- Email: [email protected]
Option pricing under joint dynamics of interest rates, dividends, and stock prices
Operations Research Letters 39 (2011) 260–264
Contents lists available at ScienceDirect
Operations Research Letters journal homepage: www.elsevier.com/locate/orl
Option pricing under joint dynamics of interest rates, dividends, and stock prices Juho Kanniainen ∗ Tampere University of Technology, Department of Industrial Management, P.O. Box 541, FI-33101 Tampere, Finland
article
info
Article history: Received 5 April 2011 Accepted 31 May 2011 Available online 15 June 2011 Keywords: Interest rates Option pricing Dividends
abstract This paper proposes a unified framework for option pricing, which integrates the stochastic dynamics of interest rates, dividends, and stock prices under the transversality condition. Using the Vasicek model for the spot rate dynamics, I compare the framework with two existing option pricing models. The main implication is that the stochastic spot rate affects options not only directly but also via an endogenously determined dividend yield and return volatility; consequently, call prices can be decreasing with respect to interest rates. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Many papers in the recent literature have aimed to improve the performance of option pricing models by incorporating stochastic interest rates (see, for example, [14,8,2]). However, the relation between stock prices and interest rates has been typically oversimplified or not modeled at all, even though the relation has been widely recognized. Empirically, especially between the 1960s and 1990s, a noticeable negative correlation emerged between the ten-year bond yield and price-earnings ratios. On the other hand, the option pricing literature has commonly assumed a deterministic dividend yield by ignoring the empirical fact that the price–dividend ratio is stochastic and time-varying, and that interest rates and price–dividend ratios can be negatively related. In this paper, I model the joint stochastic dynamics of interest rates, dividends, and stock prices by determining the underlying stock as a claim for future random continuous-time dividends with a stochastic discount rate under the transversality condition. My initial setup allows both the CIR and Vasicek model for interest rates, but because of the time-intensive computation of option prices under the CIR, option pricing analysis in this paper has been limited to the Vasicek. In the resulting framework, dividend yield and return volatility become stochastic and dependent on spot interest rates; consequently, interest rates affect the stock price dynamics not only via the expected rate of return but also via the dividend yield and return volatility. The main implication is that, due to an endogenously determined dividend yield and return volatility, an increase in interest rates can negatively affect the price of a call option, whereas the prevailing wisdom holds that the call price is increasing with respect to an instantaneous
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interest rate. In this regard, McDonald and Siegel [5] show that the call price is not necessarily increasing on the level of interest rates when the underlying stock price process is non-Markovian. In this paper, I present further arguments for a potentially negative relation between interest rates and call option prices. In the first section, I present the model setup and solve the stock price as a function of dividends and the spot rate. In addition, expressions for dividend yield and return volatility are given as functions of the spot rate. In Section 3, I compare the option prices of my model with those of two models that respectively allow for (i) constant volatility, constant dividend yield, and constant interest rates (the Black–Merton–Scholes) and (ii) constant volatility, constant dividend yield, but stochastic interest rates, whereas in my model both dividend yield and return volatility are endogenously determined by a stochastic spot interest rate and hence time-varying and stochastic. Moreover, I demonstrate a relation between interest rates and option prices. 2. Interest rates, dividends, and stock prices Consider an economy in which the underlying process of interest {xt ; t ≥ 0} and dividends {Dt ; t ≥ 0} are represented under the risk-neutral probability measure by the stochastic differential equation dxt = κ(θ − xt )dt + σx xγ dWt ,
(1)
dDt = α Dt dt + σd Dt ρ dWt + 1 − ρ 2 dZt ,
(2)
where x0 ∈ R, D0 > 0, κ, θ , σx , α , and σd are positive constants, and γ is 0 (Vasicek) or 1/2 (CIR). In addition, the correlation between dividend growth and spot interest rates ρ ∈ [−1, 1] and dWt dZt = 0. Here I also assume that the instantaneous dividends follow geometric Brownian motion. The assumption of lognormal instantaneous dividends (cash flows) is quite common
J. Kanniainen / Operations Research Letters 39 (2011) 260–264
261
Fig. 1. Price–dividend ratio and return volatility with respect to the spot interest rate. Parameters are κ = 0.136, θ = 0.070, σx = 0.0224, σd = 0.0365, α = 0.02906, and ρ = 0.0206.
