Pricing of Timer Options Carole Bernard ∗and Zhenyu Cui
‡†
First draft: November 2009. This draft: August 23, 2010
Abstract In this paper, we discuss a newly introduced exotic derivative called the “Timer Option”. Instead of being exercised at a fixed maturity date as a vanilla option, it has a random date of exercise linked to the realized variance of the underlying stock. Unlike common quadraticvariation-based derivatives, the price of a timer option generally depends on the assumptions on the underlying variance process and its correlation with the stock (unless the risk-free rate is equal to zero). In a general stochastic volatility model, we first show how the pricing of a timer call option can be reduced to a one-dimensional problem. We then propose a fast and accurate almost-exact simulation technique coupled with a powerful (model-free) control variate. Examples are derived in the Hull and White and in the Heston stochastic volatility models.
Keywords: Stochastic volatility, Volatility derivative, timer option, quadratic variation, correlation, Heston model, Hull and White model.
∗ C. Bernard is assistant professor in the department of Statistics and Actuarial Science at the University of Waterloo, 200 University Avenue West, Waterloo, N2L3G1, Canada. Tel: 001 519 888 4567 ext: 35503. Email:
[email protected]. C. Bernard acknowledges support from the Natural Sciences and Engineering Research Council of Canada and from Tata Consulting Services. † Zhenyu Cui is a PhD student at the University of Waterloo, Email:
[email protected]. ‡ Both authors thank Lorenzo Bergomi, Monique Jeanblanc, Adam Kolkiewicz, Don McLeish and David Saunders for helpful discussions. Authors are solely responsible for remaining errors.
1
Electronic copy available at: http://ssrn.com/abstract=1612014
Introduction In April 2007, Soci´et´e G´en´erale Corporate and Investment Banking (SG CIB) started to sell a new type of option that allows buyers to specify the level of volatility used to price the instrument. The standard version of this new product is called a “timer call”. A timer call is similar to a call option with a random maturity date determined by the time needed for the accumulated variance of the underlying stock to reach a prespecified level. Carr and Lee (2009) mention that they were studied by Neuberger (1990) as “mileage” options in the 1990s. Bick (1995) extended the analysis of such options to a continuous-time setting at a time such securities did not exist in the marketplace1 . Such strategies have been discussed however as early as in the 1980s with the portfolio insurance strategies developed by the company LOR (Leland, O’Brien and Rubinstein). The goal of this paper is to propose a fast and accurate technique to evaluate timer options in a general stochastic volatility model. In the case when the risk-free rate is zero, there exist robust replications for timer options as a special case of the general quadratic variation derivatives studied by Carr and Lee (2009, 2010). In this case, timer options can thus be dynamically replicated using a position in the risk-free asset and the risky asset. Recently Saunders (2010) develops an approximation for the price of timer options under fast mean reverting stochastic volatility models, and Li (2010) explains how to price and hedge these new securities in the Heston stochastic volatility model. In this paper, we propose an alternative simple and accurate simulation method to price timer call options in general stochastic volatility models. Practical implications of timer-style options seem very important. Sawyer (2007) explains that “this product is designed to give investors more flexibility and ensure they do not overpay for an option. The price of a vanilla call option is determined by the level of implied volatility quoted in the market, as well as maturity and strike price. But the level of implied volatility is often higher than realized volatility, reflecting the uncertainty of future market direction. [...] In fact, having analyzed all stocks in the Euro Stoxx 50 index since 2000, SG CIB calculates that 80% of three-month calls that have matured in-the-money were overpriced.” Timer options are linked to the realized volatility of some underlying index, stock or exchange rate. Due to their apparent complexity, “timer options” were first sold to sophisticated investors such as hedge funds but are more and more widely sold. In this paper we show that the price of timer options can be reduced to a one-dimension problem in general stochastic volatility models. We then 1
Bick (1995) writes “It should be emphasized that it is not the purpose of this paper to price options that do not exist in practice.”
2
Electronic copy available at: http://ssrn.com/abstract=1612014
develop an efficient and fast almost exact simulation technique with a powerful (model-free) control variate. In the case of the Heston model, it extends the work by Li (2010). We also investigate the Hull and White stochastic volatility model. The first section presents the standard timer call option and how to price it in a general stochastic volatility model. The second section first illustrates how a direct discretization approach fails to derive fast and accurate prices for timer options. We then develop theoretical tools needed to reduce the dimensionality of the problem and present an efficient and accurate Monte Carlo method. The last section is a numerical study. We first provide an exhaustive comparison of the efficiency of the proposed numerical schemes using a randomized set of parameters (similarly as Broadie and Detemple (1996)). We then discuss the sensitivity of timer options’ prices to the discretization step used in the simulation, to the correlation coefficient between the asset and its volatility, the risk-free interest rate and the variance budget.
1
Timer Call
In this section, we first describe a standard timer call and we derive its price using a probabilistic approach in a very general stochastic volatility model. We also include the PDE satisfied by the timer call and compare to the one satisfied by a standard call.
1.1
Standard Timer Call
A standard timer call option can simply be viewed as a call option with random maturity which depends on the time needed for a pre-specified variance budget to be fully consumed. The buyer of a timer call option specifies an investment horizon and a target volatility. A variance budget is then calculated as the target volatility squared, multiplied by the target maturity. Once the variance budget is consumed, the option expires. Let us now summarize practical details about timer options obtained from a presentation of the Soci´et´e G´en´erale (2007). The investor chooses a target volatility Σ (also called implied volatility target) and a maturity T to establish a fixed variance budget V B T arget = Σ2
NT 252
(1)
where NT is the number of trading days before the maturity date T . The
3
realized volatility for the observation period D is calculated as v u n u 1 X realized t u2 (2) ΣD = n − 1 i=1 i Si where ui = ln Si−1 , and where Si is the observed underlying stock at time ti , and 0 < t1 < ... < tn = D (the discretization step is daily in practice). √ To scale for a one-year period, we divide by D, and the annualized realized volatility becomes Σrealized σrealized = D√ . D As the stock moves daily, the variance budget is “expended” according to the realized variance consumption formula d 2 V B realized = σrealized (3) 252 where d is the number of days since the inception date. When the realized variance consumption V B realized is larger than the variance budget target V B T arget , the option is automatically exercised. If the realized volatility exactly matches the investor’s target volatility, the expiry date of the option will equal the target investment horizon. If the realized volatility is higher or lower, the option will expire respectively at an earlier or later date. A timer call option was first traded at the end of April 2007 with a hedge fund. “ At the time, the implied volatility on the plain vanilla call was slightly above 15%, but the client set a target volatility level of 12%, a little higher than the prevailing realized volatility level of around 10%. By rolling into a timer call, the hedge fund reduced its premium by 20%.” (see Sawyer (2007)). Since then, SG CIB has started to sell two new timer-style options called respectively the “timer out-performance option” and the “time swap” (see Sawyer (2008) for a detailed description of these options). Timer options are very innovative products. Traditionally investment strategies have fixed maturity. But investing in timer options is a strategy with random maturity. The investment strategy is fully driven by the realized volatility. Some thoughts about quadratic-variation-based strategies appeared already more than 10 years ago among academics (for instance Bick (1995), Geman and Yor (1993), Rendleman and O’Brien (1990)). Quadraticvariation-based strategies can be cheaper hedging strategies, and are very promising.
