equal, then also their prices coincide. In particular, the Put-Call parity relation predicts that: p t c t = Ke r( t) q( t) S t e F (r q)( t) t = S t e p t c t = e ( ) r( t) K Ft. Observe that dividend yields are are determined ex-ante considering consensus forecasts which could differ among market participants. Consistency of the consensus estimate with market expectations can be empirically tested using the quoted prices for the book of options p t, c t and future contracts with future price Ft = S t e (r q)( t) written on the same underlying stock index having different strikes. For each strike K one gets the following estimate of the dividend q (K): q (K) = 1 ( ) +Ke r( t) t log (p t c t ) q (K) corresponds to the so called implied dividend such that the Put Call Parity is verified at strike K. If q (K) is independent from K then we can reasonably conclude that No Arbitrage relations are compatible with market prices. In fact the dividend yield q cannot depend on the strike price. In the empirical test it is important to consider synchronized quotes such that arbitrage operations are easily detectable. able 1 shows an example of the implied dividend yields estimated during a normal trading day. It is possible 13 S t

18 14 CHAPER 2. B&S VALUAION APPROACH to verify that the estimate of the dividend yield is quite stable for a range of options close to the at the money strike. Implied dividends of out of the money puts are not consistent. K q (K) -6.12% 1.17% 1.1% 1.3% 1.7%.98% 1.2% Implied dividend yield on Eurostoxx5 October 2 27, the level of the underlying index was his simple test offers a good indication on the effective reliability of the absence of arbitrage opportunities. his valuation approach works for the range of contracts that are liquid and competiton among traders reduces price uncertainty. In the following, unless differently stated, we will assume that the dividend yield q =, in order to simplify the notation. In fact the modifications required to include the effects of dividends are straightforward and are left as an exercise for the reader Pricing under the historical measure. In order to understand the innovative contents of B&S theory and the relevance of the risk neutral measure in valuation, consider a "naive" valuation where price is given by the expectation under P which is the historical risk measure (assuming zero dividends) discounted with the risk-free rate. hen we get: C Hyst t = e r( t) E [ P (S K) +] = e r( t) E [ P S I {S >K}] Ke r( t) E [ ] P I {S >K} hen, using the computation carried out in the Example 1.2, one gets: C Hyst t = e r( t) S t e µ( t) N (d µ 1) Ke r( t) N (d µ 2) d µ 1 = ln (S t/k) + ( µ σ2) ( t) σ t d µ 2 = ln (S t/k) + ( µ 1 2 σ2) ( t) σ t and correspondingly for the Put payoff one gets: P Hyst t = Ke r( t) (1 N (d µ 2)) e r( t) S t e µ( t) (1 N (d µ 1))

19 2.2. B&S COMPLEE MARKE VALUAION APPROACH. 15 hence prices computed using the put and call positions imply: P Hyst t C Hyst t = e r( t) K e (µ r)( t) S t e ( ) r( t) K Ft Hence, unless r = µ, we get an arbitrage opportunity, i.e. an opportunity to gain money without risk! Observe on the contrary, that assuming a rate of return r for the risky asset or a discount rate different from r for the cash position K opens a contradiction with the assumption that investors are risk averse. In fact rational risk averse agents must necessarily receive a premium over the risk-free rate as a compensation to bear a risky asset! his is the paradox which is resolved by the B&S theory. 2.2 B&S complete market valuation approach. Let us review the derivation of the B&S pricing theory of a call option contract. A European call option contract is a contingent claim with payoff at expiration time : max (S K, ) written on an underlying asset (e.g. we will assume as reference the stock market index S&P5). he underlying asset evolves dynamically according to a continuous time stochastic process as described by a SDE. he market where the agents can trade is composed by: 1. he risky index level whose evolution is given by: ds t = (µ (S t, t) qs t ) dt + σ (S t, t) dw t S t= = s where W t is a standard Brownian motion in the probability space (Ω, F, P). It is assumed that the stock index S t pays a dividend rate q. 2. A riskless asset (cash): db t = rb t dt B = 1 3. A set of Call and Put contracts for a continuum of strikes K R, with payoffs c = max (S K, ), p = max (K S, ) is also present in the market.