in the literature (see, for example, [15,7,3,1,13]) though not extensively studied for its empirical validity. In further research, the assumption could be partly relaxed. Under the transversality condition, the underlying stock price at time t can be expressed as (note that W and Z are uncorrelated): S (xt , Dt ) = Et
∞
[∫
τ
∫ exp −
]
∞
exp (α(τ − t ))
= Dt t
τ
∫
× Et exp
1 −xu − (ρσd )2 du 2
t
τ
∫ +
ρσd dWu
1 1 − e−κ(τ −t ) − (τ − t )
κ
σx2 1 −κ(τ −t ) 2 −κ(τ −t ) − 3 1−e − 1−e x t dτ . 4κ κ
T
[∫
dτ ,
xu − y(xu ) −
ST = St exp
t
t
where Et indicates a conditional expectation taken under the risk-neutral measure Q. In general, by determining the measure ˜ by its Radon–Nikodym derivative with respect to the original Q ˜ t (Z ), where E˜ t denotes the (risk-neutral) measure Q, Et (ξ Z ) = E
where St = S (xt , Dt ) and
˜ , ξ = dQ , and E˜ t (|Z |) < ∞. conditional expectation under Q dQ Therefore, by defining (see also [1, pp. 15–16]),
η(xt )Wts =
T
∫ +
dQ
τ
[ ∫ = exp − t
1 2
(ρσd )2 du +
τ
∫
] ρσd dWu ,
γ ˜ dxt = κ θ ∗ − xt dt + σx xt dW (3) t, γ ˜, where θ ∗ = θ + ρσd σx xt /κ . Note that if ρ, γ > 0, then under Q the normal level of the short rate θ ∗ is stochastic. Therefore, using the CIR is computationally much harder than using the Vasicek, especially because the Vasicek allows simulation of the spot rate with arbitrary time steps. However, the Vasicek model is likely to produce negative interest rates, especially when interest rates are low with high implied volatilities. Despite of this shortcoming, for simplicity and fewer computations, the rest of the study assumes that γ = 0, i.e., the Vasicek model, under which θ ∗ = θ + (ρσd σx )/κ . By applying the well-known pricing formula for discount bonds under the Vasicek model, we obtain S (xt , Dt ) = Dt /y(xt ), where y(x) denotes the instantaneous dividend yield (dividend–price ˜ satisfies ratio), which under Q ∞
∫
[
τ
∫
˜ t exp − exp (α(τ − t )) E t
exp α(τ − t ) +
=1 t
] xu du
t
∞
∫
θ+
2
η (xu ) du 2
(5)
t
1 − ρ 2 σd Zt +
ρσd − σx
yx (xt ) y(xt )
Wt ,
and where yx denotes the first derivative of y with respect to x. Hence, squared total return volatility (that is, instantaneous return variance) satisfies
t
˜ are the dynamics of the spot rate under Q
y(xt ) = 1
1
] η(xu )dWus ,
˜
˜ dQ
(4)
Clearly, if κ, σx = 0, then interest rates are constant in time, and the stock price model reduces to a pure continuous-time Gordon model with a constant discount rate, resulting in a constant dividend yield of y(x) = x − α and a stock price of S (x, Dt ) = Dt /(x − α), where x is the constant spot rate. This represents the constant dividend yield case and, in fact, the Black–Merton–Scholes model. It is quite straightforward to show that under the original riskneutral measure with the Vasicek model, for T > t,
t
t
∫
xu du Dτ dτ
×
ρσd σx σ2 − x2 κ 2κ
dτ
2 yx (x) yx (x) η2 (xt ) = σd2 + σx − 2ρσd σx . y(x) y(x)
(6)
To provide a credible numerical illustration for Fig. 1, I take parameter estimates for the Vasicek model from [16], which are as follows: κ = 0.136, θ = 0.070, and σx = 0.0224. Moreover, I use monthly dividend data from 1935–2010 getting values σd = 0.0365 and ρ = 0.0206. The monthly dividend data were made available by Professor Robert Shiller on his web site http://www.econ.yale.edu/~shiller/data.htm. In addition, the riskneutral dividend growth rate α is set to 0.02906 to match the price–dividend ratio with the long-term average of the price–dividend ratio with x0 = λ. Specifically, by denoting the price–dividend ratio by q(x) = 1/y(x), with α = 0.02906, κ = 0.136, θ = 0.070, σx = 0.0224, σd = 0.0365, and ρ = 0.0206, my model gives q(x = θ ) ≈ 32.15, which is approximately equal to the empirical long-term average ratio q¯ ≈ 32.15. As Fig. 1, plot (a) shows, the stock price (with D0 = 1) is monotonically decreasing from 50 to 15 as the spot interest rate increases from zero to 0.2. This is in line with John Cochrane [11], who recently argued that ‘‘The dividend yield shock is essentially a pure discount rate shock’’. Moreover, plot (b) demonstrates that also return volatility reacts negatively to an increase in the spot
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J. Kanniainen / Operations Research Letters 39 (2011) 260–264
Fig. 2. Differences between option prices in my model and (a) the Black–Merton–Scholes model and (b) the Brigo–Mercurio model. Parameters are κ = 0.136, θ = 0.070, σx = 0.0224, σd = 0.0365, α = 0.02906, and ρ = 0.0206. For the Black–Merton–Scholes and the Brigo–Mercurio model, the (constant) dividend yield, return volatility, and correlation between returns and interest rates are y¯ = y(x0 ), η¯ = η(x0 ), and ρrx = 0, respectively. In addition, S0 = 15, T = 5, and M = 10,000.