1.2
Model
We now present the setting in which we derive the prices of timer options. We first describe the financial market and then introduce the timer call option 4
to be priced. Financial Market Consider a stock S that evolves according to a stochastic volatility model, and a constant risk-free interest rate r. Under the risk neutral measure Q that we will use for pricing purpose, one has p ( √ 1 − ρ2 dWt1 + ρdWt2 dSt = rSt dt + Vt St (4) dVt = αt dt + βt dWt2 where W 1 and W 2 are independent Brownian motions and ρ denotes the correlation between the changes in the stock returns and in the stock’s volatility process, and where αt and βt denote any measurable functions at time t with respect to theRnatural filtration generated by the market such that Vt > 0 T a.s. and ξT = 0 Vt dt exists and converges to +∞ when T → +∞. Later we will illustrate the results in two well-known stochastic volatility models that satisfy the above conditions. In the Heston model (Heston (1993)), the variance process evolves as p dVt = κ (θ − Vt ) dt + γ Vt dWt2 (5) where κ, θ and γ are the parameters of the volatility process, they are all positive and the Feller condition 2κθ − γ 2 > 0 ensures that Vt > 0 a.s.. In the Hull and White model (Hull and White (1987)), dVt = aVt dt + νVt dWt2
(6)
where a and ν are two positive parameters. Timer option Let us consider a timer call option written on the underlying asset S with strike K. Its maturity date is linked to the accumulated variance of S. Let us denote by V the “variance budget” that is chosen by the investor (corresponding to V B T arget given by (1)). The stock price evolves in a continuous time framework (see (4)), we define the realized variance in continuous time at time u as Z u Vs ds, 0
corresponding to the realized variance consumption (given by V B realized in (3)). Denote by τ the random maturity time of the option. It is defined as the first hitting time of the realized variance to the variance budget V Z u τ = inf u > 0, Vs ds = V . (7) 0
5
The payoff of a timer call option is paid at time τ and is max(Sτ − K, 0). In the above framework, the price of the timer call option is then given by (8) C0 = E Q e−rτ max (Sτ − K, 0)
where Q denotes the risk-neutral probability under which S follows (4).
1.3
Pricing Timer Options
The expression of the price (8) can be simplified using “time change techniques” (see for instance Geman (2008)). Theorem 1.1. The initial price of a timer option in the above setting is equal to h i 1 C0 = E Q max S0 eBV − 2 V − Ke−rτ , 0 . (9) p 1 2 2 where BV = Bξ(τ ) with Bu = 0 Vt 1 − ρ dWt + ρdWt is a Q-standard Brownian motion and where τ verifies (7). Ru√
Proof. To establish (9), we introduce the following martingale process N NT =
Z
0
T
p p Vt 1 − ρ2 dWt1 + ρdWt2 .
Note that NRT can be decomposed R T √ into a2linear combination of two martingale T √ 1 processes ( 0 Vt dWt and 0 Vt dWt ) with the same quadratic-variation ξT at time T Z T Vt dt. (10) ξT = 0
Note that ξt is often referred to as a stochastic clock (see Geman (2008)). We now apply Theorem 4.6 on page 174 of Karatzas and Shreve (1991) also called Dubins-Schwarz Theorem or time-change technique for martingales. Since limT →∞ ξT = ∞, there exist two standard Q-Brownian motions B 1 and B 2 such that p NT = BξT = 1 − ρ2 Bξ1T + ρBξ2T . (11)
RT √ where BξiT = 0 Vt dWti for i = 1, 2. Since W 1 and W 2 are independent, B 1 and B 2 are standard independent Brownian motions. Therefore, the process B is also a standard Brownian motion. The underlying asset St at time t can
6
1
be written as St = S0 ert eBξt − 2 ξt . By definition of τ (see (7)), we have ξτ = V. Then the stock price at the exercise time of the timer call option is 1
1
Sτ = S0 erτ eBξτ − 2 ξτ = S0 erτ eBV − 2 V . Theorem 1.1 follows from (8).
(12)
A few important implications of Theorem 1.1 follow. Remark 1.1. Note that BV and τ can be correlated. Indeed, when ρ 6= 0, it can easily be seen from formula (11) in the proof replacing T by τ that BV and B 2 are dependent. Since τ is determined by the trajectory of the variance process, and therefore of B 2 , then BV and τ may be dependent. We denote by CBS (x, K, r, σ, T ) the price of a call option in the Black and Scholes model when the initial underlying price is x, the strike is K, the interest rate is r, the maturity is T and the volatility is σ. It is given by CBS (x, K, r, σ, T ) = x N (d1 ) − Ke−rT N (d2 )
where d1 =
2 x ln( K )+ 21 r+ σ2 T √ σ T
(13)
√ and d2 = d1 − σ T .
Remark 1.2. When r = 0%, the formula (9) can be simplified and the price of a timer call option is given by i h 1 C0|r=0% = E Q max S0 eBV − 2 V − K, 0 .
The price of a timer call option has a closed-form expression equal to the Black and Scholes formula with interest rate r = 0%, and such q that the 2 volatility σ and the maturity T verify σ T = V, i.e. CBS S0 , K, 0, TV , T . Thus when r = 0%, the price of a timer call is given by b C0|r=0% = S0 N d1 − KN db2 (14) S √ ln( K0 )+ 21 V b2 = db1 − V. √ where db1 = and d V
This last remark is quite intuitive. The difference between the timer option and a standard option comes from the maturity date of the contract which is random. When the interest rate is equal to 0%, then the exact date when the cash-flow occurs does not matter. It is therefore intuitive that the price of the timer option does not depend on τ anymore. Note that the result holds in very general stochastic volatility models even if the variance V is not Markov in itself and if the correlation between the 7
risky asset S and the variance V is not constant (it even holds for stochastic correlation). In this case, there exists a robust or model-free replication by trading the risk-free asset and the risky asset (Carr and Lee (2010)). When selling timer options, the Soci´et´e Gen´erale was explaining that it was a way to buy options at a cheaper price when the realized volatility is smaller than the implied volatility in the market. When r = 0%, this 2 appears very clearly: when V 6 σimplied T , then the price of the timer option C0|r=0% is obviously smaller than the corresponding standard European call option with maturity q T and with the same strike K. Indeed C0|r=0% = CBS S0 , K, 0, T, TV 6 CBS (S0 , K, 0, T, σimplied ). Remark 1.3. When the volatility is deterministic or constant, τ is deterministic and the formula of Black and Scholes holds with T = τ , with initial underlying stock price S0 , interest rate r, the underlying’s volatility σ such that σ 2 T = V.
Remark 1.4. Prices for timer put options can be obtained using the putcall parity for timer options C0 − P0 = S0 − KE Q (e−rτ ) where C0 and P0 respectively denote the price of a timer call option and of a timer put option. Pricing a Timer Call Option at time t such that 0 < t < τ In the above paragraph, we studied how to price timer call options at the inception date. It is easy to extend the pricing formula to a later date t. The important variable is the “consumed variance” at the valuation date t which can be calculated as Z t Vs ds. CVt = 0
There are two cases, if CVt exceeds the variance budget target V, then the option has already expired and therefore has no value at time t. Otherwise, one has t < τ , and the price of the timer call option at time t can be calculated as C0 = E Q e−r(eτ −t) max (Sτe − K, 0) where τ˜ is now defined as follows Z τe = inf u > t,
t
u
Vs ds = V − CVt .
(15)
It is then clear that all computations are similar at time t with an “updated” e = V − CVt and using the values of Vt and St (remaining) variance budget V (instead of V0 and S0 ).
8
Greeks Expressions for the Greeks of timer options can be obtained similarly by conditioning techniques as their prices in Theorem 1.1. Greeks can then be expressed as expectations of the Greeks calculated in the Black and Scholes model for which there are well-known formulas2 .
1.4
PDE for timer options
In this section, we follow Gatheral (2006) (Chapter 1). Suppose that the stock price S and its variance V satisfy the following SDEs under the physical measure P p ( √ dSt = µSt dt + Vt St 1 − ρ2 dZt1 + ρdZt2 (16) dVt = α(St , Vt )dt + β(St , Vt )dZt2 where Z 1 and Z 2 are two independent standard Brownian motions under the real probability measure P , µ is the (constant) instantaneous drift of stock price returns and ρ is the correlation between random stock price returns and changes in Vt . For clarity we denote by α := α(St , Vt ) and β := β(St , Vt ). In this setting the price of a standard European volatility derivative with fixed maturity is a function of (St , Vt , t). A timer option has a random maturity linked Rto the realized variance. Therefore we also introduce as state Rt t variable ξt = 0 Vs ds. It is known that (Vt , 0 Vs ds) is a Markov process, then (St , Vt , ξt ) is also a Markov process. The price of a timer option is a priori a function f (t, St , Vt , ξt ) for any time t prior to its maturity τ . Due to its perpetuity, we omit the state variable t. Theorem 1.2. The price of a timer call option can be written as C(t ∧ τ ) = f (St∧τ , Vt∧τ , ξt∧τ ) where f (S, V, ξ) verifies the following PDE √ 1 1 V fξ + αfV + rSfS + β 2 fV,V + S 2 V fS,S + ρβS V fS,V − rf = 0 2 2
(17)
∂f where fS := ∂S (S, V, ξ), and fV , fξ , fS,S , fS,V , fV,V are defined similarly with the following boundary conditions
f (S, V, V) = (S − K)+ Proof. The proof can be found in Appendix A. 2
Greeks of timer options can be obtained from authors upon request.
