Financial Economics

Option Pricing under non constant volatility

Econ 643: Financial Economics II

Econ 643: Financial Economics II Non constant volatility 1 / 21

Department of Economics

Introduction

Attempts have been made to fix option pricing puzzles: How to be consistent with volatility smile and smirk.

The Gram-Charlier expansion is one of then but volatility is constant which is inconsistent with asset return’s dynamics

We review thre approaches that aim at integrating information embedded in past returns:

GARCH type of approach, Stochastic volatility models: Hull and White (1987), Stochastic volatility models: Heston (1993),

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The GARCH option pricing

Let St be the asset price at time t and rt = ln(St/St−1) be the log-return process. Assume that the process rt is a (G)GARCH(1,1) process:

rt = ln (

St St−1

)

= µt−1 + σt−1zt, zt ∼ NID(0, 1)

σ2t = ω + α(σt−1zt − θσt−1)2 + βσ2t−1. (1)

In this model,

µt−1 = E(rt|Jt−1) is a known function of past returns. Ex: µt = 0, µt = µ = cst, µt = µ + λσt, µt = r + λσt − 12σ

2 t , etc.

σ2t−1 = Var(rt|Jt−1) is the conditional variance of rt given the information Jt−1 available at t − 1.

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GARCH: How to price options on S?

We can rely on the risk-neutral approach:

C = e−rτ E∗ (max(ST − X, 0)) ,

where E∗ is the expectation under risk-neutral dynamics.

What is the risk-neutral dynamics of St if ln(St/St−1) is a GARCH(1,1)?

Under risk-neutral dyn., E∗ (

St St−1

)

= er and Var∗(rt|Jt−1) = σ2t−1 (same as under historical measure). Hence, if rt ∼ GARCH(1, 1) under risk-neutral, the corresponding mean has to be

µ∗t−1 = r − σ2 t−1

2 . That is:

rt = r − σ2 t−1

2 + σt−1z

∗

t , z ∗

t ∼ NID(0, 1) σ2t = ω + α(σt−1z

∗

t + r − σ2 t−1

2 − µt−1 − θσt−1)2 + βσ2t−1.

(2)

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GARCH: Simulating the option price

To obtain the price C by simulation:

Simulate B paths of stock price using the risk-neutral dynamics (2):

(S (b) t+1, S

(b) t+2, . . . , S

(b) T

) for b = 1, . . . , B (e.g. B = 5000).

Obtain the simulated price as

Ĉ = e−rτ Ê(max(ST − X, 0)),

with

Ê(max(ST − X, 0)) = 1

B

B ∑

b=1

max(S (b) T

− X, 0).

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Option pricing under stochastic volatility

GARCH option pricing is convenient but evidence are out that volatility is more likely stochastic.

Option pricing under SV is quite challenging because of the extra source of uncertainty brought by the volatility equation.

The induced PDE (by SV) for option pricing can be derived but is hard to solve.

The most common SV option pricing models are from Hull and White (1987) and Heston (1993).

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Hull and White (1987)

Consider the price process St and its instantaneous variance process V − t = σ2t obeying the dynamics:

dS = φSdt + σSdz1

dV = µVdt + ξVdz2, (3)

where φ may depend on S, σ and t µ and ξ depend on σ and t and E(dz1dz2) = ρdt.

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Hull and White (1987): PDE

Two sources of uncertainty to hedge z1 and z2.

Let f be the value of an option and Π = f − ∆sS − ∆vG be the hedging portfolio where G is another asset (maybe non investable) whose value depend on volatility. We have dΠ = df − ∆sdS − ∆vdG, with

dC = ∂f ∂t

+ ∂f ∂V

+ 1 2 ∂2f ∂V 2

(dV )2 + 1 2 ∂2f ∂S2

(dS)2 + ∂ 2f

∂V ∂S dVdS,

dG = similar.