20 16 CHAPER 2. B&S VALUAION APPROACH 2.3 B&S replication argument. he value of any call or put option contract at any time t will be determined assuming the following hypotheses: he market is free from arbitrage opportunities. he market is perfectly liquid and trading does not have any impact on the price of the underlying asset. he agent can invest in risky and riskless assets without constraints (i.e. no margins are required, any amount can be sold). he market is driven by a single source of randomness such that all the uncertainty is perfectly correlated with the index level S t and thus the price of each derivative contract at time t can be expressed as a function of a unique stochastic process S t, f (S t, t). B&S determine the PDE that has to be satisfied by the so called fair price, i.e. the price level such that two agents agree to take the long and the short position in the option contract. he derivation goes as follows: first of all, one constructs a portfolio composed by a long position in the call c t and a short position in t units of the risky asset S t : Π t = c t t S t, It is assumed that the allocation to each asset is known an instant before prices evolve randomly. hen the time variation in the value of the portfolio is given by: 1 be: dπ t = dc t t ds t And using Ito calculus one can compute: 1 Observe that according to Ito calculus the correct expression of the differential would d ( R t B t ) ( d R ) B t = dc t t ds t + d t S t + d t, ds t where R is the number of bonds in the portfolio. he self-financing condition imposes that no inflow or outflow of money occurs at any intermediate date, which implies: ( d R) B t = +d t S t + d t, ds t = ( d R) B t + (d t ) S t + d t, ds t

21 2.3. B&S REPLICAION ARGUMEN he evolution of the portfolio value for a fixed allocation: dπ t = dc t t ds [ ( t ) ct = t + ct µs t + 1 ] 2 c t σ 2 S 2 S t 2 St 2 t dt + σ Π dw t ( ) ct σ Π = σs t S t 2. he amount t needed to immunize the portfolio is computed requiring that the portfolio Π has zero volatility: ( ) ct σ Π = σs t = S t t = c t S t By construction the immunized portfolio Π can be replicated using only cash (the riskless asset), hence in order to avoid any arbitrage opportunities the evolution of the riskless portfolio must be: Hence the condition: = dπ t = rπ t dt [ ct t + 1 ] 2 c t σ 2 S 2 2 St 2 t dt rπ t dt implies = c ( t t r c t c ) t S t c t σ 2 S 2 S t 2 St 2 t (2.1) = c t t + r c t S t c t σ 2 S 2 S t 2 St 2 t rc t and at maturity the call option value must be: c = (S K) + (2.2) he PDE (2.1) with the terminal boundary condition (2.2) determines uniquely the call option price. Observe that the above derivation continues to hold also when volatility is a deterministic function of the asset price σ (S t, t). Under this hypothesis,

22 18 CHAPER 2. B&S VALUAION APPROACH according to the Feynman-Kac theorem the value of the call contract can be written like a conditional expectation: c t = e r( t) E Q [ (S K) +] that of course is compatible with the Put Call parity relation. Recall however that the measure Q such that the solution of the PDE can be written as a conditional expectation is not the historical one, it is the measure that is defined by the stochastic representation of the PDE solution. Its existence is granted by the Feynman-Kac theorem. he operational difference between the two measures can be easily stated as follows: P is the historical measures, where the probability of each event can be inferred from the historical frequency of occurrence of price fluctuations, Q is the valuation measure, i.e. the probability of each event which is implicit in quoted prices if absence of arbitrage opportunities is enforced. According to the Arrow and Debreu theory, the Q probability of an event A, Q [A], can be considered as the market undiscounted price of the bet which pays the premium in the case of occurrence of the event A itself. B&S formula is obtained and corresponds to the closed form expression of the conditional expectation obtained solving the characteristic PDE c B&S (r, q,, S t, K, σ) in terms of the observable parameters r, q,, s, K, σ under the condition that instantaneous volatility σ (S t, t) is constant: c B&S (r, q,, S t, K, σ) = S t N (d r 1) Ke r( t) N (d r 2) d r 1 = ln (S t/k) + ( r σ2) ( t) σ t d r 2 = ln (S t/k) + ( r 1 2 σ2) ( t) σ t It is important to remark that the the PDE solved by the B&S formula rationalizes the selection of the risk free discounting using a replication argument. he risk neutral measure is then the one which is associated to the resulting PDE by the Feynman-Kac theorem.