9
Remark 1.5. It is also insightful to compare the PDE (17) for the timer call option with the PDE of a standard European call option in a general stochastic volatility model. The latter can be found for instance in Gatheral (2006). The price of a standard European call option with maturity T , strike K written on S can be written as C(t) = g(St , Vt , t) where g verifies the following PDE √ 1 1 gt + αgV + rSgS + β 2 gV,V + S 2 V gS,S + ρβS V gS,V − rg = 0 2 2
(18)
with the boundary condition g(S, V, T ) = (S − K)+ . The main difference between the timer call and the standard call option is that the variable ξt plays the role of time. It is the “stochastic clock” mentioned earlier.
2
Theoretical Results
We assume that the variance process is now modeled by dVt = α(Vt ) dt + β(Vt ) dWt2
(19)
where α(.) and β(.) are twice differentiable over R∗+ . This includes the Heston and the Hull and White models. The risky asset is modeled under the risk neutral measure Q as p p dSt = rSt dt + St Vt (20) 1 − ρ2 dWt1 + ρdWt2 The following lemma is useful to price standard call options as well as timer call options in general stochastic volatility models. It is also crucial for the conditional simulation of the underlying process St .
Lemma 2.1. In the general stochastic volatility model given by (19) and (20), the value at time T of the risky asset has the following representation p ST = S0 exp{rT + aT + bT Z}, (21) where aT and bT are defined by aT = ρ(f (VT ) − f (V0 )) − ρHT − 12 ξT bT = (1 − ρ2 )ξT with HT =
Z
T
h(Vt )dt
and
0
ξT =
Z
0
10
T
Vt dt
and where Z ∼ N (0, 1) independent of VT , HT and ξT and where f and h are defined by R v √z f (v) = 0 β(z) dz (22) 1 2 ′ ′′ h(v) = α(v)f (v) + 2 β (v)f (v) Proof. The proof is given in Appendix B.
2.1
Motivation
By definition, τ depends only on the variance process, but the payoff at time τ depends on the stock St . The problem of evaluating a timer option thus involves the two dimensional process (St , Vt ). Our first attempt to price timer options was to simulate jointly the processes St and Vt using crude discretization in order to estimate (τ, Sτ ) needed to evaluate formula (8) by Monte Carlo. Assume that the dynamics of the model are given by (4) and (5) under √ the risk neutral measure used for pricing. Then α(s) = κ(θ − s), β(s) = γ s , one and f and h can be calculated from (22), f (s) = γs and h(s) = κ(θ−s) γ has for t > 0, p ξt ρ p 2 St = S0 exp rt − + (Vt − V0 − κθt + κξt ) + 1 − ρ ξt Z . (23) 2 γ
The exact transition distribution of a CIR process is well-known (see for example Cox et al. (1985) or Li (2010) Remark 7 page 67). Therefore in the Heston model the distribution of Vt+∆t |Vt is a non-central chi-square: γ 1 − e−κ∆t Vt+∆t |Vt = X (24) 4κ −κ∆t law 4κe . We proceed by a direct V with d = 4θκ where X ∼ χ′2 d γ2 γ 2 (1−e−κ∆t ) t simulation of the joint paths of the correlated processes (St , Vt ) to obtain 100,000 simulations of (τ, Sτ ) with a discretization step ∆t = V/M where M varies. For each time step along the time grids ti = i∆t , i = 1, 2, 3..., we simulate the discretized variance process Vt+∆t |Vt and approximate the R j∆t Vs ds, by the trapezoidal rule as Vj = realized variance at time t by j 0 Pj−1 V0 +Vj∆t ∆t + k=1 Vk∆t . St+∆t is simulated using a similar expression as 2 (23). We stop at the index j ∗ when the accumulated discretized variance Vj ∗ reaches the variance budget V. Then we record (j ∗ ∆t , Sj ∗ ∆t ) as an approximation to (τ, Sτ ). This method is biased because of the time discretization
11
and is very slow because of the simulation of the non-central chi square distribution for each time step and for each path. The price of a timer call option is then estimated by crude Monte Carlo using (25) E Q e−rτ max (Sτ − K, 0) .
The price obtained by this first method converges to the correct price of the timer call option when the number of simulations goes to +∞ and when the discretization step goes to 0. Assume that the parameters are given in Table 1. Table 1: Parameters S0 = 100 K = 100 r = 0.04 V = 0.0265 κ=2 V0 = 0.0625 γ = 0.1 θ = 0.0324
Results are displayed in Figure 1. We fix the number of steps M and simulate 100,000 simulations of (τ, Sτ ). With the same random numbers we calculate (25) for three possible correlation levels ρ = 0, ρ = 0.8 and ρ = −0.8. Then we change M and use new random numbers to perform a Monte Carlo simulation of (25). 9.5 ρ=−0.8 ρ=0 ρ=0.8 Price when ρ=0
9 ρ = − 0.8
Price
8.5 8 7.5 7 6.5 6 0
ρ=0 ρ = 0.8
50 100 150 Number of discretization step
200
Figure 1: Timer Call Price w.r.t. M , the number of discretization steps
12
At the right end of the graph on Figure 1 (when M = 200) the price of a call option is 7.70 (0.035) when ρ = −0.8, 7.58(0.035) when ρ = 0, 7.33(0.035) when ρ = 0.8. The number in parenthesis are the standard deviations obtained with 100,000 simulations. As can be seen from Figure 1, the discretization step has an important effect on the dependence between the price of the timer call and the correlation. The more the process is discretized, the lower is the timer call price when ρ = −0.8 but the higher it is when ρ = 0.8. As one can imagine such approach needs a very tiny discretization step and a large number of simulations. It is very slow (up to 3 hours of computation with M = 200 and N = 100000). This is simply unacceptable. The above example shows that the direct simulation of trajectories of the correlated two-dimensional process (τ, Sτ ) by discretizing the time is slow and not accurate. We now show that the pricing of a timer option can be reduced to the study of the variance process and is therefore a one-dimension problem. We then develop an efficient simulation technique that is less sensitive to the discretization step and converges quickly. Figure 1 will then be compared with Figure 2 in Section 3.
2.2
Pricing Call Options and Timer Call Options under General Stochastic Volatility model
At first, the pricing of an option in a stochastic volatility model seems to be a two-dimension problem since there are two independent Brownian motions that contribute to the payoff of the option (W1 and W2 ). However we now show how to reduce the problem to a one-dimension problem where only the dynamics of Vt needs to be simulated. This is true for standard call options as well as for timer options. Theorem 2.1. Standard European Call Option. The price of a standard call option with maturity T in the general stochastic volatility model given by (19) and (20) can be written as h i b E CBS (S0 , K, r, σ b, T ) where√CBS is the Black Scholes formula (13) with Sb0 = S0 exp aT + b2T and σ b = bT where aT and bT are defined by (22) in Lemma 2.1. Note that aT and bT depend on (VT , ξT , HT ).
Proof. Theorem 2.1 follows directly from Lemma 2.1. Under the risk neutral measure Q, the price of a standard European call option with maturity T is given by i h √ −rT −rT rT +aT + bT Z − K, 0 (26) E e max (ST − K, 0) = E e max S0 e 13
By conditioning on (VT , ξT , HT ), aT and bT become constant. + √ rT +aT + bT Z −rT S0 e E e − K VT , ξT , HT = CBS Sˆ0 , K, r, σ ˆ, T (27) 1
ˆ = where Sˆ0 = S0 eaT + 2 bT and σ iterated expectations.