interest rate. The relation between plots (a) and (b) is intuitive; because the stock price is convex with respect to the spot interest rate, the stock price is more sensitive to a low than a high spot interest rate. 3. Option prices Suppose that a derivative security makes a payoff of g (ST ) at time T > t. Under the risk-neutral measure, we can price the derivative security by computing
dSt = (rt − y¯ )St dt + η¯ St
C (t , St , xt , T , K ; ψ) = Et Λt ,T × g (ST ) ,
T
where ST = S (xT , DT ), Λt ,T = exp −
t
xu du is the stochastic
discount factor and ψ = {κ, θ , σx , σd , ρ, α} are the structural parameters. Like, for example, most GARCH models and the socalled VAR volatility model, mine requires Monte Carlo simulations to compute call option prices (see, for example, [9,10,4]). To speed up computations, I use antithetic variates. Suppose that we are pricing a vanilla call option with the Vasicek (γ = 0) model. Then g (St ) = (ST − K )+ , where ST = S (xT , DT ) can be calculated using (4) for a given xT and DT , which together with the discount factor are simulated under the original risk-neutral measure Q. The joint simulation of dividend stream, short rate, and discount factor can be performed exactly and without a discretization error with the algorithms
[
1 α − σd2 (T − t ) 2 √ + σd ρ zTx + 1 − ρ 2 zT T −t , 1 xT = xt e−κ(T −t ) + θ 1 − e−κ(T −t ) + σx 1 − e−2κ(T −t ) zTx , 2κ [ θ −κ(T −t ) xt −κ(T −t ) Λt ,T = exp − 1−e + e + κ(T − t ) − 1 κ ] κ H xH x xH 2 x + σt ,T ρt ,T zT + 1 − (ρt ,T ) z˜T , DT = Dt exp
where
2 σx 1 (T − t ) + 1 − e−2κ(T −t ) + e−κ(T −t ) − 1 , κ 2κ κ σx 1 + e−2κ(T −t ) − 2e−κ(T −t ) = √ , √ κ 2κ σtH,T 1 − e−2κ(T −t )
σtH,T = ρtxH ,T
Eq. (4) and calculate the payoff. This is done repeatedly until the option prices converge. Two models are used for comparison, the original Black– Scholes [6] model (or Black–Merton–Scholes model), which imposes a constant spot interest rate, and a model proposed by Brigo and Mercurio [8, pp. 883–889], who assume a stochastic spot interest rate but ignore stochasticity in the dividend yield. In particular, the Brigo–Mercurio model assumes that the stock price follows under the risk-neutral measure as
and where zT , zTx , and z˜Tx are independent draws from N (0, 1) (see the derivation of the exact discretization of Λt ,T in [12, pp. 115–116], note the different notation). For simulated xT and DT , we can solve the stock price S (xT , DT ) at maturity using
ρrx dWt +
2 dZ 1 − ρrx t
,
where S0 > 0, x0 ∈ R, W and Z are independent Brownian motions, and where y¯ represents a positive and constant dividend yield, η¯ the constant return volatility, and ρrx the correlation between interest rates and returns. Moreover, Brigo and Mercurio assume that spot rates follow the Hull–White model, a general version of the Vasicek dynamics (see Eq. (1)). Compared to my model, Brigo and Mercurio assume that dividend yield and return volatility are constant regardless of the stochastic spot interest rate, whereas in my model the stochastic spot interest rate determines endogenously the dividend yield and return volatility, which therefore are time-varying and stochastic under my assumptions. As Brigo and Mercurio [8, pp. 883–889] show, under their assumptions the price of a European option can be expressed as CBM (t , St , xt , T , K ; ψBM )
= St e
−¯y(T −t )
N
− KPt ,T N
ln
ln
St KPt ,T
− y¯ (T − t ) + 12 υt2,T
υ t ,T St KPt ,T
− y¯ (T − t ) − 12 υt2,T υt ,T
,
where N denotes the standard normal cumulative distribution function and where
υt2,T :=
σx2 κ2
T −t +
2 −κ(T −t ) 1 −2κ(T −t ) 3 e − e − κ 2κ 2κ
[ ] σ η¯ 1 T −t − 1 − e−κ(T −t ) . κ κ Moreover, Pt ,T := P (t , T , xt ) denotes the price of a T -maturity zero-coupon bond under the Vasicek model and ψBM the structural + η¯ 2 (T − t ) + 2ρrx
parameters of Brigo and Mercurio’s model. Note that both my model and Brigo and Mercurio’s model reduce to the Black–Merton–Scholes model with κ = σx = 0. Fig. 2 illustrates relative differences in option prices between my and (a) the Black–Merton–Scholes model with a constant
J. Kanniainen / Operations Research Letters 39 (2011) 260–264
263
Fig. 3. Effect of the spot interest rate on the price of a European call option under my model. Parameters are κ = 0.136, θ = 0.070, σx = 0.0224, σd = 0.0365, α = 0.02906, and ρ = 0.0206. In addition, T = 5 and M = 10,000. The strike price, K , is 0, 2, 4, . . . , 20 dollars. Note that the lower the strike price, the higher the curve. In plot (a), the current stock price is fixed at S0 = $15, and current dividends vary with respect to the spot rate according to D0 = S0 × y(x0 ). In plot (b), current dividends are fixed at D0 = $15/q(0) ≈ $49.47, and the current stock price varies with respect to the spot rate according to S0 = D0 × q(x0 ), where q(x) = 1/y(x) denotes the price–dividend ratio.