23 Chapter 3 From theory to practice 3.1 he forward measure In practice, if a liquid market for forward contract is available, it is possible to remove the interest rate risk from option trading using a conventional approach. he forward contract with expiration will evolve like a martingale: dft = σ ( Ft, t ) Ft dw t F = S e r where σ ( F t, t ) := σ (S t, t) and ds = df. Note that the forward price of any traded contract can be defined in the same way, by using the -forward contract as numeraire good and the price in the new numeraire evolves following a martingale process. We will denote the forward price of: the call option by: the put option by: C ( F t, K, t ) = c (S t, K, t) / exp ( r ( t)) P ( F t, K, t ) = p (S t, K, t) / exp ( r ( t)) hence the put call parity in forward units takes the form: C ( Ft, K, t ) P ( Ft, K, t ) = K Ft In practice, by using as numeraire a security with unit payoff at time, the time of liquidation of the plain vanilla contracts, all the cash flows 19

24 2 CHAPER 3. FROM HEORY O PRACICE are unaffected by interest rate risk is fully offset. expressed in units of the new numeraire: he transition density ϕ ( F, ; F t, t ) := e r( t) p (S, ; S t, t) will be called forward transition density and is determined by the equation: τ ϕ ( F, τ; Ft, t ) = 1 2 F 2 σ ( ) ( 2 F, τ; Ft ϕ F, τ; F t, t ). 2 F 2 he forward transition density determines the density of the risk neutral measure and can be used to compute the forward risk neutral price of a derivative with arbitrary integrable payoff g (F ): V ( F t, t ) = ϕ ( F, τ; F t, t ) g (F ) df 3.2 he implied volatility surface In the practice of financial markets, B&S theory plays a major role as a way to reduce the non-linear fluctuations of options prices driven by movements in underlying prices. he following definition introduces the notion of implied volatility, the quantity which is conventionally used in real markets to quote options. Definition 3.1 Suppose that an option with strike K and time to maturity t is quoted at price Π and assume (for simplicity) q =. hen the implied volatility for such option is the the level of annualized volatility σ Im which equalizes the quoted price Π with the price obtained according to the Black & Scholes model as determined by formula. In the case of a call contract c B&S (S t, K, t, r, σ) = S t N (d 1 ) Ke r( t) N (d 2 ) the level of volatility that solves the equation: d 1 = ln (S t/k) + ( r σ2) ( t) σ t d 2 = ln (S t/k) + ( r 1 2 σ2) ( t) σ t c B&S ( S t, K, t, r, σ Im) = Π and therefore it is a function with arguments: σ Im (Π, t, S t, K, r).