√
bT . The result follows from the law of
Thanks to Theorem 2.1, it is now clear that to simulate the price of a standard call option, one only needs to jointly simulate (VT , HT , ξT ). The Monte Carlo valuation of the price of a standard call option in a general stochastic volatility model is a one-dimension problem: it only requires the simulation of the variance process VT . Note that in the case of the Heston model, we will show that HT is a linear function of ξT and T , namely γHT = κθT − κξT (HT is defined in Lemma 2.1). Therefore one only needs the joint RT distribution of (VT , 0 Vs ds) to price the call option in the Heston model. Theorem 2.2. Timer Call Option. In a general stochastic volatility model given by (19) and (20), the price of a timer call option can be calculated as (1−ρ2 )V aτ + −rτ 2 C0 = E S0 e N (d1 ) − Ke N (d2 ) (28)
where aτ = ρ(f (Vτ ) − f (V0 )) − ρHτ − 12 V and S
d1 =
log( K0 )+rτ +aτ +(1−ρ2 )V
√
(1−ρ2 )V
,
d2 = d1 −
p
(1 − ρ2 )V.
Proof. The proof is very similar to the standard European case. In this case ξτ = V, and τ is random. We now condition on (τ, Hτ , Vτ ) and the price of the timer option is obtained by C0 = E E e−rτ (Sτ − K)+ |τ, Hτ , Vτ .
The conditional expectation is the Black Scholes formula for the call option, then " !# r V C0 = E CBS Se0 , K, r, ,τ τ where CBS is given by (13) with Se0 = S0 exp aτ + b2τ , where aτ and bτ are defined by (22) in Lemma 2.1 and bτ = (1 − ρ2 )V. Thus Theorem 2.2 is proved.
14
Thanks to Theorem 2.2, the simulation of the price of a timer call only requires the joint distribution of (τ, Hτ , Vτ ). Therefore the simulation of the price of the timer option is fully determined by the simulation of the variance process Vt (since (τ, Vτ , Hτ ) all only depend on the variance process). As we explain next, the result can be further simplified in the absence of correlation.
2.3
Timer option when St and Vt are not correlated
In the case when ρ = 0, the complex dependence that appeared in formula (9) between τ and BV disappears. When ρ = 0, there is no correlation between the variance process and the stock returns, and τ and BV are independent. This case is standard in credit risk modelling, see for instance Section 5 of Packham, Schl¨ogl and Schmidt (2009). The expression (9) of the price of a timer call can then be further simplified by conditioning with respect to the random variable τ , h h ii 1 (29) C0 = E Q E Q max S0 eBV − 2 V − Ke−rτ , 0 τ .
After simplifying the above expression (29), one obtains the next theorem.
Theorem 2.3. When ρ = 0, the price of a timer call option in a general stochastic volatility model is given by C0|ρ=0 = S0 E Q [N (d1 (τ ))] − KE Q e−rτ N (d2 (τ )) (30)
where
d1 (τ ) =
ln
S0 K
√ + 12 V + rτ √ , d2 (τ ) = d1 (τ ) − V. V
Note that Formula (30) may hold when ρ 6= 0 if BV conditioned with respect to τ is normally distributed N (0, V). The distribution of τ in a general stochastic volatility model can be derived from the distribution of (τ, Vτ , Hτ ).
2.4
Theoretical joint distribution of (τ, Vτ , Hτ )
The next lemma gives the joint law of (τ, Vτ , Hτ ) where τ is the first passage time of the realized variance to V, Vτ is the value of the variance process at time τ and Hτ is defined in Lemma 2.1. Lemma 2.2 is new and is a key result towards almost exact simulation of timer options using Theorem 2.2. Lemma 2.2. Assuming the general dynamic for the diffusion process Vt is given by (19). Let Z t τ := τ (V) = inf t; Vs ds = V ∈ (0, ∞) (31) 0
15
be the first passage time of the integrated functional of Vs to the fixed level V ∈ (0, ∞), then the law of (τ, Vτ , Hτ ) is given by Z V Z V law 1 h(Xs ) τ (V), Vτ (V) , Hτ (V) ∼ ds, XV , ds (32) Xs 0 Xs 0
where Xt is governed by the SDE t) df (Xt ) = h(X dt + dBt , Xt X0 = V0
(33)
where B is a standard Brownian motion, and f and h are given by (22). Proof. The proof is in Appendix C.
Recall that for a timer call option, τ is defined by (31) where V is the variance budget. From Lemma 2.2, the joint distribution of (τ, Vτ , Hτ ) can now be calculated in the Heston model and in the Hull and White model. 2.4.1
Heston Model
√ In the Heston model, α(s) = κ(θ − s) and β(s) = γ s. Then f and h can be . Lemma 2.2 implies then calculated from (22), f (s) = γs and h(s) = κ(θ−s) γ the following proposition. Proposition 2.1. Joint Distribution of (τ, Vτ ) in the Heston Model Z V ds law (τ, Vτ ) ∼ , XV , (34) 0 Xs where Xt is a Bessel process κθ − κ dt + γdBt , X0 = V0 , dXt = Xt
(35)
and where B a standard Brownian motion. This proposition is not new and can be found for example in Li (2010). More details about the properties of Bessel processes can also be found in Linetsky (2004). Theorem 2.4. In the Heston model, the price of a timer call option can be simplified as h i 1 2 C0 = S0 E Q eρc(τ,Vτ )− 2 ρ V N (d1 (τ, Vτ )) − KE Q e−rτ N (d2 (τ, Vτ )) , (36) 16
where S p ln( K0 )+rτ +ρc(τ,Vτ )+( 12 −ρ2 )V √ , d2 (τ, Vτ ) = d1 (τ, Vτ ) − (1 − ρ2 )V, d (τ, V ) = 1 τ 2 (1−ρ )V c(τ, V ) = τ
Vτ −V0 −κθτ +κV . γ
Proof. Let us apply Theorem √ 2.2 and calculate aτ from Lemma 2.1 when α(s) = κ(θ − s), β(s) = γ s, f (s) = γs and h(s) = κ(θ−s) . There are a γ κθτ −κξτ κθτ −κV = . One finally few simplifications such as ξτ = V and Hτ = γ γ obtains the price of a timer call option as in (36). 2.4.2
Hull and White Model
In the Hull and White model, α(s) = as and β(s) = νs. Then f and h can √ √ 2 s s. The following be calculated from (22), f (s) = ν and h(s) = νa − ν4 proposition follows from Lemma 2.2 in the Hull and White model. Proposition 2.2. Joint Distribution of (τ, Vτ , Hτ ) in the Hull and White Model Z V Z V 2a 1 4 ds ν 2 2 ds law , (37) , X , − (τ, Vτ , Hτ ) ∼ ν 2 0 Xs2 4 V ν2 2 0 Xs where Xt is a standard Bessel process governed by 2a 1 1 2p dXt = V0 . (38) − dt + dB , X = t 0 ν 2 2 Xt ν where B a standard Brownian motion.
Theorem 2.5. In the Hull and White model, the price of a timer call option can be simplified as (1−ρ2 )V aτ + −rτ 2 C0 = E S0 e N (d1 ) − Ke N (d2 ) (39) where aτ =
2ρ √ ( Vτ ν
−
√
V0 ) − ρHτ − 21 V and where
S
d1 =
log( K0 )+rτ +aτ +(1−ρ2 )V
√
(1−ρ2 )V
,
d2 = d1 −
p
(1 − ρ2 )V.
Proof. Apply Theorem√2.2 and calculate aτ from Lemma 2.1, when α(s) = √ as, β(s) = νs, f (s) = 2 ν s and h(s) = νa − ν4 s. There are a few simplifications such as ξτ = V and one finally obtains the price of a timer call option as in (39). 17
2.5
Monte Carlo Method
We now describe how to use the theoretical tools developed in the previous section to propose an efficient simulation technique. In the general stochastic volatility model (19) and (20), we have an expression of the price of the timer call option (Theorem 2.3 for the case ρ = 0 and Theorem 2.2 for the general case when ρ 6= 0) that involves an expectation of a function of (τ, Vτ , Hτ ). 2.5.1
General case
We will make use of Lemma 2.2 to simulate an i.i.d. sample of (τ, Vτ , Hτ ). The estimate of the timer call option price is given by C0mc
=
S0 e
(1−ρ2 )V 2
n
n X
aτi
e
i=1
n K X −rτi N (d1 (τi , Vτi , Hτi )) − e N (d2 (τi , Vτi , Hτi )) n i=1
(40)
where aτi := a(Vτi , Hτi ) = ρ(f (Vτi ) − f (V0 )) − ρHτi − 12 V and S
d1 = 2.5.2
log( K0 )+rτ +aτi +(1−ρ2 )V
√
(1−ρ2 )V
d2 = d1 −
,
p (1 − ρ2 )V.
