An Asymptotic Expansion Formula for Up-and-Out Barrier Option Price under Stochastic Volatility Model

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1 CIRJE-F-873 An Asymptotic Expansion Formula for Up-and-Out Option Price under Stochastic Volatility Model Takashi Kato Osaka University Akihiko Takahashi University of Tokyo Toshihiro Yamada Graduate School of Economics, University of Tokyo and MTEC January 23 CIRJE Discussion Papers can be downloaded without charge from: Discussion Papers are a series of manuscripts in their draft form. They are not intended for circulation or distribution except as indicated by the author. For that reason Discussion Papers may not be reproduced or distributed without the written consent of the author.

2 An Asymptotic Expansion Formula for Up-and-Out Option Price under Stochastic Volatility Model Takashi Kato Akihiko Takahashi Toshihiro Yamada November 23, 22 Abstract This paper derives a new semi closed-form approximation formula for pricing an upand-out barrier option under a certain type of stochastic volatility model including SABR model by applying a rigorous asymptotic expansion method developed by Kato, Takahashi and Yamada []. We also demonstrate the validity of our approximation method through numerical examples. Keywords: Option, Up-and-Out Call Option, Asymptotic Expansion, Stochastic Volatility Model Introduction Numerical computation schemes for pricing barrier options have been a topic of great interest in mathematical finance and stochastic analysis. One of the tractable approaches for evaluation of barrier options is to derive an analytical approximation. However, from the mathematical viewpoint, deriving an approximation formula by applying stochastic analysis is not an easy task since the Malliavin calculus approach as in Takahashi and Yamada [3] cannot be directly applied. Recently, Kato, Takahashi and Yamada [] has provided a new asymptotic expansion method for the Cauchy Dirichlet problem by developing a rigorous perturbation scheme in a partial differential equation (PDE), and as an example, derived an approximation formula for a down-and-out call option price under a stochastic volatility model. In this paper, we give a new asymptotic expansion formula for an up-and-out call option price under a stochastic volatility model which is widely used in trading practice. Moreover, we show the validity of our formula through numerical experiments. Osaka University, The University of Tokyo, The University of Tokyo & MTEC

3 2 Asymptotic expansion formula for up-and-out barrier option prices Consider the following stochastic differential equation (SDE) in a stochastic volatility model: ds ε t = (c q)s ε t dt + σ ε t S ε t db t, S ε = S, dσ ε t = ελ(θ σ ε t )dt σ ε = σ, +ενσ ε t (ρdb t + ρ 2 db 2 t ), where S, σ, c, q >, ε [, ), λ, θ, ν >, ρ [, ] and B = (B, B 2 ) is a two dimensional standard Brownian motion. This model is motivated by pricing currency options. In this case, c and q represent a domestic interest rate and a foreign interest rate, respectively. The process S ε denotes a price of the underlying currency. Our purpose is to evaluate an up-and-out barrier option with time-to-maturity T t and the upper barrier price H(> S), and its initial value is represented under a risk-neutral probability measure as follows: C SV,ε (T t, S) [ ] = E e c(t t) f(st ε t) {τ(,h) (S ε )>T t}, where f stands for a call option payoff function f(s) = max{s K, } for some K >. Here, the stopping time τ (,H) (S ε ) is defined as τ (,H) (S ε ) = inf{t [, T ]; S ε t / (, H)} (inf := ). Remark that C SV,ε (T t, S) has no closed-form solution and therefore we have to rely on some numerical method such as the Monte Carlo simulation in order to calculate C SV,ε (T t, S). However, when ε =, CSV, (T t, S) corresponds to the up-and-out barrier option price in the Black-Scholes model which is known to be solved explicitly. Then, for ε >, we are able to derive a semi closed-form expansion around C SV, (T t, S) when ε. This is our main result and hereafter we show our approximation method for C SV,ε (T t, S). Clearly, applying Itô s formula, we can derive the SDE of logarithmic process of St ε as dx ε t = (c q 2 (σε t ) 2 )dt + σ ε t db t, X ε = x := log S. Then we can rewrite C SV,ε (T t, S) as C SV,ε (T t, ex ) [ ] = E e c(t t) ε f(x T t ) {τd (X ε )>T t}, where f(x) = max{e x K, } and D = (, log H). Note that τ D (X ε ) = inf{t [, T ] ; X ε t / D} = τ (,H) (S ε ). 2

