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이학석사 학위논문

Estimation of Implied Volatilities and Interest Rate Derivative Prices

(내재변동성과 이자율 파생상품 가격 추정)

2019년 2월

서울대학교 대학원

수리과학부

류호성

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(내재변동성과 이자율 파생상품 가격 추정)

지도교수 박형빈

이 논문을 이학석사 학위논문으로 제출함

2018년 10월

서울대학교 대학원

수리과학부

류호성

류호성 의 이학 석사 학위논문을 인준함

2018년 12월

위 원 장 (인)

부 위 원 장 (인)

위 원 (인)

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A dissertation

submitted in partial fulfillment of the requirements for the degree of

Master of Science

to the faculty of the Graduate School of Seoul National University

by

Hoseong Ryu

Dissertation Director : Professor Hyungbin Park

Department of Mathematical Sciences Seoul National University

February 2019

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Abstract

The first purpose of this thesis is to derive the implied volatility of call options asymptotically under stochastic volatility models including the Heston model and the SABR model and under local volatility models including the CEV model and polynomial models. By minimizing the mean square error between the derived asymptotic implied volatility and the market implied volatility data, parameters can be estimated. Also, under the Black- Scholes model with the assumption that the short rate is zero, the asymptotic implied volatility of Asian call options can be derived such that the Asian call options can be priced without monte carlo simulations.

The second purpose of this thesis is to price interest rate derivatives under the Hull- White model. From the treasury yield data, the yield curve can be derived by the cubic spline curve method. With this curve, the formula for θ(t) in the Hull-White model is derived. By settingb= 0.5 and estimatingσfrom the historical volatility, the Hull-White model can be calibrated. Now, interest rate derivatives can be calculated with closed form formula except for swaptions. Since there is no closed form solution for swaptions, they are priced approximately.

Keywords : Heston model, SABR model, local volatility model, Black-Scholes model, Hull-White model, cubic spline curve

Student Number : 2014-21189

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Chapter 1 Introduction

As is described in the abstract, the first purpose of this paper is to derive the asymptotic implied volatility of call options under stochastic volatility models including the Heston model and the SABR model and under local volatility models including the CEV model and polynomial models. By comparing the implied volatility with the poly-fit of the market implied volatility data, we can decide parameters. This method is valid since there is a one to one correspondence between the option price and the implied volatility due to the positivity of the vega value of the call option under the Black-Scholes model.

This means the option price increases as the implied volatility increases. Also, estimating parameters is meaningful since the future behavior of the stock can be simulated by monte carlo simulations.

The strategy for deriving implied volatility asymptotically is as follow. Firstly, derive the partial differential equation of the implied volatility. Secondly, expand the implied volatility as a polynomial series with the variables x_t = log(_F^K

t) and τ =T −t. Finally, compare this with the poly-fit of the market implied volatility. By this process, we can estimate parameters.

In the case of Asian options, firstly assume that the risk-neutral interest rate is zero.

This assumption is required since deriving the partial differential equation of the implied volatility without this assumption is too complex to handle. Even though the assumption seems not realistic, it turns out that this doesn’t affect the implied volatility much.

Secondly, derive the partial differential equation of the implied volatility. Finally, derive the asymptotic implied volatility curve.

For the second purpose, we calculate the prices of interest rate derivatives such as bond options, caplets, caps, swaps, and swaptions under the Hull-White model. By deriving the cubic spline curve that fits the market yield data, setting b = 0.5, and estimating σ as historical volatility, the Hull-White model is calibrated. Then, the bond dynamics is derived under which interest rate derivative prices are calculated.

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Chapter 2

Main Ideas on the Implied Volatility

Even though there is a closed form call option price formula in a model, sometimes it is hard to estimate the market-fitting parameters. However, it is easy to estimate parameters when the asymptotic implied volatility curve is derived.

Now, let’s change the stock price S_t to the log-money x_t = log ^K_F

t

as a variable of the implied volatility whereF_t=S_te^r(T^−t) is the forward price of the stock at time t and K is the strike. This change is due to the fact that xt is closer to 0 than St is such that the implied volatility expressed as a polynomial series in terms of x_t rather than S_t is more accurate.

Let’s think of the call option with strike K and expiration time T and introduce the probability space (Ω,F,(F_t), P) withF_t=F_t^W which means that the completed filtration is generated by the standard Brownian motionW_t. At time t < T, the call price is given by

e^−r(T^−t)E((S_T −K)₊|F_t).

Since S_T =F_T holds, this can also be written as

e^−r(T^−t)E((F_T −K)₊|F_t).

Under the Black-Scholes model, the call price is of the form S_tN(d₁)−Ke^−r(T^−t)N(d₂)

where N is the cumulative normal distribution andd₁, d₂ are defined as d₁ = log(^F_K^t)

σ√

T −t + 1 2σ√

T −t d₂ =d₁−σ√

T −t

with the volatilityσ. Since S_t =F_te^−r(T^−t), this can also be written as e^−r(T^−t)(F_tN(d₁)−KN(d₂)).