25 3.2. HE IMPLIED VOLAILIY SURFACE 21 For a given set of prices we get the corresponding set of implied volatilities inverting the B&S formula with respect to the volatility parameter, as described in the definition. he explicit computation of the implied volatility require the solution of an equation that can be obtained only applying a numerical procedure. he set of prices corresponding to the same maturity with different strikes determines the implied volatility curve at that maturity. If we plot the set of implied volatilities corresponding to the grid of available strikes and available maturities, we get the implied volatility surface. It is quite important to define the main parameters which are usually considered to describe the volatility surface and its movements. As a matter of fact, all these quantities refer to the component of the book of options around the at-the-money strike for short maturities. Deviations of the strike K for a fixed maturity with respect to the forward price Ft r( t) := S t e are measured in terms of the moneyness which is given by: K/Ft 1 or by log ( ) K/Ft. Another method to parametrize moneyness is through a notion related to B&S theory, to explain this approach we need to introduce: Definition 3.2 he B&S delta of an option contract is the first order sensitivity of the option s price with respect to the current value of the underlying asset price, i.e. it is the infinitesimal variation of the option s price due to an infinitesimal variation of the underlying asset price. For a Call and a Put option we get: c B&S (s, K, t, r, σ) s p B&S (s, K, t, r, σ) s s=s t = N (d 1 ) s=s t = N ( d 1 ) In a B&S market the delta sensitivity,, corresponds to the amount of the risky asset per unit of riskless security to be invested to obtain the replication portfolio for the option s contract. As for the case of implied volatility, also the at the money option s delta has become a standard tool in the practice of trading rooms with a conventional meaning different from the original B&S one: it is conventionally used as the unit of measure of monyness. More precisely, it is assumed that the difference between the at the money (K = Ft ) implied volatility and the implied volatility at a generic strike K is a function of B&S only. he main advantage of this parametrization relies on the fact that in this way each point of the volatility surface is determined as a function Φ (x) of the combination x = ln ( F t /K ) / t. his is easily understood by observing

26 22 CHAPER 3. FROM HEORY O PRACICE that the expression of the B&S is uniquely determined by the value of the parameter d 1 which, in turn, depends only on the ratio ln (K/S t ) / t. his parametrization is particularly useful in connection with a rule of thumb used by traders, called sticky delta rule. his rule assumes that, as the level of the underlying and the time to maturity change, the level of implied volatility corresponding to a given level of B&S remains constant and is inspired by the following property: if the underlying asset price moves but the level of implied volatility for the at the money strike remains constant, then the level of the implied volatility curve remains constant at every moneyness (measured in units of B&S ). A notable market where this convention is widespread is the one for foreign exchange options. We underline that this assumption does not provide a consistent theory for the evolution of the implied volatility surface, but provides a reasonable parametrization. In fact one can prove that a market where the sticky delta rule was exact, could not be free of arbitrage opportunities. 3.3 Option strategies and PDE. It is possible to establish a tight link between differential expressions appearing in the expressions of PDE and profit and losses (P&L hereafter) of option strategies. In order to illustrate this type of connection it is useful to introduce some terminology and the most popular static trading strategies Static portfolio strategies. A static portfolio strategy involves the creation of a portfolio of fundamental and derivative securities that is created buying (for long positions) or selling (for short positions) a set of contracts at market prices at time. he portfolio is then kept until expiration when all the securities payoffs are liquidated. Liquidation time will be conventionally assumed to be fixed at time. In particular we will analyze four basic static portfolio strategies: 1. Straddle. Buy at the money (AM) call and AM put with the same strike. Notice that, following the trading conventions, Straddles are quoted by fixing the level of the AM implied volatility. It is possible to verify that for small maturities t, the value of the straddle is proportional to the at the money implied volatility S R ( K A M, t ) Σ ( K A M, t )

27 3.3. OPION SRAEGIES AND PDE Risk reversal. Buy an out of the money call with strike K + ε, sell an out of the money put with strike K ε where ε >. In terms of implied volatility, this contract price is given by: RR K ε Σ (K + ε, t) Σ (K ε, t) As ε, for K equal to the AM strike, the level of the Risk Reversal becomes proportional to the implied volatility skew, e.g. the slope of the implied volatility w.r.t. moneyness at the AM strike 3. Butterfly. buy call with strike K ε, sell 2 calls K, buy call K + ε. B F K ε Σ (K + ε, t) 2Σ (K, t) + Σ (K ε, t)