dividend yield and a constant spot interest rate and (b) the Brigo–Mercurio model with a constant dividend yield and a stochastic spot interest rate with respect to moneyness. Here I use the same parameter values as in Fig. 1, but in addition, for the Black–Merton–Scholes and the Brigo–Mercurio model (hereafter BMS and BM models), a constant dividend yield, return volatility, and a correlation between returns and interest rates of y¯ = y(x0 ), η¯ = η(x0 ), and ρrx = 0, respectively, and an initial spot stock price of S0 = 15, a time to maturity of T = 5, and M = 10,000 simulations. As plot (a) shows, under these parameter values, the BMS price, CBMS , is greater than the option price under my model, C , only if the option is deep-in-the-money. Moreover, the greater the spot interest rate, x0 , the less (more) is the relative price difference for an in-the-money (for a deep-out-of-the-money) option. Interestingly, as plot (b) shows, differences in the option price can be greater with respect to the BM model than with respect to the BMS model; especially, the greater the moneyness, the greater the difference in price (C − CBM )/CBM . Moreover, we can see that if the interest rate is low, then the BM model over-prices options in comparison to my model, and, conversely, the greater the spot interest rate, the greater the price of my model, C , with respect to the BM model, CBM . For example, if the spot interest rate is twice the long-term interest rate, that is, x0 = 2θ , then C > CBM for all 12 < K < 18. One implication of my characterization is that the option price can be a decreasing function of the spot interest rate because an increase in the spot interest rate increases the dividend yield, and thus potentially lowers the call price, which contradicts the current wisdom. These earlier arguments for a positive relation between call price and spot interest rate are, however, based on the assumption of a constant dividend yield or absence of dividends. To see how the level of the spot interest rate affects options in my model, let us, for simplicity, first consider a special case of K ↓ 0, in which, the exercise price is zero and the ‘‘option’’ holder gets the underlying stock for free at time T . Eq. (5) implies that the option price with K ↓ 0 can be expressed as C (t , St , xt , T , 0; ψ) T
[ ∫ = St Et exp t
−y(xu ) −
1 2
∫ η2 (xu ) du +
T
η(xu )dWus
]
,
(7)
t
where η(x) is given by Eq. (6). The dividend yield of the underlying stock, y(xt ), is increasing with respect to the spot interest rate, and, therefore, under these conditions, also decreases the price of the call as a greater dividend yield means a greater shortfall for the option holder. With strictly positive exercise prices, K > 0, the payoff becomes convex, and thus an increase in the spot interest rate can also increase the option price, depending on which
effects, positive or negative, dominate. Note that with K > 0, an increase in the spot rate may reduce option prices not only through dividend yield but also via return volatility (see Fig. 1). Intuitively, the greater the K , the more the spot rate can then potentially increase option prices. This is illustrated in Fig. 3, plot (a). The figure uses the same parameters as Fig. 2, except that the strike price range is 0, 2, 4, . . . , 20. As the plot illustrates, if the strike price is relatively low, the call price is monotonically decreasing in the interest rates. However, for greater strike prices (with a strike price of 14, 16, 18, or 20), the call price can be non-monotone or even monotonically increasing for the given interval. In fact, we assume in plot (a) that the current stock price level is not affected by any increase in the spot interest rate; that is, the current level of dividends must increase in response to a lower price–dividend ratio since S (xt , Dt ) = Dt × q(xt ), where q(x) = 1/y(x). The relation between the spot interest rate and option prices could also be viewed differently. We could think of a situation in which the stock price level reacts to an increase in the short rate, while the level of dividends remains an exogenous variable and unaffected. That is, we can substitute Dt × q(xt ) for St in Eq. (7). The spot interest rate then has three negative effects on option prices: via increased dividend yield, via lower return volatility, and via a lower stock price level. In Fig. 3, plot (b), we keep current dividends fixed, in which case the call price is always decreasing with respect to the spot interest rate. 4. Final remark The above framework merits further research. For example, total hedging of interest rates with equity options could be studied, especially because, depending on the moneyness, option prices can react positively or negatively to an increase in the spot rate via endogenously determined dividend yield and return volatility. Acknowledgments I am grateful to the area editor Jussi Keppo, an anonymous associate editor, and an anonymous referee for their many valuable suggestions. All remaining errors are my sole responsibility. I also gratefully acknowledge the financial support from the Academy of Finland (project 123058), the Emil Aaltonen Foundation, and the OP-Pohjola Group Foundation. References [1] A. Ang, J. Liu, Risk, return, and dividends, Journal of Financial Economics 85 (2007) 1–38. [2] G. Bakshi, C. Cao, Z. Chen, Option pricing and hedging performance under stochastic volatility and stochastic interest rates, in: C.H.C.-F. Lee (Ed.), Handbook of Quantitative Finance and Risk Management, Springer, 2010.
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[3] G. Bakshi, Z. Chen, Stock valuation in dynamic economies, Journal of Financial Markets 8 (2005) 111–151. [4] G. Barone-Adesi, R.F. Engle, L. Mancini, A GARCH option pricing model with filtered historical simulation, Review of Financial Studies 21 (2008) 1223–1258. [5] Y.Z. Bergman, B.D. Grundy, Z. Wiener, General properties of option prices, Journal of Finance 51 (1996) 1573–1610. [6] F. Black, M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81 (1973) 637–659. [7] M.J. Brennan, Y. Xia, Stock price volatility and equity premium, Journal of Monetary Economics 47 (2001) 249–283. [8] D. Brigo, F. Mercurio, Interest Rate Models—Theory and Practice, Springer, 2006. [9] P. Christoffersen, K. Jacobs, Which GARCH model for option valuation? Management Science 50 (2004) 1204–1221.