Since the implied volatility is the volatility of the Black-Scholes model that makes these two call prices equal, the equation

E((FT −K)+|Ft) =FtN(d1)−KN(d2)

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is satisfied under the implied volatility. By dividing both sides by K and expressing F_t in terms ofx_t = log(K/F_t), we get

E((e^−x^T −1)₊|F_t) = e^−x^tN(d₁)−N(d₂).

Now, define C as

C(t, x_t, ν_t) =E((e^−x^T −1)₊|F_t) for stochastic volatility models and

C(t, x_t) =E((e^−x^T −1)₊|F_t) for local volatility models. Define C^BS as

C^BS(xt, w) =e^−x^tN(d1)−N(d2).

In the above equation,d₁ andd₂ are defined asd₁ =−^x_w^t+¹₂w,d₂ =−^x_w^t−¹₂wwherex_t= log(_F^K

t) and w =σ(t, x, ν)√

T −t for stochastic volatility models and w =σ(t, x)√ T −t for local volatility models. Note that σ is the implied volatility and ν is the stochastic volatility. For t₂ > t₁, since

E(C(t₂, x_t₂, ν_t₂)|F_t₁) = E(E((e^−x^T −1)₊|F_t₂)|F_t₁) =E((e^−x^T −1)₊|F_t₁) = C(t₁, x_t₁, ν_t₁) holds, the process C is a martingale. Therefore, the dt term of dC must be zero. In stochastic volatility models,

dC(t, x, ν) = C_tdt+C_xdx+ 1

2C_xx(dx)²+C_νdν+1

2C_νν(dν)²+C_xν(dxdν).

In local volatility models,

dC(t, x) =C_tdt+C_xdx+1

2C_xx(dx)².

With these, we can derive the partial differential equation of C. Since C =C^BS, we can express the derivatives ofCin terms of the derivatives of C^BS. Now, we need to calculate the values of C_τ, C_x, C_xx, C_ν, C_νν, C_xν where τ = T −t. Before calculating these values, we need to calculateC_x^BS, C_w^BS, C_xx^BS, C_xw^BS, C_ww^BS.

C_x^BS =−e^−xN(d₁), C_w^BS =e^−xN⁰(d₁) C_xx^BS =−C_x^BS +C_w^BS1

w, C_xw^BS =C_w^BS(− x w² − 1

2), C_ww^BS =C_w^BS(x² w³ − w

4).

Now, let’s calculate C_τ, C_x, C_xx, C_ν, C_νν, C_xν.

C_τ =C_w^BSw_τ, C_x =C_x^BS+C_w^BSw_x, C_xx =C_xx^BS + 2C_xw^BSw_x+C_ww^BSw²_x+C_w^BSw_xx C_ν =C_w^BSw_ν, C_νν =C_ww^BSw²_ν +C_w^BSw_νν, C_xν =C_wx^BSw_ν +C_ww^BSw_νw_x+C_w^BSw_νx. By insertingC_xx^BS, C_xw^BS, C_ww^BS into these equations, we can expressCτ, Cx, Cxx, Cν, Cνν, Cxν

in terms ofC_x^BS and C_x^BS as below.

Cτ =C_w^BSwτ (2.1)

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C_x =C_x^BS+C_w^BSw_x (2.2) Cxx =−C_x^BS+C_w^BS(1

w + (−2x

w² −1)wx+ (x² w³ − w

4)w²_x+wxx) (2.3)

Cν =C_w^BSwν (2.4)

Cνν =C_w^BS((x² w³ − w

4)w_ν²+wνν) (2.5)

C_xν =C_w^BS((− x w² − 1

2)w_ν+ (x² w³ −w

4)w_νw_x+w_νx). (2.6) By inserting these to the partial differential equation of C, we can get the partial differential equation of the implied volatility σ which is also shown in p.5 of [7]. However, it is hard to solve the PDE. Therefore, the asymptotic method is required. In stochastic volatility models, let’s expand the series as

σ(τ, x, ν) =a(x, ν) +b(x, ν)τ +c(x, ν)τ²+· · · where

a(x, ν) =a₀(ν) +a₁(ν)x+a₂(ν)x²+a₃(ν)x³+· · · b(x, ν) = b₀(ν) +b₁(ν)x+b₂(ν)x²+b₃(ν)x³+· · · c(x, ν) =c₀(ν) +c₁(ν)x+c₂(ν)x²+c₃(ν)x³+· · · .

In local volatility models, there is noν. For both models, the partial differential equations ofa, b, and care derived by comparingτⁱ terms in the PDE ofσ. Likewise, a_n, b_n, and c_n are derived by comparing xⁱ terms in the PDE of a, b, and c, respectively.