28 24 CHAPER 3. FROM HEORY O PRACICE As ε converges to zero,, for K equal to the AM strike, the butterfly converges to a quadratic function with the curvature proportional to the curvature of the implied volatility surface in correspondence to the AM strike. 4. Calendar spread. A spread contract typically refers to a position in two similar derivatives, where one derivative has been purchased and the other one sold, so that each one acts as hedge against the other. In case of a calendar spread the two positions differ in their maturity. For example one can consider a long position in a straddle with maturity + ε and a short position in a straddle with maturity. CAL K A M ε Σ A M ( + ε) Σ A M ( ) he importance of these contracts depends on their relation with the notion of implied volatility. In fact, in the presence of a suffi ciently dense grid of traded maturities and strikes, for ε suffi ciently small, option strategies can be used to compute finite difference approximations of the partial derivatives that define the aylor expansion of the implied volatility surface with respect to moneyness measured by x := K St S t and time to maturity τ := t: σ Im (x, τ) ε S R ( K A M, ) + RRK A M ε ε x+ CALK A M ε ε τ+ B F K A M ε x 2 ε 2 2 +o ( τ, x 2) where o (τ, x 2 ) denotes terms that converge to zero faster than τ and x 2. his formula for option on exchange rates is discussed and formally derived in Durrleman and El Karoui (21). 3.4 he Breeden and Litzenberger formula A striking example of the relation between option trading strategies and the risk neutral valuation approach is given by the following relation which has been established by Breeden and Litzenberger (1978) and states: heorem 3.1 In an options s market with a continuum of quoted strikes K the risk neutral forward transition density equals the price of an infinitesimal butterfly portfolio strategy: 2 C(F, K, ) K 2 = ϕ ( F, ; F, ) F =K = lim ε B F K ε ε 2

29 3.5. HE LOCAL VOLAILIY MODEL. 25 where C(F, K, ) denotes the forward price of a call struck at K, if the time -forward price is F. B F Proof. he limiting relation lim K ε ε = 2 C(F, K, ) is simply ε 2 K 2 the definition of the second order derivative. he price of the call option will be given by the risk neutral expectation of the payoff C ( F, K, ) = K ( F K ) ϕ ( F, ; F, ) df Computation of the second order derivative leads to End Proof. 2 C ( F K 2, K, ) = 2 K 2 = K = K K [ [ K K ( F K ) ϕ ( F, ; F, ) df ( ( F K K )) ϕ ( F, ; S, ) ] df + ϕ ( F, ; F, ) ] df = ϕ ( K, ; F, ). 3.5 he Local Volatility model. he B&S derivation of the valuation equation applies in any market where there s a unique source of uncertainty that is perfectly correlated with the stock market index. In particular the level of volatility σ can be an arbitrary functions of the stock market index and of the time (S t, t) with bounded first order derivatives. Following an original intuition of Dermak and Kani (1993), that exploited the idea in the framework of discrete time binomial and trinomial trees, Dupire (1994) developed the local volatility approach under the assumption that options are traded for a continuum of maturities and strikes. hen market prices of plain vanilla options can be used as model inputs and Dupire formula provides the unique volatility function σ Loc (S t, t) that produces a B&S valuation of plain vanilla options consistent with the observed market price, this is done in the next: Definition 3.3 he Local volatility function σ Loc (S t, t) is the unique deterministic function σ (S t, t) that, inserted in eq.(2.1), generates a solution to the PDE, equal to the market price for plain vanilla options with arbitrary strike and maturity.

