Closed-form approximate solutions for stop-loss and Russian options with multiscale stochastic volatility

Min-Ku Lee; Jeong-Hoon Kim; Min-Ku Lee; Jeong-Hoon Kim

doi:10.3934/math.20231284

AIMS Mathematics

2023, Volume 8, Issue 10: 25164-25194. doi: 10.3934/math.20231284

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Closed-form approximate solutions for stop-loss and Russian options with multiscale stochastic volatility

Min-Ku Lee ^{1
,},
Jeong-Hoon Kim ^{2
,
,}

1.
Department of Mathematics, Kunsan National University, Kunsan 54150, Republic of Korea; Email: mgcorea@kunsan.ac.kr
2.
Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea; Email: jhkim96@yonsei.ac.kr

Received: 11 June 2023 Revised: 16 August 2023 Accepted: 23 August 2023 Published: 29 August 2023
MSC : 91G20, 91G60

In general, derivation of closed-form analytic formulas for the prices of path-dependent exotic options is a challenging task when the underlying asset price model is chosen to be a stochastic volatility model. Pricing stop-loss and Russian options is studied under a multiscale stochastic volatility model in this paper. Both options are commonly perpetual American-style derivatives with a lookback provision. We derive closed-form formulas explicitly for the approximate prices of these two exotic options by using multiscale asymptotic analysis and partial differential equation method. The formulas can be efficiently computed starting with the Black-Scholes option prices. The accuracy of the analytic approximation is verified via Monte-Carlo simulations and the impacts of the multiscale stochastic volatility on the corresponding Black-Scholes option prices are revealed. Also, the performance of the model is compared with that of other models.
- option pricing,
- stop-loss option,
- Russian option,
- multiscale,
- stochastic volatility,
- asymptotics
Citation: Min-Ku Lee, Jeong-Hoon Kim. Closed-form approximate solutions for stop-loss and Russian options with multiscale stochastic volatility[J]. AIMS Mathematics, 2023, 8(10): 25164-25194. doi: 10.3934/math.20231284

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Abstract

In general, derivation of closed-form analytic formulas for the prices of path-dependent exotic options is a challenging task when the underlying asset price model is chosen to be a stochastic volatility model. Pricing stop-loss and Russian options is studied under a multiscale stochastic volatility model in this paper. Both options are commonly perpetual American-style derivatives with a lookback provision. We derive closed-form formulas explicitly for the approximate prices of these two exotic options by using multiscale asymptotic analysis and partial differential equation method. The formulas can be efficiently computed starting with the Black-Scholes option prices. The accuracy of the analytic approximation is verified via Monte-Carlo simulations and the impacts of the multiscale stochastic volatility on the corresponding Black-Scholes option prices are revealed. Also, the performance of the model is compared with that of other models.

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