[10] P. Christoffersen, K. Jacobs, K. Mimouni, Models for S&P 500 dynamics: evidence from realized volatility, daily returns, and option prices, Review of Financial Studies 23 (2010) 3141–3189. [11] J.H. Cochrane, Discount rates, Working Paper University of Chicago, 2011. [12] P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, 2003. [13] J. Kanniainen, Can properly discounted projects follow geometric brownian motion? Mathematical Methods of Operations Research 70 (2009) 435–450. [14] R.S. Mamon, M.R. Rodrigo, Explicit solutions to European options in a regimeswitching economy, Operations Research Letters 33 (2005) 581–586. [15] R. McDonald, D. Siegel, Investment and the valuation of firms when there is an option of shut down, International Economic Review 28 (1985) 331–349. [16] C.Y. Tang, S.X. Chen, Parameter estimation and bias correction for diffusion processes, Journal of Econometrics 149 (2009) 65–81.
Contents lists available at ScienceDirect
Operations Research Letters journal homepage: www.elsevier.com/locate/orl
Option pricing under joint dynamics of interest rates, dividends, and stock prices Juho Kanniainen ∗ Tampere University of Technology, Department of Industrial Management, P.O. Box 541, FI-33101 Tampere, Finland
article
info
Article history: Received 5 April 2011 Accepted 31 May 2011 Available online 15 June 2011 Keywords: Interest rates Option pricing Dividends
abstract This paper proposes a unified framework for option pricing, which integrates the stochastic dynamics of interest rates, dividends, and stock prices under the transversality condition. Using the Vasicek model for the spot rate dynamics, I compare the framework with two existing option pricing models. The main implication is that the stochastic spot rate affects options not only directly but also via an endogenously determined dividend yield and return volatility; consequently, call prices can be decreasing with respect to interest rates. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Many papers in the recent literature have aimed to improve the performance of option pricing models by incorporating stochastic interest rates (see, for example, [14,8,2]). However, the relation between stock prices and interest rates has been typically oversimplified or not modeled at all, even though the relation has been widely recognized. Empirically, especially between the 1960s and 1990s, a noticeable negative correlation emerged between the ten-year bond yield and price-earnings ratios. On the other hand, the option pricing literature has commonly assumed a deterministic dividend yield by ignoring the empirical fact that the price–dividend ratio is stochastic and time-varying, and that interest rates and price–dividend ratios can be negatively related. In this paper, I model the joint stochastic dynamics of interest rates, dividends, and stock prices by determining the underlying stock as a claim for future random continuous-time dividends with a stochastic discount rate under the transversality condition. My initial setup allows both the CIR and Vasicek model for interest rates, but because of the time-intensive computation of option prices under the CIR, option pricing analysis in this paper has been limited to the Vasicek. In the resulting framework, dividend yield and return volatility become stochastic and dependent on spot interest rates; consequently, interest rates affect the stock price dynamics not only via the expected rate of return but also via the dividend yield and return volatility. The main implication is that, due to an endogenously determined dividend yield and return volatility, an increase in interest rates can negatively affect the price of a call option, whereas the prevailing wisdom holds that the call price is increasing with respect to an instantaneous
∗
Tel.: +358 40 707 4532; fax: +358 3 3115 2027. E-mail address: [email protected].
0167-6377/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2011.06.004
interest rate. In this regard, McDonald and Siegel [5] show that the call price is not necessarily increasing on the level of interest rates when the underlying stock price process is non-Markovian. In this paper, I present further arguments for a potentially negative relation between interest rates and call option prices. In the first section, I present the model setup and solve the stock price as a function of dividends and the spot rate. In addition, expressions for dividend yield and return volatility are given as functions of the spot rate. In Section 3, I compare the option prices of my model with those of two models that respectively allow for (i) constant volatility, constant dividend yield, and constant interest rates (the Black–Merton–Scholes) and (ii) constant volatility, constant dividend yield, but stochastic interest rates, whereas in my model both dividend yield and return volatility are endogenously determined by a stochastic spot interest rate and hence time-varying and stochastic. Moreover, I demonstrate a relation between interest rates and option prices. 2. Interest rates, dividends, and stock prices Consider an economy in which the underlying process of interest {xt ; t ≥ 0} and dividends {Dt ; t ≥ 0} are represented under the risk-neutral probability measure by the stochastic differential equation dxt = κ(θ − xt )dt + σx xγ dWt ,
(1)
dDt = α Dt dt + σd Dt ρ dWt + 1 − ρ 2 dZt ,
(2)
where x0 ∈ R, D0 > 0, κ, θ , σx , α , and σd are positive constants, and γ is 0 (Vasicek) or 1/2 (CIR). In addition, the correlation between dividend growth and spot interest rates ρ ∈ [−1, 1] and dWt dZt = 0. Here I also assume that the instantaneous dividends follow geometric Brownian motion. The assumption of lognormal instantaneous dividends (cash flows) is quite common
J. Kanniainen / Operations Research Letters 39 (2011) 260–264
261
Fig. 1. Price–dividend ratio and return volatility with respect to the spot interest rate. Parameters are κ = 0.136, θ = 0.070, σx = 0.0224, σd = 0.0365, α = 0.02906, and ρ = 0.0206.
