sign in sign up
<<
Home , Derivative (finance), Forward contract, Forward market, Interest rate derivative, Local volatility, Moneyness

A Derivation of the Black–Scholes Pricing Equations for Vanilla Options

Junior quant: ‘Should I be surprised that μ drops out?’ Senior quant: ‘Not if you want to keep your job.’

This appendix describes the procedure for deriving closed-form expressions for the of vanilla call and put options, by analytically performing integrals derived in Chapter 2. In that chapter, we derive two integral expressions, either of which may be used to calculate the Black–Scholes value of a European . One of the integral expressions yields the value in terms of the transformed variable X (Equation 2.64):

∞ ( − )2 1 − X X ¯ v(X,τ)= √ e 2σ2τ P(X )dX −∞ 2πσ2τ

The other integral expression yields the value in terms of the original financial variable S (spot ) (Equation 2.65):

∞ 2 − ( − ) 1 1 (lnF − lnS ) V(S,t) = e rd Ts t exp − P(S )dS 2 σ 2( − ) 0 2πσ (Te − t) S 2 Te t

We will here perform the integral in Equation 2.64 in order to obtain the pricing formulae. The transformed payoff function P¯ (X) takes the following form for a vanilla option:

¯ X P(X) = max 0,φ F0e − K (A.1)

where φ is the option trait (+1foracall;−1foraput)andF0 is the arbitrary quantity that was introduced in the transformation process for the purpose of dimensional etiquette. The zero floor in the payoff function has the result that the integrand vanishes for a semi-infinite range of X values. For φ =+1, the integrand vanishes for X < ln(K/F0), whereas for φ =−1, the integrand vanishes for X > ln(K/F0). In general, we may write the

215 216 FX Barrier Options integral in terms of lower limit a and upper limit b:

( − )2 b − X X 1 2 X v(X,τ) = √ e 2σ τ max 0,φ F0e − K dX (A.2) a 2πσ2τ where the limits a and b depend on the option trait φ in the following way:

( ln K φ =+1 a = F0 (A.3) −∞ φ =−1 ( ∞ φ =+1 b = (A.4) ln K φ =−1 F0

Let us write the two terms in Equation A.2 explicitly as such:

v(X,τ) = I1 − I2 (A.5) where ( − )2 b − X X . 1 2 X I1 = φF0 √ e 2σ τ e dX (A.6) a 2πσ2τ and ( − )2 b − X X . 1 2 I2 = φK √ e 2σ τ dX (A.7) a 2πσ2τ The integrand of Expression I2 is the probability density function (PDF) of a with mean X and variance σ 2τ. This can be written in terms of the special function N(·), which is the cumulative distribution function (CDF) of a standard normal distribution: − − = φ b √X − a √X I2 K N σ τ N σ τ (A.8)

In Expression I1, completion of the square in the exponent gives:

2 X − X+σ2τ 1 2 b − X+ σ τ 1 2 I1 = φF0e 2 √ e 2σ τ dX (A.9) a 2πσ2τ

As with I2, the integrand is again the probability density function (PDF) of a normal distribution, and the variance is again σ 2τ, but this time the mean is (X + σ 2τ). Again, this can be written in terms of N(·): 2 2 + 1 σ 2τ b − X − σ τ a − X − σ τ = φ X 2 √ − √ I1 F0e N σ τ N σ τ (A.10) Derivation of the Black–Scholes Pricing Equations for Vanilla Options 217

The closed-form expressions for I1 and I2 given by Equations A.10 and A.8 respectively can now be inserted into Equation A.5 to give a closed-form expression for v. Since the quantities a and b depend on the option trait φ (see Equations A.3 and A.4), we will separate the call and put cases. For calls (φ =+1), I1, I2 and v are given as follows:

⎛ ⎛ ⎞⎞ ln K − X − σ 2τ + 1 σ 2τ F0 = X 2 ⎝ (∞) − ⎝ √ ⎠⎠ I1 (call) F0e N N σ τ ⎛ ⎛ ⎞⎞ ln K − X − σ 2τ + 1 σ 2τ F0 = X 2 ⎝ − ⎝ √ ⎠⎠ F0e 1 N σ τ ⎛ ⎞ −ln K + X + σ 2τ + 1 σ 2τ F0 = X 2 ⎝ √ ⎠ F0e N σ τ (A.11)

⎛ ⎛ ⎞⎞ ln K − X = ⎝ (∞) − ⎝ F0√ ⎠⎠ I2 (call) K N N σ τ ⎛ ⎛ ⎞⎞ ln K − X = ⎝ − ⎝ F0√ ⎠⎠ K 1 N σ τ ⎛ ⎞ −ln K + X = ⎝ F√0 ⎠ KN σ τ (A.12)

⎛ ⎞ ⎛ ⎞ −ln K + X + σ 2τ −ln K + X + 1 σ 2τ F0 F0 ⇒ = X 2 ⎝ √ ⎠ − ⎝ √ ⎠ vcall F0e N σ τ KN σ τ (A.13)

Forputs(φ =−1), I1, I2 and v are given as follows:

⎛ ⎛ ⎞ ⎞ ln K − X − σ 2τ + 1 σ 2τ F0 =− X 2 ⎝ ⎝ √ ⎠ − (−∞)⎠ I1 (put) F0e N σ τ N ⎛ ⎞ ln K − X − σ 2τ + 1 σ 2τ F0 =− X 2 ⎝ √ ⎠ F0e N σ τ (A.14) 218 FX Barrier Options

⎛ ⎛ ⎞ ⎞ ln K − X =− ⎝ ⎝ F0√ ⎠ − (−∞)⎠ I2 (put) K N σ τ N ⎛ ⎞ ln K − X =− ⎝ F0√ ⎠ KN σ τ (A.15)

⎛ ⎞ ⎛ ⎞ ln K − X − σ 2τ ln K − X + 1 σ 2τ F0 F0 ⇒ =− X 2 ⎝ √ ⎠ + ⎝ √ ⎠ vput F0e N σ τ KN σ τ (A.16)

The similarities between Equations A.13 and A.16 allow us to recombine the call and put results into a single vanilla result, like so:

⎡ ⎛ ⎞ ⎛ ⎞⎤ X − ln K + σ 2τ X − ln K + 1 σ 2τ F0 F0 = φ⎣ X 2 ⎝φ √ ⎠ − ⎝φ √ ⎠⎦ vvanilla F0e N σ τ KN σ τ (A.17)

We now have a closed-form expression for the transformed value variable v(X,τ).To obtain an expression for the original value variable V(S,t), it only remains for us to undo the four transformations of Section 2.7.1 one by one. Undoing Transformation 4 gives us an expression for undiscounted vanilla prices in terms of the forward:

⎡ ⎛ ⎞ ⎛ ⎞⎤ ln F + 1 σ 2τ ln F − 1 σ 2τ ˜ ( τ)= φ⎣ ⎝φ K √ 2 ⎠ − ⎝φ K √ 2 ⎠⎦ U F, FN σ τ KN σ τ (A.18)

