This work provides an overview of the effect of stochastic volatility on defaulted bonds. In this work, we introduce the theoretical framework of the pricing of stochastic volatility models, which is inconsistent with the pricing of the classical Black-Scholes models.
Introduction
The current market price, determined by supply and demand, is expressed as a percentage of the principal value of the bond. A bond is called a premium bond if the price of the bond rises above its principal value, and if it does not, it is called a discount bond.
The Black-Scholes Assumptions
Hedging a derivative security is the problem faced by the financial institution that sells some contract designed to reduce risk. More recently, model-based pricing of derivative securities has become the basis of risk management.
The Black-Scholes Pricing Formula
We introduce for 0 ≤ t ≤ T the unknown price function, u(S, t), which indicates the correct price of the option at time t. As the asset price rises unbounded, it becomes increasingly likely that the option will be exercised, and the value of the utilization rate becomes less and less important.
Probabilistic Approach
Martingales
This theorem destroys the drift term of the stochastic differential equation for the stock price.
The Feynman-Kac Formula
Feynman-Kac Theorem
This can be rewritten as dSt = rS(x, t)dt +σS(x, t)dWtQ, where WtQ is the Brownian motion under the risk-neutral probability measure Q. According to the risk-neutral formula (1.4.8) ), the price of the derivative of the security at the moment is.
Pricing Biases of the Black-Scholes Model
Cash-or-Nothing Call Option
So on the line K and below the line K the value of the option is zero. If S exceeds the strike price K, the final payment of an option is equal to a fixed amount q.
Asset-or-Nothing Call Option
Assuming that the expected return is the risk-free interest rate, we obtain. Since the asset price follows the lognormal distribution, the probability density function forST is given by.
Barrier Options
The Reflection Principle (Method of Images)
For every path that ends below K but previously reached a, there is another path that goes above 2a−K: we simply reflect the path in a mirror at level w.
First Passage Times
Pricing the Knock-out Options
The integrand in equation (2.2.14) is the probability density function of the first passage time of lnH/S of a Brownian motion with drift−. Therefore, the integral can be rewritten as. Consider a European call, expiring at time T, with strike price, K, and up-and-out barrier, H. The solution to the diffusion equation, dSt =rStdt+σStdWtQ, for the asset price is given by.
Complex Barrier Options
Generalized Autoregressive Heteroskedasticity
The “(1,1)” in GARCH(1,1) indicates that σm2 is based on the most recent observation of w2 and the most recent estimate of the degree of variance. GARCH(1,1) can be extended to a GARCH(p, q) formulation in which the current conditional variance is parameterized to depend on q lags of the squared error and lags of the conditional variance, i.e.
Constant Elasticity of Variance (CEV)
Jump-Diffusion Processes
In this example, we consider a stochastic differential equation driven by a Wiener process and a Poisson random measure. The intensity Nt is denoted by λ, a positive constant, and the compensated Poisson process by Mt = Nt−λt. To derive the partial differential equation for u, we consider the stochastic differential equation (3.2.9), which can be rewritten as 3.2.13).
2The Itˆo-Doeblin formula for a jump process states that, ifX(t) is a jump process andf(x) a function for which f0(x) and f00(x) are definite and continuous, then. Equation (3.2.18) is sometimes called a differential-differential equation because it involves u at two different values of the stock price, namely St and (σ+ 1) St .
Pure Stochastic Volatility Models
Wt1 and Wt2 are correlated Brownian motion under Q with dWt1dWt2 =ρdt for some -1≤ρ ≤1, where ρ is the correlation coefficient between Wt1 and Wt2. In this model, the option price depends on two random variables which are the underlying asset and the volatility function respectively. At time t, the risk-neutral price of a call expiring at time ≤T in this stochastic volatility model is [14].
3.3.9) Since Wt1 and Wt2 are two-dimensional Brownian motions, we can derive the partial differential equation for a European call option. 3 The market price of risk is associated with the Girsanov transformation of the underlying probability measure leading to a certain martingale measure [25]. Then we have π =u−∆S−∆1u1 and the corresponding change in the value of this portfolio is given by.
The expression (3.3.21) is a linear combination of the market price of risk and the market price of volatility, b is the volatility from the stochastic process X and Λ(S, X, t) is the market price of volatility risk.
Mean-Reverting Models
- Ornstein-Uhlenbeck (OU) Model
- Cox-Ingersoll-Ross (CIR) Model
- Wiggins Model
- Hull and White Model
- Heston’s Model
Solving equation (3.3.23) is not easy, and often the only alternative is to use numerical techniques to approximate solutions. We note that while it is not possible to deal in detail with each of the above stochastic volatility models, we briefly look at a few more popular mean-reverting models. The stochastic differential equation (3.4.7) can be solved explicitly by considering for some Brownian motion Wc.
In the CIR model, a stock stand volatility, respectively, satisfies the following differential equations. where vt is the variance, a, the band-care constants, and dWt1 and dWt3 are independent Brownian motions. Using the above equations we can find the limiting distribution of vt which is a gamma distribution with mean −a/b and variance ac2/2b2. He showed that the Black-Scholes equation overprices out-of-the-money calls relative to in-the-money calls.