in the literature (see, for example, [15,7,3,1,13]) though not extensively studied for its empirical validity. In further research, the assumption could be partly relaxed. Under the transversality condition, the underlying stock price at time t can be expressed as (note that W and Z are uncorrelated): S (xt , Dt ) = Et
∞
[∫
τ
∫ exp −
]
∞
exp (α(τ − t ))
= Dt t
τ
∫
× Et exp
1 −xu − (ρσd )2 du 2
t
τ
∫ +
ρσd dWu
1 1 − e−κ(τ −t ) − (τ − t )
κ
σx2 1 −κ(τ −t ) 2 −κ(τ −t ) − 3 1−e − 1−e x t dτ . 4κ κ
T
[∫
dτ ,
xu − y(xu ) −
ST = St exp
t
t
where Et indicates a conditional expectation taken under the risk-neutral measure Q. In general, by determining the measure ˜ by its Radon–Nikodym derivative with respect to the original Q ˜ t (Z ), where E˜ t denotes the (risk-neutral) measure Q, Et (ξ Z ) = E
where St = S (xt , Dt ) and
˜ , ξ = dQ , and E˜ t (|Z |) < ∞. conditional expectation under Q dQ Therefore, by defining (see also [1, pp. 15–16]),
η(xt )Wts =
T
∫ +
dQ
τ
[ ∫ = exp − t
1 2
(ρσd )2 du +
τ
∫
] ρσd dWu ,
γ ˜ dxt = κ θ ∗ − xt dt + σx xt dW (3) t, γ ˜, where θ ∗ = θ + ρσd σx xt /κ . Note that if ρ, γ > 0, then under Q the normal level of the short rate θ ∗ is stochastic. Therefore, using the CIR is computationally much harder than using the Vasicek, especially because the Vasicek allows simulation of the spot rate with arbitrary time steps. However, the Vasicek model is likely to produce negative interest rates, especially when interest rates are low with high implied volatilities. Despite of this shortcoming, for simplicity and fewer computations, the rest of the study assumes that γ = 0, i.e., the Vasicek model, under which θ ∗ = θ + (ρσd σx )/κ . By applying the well-known pricing formula for discount bonds under the Vasicek model, we obtain S (xt , Dt ) = Dt /y(xt ), where y(x) denotes the instantaneous dividend yield (dividend–price ˜ satisfies ratio), which under Q ∞
∫
[
τ
∫
˜ t exp − exp (α(τ − t )) E t
exp α(τ − t ) +
=1 t
] xu du
t
∞
∫
θ+
2
η (xu ) du 2
(5)
t
1 − ρ 2 σd Zt +
ρσd − σx
yx (xt ) y(xt )
Wt ,
and where yx denotes the first derivative of y with respect to x. Hence, squared total return volatility (that is, instantaneous return variance) satisfies
t
˜ are the dynamics of the spot rate under Q
y(xt ) = 1
1
] η(xu )dWus ,
˜
˜ dQ
(4)
Clearly, if κ, σx = 0, then interest rates are constant in time, and the stock price model reduces to a pure continuous-time Gordon model with a constant discount rate, resulting in a constant dividend yield of y(x) = x − α and a stock price of S (x, Dt ) = Dt /(x − α), where x is the constant spot rate. This represents the constant dividend yield case and, in fact, the Black–Merton–Scholes model. It is quite straightforward to show that under the original riskneutral measure with the Vasicek model, for T > t,
t
t
∫
xu du Dτ dτ
×
ρσd σx σ2 − x2 κ 2κ
dτ
2 yx (x) yx (x) η2 (xt ) = σd2 + σx − 2ρσd σx . y(x) y(x)
(6)
To provide a credible numerical illustration for Fig. 1, I take parameter estimates for the Vasicek model from [16], which are as follows: κ = 0.136, θ = 0.070, and σx = 0.0224. Moreover, I use monthly dividend data from 1935–2010 getting values σd = 0.0365 and ρ = 0.0206. The monthly dividend data were made available by Professor Robert Shiller on his web site http://www.econ.yale.edu/~shiller/data.htm. In addition, the riskneutral dividend growth rate α is set to 0.02906 to match the price–dividend ratio with the long-term average of the price–dividend ratio with x0 = λ. Specifically, by denoting the price–dividend ratio by q(x) = 1/y(x), with α = 0.02906, κ = 0.136, θ = 0.070, σx = 0.0224, σd = 0.0365, and ρ = 0.0206, my model gives q(x = θ ) ≈ 32.15, which is approximately equal to the empirical long-term average ratio q¯ ≈ 32.15. As Fig. 1, plot (a) shows, the stock price (with D0 = 1) is monotonically decreasing from 50 to 15 as the spot interest rate increases from zero to 0.2. This is in line with John Cochrane [11], who recently argued that ‘‘The dividend yield shock is essentially a pure discount rate shock’’. Moreover, plot (b) demonstrates that also return volatility reacts negatively to an increase in the spot
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Fig. 2. Differences between option prices in my model and (a) the Black–Merton–Scholes model and (b) the Brigo–Mercurio model. Parameters are κ = 0.136, θ = 0.070, σx = 0.0224, σd = 0.0365, α = 0.02906, and ρ = 0.0206. For the Black–Merton–Scholes and the Brigo–Mercurio model, the (constant) dividend yield, return volatility, and correlation between returns and interest rates are y¯ = y(x0 ), η¯ = η(x0 ), and ρrx = 0, respectively. In addition, S0 = 15, T = 5, and M = 10,000.