Undoing Transformation 3 gives us an expression for undiscounted prices in terms of spot:

⎡ ⎛ ⎞ S + ( − + 1 σ 2)τ ( − )τ ln K rd rf 2 ( τ)= φ⎣ rd rf ⎝φ √ ⎠ U S, Se N σ τ ⎛ ⎞⎤ S 1 2 ln + (rd − rf − σ )τ − ⎝φ K √ 2 ⎠⎦ KN σ τ (A.19) Derivation of the Black–Scholes Pricing Equations for Vanilla Options 219

Undoing Transformation 2 gives us an expression for discounted prices in terms of spot:

⎡ ⎛ ⎞ S + ( − + 1 σ 2)τ − τ ln K rd rf 2 ˜ ( τ)= φ⎣ rf ⎝φ √ ⎠ V S, Se N σ τ ⎛ ⎞⎤ S + ( − − 1 σ 2)τ − τ ln K rd rf 2 − rd ⎝φ √ ⎠⎦ Ke N σ τ (A.20)

Lastly, undoing Transformation 1 gives us an expression for discounted prices in terms of spot, with explicit reference to the time variable t:

⎡ ⎛ ⎞ S + ( − + 1 σ 2)( − ) − ( − ) ln K rd rf 2 T t V(S,t) = φ⎣Se rf T t N⎝φ √ ⎠ σ T − t ⎛ ⎞⎤ S + ( − − 1 σ 2)( − ) − ( − ) ln K rd rf 2 T t −Ke rd T t N⎝φ √ ⎠⎦ (A.21) σ T − t

If it seems that we have laboured the working, it is for a reason: each of the forms of expression we have presented can be useful in its own right. All of the forms of expression shown above may be found in the literature. The expressions that form the arguments of the normal CDF are commonly given their own symbols. For example, following the conventions in Hull [2], we define: S + ( − + 1 σ 2)( − ) ln K rd rf 2 T t d1 = √ (A.22) σ T − t S + ( − − 1 σ 2)( − ) ln K rd rf 2 T t d2 = √ (A.23) σ T − t whereupon our formula for the discounted prices in terms of spot becomes: −r (T−t) −r (T−t) V(S,t) = φ Se f N(φd1) − Ke d N(φd2) (A.24)

The value V here is for an option with unit Foreign principal (Af = 1); to get the value for non-unit-principal options, we simply need to multiply V by Af. B Normal and Lognormal Probability Distributions

B.1 Normal distribution

N In the case where Z follows a normal distribution, its density function fZ has the form: ( − μ )2 N ( ) = 1 − z Z fZ z exp (B.1) πσ2 2σ 2 2 Z Z where z may take any real value. μ σ 2 The mean of the distribution equals Z , and its variance equals Z .Thestandard normal distribution is a normal distribution which has mean equal to zero and variance equal to one. We denote the PDF and CDF of the standard normal distribution by special functions n(·) and N(·) respectively: 1 z2 n(z) = √ exp − (B.2) 2π 2 z 1 x2 N(z) = √ exp − dx (B.3) −∞ 2π 2

B.2 Lognormal distribution

LN In the case where Z follows a lognormal distribution, its density function fZ has the form: ( − μ )2 LN ( ) = 1 1 − lnz ln Z fZ z exp (B.4) πσ2 z 2σ 2 2 Z Z where z > 0.

220 C Derivation of the Local Function

C.1 Derivation in terms of call prices

Our aim here is to derive an expression for the (lv) function σ(S,t) that appears in the local volatility model of Equation 4.21:

dS = (rd − rf)S dt + σ(S,t)S dWt

Central to the derivation is the probability density function (PDF) of spot. This quantity provides the crucial link between the dynamics of spot and the values of options. We introduced the PDF in the special case of the Black–Scholes model, in Section 2.7.2. With volatility equal to a constant, as we had there, we were able to write down an explicit expression for the PDF (Equation 2.69) in the form of a lognormal distribution for spot. In the context of a general surface, the PDF is not lognormal and is no longer given by Equation 2.69. The core of our derivation involves two relationships: first, the relationship between the PDF and values, and secondly, the relationship between the PDF and spot dynamics. Since the derivation involves both the time- and spot-dependence of the PDF, we will introduce the notation of a function p which depends explicitly on both variables:

. p(s,t) = fS(t)(s) (C.1)

Relationship 1 – between PDF and call option values – is the more straightforward one. To derive it, we use the fact that the value c of a call option equals the discounted risk-neutral expectation of its payoff, which can be written in terms of the risk-neutral PDF of spot at expiry:

c(K,T) = B t,T E[max(0,S(T) − K)] (C.2) ∞ = B t,T p(s,T)(s − K)ds (C.3) K

221 222 FX Barrier Options where K is the strike of the call option, T is its expiry time (dropping the subscript e for brevity), and B is the discount factor to option time. The lower bound of the integral is set to K because the payoff is zero when S(T) is below this level. Now the discounting is not of relevance to the current derivation, so we can simplify things a little by working in terms of the undiscounted call value C (the value at settlement date), defined as: . − C(K,T) = B 1 t,T c(K,T) (C.4) Relationship 1 then becomes:

∞ C(K,T) = p(s,T)(s − K)ds (C.5) K

Relationship 2 – between the PDF and spot dynamics – is given by the following equation:

∂p ∂ 1 ∂2 + r − r sp − σ 2s2p = 0 (C.6) ∂t d f ∂s 2 ∂s2

This equation is known as the Fokker–Planck equation or the Forward Kolmogorov equation, and its derivation is given in Appendix E. We now need to combine Relationships 1 and 2 (Equations C.5 and C.6). We can easily differentiate Equation C.5 with respect to T,toget:

∂C ∞ ∂p = (s − K)ds (C.7) ∂T K ∂T

∂p Equation C.6 gives us an expression for ∂T , which we can substitute into Equation C.7, to produce: ( ∂C ∞ ∂ 1 ∂2 = −(r − r ) sp + σ 2s2p (s − K)ds d f 2 ∂T K ∂s 2 ∂s We break this expression down into two integrals:

∂C 1 =−(r − r )I + I (C.8) ∂T d f 1 2 2 where ∞ ∂ I1 = (s − K) sp ds (C.9) K ∂s ∞ ∂2 I = (s − K) σ 2s2p ds (C.10) 2 2 K ∂s Derivation of the Local Volatility Function 223

To help us tackle the integrals, we revisit Relationship 1 (Equation C.5) and calculate its first and second strike-derivatives to obtain the following relationships:

∂C ∞ =− p(s,T)ds (C.11) ∂K K ∂2C = p(K,T) (C.12) ∂K2

These two relationships are not only useful for evaluating the integrals; they also have very practical interpretations. A European digital call with strike K can be structured out of two vanilla call positions with strikes closely spaced around K: a long at the lower strike and a position at the upper strike. In the limit of infinitesimal strike spacing, and with principals inversely proportional to the strike spacing, the undiscounted value of − ∂C this structure equals ∂K . Hence, using Equation C.11, we can see that the undiscounted European digital call price equals the integral of the PDF from K to infinity, which equals one minus the CDF at K. Meanwhile, the undiscounted European digital put price is precisely the CDF at K. Along similar lines, the limiting case of a butterfly with very closely spaced strikes has ∂2 undiscounted value equal to C . Equation C.12 then tells us that this butterfly value is ∂K 2 precisely the PDF. In addition to Equations C.11 and C.12, we also note the following useful result:

∂C ∞ C − K = sp(s,T)ds (C.13) ∂K K

Integrating I1 and I2 by parts gives:

∂C I =−C + K (C.14) 1 ∂K ∂2 2 2 C I2 = σ K (C.15) ∂K2 where we have made certain assumptions regarding the asymptotic behaviour of the PDF, for example that it tends to zero faster than quadratically as spot tends to infinity:

lim s2p = 0 (C.16) s→∞

Inserting Equations C.14 and C.15 into Equation C.8 gives:

∂C ∂C 1 ∂2C = (r − r )(C − K ) + σ 2K2 (C.17) ∂T d f ∂K 2 ∂K2 224 FX Barrier Options

Rearranging this equation gives us the result we need:

∂ ∂ C − (r − r )(C − K C ) σ( ) = ∂T d f ∂K K,T ∂2 (C.18) 1 K2 C 2 ∂K 2

This equation is the formula for calculating the lv model local volatility in terms of undiscounted call prices. To obtain the corresponding equation in terms of discounted call prices c, we use Equation C.4, together with its derivatives with respect to strike and maturity:

∂C − ∂c = B 1 t,T r (T )c(K,T) + ∂T d ∂T ∂C − ∂c = B 1 t,T ∂K ∂K 2 2 ∂ C − ∂ c = B 1 t,T ∂K2 ∂K2

The result is: ∂ ∂ c + r c + (r − r )K c ) σ( ) = ∂T f d f ∂K K,T ∂2 (C.19) 1 K2 c 2 ∂K 2

This equation is the formula for calculating the lv model local volatility in terms of discounted call prices. The partial derivatives with respect to strike and maturity in Equations C.18 and C.19 are the “co-” which we introduced in Section 3.6. Whilst perfectly correct, Equations C.18 and C.19 are not actually the equations best used in practice to calculate local volatilities. The reason is their numerical stability. At very high and very low strikes, the numerator and denominator of the fraction inside the square root both become very small, and the error in their quotient becomes large. It is easy to see why the numerator and denominator become small for very high strikes: the value of the call option tends to zero, and correspondingly all its co-Greeks do too. To see why it is also the case for very low strikes, we note that a call option tends to a forward as its strike tends to zero. The co-gamma in the denominator measures convexity, which is zero for a forward. A little analysis of the numerator (left as an for the reader) shows that it is zero for a forward, in fact at any strike. We should not be surprised by this asymptotic behaviour; it would after all be odd if we could somehow deduce a volatility (even an infinite one) from the price of a forward, which has no sensitivity to volatility. Derivation of the Local Volatility Function 225

C.2 Local volatility from implied volatility

The challenge of deducing volatilities from options which have vanishing volatility-dependence, as described at the end of Section C.1, arose long before the lv model was developed: the same challenge needs to be addressed in order to make vanilla option prices in the first place. A vanilla option -maker may be asked to make prices at any strikes, and therefore requires an implied volatility model which is able to produce sensible vols for very low and very high strikes. With such an implied volatility model at our disposal, if we were able to compute local volatilities from implied vols instead of from call prices, then we would expect much better numerical stability. Partly for this reason, and partly because we anyway generally prefer to work in volatility space than in price space, it is common practice to calculate local volatilities from implied volatilities. We now show how to transform Equations C.18 and C.19 to a form involving implied volatilities. Formally, the implied volatility (K,T) at strike K and expiry time T is related to call prices by the Black–Scholes pricing formula. From the results in Appendix A, we can write:

C(K,T) = FN(d1) − KN(d2) (C.20) where F ln K 1 √ d1 = √ +  T − t (C.21)  T − t 2 F ln K 1 √ d2 = √ −  T − t (C.22)  T − t 2

∂ ∂ ∂2 These relationships allow us to compute the co-Greeks C , C and C in terms of ∂T ∂K ∂K 2 derivatives of the implied volatility . Since the implied volatility always appears multiplied by the square root of the time to expiry, we can simplify the notation a little by defining a new quantity  by:

√ . (K,T) = (K,T) T − t (C.23)

We will call the quantity  the implied . The expressions for d1 and d2 now simplify to: ln F = K + 1  d1  (C.24) 2 F ln K 1 d = −  (C.25) 2  2 226 FX Barrier Options and the results for the co-Greeks are as follows:

∂C = (r − r )FN(d ) + Fn(d )˙ (C.26) ∂T d f 1 1 ∂C = Fn(d ) − N(d ) (C.27) ∂K 1 2 2 ∂ C 1 + d1K 1 + d2K = n(d2) K +  + (C.28) ∂K2 K where we have used the following shorthand forms for the derivatives of :

√ . ∂  ˙ = = ˙ T − t + √ (C.29) ∂T 2 T − t √ . ∂  = =  T − t (C.30) ∂K 2 √ . ∂   = =  T − t (C.31) ∂K2

Collecting everything together, we obtain the result: ˙ + (r − r )K σ(K,T) = 2 d f (C.32) 2 −1 K  + K +  1 + d1K 1 + d2K

This equation is the formula for calculating the lv model local volatility in terms of implied standard deviations. If we need an expression with explicit dependence on the implied volatility ,weevaluate Equations C.29–C.31 in terms of :

√ . ∂  ˙ = = ˙ T − t + √ (C.33) ∂T 2 T − t √ . ∂  = =  T − t (C.34) ∂K 2 √ . ∂   = =  T − t (C.35) ∂K2 and insert the results into Equation C.32, to get:

σ(K,T) = ˙ 1 (T−t) +  + (rd − rf)(T−t)K 2 2 √ √ (C.36) 2 −1 K  (T−t) + K (T−t) +  1 + d1K T−t 1 + d2K T−t Derivation of the Local Volatility Function 227

This equation is the formula for calculating the lv model local volatility in terms of implied volatilities.