Using his model, Heston [47] developed a new technique, based on characteristic functions, to derive a closed-form solution for the price of a European call option on an asset.
Biases of the Stochastic Volatility Models
A Solution to the Black-Scholes Equation
Representation of the Bond-Pricing Solution in
In the next section, we investigate the solution of the one-factor bond price equation under different stochastic process assumptions for rt.
One-Factor Interest-Rate Models
Bond-Option Models Based on One-Factor
The pricing models for the bond and the bond option differ only in the terminal conditions. The payoff functions for the bond price are V(r, TV) = F, where TV is the maturity date of the bond, F is the face value of the bond and the bond option value Γ(r, TΓ) = max [γ (Γ(r, TV)−K),0] with Take as the maturity date of the bond option. Here, γ is a binary variable which takes the value 1 when the option is a call and -1 when the option is a put [59].
Vasicek Models of Stochastic Volatility
Two-Factor Interest-Rate Models
Vasicek Model
Using this model Cotton et al [18] evaluated bond options. where W1t and W2t are independent Brownian motions.
Longstaff and Schwartz
4.4.31) Since X1 and X2 describe a one-factor CIR process, the Longstaff-Schwartz model can be interpreted as a two-factor CIR model.
Multifactor Interest Rate Models
The Lie Analysis
The product of the two elements x and y of a Lie algebra [x, y] = xy−yx, where [x, y] denotes their Lie bracket. We can then determine the first expansion of G to find the second expansion and then apply it to the Black-Scholes equation.
Calculation of Infinitesimal Symmetries
Reduction of Order
Group Invariant Solutions
To ensure a minimum number of reductions, one can construct the one-dimensional optimal system. Therefore, the one-dimensional subalgebra spanned by Xmeta1 6= 0 is equivalent to that spanned byX1+αX2+βX4,α, β ∈ <.
Bond-Pricing and Interest-Rate Models
- Bond Pricing
- Interest-Rate Models
- The Jarrow-Turnbull Model
- The Duffie-Singleton Models
Any Vasicek solution can be transformed into a solution of the corresponding CIR equation. Originating from the theoretical framework of Black and Scholes [7] and Merton [66], the structural or cause-and-effect approach models the cause of default. The Black-Scholes-Merton (BSM) model is limited to a zero-coupon bond, an equity value viewed as a standard option to purchase the firm's assets, and the firm's debt as a free debt less an option of sale.
However, the construction of the Black-Scholes-Merton framework excludes the possibility of default before maturity, the effect of stochastic interest rates and the valuation of coupon-paying bonds. In this case, the time of default is directly modeled as the time of the first jump of a Poisson process with stochastic intensity or compensator process. In the reduced-form models, the time of default is directly modeled as the time of the first jump of a Poisson process with random intensity.
In case of default, the recovery price is a fraction of the final price immediately before default.
Structural Approach
Merton Model
When recovery is high, ie. 1−α is small, the product is small and the credit spread is small.
Black-Cox Model
Stochastic Volatility Models
Two factor Stochastic Volatility Models
The driving volatility Xt evolves with the mean m, a mean reversion rate α > 0, and the (vol-vol) volatility volatility β. The function g is assumed to be fairly regular, nonnegative, bounded, and bounded by zero. We introduce a small parameter such that the mean rate of return defined by 1/th becomes large.
In the case of Ornstein-Uhlenbeck (OU), the variance ν2 = β2/2α must be a fixed constant o(1), which in terms of implies that β =ν√. If we substitute the rescaled parameters α and β, the partial differential equation of BS becomes. L1 contains the mixed partial derivative due to the correlation between the two Brownian movements W1 and W2 and L2, also denoted by LBS(g(x)), is the Black-Scholes operator at the volatility level g(x).
The parameters or sa and b are estimated as the slope and intercept of the line of best fit to the observed implied volatilities plotted as a function of the log-moneyness-to-maturity ratio (LMMR).
Multifactor Stochastic Volatility Models
The random volatility σt depends on the two volatility factors, Xt and Zt, and the functions Λ and Σ are given by. 6.3.27). The two stochastic volatility factors Xt and Zt are differentiated by their inherent time scales.
Pricing Defaultable Bonds
Therefore the derived calibration formulas are C0δ=σ. rewritten as the first-passage structural model for a stationary Brownian motion.
Models with Fast and Slow Volatility Factors
Fast Volatility Factor
The assumption is that the parameter α is taken as the average reversal rate of the process When one introduces the corrected effective volatilityσe byσe2 =σ2+ 2C2, the first term in the new approximation is ΠBS(t, s;σ).
6.5.5) According to [33], the accuracy of this approximation is of the order of 1/α in the case of smooth payoffs and of the order of (lnα)/α in the case of call options.
Slow Volatility Factor
Stochastic Volatility Effects in Yield Spread
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