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HW (1987): PDE

Setting dΠ = rΠdt, one obtains under some simplifying assumptions that C must satisfy the PDE:

∂f ∂t

+ 1 2

(

σ2S2 ∂ 2f

∂S2 + 2ρσ3ξS ∂

2f ∂V ∂S

+ ξ2V 2 ∂ 2f

∂V 2

)

−rf = −rS ∂f ∂S

− µσ2 ∂f ∂V

.

This PDE is hard to solve directly.

Since the PDE does not depend on risk preference, one can rely on risk-neutral approach to derive C. That is:

f = e−rτ E∗(max(ST − X, 0)), where, as usual, E∗ is expectation under risk neutral dynamics.

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HW (1987): ‘close form’ solution under restriction

Imposing ρ = 0 and letting V̄ = 1 T−t

∫ T

t σ2udu, the law of iterated

expectations ensures that:

E∗ max(ST − X, 0) = E∗ (

E∗(max(ST − X, 0)|V̄ ) )

.

HW (87) show that, if under risk neutral, (S, σ2( follow:

dS = rSdt + σSdz∗1, dσ 2 = ασ2dt + ξσ2dz∗2,

where r is the risk-free rate and α, ξ are independent of S, then:

ln S(T)

S(t) |V̄ ∼ N((r − V̄ /2)τ, V̄ τ).

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HW (1987): ‘close form’ solution under restriction

As a result, conditional of V̄ , we are in the BS world and

E∗(max(ST − X, 0)|V̄ ) = BS(V̄ )

yielding

f (St, σ 2 t ) =

∫

BS(V̄ )h(V̄ |σ2t )dV̄ ,

where h is the pdf of V̄ conditional on σ2t .

h(V̄ |σ2t ) hard to obtain even under further restrictions. . . Possible to obtain approximated value for f via the moments of V̄ .

BS(·) is convex for small values of V and concave for large values of V . As a result, by Jensen’s inequality, under SV, standard BS underprices for low V̄ and overprices for high V̄ .

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HW (1987): Numerical solution under less restriction

Keep ρ = 0 but ξ and µ may depend on σ and t.

In this case, only the process of variance needs to be simulated and then, use BS formula to obtain price along the algo. below: Repeat the follow. 3 steps, for b = 1, . . . , B (e.g. B = 5000),

1 Divide T − t into n equal subintervals, generate εi : i = 1, . . . , n ∼ NID(0, 1)

2 Let Vi be the variance at t + i T−t n

.

Vi = Vi−1e (µ−ξ2/2)∆t+ξ

√

∆tεi ; ∆t= T−t n

(If µ and ξ depend on σ, calculate their values using σ = √

Vi−1.) 3 Calculate: V̄ (b) = 1

n

∑n

i=1 Vi and BS(V̄ (b)).

Obtain the simulated derivative price:

f̂ (St, σ 2 t ) =

1

B

B ∑

b=1

BS(V̄ (b)).

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HW (1987): Numerical solution in general conditions

ρ 6= 0, and µ and ξ may possibly be functions of S, σ and t. Assume still that risk neutral valuation is valid.

We have to simulate both V and S because of correlation.

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HW (1987): Numerical solution in general conditions

Repeat the follow. 3 steps, for b = 1, . . . , B (e.g. B = 5000), 1 Divide T − t into n equal subintervals, generate

ui : i = 1, . . . , n ∼ NID(0, 1) ind. of εi : i = 1, . . . , n ∼ NID(0, 1) 2 Let Si and Vi be the price and variance at t + i

T−t n

.

Si = Si−1e (r−Vi−1/2)∆t+ui

√ Vi−1∆t

Vi = Vi−1e (µ−ξ2/2)∆t+ρξ

√

∆tui + √

1−ρ2ξ √

∆tεi ; ∆t = T−t n

(If µ and ξ depend on σ, calculate their values using σ = √

Vi−1.)