C.3 Working in space

As described in Section 2.7.3.4, it is often beneficial to work in terms of moneyness instead of strike. For that reason, we will now transform our local volatility formulae into moneyness terms. Let us take the example of the formula for local volatility in terms of discounted call prices (Equation C.19). We first define a new function in terms of moneyness:

. c˜(k,T) = c(K,T) (C.37)

Then we evaluate the partial derivatives needed for the local volatility formula:

∂c 1 ∂c˜ = (C.38) ∂K F ∂k ∂2c 1 ∂2c˜ = (C.39) ∂K2 F2 ∂k2 ∂c ∂c˜ ∂c˜ = − k(r − r ) (C.40) ∂T ∂T d f ∂k

Inserting these transformed partial derivatives into Equation C.19, we get the result: ∂˜ c + r c˜ σ( ) = ∂T f K,T ∂2 ˜ (C.41) 1 k2 c 2 ∂k2

This equation is the formula for calculating the lv model local volatility in terms of discounted call prices in moneyness space. . A similar transformation on Equation C.18 (setting C˜ (k,T) = C(K,T))gives: ∂C˜ ˜ ∂ − (rd − rf)C σ(K,T) = T (C.42) ∂2 ˜ 1 k2 C 2 ∂k2

This equation is the formula for calculating the lv model local volatility in terms of undiscounted call prices in moneyness space. . Likewise, defining (˜ k,T) = (K,T), Equation C.32 can be transformed to: ˙ ˜ σ(K,T) = 2 (C.43) 2 ˜ ˜ ˜ −1 ˜ ˜ k  + k +  1 + d1k 1 + d2k 228 FX Barrier Options

This equation is the formula for calculating the lv model local volatility in terms of implied standard deviations in moneyness space. Note that the transformation to moneyness space simplifies the expressions.

C.4 Working in log space

Another beneficial transformation is to work in terms of the logarithm of strike or moneyness. For example, we define the log-moneyness κ by:

. κ = ln(k) (C.44)

Taking the case of implied standard deviations, we define a new function as follows:

. (κ¯ ,T) = (˜ k,T) (C.45) and we then express the derivatives of ˜ in terms of the derivatives of ¯ :

∂˜ ∂¯ = −κ ∂ e ∂κ (C.46) k 2 ˜ 2 ¯ ¯ ∂  − κ ∂  ∂ = e 2 − (C.47) ∂k2 ∂κ2 ∂κ ˜ ¯ ∂ ∂ = (C.48) ∂T ∂T κ k

The expression for local volatility then becomes: ˙ ¯ σ(K,T) = 2 (C.49) ¯ ¯ −1 ¯ ¯  +  1 + d1 1 + d2

This equation is the formula for calculating the lv model local volatility in terms of implied standard deviations in log-moneyness space. Similarly, the log-strike X is given by:

. X = lnK (C.50) and we can introduce a function ˆ that gives the implied standard deviation as a function of log-strike: . (ˆ X,T) = (K,T) (C.51) Derivation of the Local Volatility Function 229

The expression for local volatility then becomes: ˙ ˆ + (r − r )ˆ σ(K,T) = 2 d f (C.52) ˆ ˆ −1 ˆ ˆ  +  1 + d1 1 + d2

This equation is the formula for calculating the lv model local volatility in terms of implied standard deviations in log-strike space. All the expressions in terms of implied volatility and implied standard deviations (Equations C.32, C.36, C.43, C.49 and C.52) are numerically better behaved than the expressions in terms of call prices (Equations C.18, C.19, C.41 and C.42).

C.5 Specialization to BSTS

It may come as a surprise to note that the formulae for the local volatility in terms of call prices (for example, Equations C.18 and C.19) are not merely applicable to the bsts model, but they are exactly the same: the formulae cannot be simplified even though the bsts model itself is much simpler than the lv model. It is only when we come to re-write the formulae in terms of implied volatility that the bsts version becomes different – and much simpler! Setting to zero the strike derivatives of the implied standard deviation  in Equation C.32, and converting back to implied volatility , we obtain: σ (K,T) = 2˙ BSTS = 2 + 2(˙ T − t) ∂ = 2 (T − t) (C.53) ∂T

This is the formula for calculating the bsts model instantaneous volatility in terms of implied volatilities. Equation C.53 can be inverted straightforwardly: T ( ) = 1 σ 2 ( ) t,T BSTS u du (C.54) T − t t

This is the formula for calculating the bsts model implied volatility in terms of its instantaneous volatility. D Calibration of Mixed Local/ Volatility (LSV) Models

This appendix describes the calibration of the local volatility factor in the lsv model introduced in Section 4.12. At this stage, we assume that the process parameters (mean-reversion parameters, volatility of volatility and spot–vol correlation) have already been determined. The key equation that provides the basis for calibration is a relationship derived in 1996 by Bruno Dupire [65] and independently in 1998 by Emanuel Derman and Iraj Kani [66]. This relationship states that the expectation of the square of the instantaneous volatility at a given time, conditional on the spot price at that time being at a particular level, equals the square of the local volatility at that time and that spot level:

2 2 E σ |S(T) = K = σLV (K,T) (D.1)

It is straightforward to see how this relationship holds for the lv model: the instantaneous volatility in this model is the deterministic local volatility function σLV (S,T), whose expectation conditional on S(T) = K is trivially σLV (K,T). Inserting instead the instantaneous volatility for the lsv model gives:

2 2 2 2 E σ0 (S(T),T) (T)|S(T) = K = σLV (K,T) (D.2)

Again the conditional expectation of the deterministic local volatility function (the local volatility factor ) is straightforwardly taken out of the expectation, as is the constant base volatility level, yielding:

2 2 2 2 σ0 (K,T)E  (T)|S(T) = K = σLV (K,T) (D.3)

is then given by the following expression:

σ 2(K,T) 2(K,T) = LV (D.4) 2 2 σ0 E  (T)|S(T) = K

230 Calibration of Mixed Local/Stochastic Volatility (LSV) Models 231

Computation of relies on evaluation of the conditional expectation of . The latter can be written in terms of the joint probability density function of S(T) and , which we will denote p : S, ξ 2p (K,ξ,T)dξ E 2(T)|S(T) = K = S, (D.5) pS,(K,ξ,T)dξ The expression for is then given by: σ 2(K,T) p (K,ξ,T)dξ 2(K,T) = LV S, (D.6) 2 2 σ0 ξ pS,(K,ξ,T)dξ

If  is modelled as the exponential of a process, as for example in the exponential Ornstein–Uhlenbeck lsv model described in Section 4.12, it is useful to write the conditional expectation in terms of the joint density function of S and Y(= ln), denoted pS,Y :

 = eY (D.7) 2y ( ) e p (K,y,T)dy E e2Y T |S(T) = K = S,Y (D.8) pS,Y (K,y,T)dy

The expression for is in that case given by: σ 2(K,T) p (K,y,T)dy 2(K,T) = LV S,Y (D.9) 2 2y σ0 e pS,Y (K,y,T)dy

We demonstrated the derivation of the Fokker–Planck equation in the case of the lv model in Appendix E. Exactly the same approach based on our chosen form of lsv model yields the joint density function p required to evaluate either of Equations D.6 and D.9. For example, for the exponential Ornstein–Uhlenbeck model, the Fokker–Planck equation is:

∂ ∂ ∂2 ∂ p 1 2 2y 2 2 ¯ + (r − r ) (sp) − σ0 e ( s p) + κ (Y − y)p ∂t d f ∂s 2 ∂s2 ∂y ∂2 ∂2 1 2 y − α (p) − ρσ0α (e sp) = 0 (D.10) 2 ∂y2 ∂s∂y