(41)
Variance reduction
It is possible to significantly improve the convergence of the Monte Carlo estimator by using the fact that when r = 0% there is a closed-form expression C0|r=0% for the timer option given by (14) in a general stochastic volatility model. We use this property to develop a powerful control variate and estimate the price by mc e C0 − λ C0 − C0|r=0% (42)
where
e0 = C
S0 e
(1−ρ2 )V 2
n
n X
aτi
e
i=1
n KX N (d1 (τi , Vτi , Hτi )) − N (d2 (τi , Vτi , Hτi )) , (43) n i=1
where λ = corr(C0mc , C˜0 ) and where aτi , d1 and d2 are defined in (41). 2.5.3
Case when ρ = 0
In the case when the correlation is equal to 0, only the distribution of τ is needed to calculate the price. The expression (40) can be simplified as mc C0|ρ=0 =
n n S0 X K X −rτi N (d1 (τi )) − e N (d2 (τi )) . n i=1 n i=1
18
(44)
Depending on the assumptions about the stochastic volatility model, an i.i.d sample of τ can be obtained by an Euler discretization of the dynamics of Xt given in Lemma 2.2. There is an alternative way when one knows the distribution of the first passage time to a given level. In the Hull and White model, the variance process evolves as in (6) and there exists a closed-form expression of the Laplace transform of the density of the stopping time τ (see Section 4 of Geman and Yor (1993)). Let g(.) denote the density of τ . Its Laplace transform is given by k Z ∞ Z 1 1 2uV0 −2uV0 −λx g(x)e dx = exp (1 − u)µ2 −k du. (45) 2 2 Γ(k) Vν Vν 0 0 where
21 µ2 − b 2a 8λ 2 ; k= b = 2 − 1; µ2 = +b . 2 ν ν 2 however this Laplace transform involves a slow decaying oscillatory function and suffers from similar issues as the Laplace transform of Asian option prices (See for example the discussion on page 51 of Fu et al. (1999)). Numerical inversion is unstable and we don’t use this approach.
3
Numerical Analysis
We now illustrate the study with the Heston model and the Hull and White model. The pricing of timer options involves the discretization of equations (35) and (38). We first discuss possible Euler schemes and compare their respective efficiency and accuracy over a randomized set of input parameters. This approach corresponds to that used by Broadie and Detemple (1996).
3.1
Comparison of Efficiency and Accuracy
There are several techniques to perform the discretization of the process Xt given in (35) or in (38). The discretization of these equations raises some issues. In both cases, the drift terms include X1t and the numerical behavior will be unstable when Xt is close to 0. Another issue is that although the continuously monitored process Xt is always positive, the discretized process by standard Euler schemes may take negative values. These two problems are actually well-known numerical problems of the Euler discretization and usually appear in the simulation of the square root diffusion process. Applying Ito’s lemma to Yt := Xt2 , we show that the dynamics of Yt is a square root diffusion process. In Heston model, from equation (35), one obtains p p 2 (46) dYt = 2κθ + γ − 2κ Yt dt + 2γ Yt dBt , Y0 = V02 . 19
In the Hull and White model, from equation (38), one gets dYt =
p 4a Yt dBt , dt + 2 ν2
Y0 =
4 V0 . ν2
(47)
Several numerical approaches have been suggested in the literature (see for example the recent survey of Lord, Koekkoek and Van Dijk (2010)). We first directly discretize (35) with scheme 1 and 2, secondly we discretize the transformed SDE (see (46)) with schemes 3 to 7: • Scheme 1: simple forward Euler scheme Xt+∆t = Xt + κθ
1 ∆t − κ∆t + γ∆Bt Xt
X0 = V0
(48)
• Scheme 2: forward Euler scheme with reflection Xt+∆t =| Xt | +κθ X0 = V0
1 ∆t − κ∆t + γ∆Bt | Xt |
(49)
• Schemes 3 to 7 are derived from equation (46). Following Lord et al. (2010), Euler schemes can be unified in a single general framework. Scheme 3 to 7 are of the form q q 2 e e e Yt+∆t = f1 (Yt ) − 2κ∆t f2 (Yt ) + (2κθ + γ )∆t + 2γ f3 (Yet )∆Bt Yt+∆t = |Yet+∆t | (50) Ye0 = Y0 = V02
where the functions f1 , f2 and f3 are given as follows Scheme Scheme Scheme Scheme Scheme
3: 4: 5: 6: 7:
f1 (y) f2 (y) Berkaoui et al. (2008) |y| |y| Higham and Mao (2005) y |y| Deelstra and Delbaen (1998) y |y| Lord et al. (2010) y (y)+ (y)+ (y)+ Absorption scheme
f3 (y) |y| |y| (y)+ (y)+ (y)+
Note that the references mentioned above for each scheme use a standard square-root diffusion process and that in our case the dynamics of the process under study is slightly different. In particular scheme 4 and 5 originally have f2 (y) = y instead of f2 (y) = |y|. However this 20
modification is needed because of the presence of the square root in the drift term. Note that unlike Lord et al. (2010), the simulation of Xt or of Yt is only an intermediate step. Our ultimate goal is to simulate (τ, Vτ ) as ! ! n n X X ∆t ∆t p √ (51) (τ, Vτ ) ≈ , Yn∆t , , Xn∆t = Xi∆t Yi∆t i=1 i=1 with ∆t = Vn . To avoid the problem when Xt or Yt takes the value 0, we avoid schemes that have a positive probability to give the value 0 to the discretized process. We then prefer to use the reflection principle (using the absolute value) instead of the absorption principle (that consists of taking the maximum between the process and 0). In particular the original schemes 5, 6 and 7 have been modified as Yt+∆t = |Yet+∆t | (on the second line of (50)) instead of Yt+∆t = (Yet+∆t )+ to avoid that the process Y takes the value 0 with a positive probability.
We assume r = 0% because the price of a timer option in this case is model-free and given by an exact Black and Scholes type formula (see remark 1.2). In this case, the price of a timer call option only depends on S0 , K and V, but not on the variance process dynamics assumed. For example when r = 0% and when S0 , K and V are such as in Table 1, the timer call price is 6.4871. We use this result to measure the bias of each Euler scheme and to understand the factors explaining this bias. To make sure the results are robust to changes in parameters, we follow the approach by Broadie and Detemple (1996) and evaluate the accuracy of the different Euler schemes over a range of randomized parameter inputs. For a given set of parameters, time step discretization and number of simulations, we simulate the price of the timer option and compute the standard deviation of the estimate as well as the root mean squared error. We describe shortly how a set of 200 timer options is generated. The root mean squared error (RMS) is defined as v u m u1 X bi − Ci )2 RM S = t (C (52) m i=1
where Ci is the “true” option value (here equal to 6.4871) and Cˆi is the estimated option value for the set of parameters of the ith option. Recall that Broadie and Detemple (1996) study the trade-off between the computation speed and the accuracy measured by the RMS relative error. In our case the computation time is quite similar for all schemes (it mainly depends 21
on the time step and the number of simulations), so we do not discuss it. Since all true prices are equal to 6.4871, we use the RMS error and not the RMS relative error as Broadie and Detemple (1996) do. Since the random numbers used to compare different schemes are the same, all schemes have similar standard deviations. Comparisons of the RMS error then provide a direct comparison of the bias. We consider a reasonably broad range of possible parameters for the variance process. For the sake of illustration, we conduct the study in Heston model, and assume that the parameters are uniformly distributed over the following ranges γ ∈ [0.1, 1.2], θ ∈ [0.01, 0.3], κ ∈ [0.5, 9], V0 ∈ [0.8θ, 1.2θ] and ρ ∈ [−0.9, 0.3]. We construct a set of 200 options based on these parameter distributions. Since the price of a timer option when the interest rate r = 0% is model-free, the true price is 6.4871 for each of these 200 sets of parameters. Table 2 summarizes our results for the 7 schemes. Implementing the schemes 1 to 7 with the set of 200 options, we observed rare cases when the bias blows up. In these cases, the absolute bias can exceed 2 (while the exact price is 6.4871!). This can easily be understood from the fact that Xt could take arbitrarily small values close to 0. This numerical issue is addressed by averaging only the simulations that lead to an absolute bias smaller than 2. This threshold is arbitrary and any number bigger than 2 would give similar results because when the algorithm diverges, it leads to a very large bias. This happens only in rare cases: fewer than 3 sets of parameters out of 200 for all schemes except for scheme 2 (that does not converge in about 10% of the parameters’ sets). Number MC Time Step 1/p Scheme Scheme Scheme Scheme Scheme Scheme Scheme
1 2 3 4 5 6 7
n=100,000 p=150 RMS Std Dev 0.11 0.0221 0.235 0.0234 0.185 0.0144 0.225 0.0162 0.213 0.0225 0.193 0.0152 0.189 0.0142
n=100,000 p=300 RMS Std Dev 0.0453 0.0136 0.232 0.0277 0.173 0.0336 0.156 0.0126 0.134 0.0127 0.168 0.0127 0.145 0.0122
n=100,000 p=600 RMS Std Dev 0.0472 0.0315 0.196 0.0384 0.163 0.0132 0.144 0.031 0.124 0.0153 0.137 0.0134 0.143 0.0132
Table 2: The standard deviations are obtained by averaging the standard deviations of the estimates obtained in each of the m = 200 sets of parameters. The RMS is given by (52).