Let the market implied volatility data be X₁, X₂,· · ·, X_n and the corresponding market log-money data and the maturity term data be x₁, x₂,· · · , x_n and τ₁, τ₂,· · · , τ_n, respectively. Then, the mean square error is

M SE =

n

X

i=1

(σ(τ_i, x_i, ν)−X_i)²

with no ν in local volatility models. By minimizing this, all the parameters can be estimated.

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Chapter 3

Main Results on the Implied Volatility

3.1 The Heston Model

The Heston model follows the stochastic volatility model dS_t=rS_tdt+√

ν_tS_tdW_1t dν_t=κ(θ−ν_t)dt+ξ√

ν_tdW_2t dW1tdW2t=ρdt.

This model was introduced in [6]. Now, let x_t = log _F^K

t

where F_t = S_te^r(T^−t) is the forward price of a stock at time t. Then,

dF_t=d(S_te^r(T^−t)) =e^r(T^−t)dS_t−re^r(T^−t)S_tdt=√

ν_tS_te^r(T^−t)dW_1t =√

ν_tF_tdW_1t dx_t=−dlogF_t=−dF_t

F_t + 1 2

dF_t F_t

2

= 1

2ν_tdt−√

ν_tdW_1t such that the Heston model can be transformed to

dx_t = 1

2ν_tdt−√

ν_tdW_1t dν_t=κ(θ−ν_t)dt+ξ√

ν_tdW_2t dW_1tdW_2t=ρdt.

Now, let’s calculate the differential of C, dC(t, x, ν) =C_tdt+C_xdx+ 1

2C_νν(dν)² +C_xν(dxdν)

= C_t+ 1

2ν(C_x+C_xx) +κ(θ−ν)C_ν +1

2ξ²νC_νν−ρξνC_xν dt

−C_x√

νdW_1t+C_νξ√

νdW_2t.

Since the dt term is zero due to the martingale property of C, we get C_t+ 1

2ξ²νC_νν−ρξνC_xν = 0.

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By changing the variable system from (t, x, ν) to (τ, x, ν) whereτ =T −t, C_τ = 1

2ξ²νC_νν −ρξνC_xν. By substituting (2.1)-(2.6) to the above equation, we get

w_τ =1

2ν(w_x+ 1

w −(2x

w² + 1)w_x+ (x² w³ − w

4)w_x²+w_xx) +κ(θ−ν)w_ν + 1

2ξ²ν((x² w³ − w

4)w_ν²+w_νν)−ρξν(−( x w² + 1

2)w_ν+ (x² w³ −w

4)w_νw_x+w_νx).

By inserting w=σ(τ, x, ν)√

τ and multiplying both sides by σ³√

τ, we get σ³σ_ττ+ 1

2σ⁴ =1

2ν(σ²−2xσσ_x+x²σ_x²− 1

4σ⁴σ_x²τ²+σ³σ_xxτ) +κ(θ−ν)σ³σ_ντ +1

2ξ²ν(x²σ_ν²− 1

4σ⁴σ²_ντ²+σ³σ_νντ)−ρξν(−xσσ_ν− 1 2σ³σ_ντ +x²σ_νσ_x− 1

4σ⁴σ_νσ_xτ²+σ³σ_νxτ).

Now, by letting σ(τ, x, ν) as

σ(τ, x, ν) =a(x, ν) +b(x, ν)τ +c(x, ν)τ²+· · · and comparing the τ⁰ coefficients, we get the PDE of a(x, ν),

1

2a⁴ = 1

2ν(a²−2xaa_x+x²a²_x) + 1

2ξ²νx²a²_ν +ρξνxaa_ν −ρξνx²a_νa_x.

By lettinga(x, ν) = a₀(ν)+a₁(ν)x+a₂(ν)x²+a₃(ν)x³+· · · and comparingxⁱ coefficients, we get

a₀(ν) =ν^1/2 a₁(ν) = 1

4ρξν^−1/2 a₂(ν) = 1

48(2−5ρ²)ξ²ν^−3/2 a₃(ν) =− 1

96ρ(5−8ρ²)ξ³ν^−5/2.

Note that in these equations,κ andθ don’t appear and they cannot be estimated. There- fore,τ¹asymptotic implied volatility terms are necessary. This meansb(x, ν) that includes κ and θ must be calculated for some extent from the PDE ofa and b,

3a³b=1

2ν(2ab−2x(ab_x+a_xb) + 2x²a_xb_x+a³a_xx) +κ(θ−ν)a³a_ν +1

2ξ²ν(2x²a_νb_ν +a³a_νν)−ρξν(−x(ab_ν +a_νb)− 1

2a³a_ν +x²(a_νb_x+a_xb_ν) +a³a_xν).

By letting b(x, ν) = b₀(ν) +b₁(ν)x+· · · and comparing xⁱ coefficients, we get b₀(ν) = 1

8ρξν^1/2+1

4κ(θ−ν)ν^−1/2 − 1

96(4−ρ²)ξ²ν^−1/2

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b₁(ν) =− 1

96ρ²ξ²ν^−1/2− 5

48ρξκ(θ−ν)ν^−3/2− 1

12ρξκν^−1/2

− 1

96ρ(5−8ρ²)ξ²ν^−3/2+ 1

384ρ(16−13ρ²)ξ³ν^−3/2.