4 Let u ε (t, x) = C SV,ε (T t, ex ) for t [, T ] and x R. Then u ε (t, x) satisfies the following PDE: ( ) t + L ε c u ε (t, x) =, (t, x) (, T ] D, u ε (T, x) = f(x), x D, u ε (t, log H) =, t [, T ], where L ε = ( c q ) 2 σ2 x + 2 σ2 2 x 2 +ερνσ ελ(θ σ) x σ σ + ε2 2 ν2 σ 2 2 σ 2. As mentioned above, when ε =, we can obtain the explicit value of u (t, x). In this case, u (t, x) = C(T BS t, e x, σ, H) represents the price of the up-and-out barrier call option under the Black Scholes model. We have where with See Hull [2] for the details. C BS = C BS Vanilla C, CVanilla BS = e x e qt N(d ) Ke ct N(d 2 ), C = e x e qt N(x ) Ke ct N(x 2 ) e x e qt ( H e x ) 2λ [N( y) N( y )] +Ke ct ( H e x ) 2λ 2 [N( y + ) N( y + )] x = x log H + (c q)t + /2σ2 T x 2 = x, λ = (c q) σ 2 + 2, y = 2 log H x log K + (c q)t + /2σ2 T y = log H x + (c q)t + /2σ2 T,., We can represent u (t, x) = P D t f(x) by using a semi-group ( P D t ) t defined as P D s g(x) = log H e cs 2(log H x)(log H y) ( e σ 2 s ) 3

5 (y x (c q 2 σ 2πσ 2 s e 2σ 2 s 2 )s) 2 g(y)dy (2.) for a continuous function g with polynomial growth rate which satisfies g(x) = on D. The main result of Kato, Takahashi and Yamada [] suggests the following approximation formula. [Asymptotic expansion formula] where u ε (t, x) = C BS +εe c(t t) T t P D s L P D T t s f(x)ds + O(ε 2 ), L = ε L ε ε= = ρσ λ(θ σ) x σ σ. Using (2.), the term T t P D s L P D T t s f(x)ds is expressed as follows: = T t P s D T t log H L P D T t s f(x)ds e cs 2(log H x)(log H y) ( e σ 2 s ) (y x (c q 2 σ 2πσ 2 s e 2σ 2 s 2 )s) 2 L P T D f(y)dyds. t s (2.2) We are able to compute the integrand of the right hand side of the above formula (2.2) as Here, σ CBS (T, e x ) and L P T D f(x) t { = e c(t t) ρσ 2 2 x σ CBS (T t, e x, σ) +λ(θ σ) σ CBS (T t, e x, σ) 2 x σ CBS (T, e x ) are concretely expressed as follows: σ CBS (T, e x ) = e qt e x n(d ) T e qt e x n(x ) T (H K)e ct n(x 2 ) x σ }. 4

6 ( ) H 2λ +e x e qt e { x 4(c q) (log H x) σ 3 [N( y) N( y )] } +[n(y) y σ n(y ) y σ ] ( ) H 2λ 2 Ke ct e { x 4(c q) (log H x) σ 3 [N( y ) N( y )] } +[n(y ) y σ n(y ) y σ ], 2 x σ CBS (T, e x ) = e qt e x n(d )( d 2 ) σ e qt e x n(x )( x 2 ) σ (H K)e ct n(x 2) σ 2 T {x x 2 } 4(c q) + σ 3 {( + 2λ)(log H x) + } e x e qt ( H e x ) 2λ [N( y) N( y )] ( ) H 2λ +e x e qt e x [n(y) y σ n(y ) y σ ] ( ( ) ) H 2λ 2λ e x ( ) H 2λ e x e qt e x (log H x) ( ( ) ( )) 4(c q) σ 3 n(y) σ n(y ) T ( ) H 2λ +e x e qt e ( x ) n(y) σ 2 T (yy ) n(y ) σ 2 T (y y ) Ke ct [N(y ) N(y )] ( ( ) ) H 2λ 2 4(c q) e x σ 3 {(2λ 2)(log H x) + } ( ) H 2λ 2 +Ke ct 4(c q) e x (log H x) σ 3 5