22
From Table 2, it already appears that Scheme 1 outperforms the other six schemes although it allows for negative values for the underlying process X. Scheme 2 is the worse scheme. We then studied the parameters from the sets when the bias becomes very large. In general it corresponds to sets of parameters for which the Feller condition (2κθ − γ 2 > 0) is not satisfied or for which 2κθ − γ 2 is positive but close to 0. Unfortunately in practice fitted parameters for the Heston model often do not satisfy the Feller condition. We now present the performance of the 7 schemes with 6 realistic sets of parameters from the literature on the Heston model. Cases (1) and (2) can be found in Table 1 of Andersen (2008), Case (3) was fitted by Smith (2008), Case (4) is given in Table 1 of Glasserman and Kim (2010), Case (5) is from Lord et al. (2010) and Case (6) is from Broadie and Kaya (2006). Set κ γ ρ V0 θ
(1) (2) (3) (4) (5) (6) 1 0.3 1.0407 6.2 2 6.21 1 0.9 0.5196 0.6 1 0.61 -0.3 -0.5 -0.6747 -0.7 -.3 -0.7 0.09 0.04 0.0194 0.02 0.09 0.010201 0.09 0.04 0.0586 0.02 0.09 0.019
Table 3: This table is based on 6 realistic sets of parameters that appear in the literature for the Heston model. bi −6.4871 For each of the 6 cases presented in Table 3, we report the bias C b (where Ci is the price estimate) in Table 4 using n = 100, 000 simulations and p = 300. Set Scheme Scheme Scheme Scheme Scheme Scheme Scheme
1 2 3 4 5 6 7
(1) 0.0026 -0.934 -0.905 -0.78 -0.696 -0.699 -0.836
(2) 0.0261 -2.66 -2.77 -2.63 -2.57 -2.57 -2.69
(3) 0.049 -2.36 -2.21 -1.6 -1.19 -1.04 -1.93
(4) -0.00876 -3.64 -3.31 -2.01 -1.37 -1.63 -2.92
(5) -0.0004 -0.836 -0.698 -0.529 -0.419 -0.424 -0.614
(6) -0.0197 -3.99 -3.75 -2.8 -1.38 -1.15 -3.38
Table 4: Results based on the 6 realistic sets of parameters from Table 3. Note that the same random numbers are used to calculate the numbers of each column of Table 2 and 4. Again, from table 4, we observe that scheme 1 outperforms the schemes 2 to 7. This was also observed for other values 23
of discretization steps. From now, we use Scheme 1 to estimate the prices of timer options in the Heston and Hull and White model and apply a control variate to improve the case when r 6= 0.
3.2
Heston model
In the Heston model (see Theorem 2.4 equation (36)), the price of a timer call option can be simulated from the distribution of (τ, Vτ ). This is a fast and accurate technique to deal with the pricing of timer options in the Heston stochastic volatility model. To simulate the price (40), we need to simulate (τ, Vτ ). It is an almost “exact” simulation approach. The only bias comes from the discretization of the process Xt defined in Lemma 2.2. Let V/p be the discretization step for the Euler discretization needed to simulate (35) and n be the number of simulations. Table 5: Prices of a timer call option in the Heston model. Parameters are given in Table 1 except for the interest rate r which is r = 0% and the correlation coefficient ρ. Standard deviations appear in parenthesis. Case when r = 0% Nb of Discret. Correl. Monte Carlo Exact Simul. n Step V/p Coeff. ρ Value (40) Value 500000 V/100 ρ = −0.8 6.722(0.011) 6.4871 500000 V/100 ρ=0 6.487(4.72e-14) 6.4871 500000 V/100 ρ = 0.8 6.349(0.011) 6.4871 500000 V/500 ρ = −0.8 6.528(0.011) 6.4871 500000 V/500 ρ=0 6.487(4.72e-14) 6.4871 500000 V/500 ρ = 0.8 6.439(0.011) 6.4871 1000000 V/500 ρ = −0.8 6.528(0.0079) 6.4871 1000000 V/500 ρ=0 6.4871(3.72e-14) 6.4871 1000000 V/500 ρ = 0.8 6.467(0.0078) 6.4871
Prices are simulated using (40) and scheme 1 when r = 0% and compared with the exact price which is equal to 6.4871 (given in Remark 1.2). The results are displayed in Table 5. It shows that our simulations are correct when ρ = 0. However we can note that the simulations when ρ 6= 0 have a higher standard deviation and depend on the discretization step. We observe that a discretization step of V/500 is not enough to obtain an accurate result in the case when the correlation is different from 0. However the fact that we know exactly the price when r = 0% under any assumptions on stochastic volatility models makes an excellent control variate. 24
It is clear from Table 5 that the Monte Carlo method performs very well when the correlation coefficient ρ = 0 and one gets a price with 4 accurate digits even when r 6= 0 as can be seen from Table 6. We now run the Monte Carlo method with r = 4% and use the case r = 0% as a control variate as it is indicated by (42). Our results are reported in Table 6. Note that this approach converges to the correct price when the number of simulations converges to +∞ and the discretization step 1/p goes to 0. Table 6: Prices of a timer call option in the Heston model. Parameters are given in Table 1. In the last columns, standard deviations are in parenthesis. Nb of Discret. Correl. Monte Carlo Monte Carlo Simul. n Step 1/p Coeff. ρ Value (40) Value (42) 100,000 V/500 ρ = −0.8 7.628(0.027) 7.6334(0.0028) 100,000 V/500 ρ=0 7.5366(0.00059) 7.5366(0.00059) 100,000 V/500 ρ = 0.8 7.433(0.026) 7.4356(0.0018) 100,000 V/5, 000 ρ = −0.8 7.646(0.027) 7.634(0.0028) 100,000 V/5, 000 ρ=0 7.534(0.00058) 7.534(0.00058) 100,000 V/5, 000 ρ = 0.8 7.395(0.026) 7.430(0.0018) 500,000 V/500 ρ = −0.8 7.667(0.012) 7.637(0.0013) 500,000 V/500 ρ=0 7.5357(0.00026) 7.5357(0.00026) 500,000 V/500 ρ = 0.8 7.391(0.011) 7.432(0.00082) 500,000 V/5, 000 ρ = −0.8 7.616(0.012) 7.631(0.0013) 500,000 V/5, 000 ρ=0 7.5338(0.00026) 7.534(0.00026) 500,000 V/5, 000 ρ = 0.8 7.431(0.012) 7.433(0.00082)
One can observe that the control variate is usually extremely efficient by reducing about 10 times the standard deviation of the simulations. The control variate (with r = 0% and ρ = 0 ) is a constant and thus is independent of the estimate of the price. So the control variate does not work for ρ = 0 and the standard deviation is then unchanged.