Now, let’s estimateν, ρ, ξ. Let the market log-money bex_iand the market implied volatility be X_i for i= 1,2,· · · , n. Since the zero order approximation of a(x, ν) is

a(x, ν) =ν^1/2+· · · , the corresponding mean square error is given by

M SE(ν) =

n

X

i=1

(ν^1/2−X_i)². By differentiating with respect to ν, we get the estimator of ν,

ˆ

ν =Pn i=1X_i

n 2

. The first order approximation of a(x, ν) is

a(x, ν) = ν^1/2 +1

4ρξν^−1/2x+· · · such that

M SE(ρ, ξ) =

n

X

i=1

(1

4ρξνˆ^−1/2x_i+ (ˆν^1/2−X_i))². From this, we get the estimator of ρξ

ρξˆ =−4 Pn

i=1νˆ^1/2(ˆν^1/2−X_i)x_i Pn

i=1x²_i . The second order approximation of a(x, ν) is

a(x, ν) = ν^1/2+ 1

4ρξν^−1/2x+ 1

48(2−5ρ²)ξ²ν^−3/2x²+· · · such that

M SE(ρ, ξ) =

n

X

i=1

(1

4ρξˆν^−1/2x_i+ 1

48(2−5ρ²)ξ²νˆ^−3/2x²_i + (ˆν^1/2−X_i))²

= 1

16ρ²ξ²νˆ⁻¹

n

X

i=1

x²_i + 1

2304(4ξ⁴−20ρ²ξ⁴+ 25ρ⁴ξ⁴)ˆν⁻³

n

X

i=1

x⁴_i +

n

X

i=1

(ˆν^1/2 −Xi)²+ 1

96(2ρξ³−5ρ³ξ³)ˆν⁻²

n

X

i=1

x³_i +1

2ρξˆν^−1/2

n

X

i=1

xi(ˆν^1/2−Xi) + 1

24(2ξ²−5ρ²ξ²)ˆν^−3/2

n

X

i=1

x²_i(ˆν^1/2 −Xi).

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Under the conditionρξ = ˆρξ, we can rewrite the MSE as M SE(ξ) = 1

16( ˆρξ)²νˆ⁻¹

n

X

i=1

x²_i + 1

2304(4ξ⁴−20( ˆρξ)²ξ²+ 25( ˆρξ)⁴)ˆν⁻³

n

X

i=1

x⁴_i +

n

X

i=1

(ˆν^1/2−Xi)²+ 1

96(2( ˆρξ)ξ²−5( ˆρξ)³)ˆν⁻²

n

X

i=1

x³_i +1

2( ˆρξ)ˆν^−1/2

n

X

i=1

xi(ˆν^1/2−Xi) + 1

24(2ξ²−5( ˆρξ)²)ˆν^−3/2

n

X

i=1

x²_i(ˆν^1/2−Xi).

Note that there is no ρ in the above equation. By differentiating this with respect to ξ, we get

ξˆ= s10

3( ˆρξ)²Pn

i=1x⁴_i −8( ˆρξ)ˆνPn

i=1x³_i −32ˆν^3/2Pn

i=1x²_i(ˆν^1/2−X_i) Pn

i=1x⁴_i .

Since ˆρξˆ= ˆρξ,

ˆ ρ=

ρξˆ ξˆ.

Now, Let’s consider the τ¹ approximation of the implied volatility. In this case, let the maturity data be τ₁, τ₂,· · · , τ_n. Since the approximation of implied volatility is

a(x, ν) +b(x, ν)τ =ν^1/2 +1

4ρξν^−1/2x+ 1

48(2−5ρ²)ξ²ν^−3/2x² + (1

8ρξν^1/2+1

4κ(θ−ν)ν^−1/2− 1

96(4−ρ²)ξ²ν^−1/2)τ +· · · , we get

M SE(κ, θ) =

n

X

i=1

1

4κ(θ−ν)ˆˆ ν^−1/2τ_i+ ˆ

ν^1/2+1

4ρˆξˆˆν^−1/2x_i+ 1

48(2−5 ˆρ²) ˆξ²νˆ^−3/2x²_i + (1

8ρˆξˆνˆ^1/2− 1

96(4−ρˆ²) ˆξ²νˆ^−1/2)τi−Xi

2

. By differentiation, we get

κ(θ\−ν) = ˆκ(ˆθ−ν) =ˆ −4Xⁿ

i=1

ˆ ν^1/2τ_i

ˆ

ν^1/2+ 1

48(2−5 ˆρ²) ˆξ²νˆ^−3/2x²_i + (1

8ρˆξˆνˆ^1/2− 1

96(4−ρˆ²) ˆξ²νˆ^−1/2)τ_i−X_i.Xⁿ

i=1

τ_i² .