We can use finite-difference methods to compute the solution of Equation D.10 numerically, as described in Chapter 6. E Derivation of Fokker–Planck Equation for the Local Volatility Model

We derive here the Fokker–Planck equation for the local volatility model. We will not be focusing on all of the mathematical conditions which the various quantities and functions need to satisfy. For a more mathematically thorough derivation, including all the conditions of behaviour we need to satisfy, see Shreve [10]. Our aim is to derive an equation of motion for the probability density function (PDF) of the spot price S(t) whose dynamics are given by the LV stochastic differential equation (SDE) of Equation 4.21: dS = (rd − rf)S dt + σ(S,t)S dWt Let g(·) be an arbitrary function, and define a stochastic variable Gt by:

. Gt = g(S(t)) (E.1)

Then, using Ito’s¯ Lemma, the SDE for Gt is given by: ∂ ∂2 ∂ g 1 2 2 g g dG = (r − r )S + σ S dt + σ(S,t)S dWt (E.2) d f ∂S 2 ∂S2 ∂S

We can then write down the following equation of expectations: ∂E ∂ ∂2 [Gt ] = E ( g 1 2 2 g rd − rf)S + σ S (E.3) ∂t ∂S 2 ∂S2 lhs rhs

The expectations on the left-hand and right-hand sides can each be written in terms of an integral involving the PDF p(S,t), and the time partial on the left-hand side can be taken inside the integral: ∂ ∞ lhs = g(s)p(s,t)ds ∂t 0 ∞ ∂p = g(s) ds (E.4) 0 ∂t

232 Derivation of Fokker–Planck Equation for the Local Volatility Model 233

∞ ∂g(s) 1 ∂2g(s) rhs = (r − r )s + σ 2s2 p(s,t)ds (E.5) d f 2 0 ∂s 2 ∂s The right-hand side can furthermore be integrated by parts, to give:

∞ ∂ 1 ∞ ∂2 rhs =−(r − r ) sp g ds + σ 2s2p g ds (E.6) d f 2 0 ∂s 2 0 ∂s

In the above step, we have assumed various quantities vanish for infinite spot. For example, we have assumed: ∂ lim σ 2s2p g = 0 s→∞ ∂s which assumes certain asymptotic properties of g. Equating LHS and RHS, we can now write that, for any function g with suitable asymptotic behaviour, the following equation holds: ( ∞ ∂p ∂ 1 ∂2 + (r − r ) sp − σ 2s2p g(s)ds = 0 (E.7) d f 2 0 ∂t ∂s 2 ∂s

We deduce that the quantity in curly braces must equal zero:

∂p ∂ 1 ∂2 + (r − r ) sp − σ 2s2p = 0 (E.8) ∂t d f ∂s 2 ∂s2

This is the Fokker–Planck equation for the local volatility (LV) model. Bibliography

[1] A. Lipton. Mathematical Methods for Foreign Exchange. World Scientific, 2001. [2] J. Hull. Options, Futures and other Derivatives (9th Ed.). Prentice Hall, 2014. [3] P. Wilmott. Paul Wilmott on Quantitative Finance. Wiley, 2006. [4] K. Pilbeam. International Finance. Palgrave Macmillan, 2013. [5] I. J. Clark. pricing. Wiley, 2011. [6] U. Wystup. FX options and structured products. Wiley, 2006. [7] R. Brown. A brief account of microscopical observations made in the months of june, july, and august, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Edinburgh new Philosophical Journal, 358–371, 1828. [8] F. Black and M. Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81(3):637–654, 1973. [9] K. Itô. On stochastic differential equations. Memoirs of the American Mathematical Society, (4):1–51, 1951. [10] S. Shreve. Stochastic Calculus for Finance II. Springer, 2004. [11] W. Doeblin. Sur l’équation de kolmogoroff. C. R. Ser. I, 331:1059–1102, 1940. [12] P. Austing. Smile Pricing Explained. Palgrave Macmillan, 2014. [13] M. Fourier. ThÃl’orie analytique de la chaleur. 1822. [14] R. C. Merton. Theory of rational option pricing. The Bell Journal of Economics and Science, 4(1):141–183, 1973. [15] Espen Gaarder Haug. The Complete Guide to Option Pricing Formulas (2nd Ed.). McGraw-Hill, 2007. [16] E. Reiner and M. Rubinstein. Breaking down the barriers. Risk Magazine, 4(8):28–35, 1991. [17] S. Shreve. Stochastic Calculus for Finance I. Springer, 2004. [18] U. Wystup. Ensuring efficient hedging of barrier options. http://www.mathfinance.de, 2002. [19] E. Reiner and M. Rubinstein. Unscrambling the binary code. Risk Magazine, 1991. [20] C. H. Hui. One-touch barrier values. Applied , 6:343–346, 1996. [21] M. Broadie, P. Glasserman, and S. Kou. A continuity correction for discrete barrier options. , 7(4):325–348, 1997. [22] R. C. Heynen and H. M. Kat. Partial barrier options. Journal of , 3:253–274, 1994. [23] F. Mercurio. A vega-gamma relationship for european-style or barrier options in the black-scholes model. Banca IMI. [24] O. Reiss and U. Wystup. Computing option price sensitivities using homogeneity and other tricks. The Journal of Derivatives, 9(2):41–53, 2001. [25] B. Dupire. Pricing with a smile. Risk Magazine, 7(1):18–20, 1994.

234 Bibliography 235

[26] P. S. Hagan, D. Kumar, A. S. Lesniewski, and D. E. Woodward. Managing smile risk. Wilmott Magazine, July:84–108, 2002. [27] S. Heston. A closed-form solution for options with stochastic volatility with applications to and options. The Review of Financial Studies, 6(2):327–343, 1993. [28] W. Feller. Two singular diffusion problems. Annals of Mathematics, 54(1):173–182, 1951. [29] J. Gatheral. The Volatility Surface. Wiley, 2006. [30] M. Jex, R. Henderson, and D. Wang. Pricing exotics under the smile. J.P.Morgan Securities Inc. Derivatives Research, 1999. [31] A. Lipton. The vol smile problem. Risk Magazine, February:61–65, 2002. [32] K. Said. Pricing exotics under the smile. Risk Magazine, November:72–75, 2003. [33] P. Henry-Labordère. Calibration of local stochastic volatility models to market smiles: A monte-carlo approach. SSRN, http://ssrn.com/abstract=1493306, 2009. [34] P. Henry-Labordère. Calibration of local stochastic volatility models to market smiles. Risk Magazine, September:112–117, 2009. [35] G. Vong. Turbo-charged local stochastic volatility models. SSRN, 2010. [36] P. Karasinski and A. Sepp. The stochastic volatility model. Risk Magazine, October:66–71, 2012. [37] Y. Tian, Z. Zhu, F. Klebaner, and K. Hamza. A hybrid stochastic volatility model incorporating local volatility. 2012 Fourth International Conference on Computational and Information Sciences (ICCIS). Available at SSRN: http://ssrn.com/abstract= 2074675, 2012. [38] C. Homescu. Local stochastic volatility models: calibration and pricing. SSRN, 2014. [39] G. E. Uhlenbeck and L. S. Ornstein. On the theory of the brownian motion. Physical Review , 36(5):823–841, 1930. [40] A. Lipton and W. McGhee. Universal barriers. Risk Magazine, May:81–85, 2002. [41] U. Wystup. The market price of one-touch options in foreign exchange markets. Derivatives Week, XII(13), 2003. [42] U. Wystup. Vanna-volga pricing. Frankfurt School of Finance and Management, (11):1–23, 2008. [43] A. Castagna and F. Mercurio. The vanna-volga method for implied volatilities. Risk Magazine, January:106–111, 2007. [44] E. Derman, D. Ergener, and I. Kani. Static options replication. Goldman Sachs Quantitative Strategy Research Notes, 1994. [45] J. Bowie and P. Carr. Static simplicity. Risk, 7(8):44–49, 1994. [46] M. Joshi. The Concepts and Practice of Mathematical Finance. Cambridge University Press, 2008. [47] Carol Alexander. Analysis. Wiley, 2009. [48] John C. Hull. and Financial Institutions. Wiley, 2012. [49] D. Duffy. Finite Difference Methods in Financial Engineering. Wiley Finance, 2006. [50] D. Tavella and C. Randall. Pricing Financial Instruments. Wiley, 2000. [51] K. W. Morton and D. F. Mayers. Numerical Solution of Partial Differential Equations. Cambridge University Press, 2005. [52] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical Recipes. Cambridge University Press, 2007. 236 Bibliography