7 where ( ) n(y ) n(y ) ( ) H 2λ 2 +Ke ct (2λ 2) e ( x n(y ) y σ n(y ) y ) σ ( ) H 2λ 2 Ke ct e ( x ) n(y ) σ 2 T (y y ) n(y ) σ 2 T (y y ), 3 Numerical Examples y = 2 log H x log K + (c q)t 2 σ2 T y = log H x + (c q)t 2 σ2 T σ. T In this section we show numerical examples for pricing European up-and-out barrier call options under SABR volatility model (λ = ) as an illustrative purpose. By the asymptotic expansion formula in the previous section, we see C SV,ε (T, S) CBS (T, S) T +εe ct Let us define AE first and AE zeroth as AE first = C(T, BS S) T +εe ct P D s P D s AE zeroth = C BS (T, S). L P D T sf(s)ds. L P D T sf(s)ds, Below we list the numerical examples, [Case ] [Case 6], where the numbers in the parentheses show the error rates (%) relative to the benchmark prices of C SV,ε (T, S) which are computed by Monte Carlo simulations with, time steps and,, trials (denoted by MC). We check the accuracy of our approximation formula by changing the model parameters. Apparently, our approximation formula AE first improves the accuracy for C SV,ε (T, S), and it is observed that the approximation term εe ct T compensates for the difference between C SV,ε the validity of our method. P D s, L P D T sf(s)ds accurately (T, S) and CBS (T, S), which confirms For all cases, we set S =, σ =.2, c =., q =., ρ =.5, ελ =., θ =. and T =.. In Case, 2 and 3, given εν =., the upper bound price is set as 6

8 H = 2, 3, 4, respectively, while in Case 4, 5 and 6, given εν =.2, H is set as 2, 3, 4, respectively. Particularly, for the case of εν =.2 (that is, higher volatility of volatility case, Case 4, 5 and 6), we remark that the errors of the approximation become slightly larger. However, as observed in comparison between AE first and AE zeroth, we are convinced that the higher order expansion improves the approximation further, which will be investigated in our next research. [Case] S =, σ =.2, c =., q =., εν =., ρ =.5, ελ =., θ =., H = 2, T =.. Table : up-and-out barrier option prices and the relative errors (Case ) (-.35%).5 (-8.25%) (-.44%).84 (-8.78%) (-.62%).463 (-9.59%) [Case2] S =, σ =.2, c =., q =., εν =., ρ =.5, ελ =., θ =., H = 3, T =.. Table 2: up-and-out barrier option prices and the relative errors (Case 2) (-.49%) (-7.78%) (-.55%) 2.46 (-8.22%) (-.69%).72 (-8.93%) [Case3] S =, σ =.2, c =., q =., εν =., ρ =.5, ελ =., θ =., H = 4, T =.. 7

9 Table 3: up-and-out barrier option prices and the relative errors (Case 3) (.5%) (-6.49%) (.6%) 4.2 (-6.77%) (.6%) 3.74 (-7.9%) [Case4] S =, σ =.2, c =., q =., εν =.2, ρ =.5, ελ =., θ =., H = 2, T =.. Table 4: up-and-out barrier option prices and the relative errors (Case 4) (-3.5%).5 (-6.2%) (-3.83%).84 (-7.5%) (-4.3%).463 (-8.65%) [Case5] S =, σ =.2, c =., q =., εν =.2, ρ =.5, ελ =., θ =., H = 3, T =.. Table 5: up-and-out barrier option prices and the relative errors (Case 5) (-.5%) (-4.66%) (-.27%) 2.46 (-5.42%) (-.48%).72 (-6.58%) 8

10 [Case6] S =, σ =.2, c =., q =., εν =.2, ρ =.5, ελ =., θ =., H = 4, T =.. Table 6: up-and-out barrier option prices and the relative errors (Case 6) (.78%) (-.59%) (.85%) 4.2 (-2.3%) (.97%) 3.74 (-2.68%) References [] T. Kato, A. Takahashi and T. Yamada, An asymptotic expansion for solutions of Cauchy-Dirichlet problem for second order parabolic PDEs and its application to pricing barrier options, arxiv preprint, (22). [2] J.C. Hull, Options, Futures, and Other Derivatives, 6-th Edition, Prentice Hall, 25 [3] A.Takahashi and T.Yamada, An asymptotic expansion with push-down of Malliavin weights, SIAM Journal on Financial Mathematics, 3, (22),