Since we have already estimated κ(θ−ν) as κ(θ\−ν) = ˆκ(ˆθ−ν) andˆ b₁(ν) has κ and θ terms as a form of κ(θ−ν) and κ, the mean square error of a₀(ν) +a₁(ν)x+a₂(ν)x²+ b₀(ν) +b₁(ν)xτ to the market implied volatility data is actually a function of κ only.

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By differentiating the mean square error of κ with respect to κ, ˆκ is derived. The mean square error of κ is

M SE(κ) =

n

X

i=1

− 1

12κρˆξˆνˆ^−1/2x_iτ_i+ ˆ

ν^1/2+1

48(2−5 ˆρ²) ˆξ²νˆ^−3/2x²_i +1 8ρˆξˆˆν^1/2 + 1

4κ(θ\−ν)ˆν^−1/2− 1

96(4−ρˆ²) ˆξ²νˆ^−1/2

τi+

− 1

96ρˆ²ξˆ²νˆ^−1/2 − 5

48ρˆξˆκ(θ\−ν)ˆν^−3/2

− 1

96ρ(5ˆ −8 ˆρ²) ˆξ²νˆ^−3/2+ 1

384ρ(16ˆ −13 ˆρ²) ˆξ³νˆ^−3/2

xiτi−Xi

2

such that ˆ

κ=12 ˆρ⁻¹ξˆ⁻¹νˆ^1/2 Xⁿ

i=1

xiτi

ˆ

ν^1/2+1

4ρˆξˆˆν^−1/2xi+ 1

48(2−5 ˆρ²) ˆξ²νˆ^−3/2x²_i + 1

8ρˆξˆνˆ^1/2 + 1

4κ(θ\−ν)ˆν^−1/2− 1

96(4−ρˆ²) ˆξ²νˆ^−1/2 τ_i+

− 1

96ρˆ²ξˆ²νˆ^−1/2− 5

48ρˆξˆκ(θ\−ν)ˆν^−3/2

− 1

96ρ(5ˆ −8 ˆρ²) ˆξ²νˆ^−3/2+ 1

384ρ(16ˆ −13 ˆρ²) ˆξ³νˆ^−3/2

x_iτ_i−X_i.Xⁿ

i=1

x²_iτ_i² . Then, ˆθ is of the form

θˆ= ˆν+κ(θ\−ν) ˆ κ

Another way is to derive the poly-fit of the implied volatility with M SE(m₀, m₁, m₂, m₃, m₄) =

n

X

i=1

m₀+m₁x_i+m₂x²_i +m₃τ_i+m₄τ_ix_i−X_i2

.

Let ˆm₀,mˆ₁,mˆ₂,mˆ₃,mˆ₄be the corresponding values that minimize the MSE. Then ˆν,ρ,ˆ ξ,ˆ ˆκ,θˆ is

ˆ ν= ˆm²₀ ˆ

ρ= 2 ˆm₀mˆ₁

p6 ˆm³₀mˆ2 + 10 ˆm²₀mˆ²₁ ξˆ= 2

q

6 ˆm³₀mˆ₂+ 10 ˆm²₀mˆ²₁ ˆ

κ= 1 ˆ m₁

−5mˆ₁mˆ₃ ˆ

m₀ + 2 ˆm₀mˆ²₁−5 ˆm₀mˆ₁mˆ₂− 15

2 mˆ³₁−3 ˆm₄

− 1

32ρ(5ˆ −8 ˆρ²) ˆξ²νˆ^−3/2+ 1

128ρ(16ˆ −13 ˆρ²) ˆξ²νˆ^−3/2

θˆ= ˆν+ 1 ˆ

κ(4 ˆm₀mˆ₃−2 ˆm³₀mˆ₁+ 4 ˆm³₀mˆ₂+ 6 ˆm²₀mˆ²₁).

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3.2 The SABR Model

The SABR model follows the stochastic volatility model dF_t =ν_tF_t^βdW_1t

dν_t=αν_tdW_2t dW_1tdW_2t=ρdt.

This model was introduced in [4]. The estimation of parametersα, β, ρ, νis required. The dynamics of x_t= log(_F^K

t) is dx_t=−dFt

F_t + 1 2

dFt

F_t 2

= 1

2ν_t²F_t^2β−2dt−ν_tF_t^β−1dW_1t. Since F_t =Ke^−x^t, we get

dx_t = 1

2ν_t²K^2β−2e^(2−2β)x^tdt−ν_tK^β−1e^(1−β)x^tdW_1t dνt=ανtdW2t

dW_1tdW_2t =ρdt such that

dC(t, x, ν) =C_tdt+C_xdx+ 1

2C_νν(dν)²+C_xν(dxdν)

= C_t+ 1

2ν²K^2β−2e^(2−2β)x(C_x+C_xx) + 1

2α²ν²C_νν −ραν²K^β−1e^(1−β)xC_xν dt

−νK^β−1e^(1−β)xC_xdW_1t+ανC_νdW_2t.