[53] J. Crank and P. Nicolson. A practical method for numerical evaluation of solu- tions of partial differential equations of the heat-conduction type. Advances in Computational Mathematics, 6(1):207–226, 1996. [54] I. J. D. Craig and A. D. Sneyd. An alternating-direction implicit scheme for parabolic equations with mixed derivatives. Computers and Mathematics with Applications, 16(4):341–350, 1988. [55] P. Jäckel. Monte Carlo Methods in Finance. Wiley, 2002. [56] P. Glasserman. Monte Carlo Methods in Financial Engineering. Springer, 2003. [57] M. Giles and P. Glasserman. Smoking adjoints: fast monte carlo greeks. Risk Magazine, January:88–92, 2006. [58] Suisse. Emerging markets currency guide. Credit Suisse, www.credit-suisse. com, 2013. [59] Swiss National . Swiss National Bank sets minimum at CHF 1.20 per euro. Swiss National Bank Press Release, 2011. [60] K. Amin and R. Jarrow. Pricing foreign currency options under stochastic interest rates. Journal of International Money and Finance, 10:310–329, 1991. [61] A. Brace, D. Gatarek, and M. Musiela. The market model of dynamics. Mathematical Finance, 7(2):127–155, 1997. [62] J. Hull and A. White. Pricing securities. Review of Financial Studies, 3(4):573–592, 1990. [63] S. Gurrieri, M. Nakabayashi, and T. Wong. Calibration methods of hull–white model. SSRN, http://ssrn.com/abstract=1514192, 2009. [64] Bank of England, HM Treasury, and Financial Conduct Authority. How fair and effective are the fixed income, foreign exchange and markets? Fair and Effective Markets Review, 2014. [65] B. Dupire. A unified theory of volatility. Paribas Capital Markets discussion paper, 1996. [66] E. Derman and I. Kani. Stochastic implied trees: pricing with stochastic term and strike structure of volatility. International Journal of Theoretical and Applied Finance, 1(01):61–110, 1998. Index

10%-TV double no-touch, 108, 110, 134, barrier survival probability, see survival 165, 168, 200 probability 2-way price, 210 barrier trigger probability, see trigger 25-delta, see delta, strike quotation method probability 3-factor models, 207–210 barrier types, 26–27 continuously monitored, 26 discretely monitored, 26–27 AAD, see adjoint algorithmic Parisian, 27 differentiation partial, 26 accumulators, 1, 27 re-setting, 26 adjoint algorithmic differentiation, 204 time-dependent, 26 adjusted barriers, 119 window, 26 adjusted drift rate, 37 barrier-contingent payments, 23–25 American bets, 23, 94 Black–Scholes pricing, 73–80 American binaries, 23, 94 local volatility model pricing, 144–150 American options, 23 local/stochastic volatility model pricing, analytic Greeks, 86 168 analytical methods, 80–81 barrier-contingent vanilla options, 16–23 antithetic variables Black–Scholes pricing, 64–73 Monte Carlo, 197–199 local volatility model pricing, 150–154 arbitrage local/stochastic volatility model pricing, calendar, 124, 143 168 distributional, 143 base volatility, 163 falling variance, 124 basis points, 119, 154, 209 no-arbitrage conditions, 143 definition, 4 no-arbitrage principle, 15, 41 benchmark Asian options, 19 spot, 3, 4 at-the-money conventions, 129–131, volatility surfaces, 126 see also moneynesses bid–offer spreads, 42, 210–212 at-the-money forward, 129 big figures, 4, 30 delta-neutral , 129–130 bips, 4 at-the-money strikes, 58–59, 134 Black–Scholes model, 33–81 at-the-money volatility, 128–131, 137 barrier-contingent payments, 73–80 relationship to smile level, 131 barrier-contingent vanilla options, 64–73 barrier bending, 119 conceptual inputs and outputs, 38 barrier continuity correction, 80, 195 discrete barrier options, 80 barrier over-hedging, 91 equation for spot price, 33 barrier radar reports, 119 numerical pricing methods, 81