30 26 CHAPER 3. FROM HEORY O PRACICE Dupire formula provides an explicit expression of the Local Volatility function σ Loc (t, S t ) from and states: heorem 3.2 Assume that in the market are traded a continuum of options with arbitrary strike and maturity. hen the Local Volatility function is given by: σ Loc (, K) = 2 C(,K) K 2 2 C(,K) K 2 Proof. Let ϕ be the forward RN density of the final spot F then: C(F, K, ) = K df ϕ(f, ; F )(F K) which corresponds to the martingale evolution of the price (recall that dτ = dt). Starting from: C (F, K, τ) = K (F K) ϕ ( F, τ; F, ) ds (3.1) Substitute left hand side of (3.1) in (??) and exchange differential operators with integration, then: = C (F, K, τ) τ K (F K) τ ϕ ( F, τ; F, ) df Note that the possibility to exchange differential operators and integration is granted in this case by the definition of the forward transition probability density. he forward transition density ϕ ( F, τ; F, ) evolves according to the Forward Kolmogorov Equation: τ ϕ ( F, τ; F, ) = 1 2 F 2 σ ( 2 F, τ; F 2 F 2 Using this equality, one gets: = 1 2 C (F, K, τ) τ K (F K) 2 F 2 σ ( 2 F, τ; F F 2 ) ϕ ( F, τ; F, ) ) ϕ ( F, τ; F, ) df

31 3.5. HE LOCAL VOLAILIY MODEL. 27 and integrating by parts (easy but cumbersome): C (F, K, τ) τ = 1 (F K) 2 [ F 2 σ ( ) ( 2 F, τ; F 2 K F 2 ϕ F, τ; F, )] df = 1 (F K) F 2 σ ( ) ( 2 F, τ; F ϕ F, τ; F, ) df 2 K F F = 1 [ F 2 σ ( ) ( 2 F, τ; F ϕ F, τ; F 2 K F, )] df = 1 [ F 2 σ ( ) ( 2 F, τ; F ϕ F, τ; F 2, )] + K = 1 [ σ ( ) 2 K, τ; F K K C ( F 2, K, τ ) ] where, in the last equality, we used the property ϕ ( F, τ; F, ) F + and the Breeden and Litzenberger formula. he payoff corresponding to the price 2 C ( F K 2, K, ) around strike K is well approximated by an infinitesimal butterfly contract around K. he payoff corresponding to C ( F τ, K, ) is well approximated by a calendar (time) spread. Hence the Dupire formula states that locally in a neighborhood of a pair (K, ) the local volatility function is proportional to the ratio between a calendar spread and a butterfly centered at (K, ). It is important to stress that the notion of local volatility function is radically different from that of implied volatility. In fact in the local volatility approach, a unique function σ Loc (t, S t ) is selected by a multiplicity of plain vanilla option prices and is constructed in such a way that the value function that solves the B&S PDE eq.(2.1) is consistent with option prices for any strike and any maturity. On the contrary, implicit volatility is the level of volatility which inserted in the B&S formula produces a price which is equal to the observed one. he critical difference relies on the observation that B&S formula solves eq.(2.1) only if the volatility function is assumed to be a constant, i.e. independent from the strike and the maturity. Hence, strictly speaking, the B&S formula provides the same prices of a local volatility model only in the trivial case, when the implied volatility function is a constant flat surface, independent from maturity and moneyness. In reality, only a finite number of options generating a grid of strikes and maturities is traded. hen the reconstruction of the local volatility function is obtained by interpolation methods. Historically, the local volatility approach has been the first one that allowed a profitable use of B&S theory in the

32 28 CHAPER 3. FROM HEORY O PRACICE practice of investment banking. In fact, as shown above, once a local volatility function has been selected, it is possible to value and hedge exotic derivatives with arbitrary payoff function by solving the boundary value problem for the B&S PDE eq.(2.1) with the appropriate terminal condition. In the next section wee show how it is possible to identify an hedging portfolio for any exotic derivative contract in terms appropriate portfolio of liquid securities, that are positions in stock, cash and plain vanilla options. 3.6 Static replication of exotic derivatives Let g (S ) the payoff of a derivative contract. Using Bredeen and Litzenberger one can prove that the (forward) price of the contract at time t for an initial (forward) price Ft is given by: V ( g Ft, t ) = E [ ] Q g (S ) Ft = g ( ) F Ft t + C(F t, K, )g (K) dk + F t P (F t, K, )g (K) dk Proof. Using the forward probability density and Breeden Litzenberger formula, the price of the claim can be written in terms of a weighted combination of butterfly spreads: E Q [g (S ) S t ] = = +Ft + g ( F ) ϕ ( F, ; F t, t ) ds g (K) 2 C(Ft, K, ) dk K 2 g (K) 2 P (Ft, K, ) dk +F t K 2 In fact recall the Put Call parity relation: hence: C(Ft, K, ) P (Ft, K, ) = Ft K C(Ft, K, ) K 2 C(Ft, K, ) P (Ft, K, ) = 1 + K t, K, ) K 2 K 2 = 2 P (F