interest rate. The relation between plots (a) and (b) is intuitive; because the stock price is convex with respect to the spot interest rate, the stock price is more sensitive to a low than a high spot interest rate. 3. Option prices Suppose that a derivative security makes a payoff of g (ST ) at time T > t. Under the risk-neutral measure, we can price the derivative security by computing
dSt = (rt − y¯ )St dt + η¯ St
C (t , St , xt , T , K ; ψ) = Et Λt ,T × g (ST ) ,
T
where ST = S (xT , DT ), Λt ,T = exp −
t
xu du is the stochastic
discount factor and ψ = {κ, θ , σx , σd , ρ, α} are the structural parameters. Like, for example, most GARCH models and the socalled VAR volatility model, mine requires Monte Carlo simulations to compute call option prices (see, for example, [9,10,4]). To speed up computations, I use antithetic variates. Suppose that we are pricing a vanilla call option with the Vasicek (γ = 0) model. Then g (St ) = (ST − K )+ , where ST = S (xT , DT ) can be calculated using (4) for a given xT and DT , which together with the discount factor are simulated under the original risk-neutral measure Q. The joint simulation of dividend stream, short rate, and discount factor can be performed exactly and without a discretization error with the algorithms
[
1 α − σd2 (T − t ) 2 √ + σd ρ zTx + 1 − ρ 2 zT T −t , 1 xT = xt e−κ(T −t ) + θ 1 − e−κ(T −t ) + σx 1 − e−2κ(T −t ) zTx , 2κ [ θ −κ(T −t ) xt −κ(T −t ) Λt ,T = exp − 1−e + e + κ(T − t ) − 1 κ ] κ H xH x xH 2 x + σt ,T ρt ,T zT + 1 − (ρt ,T ) z˜T , DT = Dt exp
where
2 σx 1 (T − t ) + 1 − e−2κ(T −t ) + e−κ(T −t ) − 1 , κ 2κ κ σx 1 + e−2κ(T −t ) − 2e−κ(T −t ) = √ , √ κ 2κ σtH,T 1 − e−2κ(T −t )
σtH,T = ρtxH ,T
Eq. (4) and calculate the payoff. This is done repeatedly until the option prices converge. Two models are used for comparison, the original Black– Scholes [6] model (or Black–Merton–Scholes model), which imposes a constant spot interest rate, and a model proposed by Brigo and Mercurio [8, pp. 883–889], who assume a stochastic spot interest rate but ignore stochasticity in the dividend yield. In particular, the Brigo–Mercurio model assumes that the stock price follows under the risk-neutral measure as
and where zT , zTx , and z˜Tx are independent draws from N (0, 1) (see the derivation of the exact discretization of Λt ,T in [12, pp. 115–116], note the different notation). For simulated xT and DT , we can solve the stock price S (xT , DT ) at maturity using
ρrx dWt +
2 dZ 1 − ρrx t
,
where S0 > 0, x0 ∈ R, W and Z are independent Brownian motions, and where y¯ represents a positive and constant dividend yield, η¯ the constant return volatility, and ρrx the correlation between interest rates and returns. Moreover, Brigo and Mercurio assume that spot rates follow the Hull–White model, a general version of the Vasicek dynamics (see Eq. (1)). Compared to my model, Brigo and Mercurio assume that dividend yield and return volatility are constant regardless of the stochastic spot interest rate, whereas in my model the stochastic spot interest rate determines endogenously the dividend yield and return volatility, which therefore are time-varying and stochastic under my assumptions. As Brigo and Mercurio [8, pp. 883–889] show, under their assumptions the price of a European option can be expressed as CBM (t , St , xt , T , K ; ψBM )
= St e
−¯y(T −t )
N
− KPt ,T N
ln
ln
St KPt ,T
− y¯ (T − t ) + 12 υt2,T
υ t ,T St KPt ,T
− y¯ (T − t ) − 12 υt2,T υt ,T
,
where N denotes the standard normal cumulative distribution function and where
υt2,T :=
σx2 κ2
T −t +
2 −κ(T −t ) 1 −2κ(T −t ) 3 e − e − κ 2κ 2κ
[ ] σ η¯ 1 T −t − 1 − e−κ(T −t ) . κ κ Moreover, Pt ,T := P (t , T , xt ) denotes the price of a T -maturity zero-coupon bond under the Vasicek model and ψBM the structural + η¯ 2 (T − t ) + 2ρrx
parameters of Brigo and Mercurio’s model. Note that both my model and Brigo and Mercurio’s model reduce to the Black–Merton–Scholes model with κ = σx = 0. Fig. 2 illustrates relative differences in option prices between my and (a) the Black–Merton–Scholes model with a constant
J. Kanniainen / Operations Research Letters 39 (2011) 260–264
263
Fig. 3. Effect of the spot interest rate on the price of a European call option under my model. Parameters are κ = 0.136, θ = 0.070, σx = 0.0224, σd = 0.0365, α = 0.02906, and ρ = 0.0206. In addition, T = 5 and M = 10,000. The strike price, K , is 0, 2, 4, . . . , 20 dollars. Note that the lower the strike price, the higher the curve. In plot (a), the current stock price is fixed at S0 = $15, and current dividends vary with respect to the spot rate according to D0 = S0 × y(x0 ). In plot (b), current dividends are fixed at D0 = $15/q(0) ≈ $49.47, and the current stock price varies with respect to the spot rate according to S0 = D0 × q(x0 ), where q(x) = 1/y(x) denotes the price–dividend ratio.