237 238 Index

Black–Scholes model – continued DNI, 75 option pricing PDE, 37–42 DNI_H, 76 boundary conditions, 45 DNN, 45, 115 derivation, 39 DNO, 75 payoff solution, 66 DON, 75 supplementary solution, 66 DOO, 75 transformation, 48 FNI, 77 role in options markets, 133 FNN, 46 vanilla options, 47–59 FNO, 76 derivation of formulae, 215–219 FOO, 78 formulae, 57–59 WNN, 47 window barrier options, 80 control variates Black–Scholes with term structure, 123 Monte Carlo, 197–199 boundary conditions Crank–Nicolson scheme, 192 barrier-contingent vanilla, 65 inversion, 29, 31, 76 vanilla options, 53 currency pair symbols, 2–4 broker markets, 135–136 currency pairs Brownian Bridge, 195, 199–200 AUDJPY, 3, 4, 29, 33, 59, 60, BS, see Black–Scholes model 86, 126, 131, 138, BSTS, see Black–Scholes with term 146, 150 structure BRLJPY, 130 bumped Greeks, 86 Domestic vs Foreign, 3–4 butterflies, 128–137 EURCHF, 172, 185, 206 relationship to smile convexity, 133 EURGBP, 3 relationship with risk-neutral PDF, 223 EURUSD, 1–4, 6–14, 16–25, 29, 33, 60, smile vs market, 137 66, 70, 76, 86, 87, 91–94, 97, 98, 104–110, 126, 128, 131, 132, 136, calculation agents, 214 138, 145, 150, 166, 168 call options, see vanilla options quote order, 2–5 call spreads, 16 USDTRY, 3, 29, 60, 77–80, 95, 126, 131, carry , 44 138, 146, 150, 209 CDF normal distribution, 220 d-Vega-d-Vol, see volgamma co-Delta, 120 Danish krone, 205 co-Gamma, 120 datetime, 5 co-Greeks, 119–120, 138, 224 de-peg event, 206 co-Theta, 120 de-peg risk, 206 common misconceptions, 90, 92–94, 99, delta, 39–41, 44–45, 83–95 126, 136 Forward-Delta-in-Domestic, 84 continuously monitored barriers, 19, 26, Forward-Delta-in-Foreign, 84 194 , 40–41 type codes premium adjustment, 84–85 CNN, 86 Spot-Delta-in-Domestic, 84 complete list, xxviii Spot-Delta-in-Foreign, 39–41, 84 DII, 75 strike quotation method, 88–90 DII_H, 76 delta exchange, 136 DIN, 75 delta gap, 91, 201 DIN_H, 76 delta hedging, 83 Index 239 delta-neutral straddle, 129 definition, 42 derivatives, 11 Feller Score, 162 diffusion equation, 52 finite-difference methods, 30, 81, 186–193 discount factors, 4–5 algorithms, 189 definition, 5 explicit scheme, 189–191 discretely monitored barriers, 26–27, 194 implicit scheme, 191–192 DNS, see delta-neutral straddle implicit-explicit schemes, 193 Domestic currency, 3–4, see also Foreign local/stochastic volatility models, 167 currency operator splitting, 193 double knock-ins, 22 first exit time, see first passage time double knock-outs, 21 first passage time, xxvii, 73 double no-touches, 134 first-generation exotic options, 26 vega, 108–110 flat volatility curve volgamma, 108–110 bsts, 124 downward-sloping volatility curve flies, see butterflies bsts, 124 flow products, 26 drift rate, 34 Fokker–Planck equation, 222 dual Greeks, see co-Greeks local volatility model, 232–233 Dupire local volatility, 164, see also local local/stochastic volatility models, 231 volatility model Foreign currency, 3–4, see also Domestic dVega, see volgamma currency dynamic hedging, 39, 183 forward , 6–12, 208 payoff, 10 eCommerce, 212–213 replication, 9, 31 electronic price distribution, 212–213 forward curve, 7 euro, see EUR forward Kolmogorov equation, 222 European derivatives, 14 , 6–12 European digitals, 15–16, 19, 31, 213 forward points, 7–8 relationship with risk-neutral CDF, 223 quotation convention, 8 European options, 14, 53 scaling factor, 8 exchange houses, 135 forward rate, 7 exchange rate, see spot rate, forward rate forward smile, 155 exotic contracts, 19 forward volatility agreements, 155 exotic options, 19 free-floating , 205–206 expected , 34 frown, see implied volatility frown expiry cuts, 14 funding valuation adjustment, 5 expiry times FVA, see forward volatility agreements, see standardized, 129 funding valuation adjustment explicit scheme finite-difference methods, 189–191 gamma, 44–45, 83–95 mathematical, 85 F, see fair forward rate practitioner, 86 fair forward rate, see also forward points, gamma of vega, see volgamma see forward rate gamma of vol, see volgamma formula, 8–12 geometric Brownian Motion, 34 geometry as risk-neutral expectation, 55 finite-difference grid, 186–189 240 Index

Greeks, 40, 83, 99, see also individual LV, 141 Greeks SABR, 157 analytic, 86 inter-bank markets, 135 bumped, 86 interest rates assumed deterministic, 5 heat equation, 52 interventions, 205 hedge ratios, 83, see also Greeks intrinsic value, 57–58 Heston model, 158 inversion method, see currency pair gamma, 162 inversion Hong Kong dollar, 205 Itô’s lemma, xxi, 36–37 Hull–White model, 207–210 Itô process, 36

IMEX, see implicit/explicit KIKOs, 25–26 implicit scheme sequential vs non-sequential, 26 finite-difference methods, 191–192 structurable vs non-structurable, 26 implicit/explicit schemes finite-difference methods, 193 implicitness parameter lagless approach, 38–39 finite-difference, 192 leptokurtic distributions, 138 implied standard deviation, 225 local variance, 142–143, 230 implied variance, 142–143 local volatility component implied volatility, 121 of lsv model, 163 at-the-money volatility, 129–131 local volatility factor butterfly, 132–133 of lsv model, 164 curve, 123 local volatility model, 141–154 definition, 125–126 barrier-contingent payments, 144–150 fly, 132 barrier-contingent vanilla options, frown, 133, 166, 170 150–154 market, 122 calibration, 221–229 models, 136–137 Fokker–Planck equation, 232–233 smile risk reversal, 131–132 Monte Carlo methods, 144 smiles, 126 option pricing PDE, 144 surface spot dynamics, 179–182 risk-neutral process, 144 surfaces, 126 local/stochastic volatility models, 162–171 term structure, 123 barrier-contingent payments, 168 interpolation model, 137 barrier-contingent vanilla options, 168 in-the-money strikes, 58–59 calibration, 164–165, 230–231 incremental bumping, 202 EURUSD, 166 industry parlance, 1, 3, 7, 131, 156 finite-difference methods, 167 infinitesimal-difference limit Fokker–Planck equation, 231 finite-difference grid, 190 generic form, 163 initial-value problem, 53 Monte Carlo methods, 167 instantaneous variance, 158 option pricing PDE, 167–168 instantaneous volatility log-moneyness, 228 BSTS, 123 log-spot, 35 Heston, 158, 159 lognormal distribution, 220 Index 241