33 3.7. HE REPLICAION OF VOLAILIY INDEX 29 and integrating by parts twice one gets: = +F t = g ( F t g (K) 2 C(Ft, K, ) dk + K 2 ) [ C(Ft, K, ) K g ( ) [ Ft C(F t, K, ) P (Ft, K, ) ] K=Ft + +F t = g ( F t + +Ft = g (F t ) + g (K) C(Ft, K, )dk + ) ( ) [ 1 g Ft K F t ]K=Ft g (K) C(Ft, K, )dk + +Ft g (K) 2 P (Ft, K, ) dk +Ft K 2 ] P (Ft, K, ) K +F t + g (K) C(F t, K, )dk + K=F t + g (K) P (F t, K, )dk g (K) P (Ft, K, )dk +F t +F t g (K) P (F t, K, )dk ( ) S Example 3.1 Consider a contract with payoff g (S ) = log, then: Ft E Q t [ ( )] F log F t = E Q [log = Ft Ft ( Ft F t )] C(F t, K, ) dk K 2 C(F t, K, ) dk K 2 F t F t P (F t, K, ) dk K 2 P (F t, K, ) dk K 2 he Log contract plays an important role in the construction of the VIX index, which is discussed in the next section. 3.7 he replication of volatility index Define the square of the VIX level as the risk neutral expected annualized integrated variance over the next 3 days: then one can state: V IX t = EQ t [ t ] σ 2 τ (ω) dτ t 3dd

34 3 CHAPER 3. FROM HEORY O PRACICE heorem 3.3 Assume that stock price index follows a continuous process, r = q = then: [ V IX 2 = 252 +F 3 2 t C(Ft, K, ) dk ] + K + P (F 2 t, K, ) dk K 2 i.e. the level of the expected variance can be replicated using a portfolio of (out of the money) plain vanilla options. Proof. In order to compute the integrated variance, one observes that by Ito Lemma: ( ) S ds u log = d log (S t ) = 1 σ 2 S t t t S u 2 udu t hence: t +F t ( ) σ 2 S udu = 2 log + 2 Ft t ds u S u (3.2) and taking the risk neutral expectation on both sides, one gets: V IXt 2 = 252 [ ] 3 EQ σ 2 udu = [ ( )] S t 3 EQ log Ft Application of the static replication formula to a contract with payoff 2 log one gets: ( )] [ +F 2E [log Q S t = 2 C(F Ft t, K, ) dk ] + K + P (F 2 t, K, ) dk K 2 and one concludes inserting this expression in eq.(3.2) he above procedure is the one which is currently implemented by the CBOE to compute the VIX index. he details on the discretization procedure of the integral expressions can be found in the CBOE technical document available on the CBOE website. +F t ( ) S, Ft

35 Chapter 4 Stochastic Volatility 4.1 Volatility, Subordination and Stochastic Clocks By the fundamental theorem of asset pricing, any forward price under the risk neutral measure evolves like a martingale. A famous theorem, the Dambis- Dubins-Schwarz theorem, see Revuz and Yor (213), states that any continuous martingale M t can be written as a time changed Brownian motion: M t = Ŵτ t where the time-change process τ t is given by the quadratic variation process: τ t = M, M c (t) := t t/ ( ) σ 2 2 (s) ds = lim Mn M (n 1) Hence the integrated variance can be considered as a stochastic clock. In this interpretation forward prices evolve as a Brownian motion with respect to a stochastic clock whose velocity is determined by market trading activity. he standard B&S framework, where market volatility, hence the time change τ t, must be a deterministic function of time t and of the price level S t. A stochastic volatility market is a market where the speed of this trading clock can be driven by an additional source of uncertainty which is independent from the level of the price. Investors willingness to insure their portfolios from the fluctuations induced by unexpected changes in the speed of this clock drive the so called volatility risk premium. Consider for example a variance swap, its payoff depends on the difference between expected and realized 3 days integrated 31 n=