dividend yield and a constant spot interest rate and (b) the Brigo–Mercurio model with a constant dividend yield and a stochastic spot interest rate with respect to moneyness. Here I use the same parameter values as in Fig. 1, but in addition, for the Black–Merton–Scholes and the Brigo–Mercurio model (hereafter BMS and BM models), a constant dividend yield, return volatility, and a correlation between returns and interest rates of y¯ = y(x0 ), η¯ = η(x0 ), and ρrx = 0, respectively, and an initial spot stock price of S0 = 15, a time to maturity of T = 5, and M = 10,000 simulations. As plot (a) shows, under these parameter values, the BMS price, CBMS , is greater than the option price under my model, C , only if the option is deep-in-the-money. Moreover, the greater the spot interest rate, x0 , the less (more) is the relative price difference for an in-the-money (for a deep-out-of-the-money) option. Interestingly, as plot (b) shows, differences in the option price can be greater with respect to the BM model than with respect to the BMS model; especially, the greater the moneyness, the greater the difference in price (C − CBM )/CBM . Moreover, we can see that if the interest rate is low, then the BM model over-prices options in comparison to my model, and, conversely, the greater the spot interest rate, the greater the price of my model, C , with respect to the BM model, CBM . For example, if the spot interest rate is twice the long-term interest rate, that is, x0 = 2θ , then C > CBM for all 12 < K < 18. One implication of my characterization is that the option price can be a decreasing function of the spot interest rate because an increase in the spot interest rate increases the dividend yield, and thus potentially lowers the call price, which contradicts the current wisdom. These earlier arguments for a positive relation between call price and spot interest rate are, however, based on the assumption of a constant dividend yield or absence of dividends. To see how the level of the spot interest rate affects options in my model, let us, for simplicity, first consider a special case of K ↓ 0, in which, the exercise price is zero and the ‘‘option’’ holder gets the underlying stock for free at time T . Eq. (5) implies that the option price with K ↓ 0 can be expressed as C (t , St , xt , T , 0; ψ) T
[ ∫ = St Et exp t
−y(xu ) −
1 2
∫ η2 (xu ) du +
T
η(xu )dWus
]
,
(7)
t
where η(x) is given by Eq. (6). The dividend yield of the underlying stock, y(xt ), is increasing with respect to the spot interest rate, and, therefore, under these conditions, also decreases the price of the call as a greater dividend yield means a greater shortfall for the option holder. With strictly positive exercise prices, K > 0, the payoff becomes convex, and thus an increase in the spot interest rate can also increase the option price, depending on which
effects, positive or negative, dominate. Note that with K > 0, an increase in the spot rate may reduce option prices not only through dividend yield but also via return volatility (see Fig. 1). Intuitively, the greater the K , the more the spot rate can then potentially increase option prices. This is illustrated in Fig. 3, plot (a). The figure uses the same parameters as Fig. 2, except that the strike price range is 0, 2, 4, . . . , 20. As the plot illustrates, if the strike price is relatively low, the call price is monotonically decreasing in the interest rates. However, for greater strike prices (with a strike price of 14, 16, 18, or 20), the call price can be non-monotone or even monotonically increasing for the given interval. In fact, we assume in plot (a) that the current stock price level is not affected by any increase in the spot interest rate; that is, the current level of dividends must increase in response to a lower price–dividend ratio since S (xt , Dt ) = Dt × q(xt ), where q(x) = 1/y(x). The relation between the spot interest rate and option prices could also be viewed differently. We could think of a situation in which the stock price level reacts to an increase in the short rate, while the level of dividends remains an exogenous variable and unaffected. That is, we can substitute Dt × q(xt ) for St in Eq. (7). The spot interest rate then has three negative effects on option prices: via increased dividend yield, via lower return volatility, and via a lower stock price level. In Fig. 3, plot (b), we keep current dividends fixed, in which case the call price is always decreasing with respect to the spot interest rate. 4. Final remark The above framework merits further research. For example, total hedging of interest rates with equity options could be studied, especially because, depending on the moneyness, option prices can react positively or negatively to an increase in the spot rate via endogenously determined dividend yield and return volatility. Acknowledgments I am grateful to the area editor Jussi Keppo, an anonymous associate editor, and an anonymous referee for their many valuable suggestions. All remaining errors are my sole responsibility. I also gratefully acknowledge the financial support from the Academy of Finland (project 123058), the Emil Aaltonen Foundation, and the OP-Pohjola Group Foundation. References [1] A. Ang, J. Liu, Risk, return, and dividends, Journal of Financial Economics 85 (2007) 1–38. [2] G. Bakshi, C. Cao, Z. Chen, Option pricing and hedging performance under stochastic volatility and stochastic interest rates, in: C.H.C.-F. Lee (Ed.), Handbook of Quantitative Finance and Risk Management, Springer, 2010.
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