Since the dt term is zero due to the martingale property of C, we get C_t+1

2ν²K^2β−2e^(2−2β)x(C_x+C_xx) + 1

2α²ν²C_νν −ραν²K^β−1e^(1−β)xC_xν = 0.

By changing the variables from (t, x, ν) to (τ, x, ν) whereτ =T −t, we get C_τ = 1

2ν²K^2β−2e^(2−2β)x(C_x+C_xx) + 1

2α²ν²C_νν −ραν²K^β−1e^(1−β)xC_xν. By substituting (2.1)-(2.6) to the above equation, we get

w_τ =1

2ν²K^2β−2e^(2−2β)x(1

w −2xw_x w² + (x²

w³ − w

4)w_x²+w_xx) + 1

2α²ν²((x² w³ − w

4)w²_ν +w_νν)

−ραν²K^β−1e^(1−β)x((− x w² −1

2)w_ν + (x² w³ − w

4)w_νw_x+w_νx).

By inserting w=σ(τ, x, v)√

τ and multiplying by σ³√

τ, we get σ³σ_ττ+ 1

2σ⁴ =1

2ν²K^2β−2e^(2−2β)x(σ²−2xσσ_x+x²σ²_x−1

4σ⁴σ²_xτ²+σ³σ_xxτ) +1

2α²ν²(x²σ²_ν −1

4σ⁴σ²_ντ²+σ³σ_νντ)−ραν²K^β−1e^(1−β)x(−xσσ_ν

−1

2σ³σ_ντ +x²σ_νσ_x− 1

4σ⁴σ_νσ_xτ²+σ³σ_νxτ).

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From σ(τ, x, ν) =a(x, ν) +b(x, ν)τ +c(x, ν)τ²+· · ·, we get the PDE of a(x, ν), 1

2a⁴ = 1

2ν²K^2β−2e^(2−2β)x(a²−2xaa_x+x²a²_x)+1

2α²ν²x²a²_ν+ραν²K^β−1e^(1−β)x(xaa_ν−x²a_νa_x).

Note that the above equation includesα, β, ρ, ν all such that there is no need to calculate the b terms. By a(x, ν) =a₀(ν) +a₁(ν)x+a₂(ν)x²+a₃(ν)x³+· · ·, we get

a₀(ν) = νK^β−1 a₁(ν) = 1

2(1−β)a₀(ν) + 1

2αρ= 1

2(1−β)νK^β−1+1 2αρ a₂(ν) = 1

12(1−β)²νK^β−1+ 1

12α²(2−3ρ²)ν⁻¹K^1−β a₃(ν) = − 1

24α²(1−β)(2−3ρ²)ν⁻¹K^1−β − 1

24α³ρ(5−6ρ²)ν⁻²K^2−2β. The log version MSE on a₀(ν) is

M SE(ν, β) =

n

X

i=1

(logν+ (β−1) logK_i−logX_i)². By differentiation, we get

ˆ

ν= exp(Pn i=1

logKi

n )(Pn i=1

logKilogXi

n )−(Pn i=1

logXi

n )(Pn i=1

(logKi)²

n )

(Pn i=1

logKi

n )²−Pn i=1

(logKi)² n

βˆ= 1 + (Pn i=1

logKi

n )(Pn i=1

logXi

n )−Pn i=1

logKilogXi

n

(Pn i=1

logKi

n )²−Pn i=1

(logKi)² n

. An alternative way is to consider

M SE(m₀, m₁) =

n

X

i=1

(m₀+m₁logK_i−logX_i)². We can get ˆm₀,mˆ₁ that minimize the mean square error such that

ˆ ν=e^m^ˆ⁰ βˆ= ˆm₁+ 1.

Fora(x, ν) with

a(x, ν) =νK^β−1+ (1

2(1−β)νK^β−1+1

2αρ)x+· · · , the mean square error is

M SE(α, ρ) =

n

X

i=1

1

2αρx_i+ ˆ

νK_i^β−1^ˆ +1

2(1−β)ˆˆ νK_i^β−1^ˆ x_i−X_i2

.

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From this, we get

ˆ

αρ=−2 Pn

i=1

ˆ

νK_i^β−1^ˆ +¹₂(1−β)ˆˆ νK_i^β−1^ˆ x_i−X_i x_i Pn

i=1x²_i .

Fora(x, ν) with

a(x, ν) =νK^β−1+ (1

2(1−β)νK^β−1 +1 2αρ)x + ( 1

12(1−β)²νK^β−1+ 1

12α²(2−3ρ²)ν⁻¹K^1−β)x²+· · · , the mean square error is

M SE(α, ρ) =

n

X

i=1

1

6α²νˆ⁻¹K_i¹⁻^β^ˆx²_i + ˆ

νK_i^β−1^ˆ + (1

2(1−β)ˆˆ νK_i^β−1^ˆ + 1

2( ˆαρ))x_i + ( 1

12(1−β)ˆ ²νKˆ _i^β−1^ˆ − 1

4( ˆαρ)²νˆ⁻¹K_i¹⁻^β^ˆ)x²_i −X_i2

.