36 32 CHAPER 4. SOCHASIC VOLAILIY variance. Hence, if there s a risk premium attached to the bets on the trading clock the premium on these bets is quantified by the so called variance risk premium. he valuation of options in this extended setup requires an extension of the replication argument that accounts for the presence of multiple sources of uncertainty, which is discussed below. In order to reduce the computational complexity, the dynamics of the trading clock exogenously specified. In particular the instantaneous variance is assumed to evolve as the Feller square root process. his assumption has been originally proposed in the formulation of a famous stochastic volatility model, the Heston (1993) model. 4.2 he Heston Model he Heston (1993) specification of the stochastic volatility model is given by: ds t = µdt + v t dzt 1 S t dv t = b (v v t ) dt + η ( v t ρdzt 1 + ( 1 ρ 2) ) 1/2 dz 2 t dz 1 t dzt 2 =, S = s, v = v, the parameter η is termed vol of vol, v long run mean variance, b rate of mean reversion, ρ variance-return correlation. Note that the diffusion coeffi cient of the process v t is well defined only in the state space of non-negative values. A suffi cient condition for the process started at a positive value to remain non negative for all future times is given by the so called Feller Condition: bv η Note that in the limit η the model is equivalent to a B&S model with time dependent volatility: v t = v ( 1 e bt ) + v e bt Note that for ρ = ±1, the level of the price is uniquely determined given the level of the volatility, like in a local volatility model. his model is the Heston-Nandi model and B&S pricing applies. For 1 < ρ < 1 the market composed by the underlying risky security and the risk free is incomplete and a new traded security is necessary to hedge volatility risk. In the following sections the replication approach is extended to this more general case.

37 4.3. MULIDIMENSIONAL IO LEMMA Multidimensional Ito Lemma. he assumption that the variance of the returns follows a process driven by an independent source of noise implies that the state of the system is described by two state variables S t, v t. hen the stochastic evolution is described by two equations driven by two sources of noise and the value of a generic derivative asset is described by a function that depends on the current level of S t and of v t, f (S t,v t ). Hence the evolution of such price is then determined by the two dimensional extension of the Ito lemma. In vector notation this extension can be stated as follows: consider the diffusion: then df (X t ) 1 1 = dx t n 1 { f (X t ) µ t (X t ) 1 n n 1 + f (X t ) 1 n σ t (X t ) n n = µ t (X t ) dt + σ t (X t ) dz t n 1 n n n r [Hess (f (X t )) n n dz t n 1 ]} σ t (X t ) σ t (X t ) dt + n n n n (4.1) where f (X t ) is the gradient, i.e. the line vector of first order partial derivatives f(xt) X i w.r.t. the elements of the vector X t, and Hess (f (X t )) is the Hessian matrix, i.e. the symmetric matrix of second order partial derivatives 2 f(x t) X i X j w.r.t. the elements of the vector X t and the trace denotes the sum of the elements of a matrix along a diagonal. As an example, consider the dynamics of the Heston model, then: [ ] [ ] S dst X = and dx v t =, dv t and: dx t = µ (X t ) dt + σ (X t ) dz t [ ] [ ( dst µ v t ) ] [ ] [ St vt = 2 S dt + t dz dv t bv bv t η v t ρ η 1 t v t ρ where ρ = 1 ρ 2. Hence the computation of the terms appearing in the multidimensional Ito formula eq.(4.1) goes as follows: the first order differential contribution: ( f (X) µ (X) = s f (s, v) µ v ) s + v f (s, v) (bv bv) 2 dz 2 t ]