3.3
Hull and White Model
In table 7, we provide an example in the Hull and White setting. We note that the order of magnitude are similar. When the interest rate r = 0% the result is similar to the Heston case. This confirms that the price of a timer option is model-free when r = 0%. In the case when r = 4%, the results between table 6 and 7 are quite different. This also confirms that the price of a timer option is not modelfree in the presence of a positive interest rate. 25
Table 7: Prices of a timer call option in the Hull and White model. The constant interest rate is denoted by r, the strike of the call option is equal to K = 100. The initial stock price is S0 = 100, and the parameters of the volatility dynamics (6) are given by V0 = 0.0625, ν = 0.1, a = 0.2. The variance budget is determined by V = 0.0265. Interest Nb of Rate r Simul. n 0% 500,000 0% 500,000 0% 500,000 0% 500,000 0% 500,000 0% 500,000 4% 500,000 4% 500,000 4% 500,000 4% 500,000 4% 500,000 4% 500,000
Discret. Step 1/p V/300 V/300 V/300 V/3, 000 V/3, 000 V/3, 000 V/300 V/300 V/300 V/3, 000 V/3, 000 V/3, 000
Correl. Coeff. ρ ρ = −0.8 ρ=0 ρ = 0.8 ρ = −0.8 ρ=0 ρ = 0.8 ρ = −0.8 ρ=0 ρ = 0.8 ρ = −0.8 ρ=0 ρ = 0.8
Monte Carlo Monte Carlo Value (40) with CV (42) 6.543(0.011) 6.4871(exact) 6.4871(4.7e-14) 6.4871(exact) 6.474(0.011) 6.4871(exact) 6.464(0.011) 6.4871(exact) 6.4871(4.7e-14) 6.4871(exact) 6.489(0.011) 6.4871(exact) 7.333(0.012) 7.297(0.0008) 7.2786(4.1e-5) 7.2786(4.14e-5) 7.241(0.012) 7.262(0.0007) 7.277(0.0117) 7.291(0.0007) 7.276(4.1e-5) 7.276(4.14e-5) 7.256(0.0117) 7.260(0.0007)
The control variate for the Hull and White model works extremely well as can be seen from the reduction in the standard deviation from 0.01 to 0.0007.
3.4
Sensitivity Analysis
Finally, we illustrate the sensitivity of timer call option prices to the variance budget V, the interest rate r and the correlation ρ. Graphs are obtained using the Monte Carlo simulation approach with control variate in the Heston model. Parameters are given in Table 1 unless specified. 3.4.1
Discretization step 1/M
Figure 2 represents the price of a timer call obtained by the Monte Carlo technique developed in this paper. It takes less than 1 minute to get the graph. There are several important observations. First the convergence is very quick to the true values. Second the graph is obtained in less than 1 minute. This is a better convergence than the one obtained by crude Monte Carlo as shown in Figure 1.
26
7.85 7.8 ρ=−0.8
7.75 7.7 Price
7.65 ρ=0
7.6 7.55 7.5 7.45 7.4 7.35 0
ρ=0.8 100
200
300 400 500 600 700 800 Number of discretization step
900 1000
Figure 2: Timer Call Price w.r.t. M , the number of discretization steps. The graph is based on 50,000 simulations for each value of M .
In the case when r=0%, we can calculate the standard deviations and the RMS errors of the crude MC presented in Section 2.1 and compare it with the scheme 1 developed in the previous section. Assume p = 100, and M = 20, 000. We then simulate the price of the timer option, calculate the standard deviation of the estimate of the price and the bias using the crude method from Section 2.1, respectively the scheme 1 from the previous section. We repeat this experiment 250 times to obtain the average standard deviation for each method, as well as the RMSE. Table 8: Comparison of the earlier method and the one developed in the previous section using 250 times a Monte Carlo simulation with 20,000 trials and a time step p = 100. Method Crude MC from Section 2.1 Scheme 1 from Section 3.1
ρ = −0.8 ρ=0 ρ = 0.8 0.25 (0.075) 0.09 (0.074) 0.19 (0.074) 0.22 (0.057) 1.9e-12 (1.3e-14) 0.15 (0.05)
We should first note that the algorithm from Section 2.1 is much slower: it took about 24 hours to run 250 times whereas the one presented in the previous section (see Section 3.1)) took only a few minutes. it is also clear from Table 8 that our method (scheme 1) presented in Section 3.1 is far more accurate than the crude Monte Carlo method when the correlation coefficient is small and also outperforms this crude MC method when the absolute value 27
of the correlation coefficient is large. 3.4.2
Sensitivity to the variance budget V
Figure 3 displays the sensitivity to the variance budget. It illustrates a risk that we have not taken into account. It is the risk of using a continuoustime framework to price a contract that in practice is based on the discrete measurement of the realized variance. In practice, the investor pays for a variance budget V but on average gets a timer option with a variance budget V+A where A is the average amount by which the variance process overshoots the level V. Figure 3 shows that continuously monitored timer call options are underpriced. 18 Timer Call Option when r=4% Timer Call Option when r=0%
16
Timer Call Price
14 12 10 8 6 4 2 0
0.02
0.04 0.06 Variance Budget
0.08
0.1
Figure 3: Timer Call Price w.r.t. V This graph is obtained with a time step of 1/3000 and 1, 000, 000 Monte Carlo simulations for each value of the variance budget “V”. The correlation is ρ = −0.5.
3.4.3
Sensitivity to the interest rate r
Figure 4 represents the sensitivity to the risk-free rate for different values of the correlation coefficient. Figure 4 shows that timer call options are more valuable when the riskfree rate increases. Recall that the only reason why timer options cannot be priced by robust replication in a model-free setting is the presence of a positive risk-free rate. Hedging a timer call option requires to hold a positive number of shares of stocks (because the delta of a call is positive) and to borrow money to implement the hedging strategy. Hedging will therefore be more expensive if interest rates are high. 28
9
Timer Call Price
8.5
8
7.5
7 Timer Call Option when ρ=−0.8 Timer Call Option when ρ=0 Timer Call Option when ρ=0.8
6.5
6 0
0.01
0.02
0.03 0.04 0.05 Interest rate r
0.06
0.07
0.08
Figure 4: Timer Call Price w.r.t. r This is done with a time step of 1/3000 and 1, 000, 000 Monte Carlo simulations for each value of the correlation coefficient “r”.
In the presence of stochastic interest rates correlated with the variance process, an important risk for the timer option is negative correlation between the risk-free interest rate and the variance process. The worst case scenario is obtained when interest rates are high but when the variance is low. Indeed if the variance stays low, the maturity of the timer call option is postponed to a later time, and one needs to borrow during a longer period. There is therefore a real risk that the cost of hedging will significantly increase if interest rates become high when volatility is low. Introducing stochastic interest rates and studying the effects of negative correlation between the interest rate process and the variance process are left for future research. 3.4.4
Sensitivity to the Correlation Coefficient ρ
The price of a timer option depends in a complicated way on the correlation coefficient between the underlying’s asset price and its volatility. Figure 5 gives the sensitivity of the price in our continuous setting given by (40) to the correlation coefficient ρ. In Figure 5, prices are displayed as a function of the correlation coefficient. The Black andq Scholes value is calculated with a maturity equal to EQ [τ ], a
volatility σ = TV , the same strike K, the initial price S0 and the interest rate r = 4%. Since the volatility is constant, it does not depend on the correlation coefficient ρ and we obtain an horizontal line as can be seen from Figure 5. It is a good approximation of the price when ρ = 0. However it is 29
not equal to the price of a timer call when ρ = 0. 7.65 Black Scholes Monte Carlo with CV
Timer Call Price
7.6
7.55
7.5
7.45
7.4 −0.8
−0.6
−0.4
−0.2 0 0.2 Correlation ρ
0.4
0.6
0.8
Figure 5: Timer Call Price w.r.t. ρ This is done with a time step of 1/3000 and 1, 000, 000 Monte Carlo simulations for each value of the correlation coefficient “ρ”.
Conclusions We have studied the theoretical properties of the pricing of timer options under very general stochastic volatility models and have proposed an efficient almost exact Monte Carlo simulation scheme for pricing timer options. Because of the booming of derivative markets for the realized variance of stock returns (see Carr and Lee [2009], Carr and Madan [1998]), such as variance swaps, volatility swaps, corridor gamma swaps, the market for volatility products is more and more important. These timer products can probably serve as hedging or replication tools for variance swaps or volatility swaps or vice versa and help to complete the market. In this study, we neglected the interest rate risk of timer options because we assumed that the risk-free rate is constant. We also neglected the risk that the variance budget is discretely monitored and will likely be strictly bigger than the original target at expiry. Finally as mentioned in the numerical analysis the effect of the correlation between the variance process and the interest rate process may also be an important risk factor for timer options. These are left for future research.