3 Calculate: f̂ (b) = e−rτ max(S (b) n − X, 0), with S(b)n ≡ Sn.

Obtain the simulated derivative price:

f̂ (St, σ 2 t ) =

1

B

B ∑

b=1

f̂ (b).

In both cases, it is possible to optimize the simulation approach by antithetic simulation designs. See HW (87).

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Heston (1993)

The spot asset price at time t, S(t) and its spot variance V (t) follow

dS = µSdt + √ V Sdz1

dV = κ(θ − V )dt + σ √ V dz2,

where z1 and z2 are standard Brownian motions with correlation E(dz1dz2) = ρdt, µ: the rate of return on the asset, θ: long run variance, κ: rate of mean reversion, σ: volatility of volatility.

2κθ > σ2: to guarantee that Vt is always positive.

Vt is a square root process or a CIR process (Cox-Ingersoll-Ross, 1985).

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Heston (1993): The PDE

If C(S, V , t) is the value of an option on S, under Heston’s model, C must follow the PDE:

1 2 VS2 ∂

2C ∂S2

+ ρσVS ∂ 2C

∂S∂V + 1

2 σ2V ∂

2C ∂V 2

+ rS ∂C ∂S

+ (κ(θ − V ) − λ(S, V , t))∂C ∂V

− rC + ∂C ∂t

= 0.

λ is the price of volatility risk.

The solution C(S, V , t) to this PDE has a closed form expression.

Econ 643: Financial Economics II Non constant volatility 16 / 21

Heston (1993): Solution to the PDE

Using some relevant boundary conditions, the PDE is solved by:

C(S, V , t) = SP1 − XP(t, T)P2, with P(t, t + τ) = e−rτ and

Pj = 1

2 +

1

π

∫

∞

0 Re

(

e−iφ ln(K)fj iφ

)

dφ, i2 = −1, j = 1, 2,

fj = e cj +Dj V +iφ ln(S).

cj = rφiτ + a σ2

(

(bj − ρσφi + dj)τ − 2 ln 1−gj e

dj τ

1−gj

)

Dj = bj −ρσφi+dj

σ2

(

1−e dj τ

1−gj e dj τ

)

gj = bj −ρσφi+dj bj −ρσφi−dj

dj = √

(ρσφi − bj)2 − σ2(2ujφi − φ2) u1 = 1/2, u2 = −1/2, , a = κθ, b1 = κ + θ − ρσ, b2 = κ + λ.

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Some exotic options

There are many types of options that financial engineers are expected to provide tools to value.

European and American put call options are called plain vanilla options.

Beside them are the exotic options.

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Examples of exotic options

Package options: A package option is a collection of standard European call, put options: Bull spread, bear spread, butterfly, etc. We know how to price such packages!

Bermuda options: A Bermuda option is a standard European put/call option but that can be exercised before maturity at a finite number of predetermined dates. We don’t know how to price them in this course!

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Barrier options

They are options with payoff depending on whether the underlying asset’s price reaches a certain level during a certain period of time. They are 4 sorts of barrier options:

Down and out call: Regular call that ceases to exist if mT = min(St) over [0, T] is smaller than a certain predetermined amount H. Its payoff at T is: max(ST − X, 0) · 1(mT > H). Down and in call: Regular call that has no value if mT is larger than predetermined amount H. Its payoff at T is: max(ST − X, 0) · 1(mT ≤ H). Up and out call: Regular call that ceases to exist if Mt = max(St) over [0, T] is larger than a certain predetermined amount H. Its payoff at T is: max(ST − X, 0) · 1(MT ≤ H). Up and in call: Regular call that has value only if Mt = max(St) over [0, T] is larger than a certain predetermined amount H. Its payoff at T is: max(ST − X, 0) · 1(MT > H).

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Barrier options

The put version of these call options are defined analogously with there payoffs expressed similarly with ST − X replaced by X − ST . All these barrier options can be priced by the risk neutral approach using, BS, GARCH, Hull and White etc.

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