LSV, see local/stochastic volatility models Monte Carlo simulation, see Monte Carlo LV, see local volatility model methods managed currencies, 205–206 netting market abuse, 213–214 of bid–offer spreads, 212 market roll, 122 New York expiry cut, 14 maturity, 31 no-arbitrage principle, 15, 41 of forward, 6 non-deliverable currencies, 130 of option, 39 non-sequential KIKOs, 26 mean reversion, 159 normal distribution, 220 mean-reversion level standard, xxvii, 194, 220 Heston, 159 normal knock-outs, 19–21, 31, 71, see also Ornstein–Uhlenbeck, 164 reverse knock-outs mean-reversion speed notional amounts, 1 Heston, 159 numerical methods, 80–81 Ornstein–Uhlenbeck, 164 method of images, 67 offer prices, 210 mid prices, 210 operator splitting mio (million), 29 finite-difference methods, 193 mixed Dupire model, 171 option inversion, 29 mixed local/stochastic volatility, see option pricing PDEs local/stochastic volatility model local volatility model, 144 mixing factor, 170, 182 local/stochastic volatility models, mixture risk 167–168 local/stochastic volatility models, options 182–183 holder, 31 mixxa, 183 premium, 13 moneyness, 58–59, 227–228 vanilla, 12–15 lines of constant moneyness, 143 writer, 31 Monte Carlo methods, 81, 125, 186, Ornstein–Uhlenbeck process, 164 193–200 exponential, 164 adjoint algorithmic differentiation, 204 orthogonality of risk factors, 179 antithetic variables, 197–199 OTC, see over-the-counter Brownian Bridge, 199–200 out-of-the-money strikes, 58–59 compute grids, 203 outright forward rate, 8, 31, see also fair contract schedule, 194–195 forward rate control variates, 197–199 over-hedging, 117–119 early termination, 200 over-the-counter markets, 135 estimators, 195 farms, 203 Greeks, 203 P&L, 185 local volatility model, 144 Parisian barriers, 27, 31 local/stochastic volatility models, 167 Parisian options, 27, 32 pathwise method, 203 partial barriers, see window barriers simulation schedule, 194–195 pay-at-hit, 24, see also pay-at-maturity variance reduction, 197–199 pay-at-maturity, 24, see also pay-at-hit 242 Index payoff profiles reverse knock-outs, 19–21, 32, see also definition, 11 normal knock-outs payoff spike, 19 similarity to barrier-contingent payoffs payments, 94 accumulators, 27–28 rho, 113–115 barrier-contingent payments, 23–25 bucketed, 179 barrier-contingent vanilla options, discounting effect, 115 16–23 forward effect, 115 , 10 parallel, 179 KIKOs, 25–26 weighted, 179 vanilla options, 12–14 ringing, 177 PDEs risk analysis option pricing, see option pricing PDEs spot, 83–97 PDF local, 83–95 lognormal distribution, 220 non-local, 83, 96–97 normal distribution, 220 risk ratios, 83, see also Greeks risk-neutral, see risk-neutral risk reports skew, 138 spot, 97 pegged currencies, 205–206 risk reversal gamma, 156 pips, 4, 32 Heston model, 162 pivot maturity, 178 local volatility model, 156 pre-hedging, 213 local/stochastic volatility models, premium 162–163 of option, 13 risk reversals, 128–137, 156, 184 premium currency, 130 option structure, 131–132 premium-adjusted delta, 84–85 relationship to smile skew, 132 price quotation styles, 59–60 risk-neutral CDF pricing rules, see rules-based pricing relationship with European digitals, 223 methods risk-neutral distributions, 55, 76, 100, 137, principal amounts, 1, 32 138, 143, 221 probability distributions risk-neutral drift rate, 55 lognormal, 220 risk-neutral expectation, 47, 55, 56, 115, normal, 220 221 pseudo term sheet, 28–29 risk-neutral measure, 55 put options, see vanilla options risk-neutral PDEs, 42, 55 put spreads, 16 bs,42 put–call parity, 14–15, 32 risk-neutral PDF quantitative analysts, xx relationship with butterflies, 223 quote order convention, 3 risk-neutral processes bs,55 ranges, 23, 32, 78 bsts, 125 ratchet option structure, 212 local volatility model, 144 re-setting barriers, 26 lv, 141 rebates, 25 risk-neutral valuation, 4, 42, 55, 83 reflection principle, 69 definition, 42 regulation, 213–214 rules-based pricing methods, 212–213 return, 34 rungs, 97 Index 243

SABR model, 157 standard normal distribution, see normal schedules, 27, 194–195 distribution schemes static hedging, 183–184 finite-difference, 189 static replication, 183–184 sequential KIKOs, 26 sticky local volatility, 202 settlement date sticky moneyness, 156 of spot trade, 1 sticky strike, 156 settlement lags, 2, see also settlement rules stochastic interest rates, 206–210 settlement rules, 2–32 stochastic processes, 35 short rates, 5 stochastic volatility component lsv short-dated forwards, 7 of model, 163 short-term interest rate trading, 10 stochastic volatility factor lsv sibling options, 22, 24, 71 of model, 164 stochastic volatility models, 157 simulation Heston, 158 Monte Carlo, see Monte Carlo methods SABR, 157 SIR, see stochastic interest rates stochastic/local volatility model, see skew to tv, 134, 144–154 local/stochastic volatility models vanilla options, 151 straddle, 129 SLV, see local/stochastic volatility models delta-neutral, see delta-neutral straddle small figures, 4, 32 straight dates, 129, 137 spot move, 191 smile convexity smile vs market, 137 relationship to butterfly, 133 strike, 32 smile dynamics, 154–156, 184, 202 of forward, 7 re-location, 179–182 strike rate, see strike smile level structured products, 27–28, 212 relationship to at-the-money volatility, survival probability, see also trigger 131 probability, 73–76 smile re-location, 156 SV, see stochastic volatility models smile skew points, see forward points relationship to risk reversal, 132 spikes in payoff, 19 tenors, 129 spot dates, 2, 32 term sheet, 28–29 spot dynamics, 155 terminal spot rate, 14, 32 spot exchange rate, see spot rate theoretical value, 133–136 spot ladders, 97 theta, 115–117, see also implicitness spot lags, 2, 32, see also settlement rules parameter lagless approach, 38–39 mathematical, 115 , 1–5 scaled mathematical, 116 spot price, see spot rate theta scheme spot rate, 2, 32 finite-difference, 192 spot , 1–5, 32 time value, 61–64 spot–vol correlation time-dependent barriers, 26, 32 Heston, 159 trade date, 2 Ornstein–Uhlenbeck, 164 trade time, 2 spot-vol matrix, 113 trading book, 82 244 Index trait, xxvii, 53 mathematical, 99 trigger probability, see also survival parallel, 177–178 probability, 73–76 practitioner, 99 TV, see theoretical value term of local/stochastic volatility model PDE, 167–168 uncertain volatility models, 171–172 weighted, 178 , 11 vol convexity, see volgamma units volatility drift rate, 34 Black–Scholes, 35 interest rate, 5 cone, 171 time, 5 ladders, 112 volatility, 35 risk reports, 112–113 upward-sloping volatility curve volatility Greeks, 173–174 bsts, 124 volatility of volatility, 182 Heston, 159 Ornstein–Uhlenbeck, 164 (VaR), 185 term structure, 170–171 vanilla options, 12–15 volatility swaps, 1 boundary condition, 53 volatility term structure risk, 175 definition, 13 volga, see volgamma market, 126–137 volgamma, 99–110, 173–174 standardized quotes, 128 heuristic, 102 structures, 128 other names, 99 trait, 53 term of local/stochastic volatility model vanna, 110–112, 173–174 PDE, 167–168 gap, 112 vomma, see volgamma term of local/stochastic volatility model PDE, 167–168 vanna–volga methods, 173–174 weighted vega, see vega, weighted VaR, see Value at Risk Wiener processes, 34–35 variance reduction window barriers, 26 Monte Carlo, 197–199 vega, 97–110, 129, 173–174 zero- bond, 135 bucketed, 176–178 zero-delta straddle, see delta-neutral ladders, 112 straddle