30
A
PDE for the timer option
Recall that under the physical probability measure, p ( √ 1 − ρ2 dZt1 + ρdZt2 dSt = µSt dt + Vt St dVt = αP (St , Vt )dt + βP (St , Vt )dZt2
(53)
where µ, αP and βP are the respective parameters under the physical probability. We write α and β and assume that the market price of risk of the volatility is therefore equal to 0. We follow the method in Gatheral (2006) and Li (2010) to derive the PDE for the timer call option and its associated boundary conditions. Due to the perpetuity property of the timer option, its price does not depend on t. At time t ∧ τ , the price of a timer call option can be represented as a function of the three dimensional Markov process (S, V, ξ) Ct∧τ = u (St∧τ , Vt∧τ , ξt∧τ )
(54)
Rt where ξt = 0 Vs ds. We have two sources of randomness here, one introduced by the stock process and the other introduced by the stochastic variance process. Thus similar to the idea in Gatheral (2006), consider another risky asset whose value depends only on the volatility process with maturity T . Denote its price by Gt = g (t, Vt , ξt ) at time t < T . Without loss of generality, we consider the following dynamically rebalanced portfolio on the time interval [0, τ ∧ T ]: the portfolio contains a timer call option with value Ct = f (St , Vt , ξt ), a quantity −∆1t of the stock St and −∆2t of the volatility derivative Gt . Denote the value of the portfolio as Pt , then Pt = Ct − ∆1t St − ∆2t Gt Applying Ito’s formula, we have (here S stands for St , V stands for Vt , ξ stands for ξt ) that dPt = dCt − ∆1t dSt − ∆2t dGt , then √ 1 1 dPt = µSfS + αfV + V fξ + V S 2 fSS + β 2 fV V + ρ V βSfS,V − µ∆1t S 2 2 1 2 2 dt −∆t gt + αgV + V gξ + β gV V 2 √ + ρ V S fS − ∆1t + β fV − gV ∆2t dZt1 p √ + 1 − ρ2 V S fS − ∆1t dZt2 (55)
31
Then to make the portfolio instantaneously risk-free, we must choose fS = ∆1t fV = gV ∆2t
(56)
Using the fact that the return on a risk-free portfolio must equal to the risk-free rate r and that ∆1 and ∆2 are now known in (56), √ 1 1 αfV + V fξ + V S 2 fSS + β 2 fV V + ρ V βSfSV 2 2 1 2 fV 2 − ∆t gt + αgV + V gξ + β gV V = r f − fS S − g 2 gV Assume fV and gV are different from 0, then we rearrange the above equation and have √ 1 1 1 2 2 αfV + V fx + V S fSS + β fV V + ρ V βSfV S − rf + rSfS fV 2 2 1 2 1 gt + αgV + V gξ + β gV V − rg (57) = gV 2 The left hand side is a function of f only and the right hand side is a function of g only. The right hand side does not depend on S. the left-hand side does not depend on t. Thus for them to be equal, according to the same reasoning as in Gatheral (2006), we have that both sides should be equal to some function k(.) of the independent variables V and ξ. Φ(v, ξ) k(v, ξ) = − α(v) − β(v) (58) σv where σv is the volatility of volatility and Φ is called the market price of volatility risk. Under the risk neutral dynamics, then Φ(v, ξ) = 0 and we obtain the following PDE for the timer call option price √ 1 1 V fξ + αfV + V S 2 fSS + β 2 fV V + ρ V βSfSV − rf + rSfS = 0. 2 2
(59)
The main boundary condition is f (s, v, V) = max{s − K, 0}. √ In the case of the Heston model, under P , α(v) = κ (θ − v), β(v) = γ v, √ the volatility of volatility is σv = γ and β(v) = σv v. For the Hull-White model, α(v) = av, β(v) = νv.
32
B
Proof of Lemma 2.1
Proof. Recall that f and h are defined by (22). By Ito’s lemma, we have p (60) df (Vt ) = h(Vt )dt + Vt dWt2 . After integrating from 0 to T on both sides of the above equation (60), we have Z Tp Z T 2 Vt dWt = f (VT ) − f (V0 ) − h(Vt )dt. (61) 0
0
Since
d (log (St )) =
Vt r− 2
dt + ρ
p
Vt dWt2 +
p
Now we plug equation (61) into equation (62), Z p 2 ST = S0 exp rT + aT + 1 − ρ
0
1 − ρ2
T
p Vt dWt1 .
p 1 Vt dWt ,
RT RT where aT = ρ(f (VT ) − f (V0 )) − ρ 0 h(Vt )dt − 12 0 Vt dt. Then p ST = S0 exp rT + aT + bT Z , Z ∼ N (0, 1)
(62)
(63)
(64)
RT √ with bT = (1 − ρ2 ) 0 Vt dt because of the independence of Vt and Wt1 , thus we can make use of the Dubins-Schwartz theorem (Karatzas and Shreve (1991)) and represent the stochastic integral as a time-changed standard Brownian motion.
C
Proof of Lemma 2.2
Rt Proof. Denote by θ(t) = 0 Vs ds < ∞ for any t < ∞. θ(t) is a continuous and increasing function on R. It is therefore invertible. Let τ (y) be the inverse function of θ(t) for t ∈ (0, ∞), i.e. τ (θ(t)) = t. On one hand, since θ(τ (y)) = y, we have θ′ (τ (y)) = τ ′1(y) . On the other hand by definition of θ(.), one has also θ′ (s) = Vs , therefore ( Ry Ry Ry τ (y) = 0 τ ′ (s)ds = 0 θ′ (τ1(s)) ds = 0 Vτ1(s) ds θ′ (τ (y)) = τ ′1(y) = Vτ (y)
33
Consider f and h are defined by (22). By Ito’s lemma, we have df (Vt ) = h(Vt )dt + f ′ (Vt )β(Vt )dWt = h(Vt )dt + m(Vt )dWt . (65) Ru Define τ (t) = inf{u > 0, 0 m2 (Vs )ds = t}, which is the R ∞ inverse of θ(u). It is well-defined, because for any fixed t, we have P ( 0 Vs ds > t) = 1 since lim(ξt ) = +∞. Then by the Dubins-Schwartz theorem (Karatzas and Shreve(1991), p174), we have M (τ (t)) =
Z
p Vs dWs = Bt .
τ (t)
0
(66)
Now if we integrate both sides of the equation (65) from 0 to τ (t), we have Z
τ (t)
0
p
Vs dWs = f (Vτ (t) ) − f (V0 ) −
Z
τ (t)
h(Vs )ds.
(67)
0
From equation (66) and (67), we have Bt =
Z
τ (t)
0
Z p Vs dWs = f (Vτ (t) ) − f (V0 ) −
τ (t)
h(Vs )ds.
(68)
0
Then we take the differentiate representation of (68): df (Vτ (t) ) = h(Vτ (t) )τ ′ (t)dt + dBt .
(69)
Also from equation (65), we have τ (t) =
Z
t
0
so τ ′ (t) =
1 . m2 (Vτ (t) )
1 Vτ (s)
ds,
(70)
Then we can rewrite (69) as df (Vτ (t) ) =
Note that by definition, Hτ (t) =
h(Vτ (t) ) dt + dBt . Vτ (t)
R τ (t) 0
(71)
h(Vs )ds, so we have
dHτ (t) = h(Vτ (t) )τ ′ (t)dt =
34
h(Vτ (t) ) dt Vτ (t)
(72)
If we integrate both sides of equation (72) from 0 to V, we have Z
Hτ (V) =
V
0
h(Vτ (s) ) ds Vτ (s)
(73)
Then if we let Xt = Vτ (t) , using (70), (71) and (73), we finally have law τ (V), Vτ (V) , Hτ (V) ∼
Z
0
V
1 ds, XV , Xs
Z
V 0
h(Xs ) ds , Xs
(74)
where Xt is governed by
df (Xt ) =
h(Xt ) dt + dBt , Xt
X0 = V0 .
(75)
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