This equation is earned with the condition thatαρ= ˆαρ. Since ˆαρ is not a variable, this is actually a function of α only. Thus, by differentiating this with respect to α, we get

ˆ α=

−6Xⁿ

i=1

ˆ

νK_i^β−1^ˆ x²_i ˆ

νK_i^β−1^ˆ + (1

2(1−β)ˆˆ νK_i^β−1^ˆ +1

2( ˆαρ))x_i + ( 1

12(1−β)ˆ ²νKˆ _i^β−1^ˆ −1

4( ˆαρ)²νˆ⁻¹K_i¹⁻^β^ˆ)x²_i −X_i.Xⁿ

i=1

x⁴_i¹₂ . Since ˆαρ= ˆαρ,ˆ

ˆ ρ= αρˆ

ˆ α . By considering

M SE(m2, m3) =

n

X

i=1

m2xi+m3νˆ⁻¹K_i¹⁻^β^ˆx²_i +

ˆ

νK_i^β−1^ˆ +1

2(1−β)ˆˆ νK_i^β−1^ˆ xi

+ 1

12(1−β)ˆ ²νKˆ _i^β−1^ˆ x²_i −X_i2

,

we can get ˆm₂,mˆ₃ that minimize the mean square error such that ˆ

α= q

6 ˆm²₂+ 6 ˆm₃ ˆ

ρ= 2 ˆm₂ p6 ˆm²₂+ 6 ˆm₃.

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3.3 Local Volatility Models

Local volatility model follows the stochastic process dSt=rSt+ξ(t, St)StdWt. The dynamics of x_t= log(_F^K

t) is dx_t= 1

2ξ(t, x_t)²dt−ξ(t, x_t)dW_t such that

dC(t, x) =C_tdt+C_xdx+1

2C_xx(dx)²

= (C_t+ 1

2ξ(t, x)²(C_x+C_xx))dt−ξ(t, x)C_xdW_t. Since dt term is zero due to the martingale property of C, we get

C_t+1

2ξ(t, x)²(C_x+C_xx) = 0 such that

C_τ = 1

2ξ(τ, x)²(C_x+C_xx)

with τ =T −t. By substituting (2.1)-(2.3) to the above equation, we get wτ = 1

2ξ(τ, x)²(1 w − 2x

w²wx+ (x² w³ − w

4)w²_x+wxx).

Since w(τ, x) = σ(τ, x)√ τ,

2σ³σ_ττ +σ⁴ =ξ(τ, x)²(σ²−2xσσ_x+x²σ_x²−1

4σ⁴σ²_xτ²+σ³σ_xxτ).

The squared local volatility ξ(τ, x)² can be expressed as ξ(τ, x)² = 2σ³σ_ττ+σ⁴

σ²−2xσσ_x+x²σ_x²− ¹₄σ⁴σ_x²τ²+σ³σ_xxτ

= 2σ³σττ+σ⁴

(σ−xσ_x)²− ¹₄σ⁴σ_x²τ²+σ³σ_xxτ

= 2σσ_ττ+σ²

(1−x^σ_σ^x)²−¹₄σ²σ_x²τ²+σσ_xxτ

= 2σσ_ττ+σ²

(1−x(logσ)_x)²−¹₄σ²σ_x²τ²+σσ_xxτ. Let the local volatility ξ(τ, x) as

ξ(τ, x) =p(x) +q(x)τ +r(x)τ²+· · ·

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where each p(x), q(x), r(x) is of the form

p(x) = p₀+p₁x+p₂x²+p₃x³ +· · · q(x) = q₀+q₁x+q₂x²+q₃x³+· · · r(x) =r₀+r₁x+r₂x²+r₃x³+· · · . Letσ(τ, x) be

σ(τ, x) =a(x) +b(x)τ+c(x)τ²+· · · and compare the coefficients ofτⁱ. Then, the PDE of a(x) is

a⁴ =p²(a²−2xaa_x+x²a²_x) =p²(a−xa_x)² such that

a² =p(a−xax).

By dividing both sides by a², this can be transformed to 1 =p(x)( x

a(x))⁰ such that

a(x) = x Rx

0 1 p(x)dx.

Even though this is a closed form formula, sometimes the integral of _p(x)¹ is hard. Even in the case that the integral is easy, it might be hard to apply the MSE method. Therefore, considering another way is meaningful. One way is to regarda(x) asp(^x₂) sinceRx

0 1 p(x)dx'

x

p(^x₂). Another way is as usually letting a(x) = a₀+a₁x+a₂x²+· · · and comparing the xⁱ coefficients. Then,

a₀ =p₀ a1 = 1

2p1

a₂ = 1

3p₂− 1 12

p²₁ p₀ a₃ = 1

4p₃ −1 6

p1p2

p₀ + 1 24

p³₁ p²₀ a₄ = 1

5p₄− 3 20

p₁p₃ p₀ − 4

45 p²₂ p₀ + 23

180 p²₁p₂

p²₀ − 19 720

p⁴₁ p³₀ a₅ = 1

6p₅− 2 15

p₁p₄ p₀ −1

6 p₂p₃

p₀ + 29 240

p²₁p₃

p²₀ + 133 810

p₁p²₂

p²₀ − 401 3240

p³₁p₂

p³₀ + 67 3240

p⁵₁ p⁴₀. Equivalently, there is a relation between the derivatives of a and p

a(0) =p(0) a⁽¹⁾(0) = 1

2p⁽¹⁾(0)

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a⁽²⁾(0) = 1

3p⁽²⁾(0)− 1 6

p⁽¹⁾(0)² p(0) a⁽³⁾(0) = 1

4p⁽³⁾(0)− 1 2

p⁽¹⁾(0)p⁽²⁾(0) p(0) +1

4

p⁽¹⁾(0)³ p(0)² a⁽⁴⁾(0) = 1

5p⁽⁴⁾(0)−3 5

p⁽¹⁾(0)p⁽³⁾(0) p(0) − 8

15

p⁽²⁾(0)² p(0) + 23

15

p⁽¹⁾(0)²p⁽²⁾(0) p(0)² − 19

30

p⁽¹⁾(0)⁴ p(0)³ a⁽⁵⁾(0) =1

6p⁽⁵⁾(0)− 2 3

p⁽¹⁾(0)p⁽⁴⁾(0) p(0) − 5

3

p⁽²⁾(0)p⁽³⁾(0) p(0) +29

12

p⁽¹⁾(0)²p⁽³⁾(0) p(0)² + 133

27

p⁽¹⁾(0)p⁽²⁾(0)²

p(0)² − 401 54

p⁽¹⁾(0)³p⁽²⁾(0) p(0)³ + 67

27

p⁽¹⁾(0)⁵ p(0)⁴ . Forb terms, we get the PDE of the form

6a³b=p²(2ab−2x(ab_x+a_xb) + 2x²a_xb_x+a³a_xx) + 2pq(a²−2xaa_x+x²a²_x).

Withφ(x) = ^b(x)_a(x), another equation can be derived. Let’s go back to the equation 2σσ_ττ +σ² =ξ(τ, x)²

(1−x(logσ)_x)²−1

4σ²σ²_xτ²+σσ_xxτ . Since

logσ= log(a+bτ +· · ·) = loga+ log(1 +φτ +· · ·) = loga+φτ +· · · x(logσ)_x =x(a_x

a +φ_xτ +· · ·) holds, (1−x(logσ)x)² is

(1−x(logσ)x)² = (1−xa_x

a −xφxτ +· · ·)²

= (1−xax

a )²−2xφ_x(1−xax

a )τ+· · ·

= a²

p² −2xφ_xa

pτ +· · · .

The last equality is from the equation 1−x^a_a^x = ^a_p. Now, the comparison of coefficients of τ¹ induces

4a²φ=p²(−2xφ_xa

p +aa_xx) + 2pqa² p²

=−2xφ_xpa+p²aa_xx+ 2qa² p . Therefore, we get

4apφ+ 2xp²φ_x =p³a_xx+ 2qa.

Equivalently,

xφ_x+ 2a pφ= 1

2pa_xx+ q p²a.

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According to p.144 of [5], this equation has a solution of the form φ(x) = −1

2 a(x)

x 2

log a(x)² p(x)p(0)

− Z x

0

q(y) p(y)( y²

a(y)²)⁰dy .

Even though φ(x) has a closed form formula, it is useful to derive a polynomial series of φ(x),

φ(x) =φ₀ +φ₁x+φ₂x²+φ₃x³+· · · , with

φ0 = 1

6p0p2− 1

24p²₁+1 2

q₀ p₀ φ₁ = 1

4p₀p₃ −1 3

p₁q₀ p²₀ + 1

3 q₁ p₀.

When it is the case that the local volatility does not depend on τ, the local volatility is σ(τ, x) =p(x) and the calculation of φ0, φ1, φ2, and φ3 is easier. In this case, q= 0 such that the partial differential equation is

4apφ+ 2xp²φ_x=p³a_xx with

φ₀ = 1

6p₀p₂− 1 24p²₁ φ₁ = 1

4p₀p₃ φ2 = 3

10p0p4+ 1

40p1p3+ 1

180p²₂− 1 360

p²₁p₂ p₀ + 23

576 p⁴₁ p²₀ φ₃ = 1

3p₀p₅+ 1

30p₁p₄− 1 240

p²₁p₃

p<

저작자표시

이학석사 학위논문

지도교수 박형빈

2018년 12월

Hoseong Ryu

Contents

Chapter 1 Introduction

Chapter 2

Chapter 3

3.2 The SABR Model

3.3 Local Volatility Models